Articles in Journals (In reverse chronological order):
C.D. Charalambous, Jan van Schuppen, "A Realization Approach to Lossy Network Compression of a Tuple of Correlated Multivariate Gaussian RVs.", Entropy 2022
Abstract
Examined in this paper is the Gray and Wyner source coding for a simple network of correlated multivariate Gaussian random variables, Y1:Ω→Rp1 and Y2:Ω→Rp2 . The network consists of an encoder that produces two private rates R1 and R2 , and a common rate R0 , and two decoders, where decoder 1 receives rates (R1,R0) and reproduces Y1 by Yˆ1 , and decoder 2 receives rates (R2,R0) and reproduces Y2 by Yˆ2 , with mean-square error distortions E||Yi−Yˆi||2Rpi≤Δi∈[0,∞],i=1,2 E | | Y i − Y ^ i | | R p i 2 ≤ Δ i ∈ [ 0 , ∞ ] , i = 1 , 2 . Use is made of the weak stochastic realization and the geometric approach of such random variables to derive test channel distributions, which characterize the rates that lie on the Gray and Wyner rate region. Specific new results include: (1) A proof that, among all continuous or finite-valued random variables, W:Ω→W W : Ω → W , Wyner’s common information, C(Y1,Y2)=infPY1,Y2,W:PY1,Y2|W=PY1|WPY2|WI(Y1,Y2;W) C ( Y 1 , Y 2 ) = inf P Y 1 , Y 2 , W : P Y 1 , Y 2 | W = P Y 1 | W P Y 2 | W I ( Y 1 , Y 2 ; W ) , is achieved by a Gaussian random variable, W:Ω→Rn W : Ω → R n of minimum dimension n, which makes the two components of the tuple (Y1,Y2) ( Y 1 , Y 2 ) conditionally independent according to the weak stochastic realization of (Y1,Y2) ( Y 1 , Y 2 ) , and a the formula C(Y1,Y2)=12∑nj=1ln(1+dj1−dj), C ( Y 1 , Y 2 ) = 1 2 ∑ j = 1 n ln 1 + d j 1 − d j , where di∈(0,1),i=1,…,n d i ∈ ( 0 , 1 ) , i = 1 , … , n are the canonical correlation coefficients of the correlated parts of Y1 Y 1 and Y2 Y 2 , and a realization of (Y1,Y2,W) ( Y 1 , Y 2 , W ) which achieves this. (2) The parameterization of rates that lie on the Gray and Wyner rate region, and several of its subsets. The discussion is largely self-contained and proceeds from first principles, while connections to prior literature is discussed. Keywords: Gray–Wyner network; Wyner’s lossy common information; weak realizations of conditional independence; canonical variable form of multivariate Gaussian random variables; multi-user communication
S. Loyka and C. D. Charalambous, "On the Capacity of Gaussian MIMO Channels With Memory," in IEEE Communications Letters, vol. 26, no. 8, pp. 1760-1763, Aug. 2022
Abstract
The operational capacity of Gaussian MIMO channels with memory was obtained by Brandenburg and Wyner (1974) under certain mild assumptions on the channel impulse response and its noise covariance matrix, which essentuially require channel memory to be not too strong. This channel was also considered by Tsybakov (2006) and its information capacity was obtained in some cases. It was further conjectured, based on numerical evidence, that these capacities are the same in all cases. This conjecture is proved here. An explicit closed-form expression for the optimal input power spectral density matrix is also given. The obtained result is further extended to the case of joint constraints, including per-antenna and interference power constraints as well as energy harvesting constraints. These results imply the information-theoretic optimality of OFDM-type transmission systems for such channels with memory.
C. D. Charalambous, T. Charalambous, C. Kourtellaris and J. H. van Schuppen, "Complete Characterization of Gorbunov and Pinsker Nonanticipatory Epsilon Entropy of Multivariate Gaussian Sources: Structural Properties," in IEEE Transactions on Information Theory, vol. 68, no. 3, pp. 1440-1464, March 2022
Abstract
This paper derives the optimal test channel distribution and the complete characterization of the classical Gorbunov and Pinsker (1973), Gorbunov and Pinsker (1974) nonanticipatory epsilon entropy of multivariate Gaussian Markov sources with square-error fidelity, which remained an open problem since 1974. The paper also formulates a state dependent nonanticipatory epsilon entropy, in which past reproductions are available to the decoder and not to the encoder, the test channel is specified with respect to an auxiliary (state) random process, and the reproduction process is a causal function of past reproduction and the auxiliary random process. This variation is analogous to the Wyner and Ziv (1976) and Wyner (1978) rate distortion function (RDF), of memoryless sources. It is shown that the operational rate of zero-delay codes, with past reproductions available to the decoder but not to the encoder is bounded below by the state dependent nonanticipatory epsilon entropy rate. For the case of multivariate Gaussian Markov sources with square-error fidelity, the optimal test channel distribution and the complete characterization of the state dependent of nonanticipatory epsilon entropy are derived, and also shown that that the two nonanticipatory epsilon entropies coincide. The derivations are new; they are based on structural properties of the stochastic realizations of the reproduction process that induce the optimal test channel distributions. They are derived using, achievable lower bounds on information theoretic measures, properties of mean-square estimation theory, Hadamard’s inequality, and canonical correlation coefficients of a tuple of multivariate jointly Gaussian random processes. Applications of the nonanticipatory epsilon entropy and its state dependent variation are discussed to the areas of control of unstable Gaussian systems over limited memory channels, design of causal estimators for Gaussian Markov sources with a fidelity criterion, computation of the rate loss of causal and zero-delay codes of Gaussian Markov sources with respect to non-causal codes.
I. Tzortzis, C. D. Charalambous and C. N. Hadjicostis, "Jump LQR Systems With Unknown Transition Probabilities," in IEEE Transactions on Automatic Control, vol. 66, no. 6, pp. 2693-2708, June 2021
Abstract
This article develops a robust linear quadratic regulator (LQR) approach applicable to nonhomogeneous Markov jump linear systems with uncertain transition probability distributions. The stochastic control problem is investigated under two equivalent formulations, using i) minimax optimization theory, and ii) a total variation distance metric as a tool for codifying the level of uncertainty of the jump process. By following a dynamic programming approach, a robust optimal controller is derived, which in addition to minimizing the quadratic cost, it also restricts the influence of uncertainty. A solution procedure for the LQR problem is also proposed, and an illustrative example is presented. Numerical results indicate the applicability and effectiveness of the proposed approach.
C.D. Charalambous, T. Charalambous, C. Kourtellaris, and Jan H. van Schuppen, “Complete Characterization of Gorbunov and Pinsker Nonanticipatory Epsilon Entropy of Multivariate Gaussian Sources: Structural Properties,” IEEE Transactions on Information Theory, pages-48, 26 June, 2020
Abstract
The complete characterization of the Gorbunov and Pinsker [1], [2] nonanticipatory epsilon entropy of multivariate Gauss-Markov sources with square-error fidelity is derived, which remained an open problem since 1974. Specifically, it is shown that the optimal matrices of the stochastic realization of the optimal test channel or reproduction distribution, admit spectral representations with respect to the same unitary matrices, and that the optimal reproduction process is generated, subject to pre-processing and post-processing by memoryless parallel additive Gaussian noise channels. The derivations and analyses are new and bring out several properties of such optimization problems over the space of conditional distributions and their realizations.
T. Charalambous and C.D. Charalambous, “Characterizations of Nonanticipative Rate Distortion Function for Multivariate Nonstationary Gaussian Finite-Memory Autoregressive Processes,” IEEE Journal on Selected Areas in Information Theory, Special Issue on Estimation and Inference pages-34, May 15, 2020
C.D. Charalambous and C. Kourtellaris and I. Tzortzis, “Optimal Control and Signalling Strategies of Control-Coding Capacity of General Decision Models: Applications to Gaussian Models and Decentralized Strategies,” SIAM Journal on Control and Optimization, pages-28, 3 February 2020.
C.D. Charalambous and C.Kourtellaris and I. Tziortzis, "Ergodic Control Coding Capacity of Stochastic Control Systems:" Information Signalling and Hierarchical Optimality of Gaussian Systems," SIAM Journal on Control And Optimization, vol.58, no.1, pages 104-135, January 2020
Abstract
The control-coding (CC) capacity of dynamical decision models (DMs) is defined as the maximum amount of information transfer per unit time from its inputs to its outputs, called CC rate $R$ in bits/second, which is operational with the aid of a controller-encoder and a decoder, as in Shannon's mathematical theory of communication over noisy channels, with the encoder replaced by a controller-encoder [C. Kourtellaris and C. D. Charalambous, IEEE Trans. Inform. Theory, 64 (2018), pp. 4962--4992]. In the first part of the paper, data processing inequalities and information structures of optimal controllers-encoders are derived for controlled and uncontrolled information processes. Further, the ergodic theory of a Markov decision is applied to establish direct and converse CC theorems. In the second part of the paper, the problem of signalling information via a Gaussian decision model (G-DM) subject to a quadratic cost constraint to another Gaussian control system (G-CS) with a quadratic pay-off is investigated. A hierarchical decomposition and decentralized optimality of linear quadruple of strategies, \controller-encoder, decoder, controller\, is shown by construction as follows: (i) the controller-encoder simultaneously controls the output of the G-DM and encodes the state of the G-CS, and operates at the CC capacity of the G-DM; (ii) the decoder is optimal with respect to a mean square error (MSE) criterion; and (iii) the controller of the G-CS which acts on the decoder's information is optimal.
I.Tziortzis, C.D. Charalambous and T.Charalambous, "Infinite Horizon Average Dynamic Programming Subject to Total Variation Ambiguity" SIAM Journal on Control and Optimization, Vol.57, No4, pp.2843-2872, 2019
Abstract
This paper addresses the optimality of stochastic control strategies based on the infinite horizon average cost criterion, subject to total variation distance ambiguity on the conditional distribution of the controlled process. This stochastic optimal control problem is formulated using minimax theory, in which the minimization is over the control strategies and the maximization is over the conditional distributions. Under the assumption that, for every stationary Markov control law the maximizing conditional distribution of the controlled process is irreducible, we derive a new dynamic programming recursion which minimizes the future ambiguity, and we propose a new policy iteration algorithm. The new dynamic programming recursion includes, in addition to the standard terms, the oscillator semi-norm of the cost-to-go. The maximizing conditional distribution is found via a water-filling algorithm. The implications of our results are demonstrated through an example.
P. Stavrou and J. Ostergaard and C.D. Charalambous, “Zero-Delay Rate Distortion via Filtering for Vector-Valued Gaussian Sources,” IEEE Journal on Selected Topics on Signal Processing, vol.12, no.5, pp.841-856, 2018
Abstract
We deal with zero-delay source coding of a vector-valued Gauss-Markov source subject to a mean-squared error (MSE) fidelity criterion characterized by the operational zero-delay vector-valued Gaussian rate distortion function (RDF). We address this problem by considering the nonanticipative RDF (NRDF), which is a lower bound to the causal optimal performance theoretically attainable function (or simply causal RDF) and operational zero-delay RDF. We recall the realization that corresponds to the optimal “test-channel” of the Gaussian NRDF, when considering a vector Gauss-Markov source subject to a MSE distortion in the finite time horizon. Then, we introduce sufficient conditions to show existence of solution for this problem in the infinite time horizon (or asymptotic regime). For the asymptotic regime, we use the asymptotic characterization of the Gaussian NRDF to provide a new equivalent realization scheme with feedback, which is characterized by a resource allocation (reverse-waterfilling) problem across the dimension of the vector source. We leverage the new realization to derive a predictive coding scheme via lattice quantization with subtractive dither and joint memoryless entropy coding. This coding scheme offers an upper bound to the operational zero-delay vector-valued Gaussian RDF. When we use scalar quantization, then for r active dimensions of the vector Gauss-Markov source the gap between the obtained lower and theoretical upper bounds is less than or equal to 0.254r + 1 bits/vector. However, we further show that it is possible when we use vector quantization, and assume infinite dimensional Gauss-Markov sources to make the previous gap to be negligible, i.e., Gaussian NRDF approximates the operational zero-delay Gaussian RDF. We also extend our results to vector-valued Gaussian sources of any finite memory under mild conditions. Our theoretical framework is demonstrated with illustrative numerical experiments.
Photios Stavrou and T. Charalambous, C.D. Charalambous, and S. Loyka, “Optimal Estimation via Nonanticipative Rate Distortion Function and Applications for Time-Varying Gauss-Markov Processes,” SIAM Journal on Control and Optimization, vol.56, no.5, pp.3731-3765, 2018
Abstract
In this paper, we develop finite-time horizon causal filters for general processes taking values in Polish spaces using the nonanticipative rate distortion function ($NRDF$). Subsequently, we apply the $NRDF$ to design optimal filters for time-varying vector-valued Gauss--Markov processes, subject to a mean-squared error ($MSE$) distortion. Unlike the classical Kalman filter design, the developed filters based on the $NRDF$ are characterized parametrically by a dynamic reverse-waterfilling optimization problem obtained via Karush--Kuhn--Tucker conditions. We develop algorithms that provide, in general, tight upper bounds to the optimal solution to the dynamic reverse-waterfilling optimization problem subject to a total and per-letter $MSE$ distortion constraint. Under certain conditions, these algorithms produce the optimal solutions. Further, we establish a universal lower bound on the total and per-letter $MSE$ of any estimator of a Gaussian random process. Our theoretical framework is demonstrated via simple examples.
P. Stavrou and T. Charalambous and C.D. Charalambous, “Finite-Time Nonanticipative Rate Distortion Function for Time-Varying Scalar-Valued Gauss-Markov Sources,” IEEE Control Systems Letters, vol.2, issue 1, pages 175-180, January 2018
Abstract
We derive the finite-time horizon nonanticipative rate distortion function (NRDF) of timevarying scalar Gauss-Markov sources under an average mean squared-error (MSE) distortion fidelity. Further, we show that a conditionally Gaussian reproduction process realizes the optimal reproduction distribution, and this is determined from the solution of a dynamic reversewaterfilling optimization problem. We provide an iterative algorithm that approximates the solution of the dynamic reverse-waterfilling problem. From the above results, we also obtain, as a special case, the NRDF under a per-letter orpointwise MSE distortion fidelity, and we draw connections to the classical RDF of Gaussian processes. Our results are corroborated with illustrative examples.
C.D. Charalambous and C. Kourtellaris and S. Loyka, “Capacity Achieving Distributions and Separation Principle for Feedback Channels with Memory: The LQG Theory of Directed Information,” IEEE Transactions on Information Theory, vol.64, no.9, pages 6384-6418, September 2018
Abstract
A method is developed to realize optimal channel input conditional distributions, which maximize the finite transmission feedback information (FTFI) capacity, often called n -block length feedback capacity, by information lossless randomized strategies. The method is applied to compute closed form expressions for the FTFI capacity and feedback capacity, of nonstationary, nonergodic, unstable, multiple input multiple output Gaussian channels with memory on past channel outputs, subject to average transmission cost constraints of quadratic form in the channel inputs and outputs. It is shown that randomized strategies decompose into two orthogonal parts-an deterministic part, which controls the channel output process, and an innovation part, which transmits new information over the channel. Then a separation principle is shown between the computation of the optimal deterministic part and the random part of the optimal randomized strategies. Finally, the ergodic theory of linear-quadratic-Gaussian stochastic optimal control theory, is applied to identify sufficient conditions, expressed in terms of solutions to matrix difference and algebraic Riccati equations, so that the optimal control part of randomized strategies induces asymptotic stationarity and ergodicity, and feedback capacity is characterized by the per unit time limit of the FTFI capacity. The method reveals an interaction of the control and the information transmission parts of the optimal randomized strategies, and that whether feedback increases capacity, is directly related to the channel parameters and the transmission cost function, through the solutions of the matrix Riccati equations. For unstable channels, it is shown that feedback capacity exists and it is strictly positive, provided the power exceeds a critical threshold.
C. Kourtellaris and C.D. Charalambous, “Information Structures of Capacity Achieving Distributions for Feedback Channels with Memory and Transmission Cost: Stochastic Optimal Control & Variational Equalities,” IEEE Transactions on Information Theory, Vol.64, No.7, pp.4962-4992, July, 2018
Abstract
The information structures of the optimal channel input distributions P [0,n] = Δ {P Ai|Ai-1, Bi-1 : i = 0,1,..., n}, which correspond to the extremum problem of feedback capacity C An→Bn FB = Δ sup P [0,n] Σ i=0 n I(A i ;B i |B i-1 ) are identified, for any class of channel distributions {P Bi|Bi-1,Ai : i = 0,1,...,n} and {P Bi|B i-M i-1 ,Ai : i = 0,1,...,n}, where B n = Δ {B j : j = 0,1,...,n} are the channel output RVs, A n = Δ {A j : j = 0,1,...,n} are the channel inputs RVs, and M is a finite nonnegative integer. The methodology utilizes stochastic optimal control theory, to identify the control process, the controlled process, and a variational equality of directed information, to derive upper bounds on I(A n → B n )= Δ Σ i=0 n I(A i ;B i |B i-1 ), which are achievable over specific subsets of P [0,n] , which satisfy conditional independence. The main theorem states, that for any channel with memory M, the optimal channel input conditional distribution occur in the subset P [0,n] = Δ {P Ai|B i-M i-1 : i = 1,...,n} ⊂ P [0,n] , and the corresponding extremum problem simplifies to the following characterization C An→Bn FB,M = Δ sup P [0,n] Σ i=0 n I(A i ;B i |B i-M i-1 ).
C.D. Charalambous and N.U. Ahmed, “Centralized Versus Decentralized Team Optimality Conditions of Distributed Stochastic Differential Decision Systems with Different Information Structures-Part II: Applications," IEEE Transactions on Automatic Control, vol.63, no.7, pp.1913-1928, 2018
Abstract
In this second part of the two-part paper, a stochastic maximum principle, conditional Hamiltonians, and the coupled backward-forward stochastic differential equations of the first part [1] are employed to derive decentralized team optimal strategies for distributed decision systems with different information structures. Examples of such team problems are presented in nonlinear and linear quadratic forms. In many cases, the expressions of the optimal decentralized strategies are obtained. An interesting feature of any one optimal decentralized strategy is the dependence on the conditional estimates of the other optimal responses. For team problems of linear quadratic form and independent nonanticipative information structures, any optimal strategy depends linearly on the mean values of the other optimal responses. This property makes their computation feasible, even for large-scale distributed systems with many decision makers (DMs). It is also related to mean field stochastic optimal control problems, with finite number of DMs. The examples presented illustrate the effect of information signaling among the DMs in reducing the computational complexity of optimal decentralized strategies.
C.D. Charalambous and C. Kourtellaris and I. Tzortzis, “Information Transfer of Control Strategies: Dualities of Stochastic Optimal Control Theory and Feedback Capacity of Information Theory,” IEEE Transactions on Automatic Control, vol.62, no.10, pp.5010-5025, November 2017
Abstract
The control-coding capacity of stochastic control systems is introduced, and its operational meaning is established using randomized control strategies, which simultaneously control output processes encode information, and communicate information from control processes to output processes. The control-coding capacity is the analog Shannon's coding-capacity of noisy channels. Furthermore, duality relations to stochastic optimal control problems with deterministic and randomized control strategies are identified including the following. First, extremum problems of stochastic optimal control with directed information payoff are equivalent to feedback capacity problems of information theory, in which the control system act as a communication channel. Second, for Gaussian linear decision models with average quadratic constraints, it is shown that optimal randomized strategies are Gaussian, and decompose into a deterministic part and a random part. The deterministic part is precisely the optimal strategy of the linear quadratic Gaussian stochastic optimal control problem, whereas the random part is the solution of an water-filling information transmission problem that encodes information, which is estimated by a decoder.
Photios Stavrou and C.D. Charalambous and C. Kourtellaris, “Sequential Necessary and Sufficient Conditions for Capacity Achieving Distributions of Channels with Memory and Feedback,” IEEE Transactions on Information Theory, vol.63, no.11, pp.7095-7115, July 2017
Abstract
We derive sequential necessary and sufficient conditions for any channel input conditional distribution P 0,n =Δ {P Xt|Xt-1,Yt-1 : t = 0, ..., n} to maximize the finite-time horizon directed information defined by C Xn→Yn FB =Δ sup P0,n I(X n → Y n ), where I(X n → Y n ) = Σ t=0 n I(X t ; Y t |Y t-1 ), for channel distributions {P Yt |Y t-1 ,Xt: t = 0, ..., n} and {P Yt |Y t-M t-1 ,X t : t = 0, ..., n}, where Y t =Δ {Y -1 , Y 0 , ..., Y t } and X t =Δ {X 0 , ..., X t } are the channel input and output random processes, and M is a finite non-negative integer. We apply the necessary and sufficient conditions to application examples of time-varying channels with memory to derive recursive closed form expressions of the optimal distributions, which maximize the finite-time horizon directed information. Furthermore, we derive the feedback capacity from the asymptotic properties of the optimal distributions by investigating the limit C X∞→Y∞ FB =Δ lim n→∞ (1/(n + 1))C Xn→Yn FB without any á priori assumptions, such as stationarity, ergodicity, or irreducibility of the channel distribution. The framework based on sequential necessary and sufficient conditions can be easily applied to a variety of channels with memory, beyond the ones considered in this paper.
C.D. Charalambous and N.U. Ahmed, “Team Games Optimality Conditions of Distributed Stochastic Differential Decision Systems with Decentralized Noisy Information Structures," IEEE Transactions on Automatic Control, vo.62, no.2, pp.708-723, June 2017
Abstract
We derive team and Person-by-Person (PbP) optimality conditions using the stochastic Pontryagin's maximum principle, for distributed stochastic differential decision systems with decentralized noisy information structures. The optimality conditions are given by a Hamiltonian system consisting of forward and backward stochastic differential equations, and conditional Hamiltonians. We also show that, under global convexity conditions, PbP optimality implies team optimality. Finally, we apply the stochastic maximum principle to examples from communications, filtering and control.
C.D. Charalambous and N.U. Ahmed, “Centralized Versus Decentralized Team Optimality of Distributed Stochastic Differential Decision Systems with Different Information Structures-Part I: General Theory," IEEE Transactions on Automatic Control, vol.62, issue, 3, pages 1194-1209, March 2017
Abstract
Decentralized optimization of distributed stochastic dynamical systems with two or more controls of the decision makers (DMs) has been an active area of research for over half a century. Although, such decentralized optimization problems are often formulated utilizing static team and person-by-person (PbP) optimality criteria, the corresponding static team theory results have not been extended to dynamical systems. In this first part of the two-part paper, we derive team and PbP optimality conditions for distributed stochastic differential systems, when the controls of the DMs generate actions based on different information structures. The necessary conditions are given by a Hamiltonian System described by coupled backward and forward stochastic differential equations (SDEs) and a conditional Hamiltonian, conditioned on the information structures available to the controls of the DMs. The sufficient conditions state that PbP optimality implies team optimality, if the Hamiltonian is convex in the state and/or actions spaces of the controls of the DMs. We show existence of relaxed team optimal strategies, when the information structures are not affected by the controls of the DMs. Throughout the paper we discuss similarities to analogous optimality conditions of centralized decision or control of stochastic systems, and we note a connections to mean field stochastic optimal control problems.
I. Tziortzis, C.D. Charalambous, T. Charalambous, C. Hadjicostis, M. Johansson, “Approximation of Markov Processes by Lower Dimensional Processes via Total Variation Metrics," IEEE Transactions on Automatic Control, vol.62, no.3, pp.1030-1045, March 2017
Abstract
The aim of this paper is to approximate a Finite-State Markov (FSM) process by another process defined on a lower dimensional state space, called the approximating process, with respect to a total variation distance fidelity criterion. The approximation problem is formulated as an optimization problem using two different approaches. The first approach is based on approximating the transition probability matrix of the FSM process by a lower-dimensional transition probability matrix, resulting in an approximating process which is a Finite-State Hidden Markov (FSHM) process. The second approach is based on approximating the invariant probability vector of the original FSM process by another invariant probability vector defined on a lower-dimensional state space. Going a step further, a method is proposed based on optimizing a Kullback-Leibler divergence to approximate the FSHM processes by FSM processes. The solutions of these optimization problems are described by optimal partition functions which aggregate the states of the FSM process via a corresponding water-filling solution, resulting in lower-dimensional approximating processes which are FSHM or FSM processes. Throughout the paper, the theoretical results are justified by illustrative examples that demonstrate our proposed methodology.
S. Loyka and C.D. Charalambous, “Optimal Signaling for Secure Communications Over Gaussian MIMO Wiretap Channels,” IEEE Transactions on Information Theory, vol.62, no.12, pp. 7207-7215, December 2016
Abstract
Optimal signaling over the Gaussian multiple-input multiple-output wire-tap channel is studied under the total transmit power constraint. A closed-form solution for an optimal transmit covariance matrix is obtained when the channel is strictly degraded. In combination with the rank-1 solution, this provides the complete characterization of the optimal covariance for the case of two transmit antennas. The cases of weak eavesdropper and high SNR are considered. It is shown that the optimal covariance does not converge to a scaled identity in the high-SNR regime. Necessary optimality conditions and a tight upper bound on the rank of an optimal covariance matrix are established for the general case, along with a lower bound to the secrecy capacity, which is tight in a number of scenarios.
C.D. Charalambous and Photios A. Stavrou, “Directed Information on Abstract Spaces: Properties and Extremum Problems,” IEEE Transactions on Information Theory, vol.62, no.11, pages 6019-6052, November 2016
Abstract
Directed information or its variants are utilized extensively in the characterization of the capacity of channels with memory and feedback, nonanticipative lossy data compression, and their generalizations to networks. In this paper, we derive several functional and topological properties of directed information, defined on general abstract alphabets (complete separable metric spaces), using the topology of weak convergence of probability measures. These include the convexity of the set of consistent distributions, which uniquely define causally conditioned distributions, convexity, and concavity of directed information with respect to the sets of consistent distributions, weak compactness of such sets of distributions, their joint distributions, and their marginals. Furthermore, we show lower semicontinuity of directed information, and under certain conditions, we also establish continuity. Finally, we derive variational equalities for directed information, including sequential versions. These may be viewed as the analog of the variational equalities of mutual information (utilized in Blahut-Arimoto algorithms). In summary, we extend the basic functional and topological properties of mutual information to directed information. These properties are discussed throughout this paper, in the context of extremum problems of directed information.
S. Loyka and C.D. Charalambous, “A General Formula for Compound Channel Capacity,” IEEE Transactions on Information Theory, vol.62, no.7, pp.3971-3991, July 2016
Abstract
A general formula, for the capacity of arbitrary compound channels with the receiver channel state information, is obtained using the information density approach. No assumptions of ergodicity, stationarity, or information stability are made and the channel state set is arbitrary. A direct (constructive) proof is given. To prove achievability, we generalize Feinstein Lemma to the compound channel setting, and to prove converse, we generalize Verdu-Han Lemma to the same compound setting. A notion of a uniform compound channel is introduced and the general formula is shown to reduce to the familiar sup-inf expression for such channels. As a by-product, the arbitrary varying channel capacity is established under maximum error probability and deterministic coding. Conditions are established under which the worst-case and compound channel capacities are equal, so that the full channel state information at the transmitter brings in no advantage. The compound inf-information rate plays a prominent role in the general formula. Its properties are studied and a link between the information-unstable and information-stable regimes of a compound channel is established. The results are extended to include ε-capacity of compound channels. Sufficient and necessary conditions for the strong converse to hold are given.
S. Loyka and C.D. Charalambous, “Rank Deficient Solutions for Optimal Signaling Over Wiretap MIMO Channels,” IEEE Transactions on Communications, vol.64, no.6, pp.2400-2411, May, 2016
Abstract
Capacity-achieving signaling strategies for the Gaussian wiretap MIMO channel are investigated without the degradedness assumption. In addition to known solutions, a number of new rank-deficient solutions for the optimal transmit covariance matrix are obtained. The case of weak eavesdropper is considered in details and the optimal covariance is established in an explicit, closed-form with no extra assumptions. The conditions for optimality of zero-forcing signaling are established, and the standard water-filling is shown to be optimal under those conditions. No wiretap codes are needed in this case. The case of identical right singular vectors for the required and eavesdropper channels is studied and the optimal covariance is established in an explicit closed form. As a by-product of this analysis, we establish a generalization of celebrated Hadamard determinantal inequality using information-theoretic tools.
C.D. Charalambous, “Decentralized Optimality Conditions of Stochastic Differential Decision Problems via Girsanov’s Measure Transformation," Mathematics of Control, Signals, and Systems, pp.1-55, 2016
Abstract
In this paper, we apply two methods to derive necessary and sufficient decentralized optimality conditions for stochastic differential decision problems with multiple Decision Makers (DMs), which aim at optimizing a common pay-off, based on the notions of decentralized global optimality and decentralized person-by-person (PbP) optimality. Method 1: We utilize the stochastic maximum principle to derive necessary and sufficient conditions which consist of forward and backward Stochastic Differential Equations (SDEs), and conditional variational Hamiltonians, conditioned on the information structures of the DMs. The sufficient conditions for decentralized PbP optimality are local conditions, closely related to the necessary conditions for decentralized PbP optimality. However, under certain convexity condition on the Hamiltonian, and a global version of the sufficient conditions for decentralized PbP optimality, we show decentralized global optimality. Method 2: We utilize the value processes of decentralized PbP optimal policies, we relate them to solutions of backward SDEs, we identify sufficient conditions for decentralized PbP optimality, and we show these are precisely those derived via the maximum principle. For both methods, as usual, we utilize Girsanov’s theorem to transform the initial decentralized stochastic optimal decision problems, to equivalent decentralized stochastic optimal decision problems on a reference probability space, in which the controlled process and the information processes which generate part of the information structures of the DMs, are independent of any of the decisions.
S. Loyka and C.D. Charalambous, “Novel Matrix Singular Value Inequalities and their Applications to Uncertain MIMO Channels,” IEEE Transactions on Information Theory, Vol.61, No.12. pp.6623-6634, December 2015
Abstract
Novel matrix singular value inequalities are established for a sum/product of three matrices. Their application to the uncertain (compound) multiple-input multiple-output (MIMO) channel subject to normed additive uncertainty establishes the saddle-point property for a wide range of performance metrics monotonic in the channel singular values, including, among others, the mutual information, MMSE, error exponent, and pairwise error probability. This, in turn, implies that the transmission on the eigenmodes of the nominal (or worst case) channel is also optimal for the whole set of channels under a general power constraint and hence achieves the compound channel capacity. The worst case channel turns out to be antiparallel of the nominal one for all these performance metrics. An application of these results to beamforming over compound MIMO channels is discussed. An optimal robust precoder for the uncertain MIMO channel is obtained in a closed-form under the sum-MSE criterion and the total power constraint. The saddle-point property is shown to hold and the optimal strategy is to diagonalize the nominal (or worst case) channel.
S. Loyka and C.D. Charalambous, “An Algorithm for Global Maximization of Secrecy Rates Maximization in Gaussian MIMO Wiretap Channel," IEEE Transactions on Communications, Vol.63, No.6, pp.2288-2299, 2015
Abstract
Optimal signaling for secrecy rate maximization in Gaussian MIMO wiretap channels is considered. While this channel has attracted a significant attention recently and a number of results have been obtained, including the proof of the optimality of Gaussian signalling, an optimal transmit covariance matrix is known for some special cases only and the general case remains an open problem. An iterative custom-made algorithm to find a globally-optimal transmit covariance matrix in the general case is developed in this paper, with guaranteed convergence to a global optimum. While the original optimization problem is not convex and hence difficult to solve, its minimax reformulation can be solved via the convex optimization tools, which is exploited here. The proposed algorithm is based on the barrier method extended to deal with a minimax problem at hand. Its convergence to a global optimum is proved for the general case (degraded or not) and a bound for the optimality gap is given for each step of the barrier method. The performance of the algorithm is demonstrated via numerical examples. In particular, 20 to 40 Newton steps are already sufficient to solve the sufficient optimality conditions with very high precision (up to the machine precision level), even for large systems. Even fewer steps are required if the secrecy capacity is the only quantity of interest. The algorithm can be significantly simplified for the degraded channel case and can also be adopted to include the per-antenna power constraints (instead or in addition to the total power constraint). It also solves the dual problem of minimizing the total power subject to the secrecy rate constraint.
I.Tziortzis, C.D. Charalambous and T.Charalambous, "Dynamic Programming Subject to Total Variation Distance Ambiguity", SIAM Journal on Control and Optimization, Vol.53, No4, pp.2040-2075, 2015
Abstract
The aim of this paper is to address optimality of stochastic control strategies via dynamic programming subject to total variation distance ambiguity on the conditional distribution of the controlled process. We formulate the stochastic control problem using minimax theory, in which the control minimizes the payoff while the conditional distribution, from the total variation distance set, maximizes it. First, we investigate the maximization of a linear functional on the space of probability measures on abstract spaces, among those probability measures which are within a total variation distance from a nominal probability measure, and then we give the maximizing probability measure in closed form. Second, we utilize the solution of the maximization to solve minimax stochastic control with deterministic control strategies, under a Markovian and a non-Markovian assumption, on the conditional distributions of the controlled process. The results of this part include (1) minimax optimization subject to total variation distance ambiguity constraint; (2) new dynamic programming recursions, which involve the oscillator seminorm of the value function, in addition to the standard terms; and (3) a new infinite horizon discounted dynamic programming equation, the associated contractive property, and a new policy iteration algorithm. Finally, we provide illustrative examples for both the finite and infinite horizon cases. For the infinite horizon case, we invoke the new policy iteration algorithm to compute the optimal strategies. Read More: https://epubs.siam.org/doi/abs/10.1137/140955707
M.M. Olama S. Djouadi, and C.D. Charalambous, “Stochastic Modeling and Power Control of Time-Varying Wireless Communication, “Communications and Networks, Vol.6, pp.155-164, August, 2014
Abstract
Wireless networks are characterized by nodes mobility, which makes the propagation environment time-varying and subject to fading. As a consequence, the statistical characteristics of the received signal vary continuously, giving rise to a Doppler power spectral density (DPSD) that varies from one observation instant to the next. This paper is concerned with dynamical modeling of time-varying wireless fading channels, their estimation and parameter identification, and optimal power control from received signal measurement data. The wireless channel is characterized using a stochastic state-space form and derived by approximating the time-varying DPSD of the channel. The expected maximization and Kalman filter are employed to recursively identify and estimate the channel parameters and states, respectively, from online received signal strength measured data. Moreover, we investigate a centralized optimal power control algorithm based on predictable strategies and employing the estimated channel parameters and states. The proposed models together with the estimation and power control algorithms are tested using experimental measurement data and the results are presented.
T. Charalambous, C.D. Charalambous, and F. Rezaei “Optimal Merging Algorithms for Lossless Codes with Generalized Criteria,” IEEE Transactions on Information Theory, Vol.50, No. 9, pp.5486-5499, Sept 2014
Abstract
This paper presents lossless prefix codes optimized with respect to a payoff criterion consisting of a convex combination of maximum codeword length and average codeword length. The optimal codeword lengths obtained are based on a new coding algorithm, which transforms the initial source probability vector into a new probability vector according to a merging rule. The coding algorithm is equivalent to a partition of the source alphabet into disjoint sets on which a new transformed probability vector is defined as a function of the initial source probability vector and scalar parameter. The payoff criterion considered encompasses a tradeoff between maximum and average codeword length; it is related to a payoff criterion consisting of a convex combination of average codeword length and average of an exponential function of the codeword length, and to an average codeword length payoff criterion subject to a limited length constraint. A special case of the first related payoff is connected to coding problems involving source probability uncertainty and codeword overflow probability, whereas the second related payoff compliments limited length Huffman coding algorithms.
C.D. Charalambous, I.Tziortzis, S. Loyka, and T. Charalambous, "Extremum Problems with Total Variation Distance and Their Applications," IEEE Transactions on Automatic Control, Vol.33, No.9, pp.2353-2368, September 2014
Abstract
The aim of this paper is to investigate extremum problems with pay-off being the total variation distance metric defined on the space of probability measures, subject to linear functional constraints on the space of probability measures, and vice-versa; that is, with the roles of total variation metric and linear functional interchanged. Utilizing concepts from signed measures, the extremum probability measures of such problems are obtained in closed form, by identifying the partition of the support set and the mass of these extremum measures on the partition. The results are derived for abstract spaces; specifically, complete separable metric spaces known as Polish spaces, while the high level ideas are also discussed for denumerable spaces endowed with the discrete topology. These extremum problems often arise in many areas, such as, approximating a family of probability distributions by a given probability distribution, maximizing or minimizing entropy subject to total variation distance metric constraints, quantifying uncertainty of probability distributions by total variation distance metric, stochastic minimax control, and in many problems of information, decision theory, and minimax theory.
C.D. Charalambous and Photios A. Stavrou and N.U. Ahmed, “Nonanticipative Rate Distortion Function and Relations to Filtering Theory,” IEEE Transactions on Automatic Control, Vol.59, No.4, pp. 937-952, 2014
Abstract
The relation between nonanticipative rate distortion function (RDF) and filtering theory is discussed on abstract spaces. The relation is established by imposing a realizability constraint on the reconstruction conditional distribution of the classical RDF. Existence of the extremum solution of the nonanticipative RDF is shown using weak *-convergence on appropriate topology. The extremum reconstruction conditional distribution is derived in closed form, for the case of stationary processes. The realization of the reconstruction conditional distribution which achieves the infimum of the nonanticipative RDF is described. Finally, an example is presented to illustrate the concepts.
N.U. Ahmed and C.D. Charalambous, " Stochastic Minimum Principle for Partially Observed Systems Subject to Continuous and Jump Processes and Driven by Relaxed Controls," SIAM Journal on Control and Optimization, Vol.51, No4, pp.3235-3257, June 2013
Abstract
In this paper, we consider nonconvex control problems of stochastic differential equations driven by relaxed controls adapted, in the weak star sense, to a current of sigma algebras generated by observable processes. We cover in a unified way both continuous diffusion and jump processes. We present existence of optimal controls before we construct the necessary conditions of optimality (unlike some papers in this area) using only functional analysis. We develop a stochastic Hamiltonian system of equations on a rigorous basis using the semimartingale representation theory and the Riesz representation theorem, leading naturally to the existence of the adjoint process which satisfies a backward stochastic differential equation. In other words, our approach predicts the existence of the adjoint process as a natural consequence of Riesz representation theory ensuring at the same time the (weak star) measurability. This is unlike other papers, where the adjoint process is introduced before its existence is proved. We believe this is one of our major contributions in this paper. We also discuss the realizability of relaxed controls by regular controls using the Krein--Millman theorem. We believe this is another major contribution of this paper. We also believe that our approach is direct and easy to understand following simply the precise logic of functional analysis.
I. Ioannou, C.D. Charalambous and S. Loyka, “Outage Probability Under Channel Distribution Uncertainty,” IEEE Transactions on Information Theory, Vol.58, No.11, pp. 6825-6538, November 2012
Abstract
Outage probability and capacity of a class of block fading MIMO channels are considered under partial channel distribution information. Specifically, the channel or its distribution is not known but the latter is known to belong to a class of distributions where each member is within a certain distance (uncertainty) from a nominal distribution. Relative entropy is used as a measure of distance between distributions. Compound outage probability defined as min (over the transmitted signal distribution) -max (over the channel distribution class) outage probability is introduced and investigated. This generalizes the standard outage probability to the case of partial channel distribution information. Compound outage probability characterization (via 1-D convex optimization and in a closed form), its properties, and approximations are given. It is shown to have two-regime behavior: when the nominal outage probability decreases (e.g., by increasing the SNR), the compound outage first decreases linearly down to a certain threshold (related to the relative entropy distance; this is the nominal outage-dominated regime) and then only logarithmically (i.e., very slowly; this is the uncertainty-dominated regime) so that no significant further decrease is possible. This suggests the following design guideline: the outage probability is decreased by increasing the SNR or optimizing the transmitted signal distribution (both decrease nominal outage) in the first regime and by reducing the channel distribution uncertainty (e.g., via better estimation) in the second one. The compound outage depends on the relative entropy distance and the nominal outage only, all other details (nominal fading and noise distributions) being irrelevant. The transmit signal distribution optimized for the nominal channel distribution is shown to be also optimal for the whole class of distributions. The effect of swapping the distributions in relative entropy is investigated and an error floor effect is established. The compound outage probability under L p distance constraint is also investigated. The obtained results hold in full generality, i.e., for the general channel model with arbitrary nominal fading and noise distributions.
S. Loyka and C.D. Charalambous, “On the Compound Capacity of a Class of MIMO Channels Subject to Normed Uncertainty,” IEEE Transactions on Information Theory, Vol.58, No.4, pp.2048-2063, April 2012
Abstract
The compound capacity of uncertain multiple-input multiple-output channels is considered, when the channel is modeled by a class described by a (known) nominal channel and a constrained-norm (unknown) uncertainty. Within this framework, two types of classes are investigated with additive and multiplicative uncertainties subject to a spectral norm constraint, using the singular value decomposition and related singular value inequalities as the main tools. The compound capacity is a maxmin mutual information, representing the capacity of the class, in which the minimization is done over the class of channels while the maximization is done over the transmit covariance. Closed-form solutions for the compound capacity of the classes are obtained and several properties related to transmit and receive eigenvectors are presented. It is shown that, under certain conditions, the compound capacity of the class is equal to the worst-case channel capacity, thus establishing a saddle-point property. Explicit closed-form solutions are given for the worst-case channel uncertainty and the capacity-achieving transmit covariance matrix: the best transmission strategy achieving the compound capacity is a multiple beamforming on the nominal (known) channel eigenmodes with the beam power distribution via the water filling at a degraded SNR. As the uncertainty increases, fewer eigenmodes are used until only the strongest one remains active so that transmit beamforming is an optimal robust transmission strategy in this large-uncertainty regime, for which explicit conditions are given. Using these results, upper and lower bounds of the compound capacity are constructed for other bounded uncertainties and some generic properties are pointed out. The results are extended to compound multiple-access and broadcast channels. In all considered cases, the price to pay for channel uncertainty is an SNR loss (or, equivalently, the nominal channel degradation) commensurate with the uncertainty set radius measured by the spectral norm and the optimal signaling strategy is the transmission on the degraded nominal channel.
F. Rezaei, C.D. Charalambous and N.U. Ahmed, “Optimal Control of Uncertain Stochastic Systems Subject to Total Variation Distance Uncertainty,” SIAM Journal on Control and Optimization, Vol.50, No.5, pp.2683-2725, January 2012
Abstract
This paper is concerned with optimization of uncertain stochastic systems, in which uncertainty is described by a total variation distance constraint between the measures induced by the uncertain systems and the measure induced by the nominal system, while the payoff is a linear functional of the uncertain measure. Robustness at the abstract setting is formulated as a minimax game, in which the control seeks to minimize the payoff over the admissible controls while the uncertainty aims at maximizing it over the total variation distance constraint. It is shown that the maximizing measure in the total variation distance constraint exists, while the resulting payoff is a linear combination of L 1 and L ∞ norms. Further, the maximizing measure is characterized by a linear combination of a tilted measure and the nominal measure, giving rise to a payoff which is a nonlinear functional on the space of measures to be minimized over the admissible controls. The abstract formulation and results are subsequently applied to continuous-time uncertain stochastic controlled systems, in which the control seeks to minimize the payoff while the uncertainty aims to maximize it over the total variation distance constraint. The minimization over the admissible controls of the nonlinear functional payoff is addressed by developing a generalized principle of optimality or dynamic programming equation satisfied by the value function. Subsequently, it is proved that the value function satisfies a Hamilton-Jacobi-Bellman (HJB) equation. It is also shown that the value function is also a viscosity solution of the HJB equation. Finally, the linear quadratic case is studied, and it is shown that the infinity norm of a quadratic payoff is well defined and finite. Throughout the paper the formulation and conclusions are related to previous work found in the literature. © 2012 Society for Industrial and Applied Mathematics.
I. Krikidis, Z. Ding, and C.D. Charalambous, “Non-Coherent Energy Detection with Orthogonal Signalling for an Uncoded Two-Way Relay Channel,” IEEE Transactions on Vehicular Technologies, Vol.61, No.1, pp.404-409, January 2012
Abstract
In this correspondence, we investigate a two-phase two-way relay protocol for a scenario without channel coding or channel estimation. The considered configuration represents practical sensor networks, where the nodes wish to exchange some low-rate messages (e.g., 1 bit) via a shared relay under strict complexity or latency constraint (corresponding to our assumptions of no-coding and no-channel estimation). A new scheme that incorporates single-threshold noncoherent energy detection and a digital network coding process at the relay node is proposed. Energy detection is optimized for both phases of the protocol, and a threshold that exhibits no error floors is derived in closed form. Numerical results validate the efficiency of the proposed scheme and introduce the two-way relay channel as an efficient structure for noncoherent and uncoded systems as well.
X. Ma, S.M. Djouadi, C.D. Charalambous “Optimal Filtering Over Uncertain Wireless Communication Channels,” IEEE Signal Processing Letters, Vol.18, No.6, pp.359-362, June 2011
Abstract
In this letter, filtering over wireless communication channels subject to packet losses is considered. The packet losses are assumed to follow a Bernoulli distribution. The latter is interpreted as a special case of a Markov process for which hybrid filtering theory is shown to provide an exact solution. Unlike existing works where only the linear optimal state estimate is derived as a linear function of measurements, this paper shows that the optimal state estimate is in fact a nonlinear function of measurements. In addition, the optimal estimator is derived explicitly. Illustrative examples compare the performance of the linear optimal estimator and the optimal estimator, and show that the latter offer superior performance, in particular for unstable systems.
I. Ioannou and C.D. Charalambous “Mutual Information Expansions for MIMO Systems and Capacity Formulae at Low SNR,” IEEE Communication Letters, Vol.15, No.3, pp.329-331, March 2011
Abstract
The paper introduces a new asymptotic series expansion for the mutual information of multiple-input multiple-output (MIMO) channels. The expansion approaches mutual information from below, and for low signal to noise ratio (SNR) first and second order terms give a tight lower bound. The low SNR approximation is used to derive a new water-filling power allocation at the transmitter for the case when the receiver and the transmitter have perfect channel state information. The low SNR approximation is also compared against existing approximations, while the new water-filling policy is related to minimum mean square decoding subject to power constraint.
S.Z. Denic, B. Basic, and C.D. Charalambous, J. Cheng, J.M. Wang, “Information Theoretic Modeling and Analysis for Global Interconnects with Process Variations,” IEEE Transactions on VLSI Systems, Vol. 19, No.3, pp.397-410, March 2011
Abstract
As the CMOS semiconductor technology enters nanometer regime, interconnect processes must be compatible with device roadmaps and meet manufacturing targets at the specified wafer size. The resulting ubiquitous process variations cause errors in data delivering through interconnects. This paper proposes an Information Theory based design method to accommodate process variations. Different from the traditional delay based design metric, the current approach uses achievable rate to relate interconnect designs directly to communication applications. More specifically, the data communication over a typical interconnect, a bus, subject to process variations (“uncertain” bus), is defined as a communication problem under uncertainty. A data rate, called the achievable rate, is computed for such a bus, which represents the lower bound on the maximal data rate attainable over the bus. When a data rate applied over the bus is smaller than the achievable data rate, a reliable communication can be guaranteed regardless of process variations, i.e., a bit error rate arbitrarily close to zero is achievable. A single communication strategy to combat the process variations is proposed whose code rate is equal to the computed achievable rate. The simulations show that the variations in the interconnect resistivity could have the most harmful effect regarding the achievable rate reduction. Also, the simulations illustrate the importance of taking into account bus parasitic parameters correlations when measuring the influence of the process variations on the achievable rates.
A. Farhadi and C.D. Charalambous, “Stability and Reliable Data Reconstruction of Uncertain Dynamical Systems over Finite Capacity Channels,” Automatica, Vol.46, No.5, pages 889-896, May 2010
Abstract
In this paper we present an encoder, decoder and a stabilizing controller for reliable data reconstruction and robust stability of uncertain dynamic systems controlled over Additive White Gaussian Noise (AWGN) channels. The uncertainty in the dynamic system is described by a relative entropy constraint. Such an uncertainty description is a natural stochastic generalization of the sum quadratic uncertainty description. This paper complements the results of Farhadi and Charalambous (2008) by showing that the necessary condition presented there can be tight. This is shown by designing an encoder, decoder and a stabilizing controller. © 2010 Elsevier Ltd. All rights reserved.
Articles in Conferences (In reverse chronological order):
C. D. Charalambous, C. K. Kourtellaris and S. Louka, "Signalling and Control of Nonlinear Partially Observable Stochastic Control Models," 2022 European Control Conference (ECC), 2022, pp. 1152-1157
Abstract
We characterize the signalling and control rate of nonlinear partially observable stochastic control or decision models (DMs), called operational control-coding (CC) capacity. This is defined as the maximum signalling rate in bits/second, of encoding information signals into randomized control strategies, and reproducing them asymptotically at the output of the DM, with arbitrary small error probability. We show that the CC capacity is characterized by an extremum problem of an information theoretic pay-off, with information state the posteriori distribution of nonlinear filtering, subject to an average cost constraint. The dual of the CC capacity is characterized by the minimization of the average cost subject to a rate constraint. As an application example, we consider the partially observable, linear-quadratic Gaussian (LQG)-DM. We show that optimal randomized control strategies consist of an estimation and control part which controls the unobserved state of the DM, and an information transmission/signalling part which signals information through the DM.
I. Tzortzis, C. D. Charalambous and C. N. Hadjicostis, "A Distributionally Robust LQR for Systems with Multiple Uncertain Players," 2021 60th IEEE Conference on Decision and Control (CDC), 2021, pp. 3972-3977
Abstract
In this paper, we study the robust linear quadratic regulator (LQR) problem for a class of discrete-time dynamical systems composed of several uncertain players with unknown or ambiguous distribution information. A distinctive feature of the assumed model is that each player is prescribed by a nominal probability distribution and categorized according to an uncertainty level of confidence. Our approach is based on minimax optimization. By following a dynamic programming approach a closed-form expression of the robust control policy is derived. The effect of ambiguity on the performance of the LQR is studied via a sequential hierarchical game with one leader and several followers. The equilibrium solution is obtained through a maximizing, time-varying probability distribution characterizing each player’s optimal policy. The behavior of the proposed method is demonstrated through an application to a drop-shipping retail fulfillment model.
C. D. Charalambous and E. Stylianou, "Joint Nonanticipative Rate Distortion Function for a Tuple of Random Processes with Individual Fidelity Criteria," 2021 60th IEEE Conference on Decision and Control (CDC), 2021, pp. 5952-5957
Abstract
The joint nonanticipative rate distortion function (NRDF) for a tuple of random processes with individual fidelity criteria is considered. Structural properties of optimal test channel distributions are derived. For the application example of the joint NRDF of a tuple of jointly multivariate Gaussian Markov process with individual square-error fidelity criteria, a realization of the reproduction processes which induces the optimal test channel distribution is derived, and the corresponding joint NRDF is characterized. The analysis of the simplest example, of a tuple of scalar correlated Markov processes, illustrates many of the challenging aspects of such problems.
I. Tzortzis, C. N. Hadjicostis and C. D. Charalambous, "Linear Quadratic Tracking Control of Hidden Markov Jump Linear Systems Subject to Ambiguity," 2021 60th IEEE Conference on Decision and Control (CDC), 2021, pp. 2336-2341
Abstract
The linear quadratic tracking control problem is studied for a class of discrete-time uncertain Markov jump linear systems with time-varying conditional distributions. The controller is designed under the assumption that it has no access to the true states of the Markov chain, but rather it depends on the Markov chain state estimates. To deal with uncertainty, the transition probabilities of Markov state estimates between the different operating modes of the system are considered to belong in an ambiguity set of some nominal transition probabilities. The estimation problem is solved via the one-step forward Viterbi algorithm, while the stochastic control problem is solved via minimax optimization theory. An optimal control policy with some desired robustness properties is designed, and a maximizing time-varying transition probability distribution is obtained. A numerical example is given to illustrate the applicability and effectiveness of the proposed approach.
S. Louka, C. Kourtellaris and C. D. Charalambous, "Qualitative Analysis of Feedback Capacity of AGN Channels Driven by Unstable Versus Stable Autoregressive Moving Average Noise," 2021 IEEE Information Theory Workshop (ITW), 2021, pp. 1-6
Abstract
Presented are closed-form feedback capacity formulas, and lower bounds on nonfeedback achievable rates, for additive Gaussian noise (AGN) channels, driven by unstable, i.e., nonstationary, and stable, autoregressive moving average noise, ARMA (a,c),a∈(−∞,∞),c∈(−∞,∞) (with one pole at c and one zero at a), which are independent of the distributions of the initial random variables, i.e., the initial state of the noise. The feedback capacity exhibits multiple regimes of capacity; (i) for the regime that includes the stable noise, ARMA (a,c),a∈(−1,1),c∈(−1, 1) and for all transmit powers κ∈(0,∞), feedback does not increase capacity, (ii) for the regime that includes the unstable noise, ARMA (a,c),|c|>1,a∈(−1, 1), and for transmit power above a threshold κ>κmin, feedback increases capacity, and the higher the |c| the higher the capacity. The unstable regime is verified by the semi-definite program of [1]. Although, our answer} feedback does not increase capacity for stable noise, ARMA (a,c), a∈(−1, 1), c∈(−1, 1) contradicts the feedback capacity, CFB, in [2, Theorem 6.1,] CFB], [1] and the maximal information rate, Imax in [3, Theorem 7,] Imax, this is attributed to the fact that rates in [1] –[3], depend on the initial state of the noise, and these rates are not achievable, for asymptotically stationary noise. This paper uses results from [4].
C. D. Charalambous, C. Kourtellaris and S. Louka, "Sequential Characterizations of Cover and Pombra Gaussian Feedback Capacity: Generalizations to MIMO Channels via Sufficient Statistic," 2021 IEEE Information Theory Workshop (ITW), 2021, pp. 1-6
Abstract
The multiple-input multiple-output (MIMO) generalization of Cover’s and Pombra’s Gaussian feedback capacity [1] is considered. A sequential characterization of the finite block or transmission feedback information (FTFI) capacity is derived, with the optimal channel input process expressed as functional of a sufficient statistic and a Gaussian orthogonal innovations process. From the new representations follows that the MIMO version of the Cover and Pombra characterization of the FTFI capacity is expressed as a functional of two generalized matrix difference Riccati equations (DRE) of filtering theory of Gaussian systems. Analogous expressions for nonfeedback capacity are also derived, which involve a Lyapunov equation. The derivations follow directly from [2]; application examples to autoregressive moving average noise are found in [3], and to autoregressive noise in [4], [5]. Asymptotic formulas of feedback, nonfeedback capacity, and achievable lower bound incurred by asymptotically stationary channel inputs follow from the analysis of [2, Section III].
E. Stylianou, C. D. Charalambous and T. Charalambous, "Joint Rate Distortion Function of a Tuple of Correlated Multivariate Gaussian Sources with Individual Fidelity Criteria," 2021 IEEE International Symposium on Information Theory (ISIT), 2021, pp. 2167-2172
Abstract
In this paper we analyze the joint rate distortion function (RDF), for a tuple of correlated sources taking values in abstract alphabet spaces (i.e., continuous) subject to two individual distortion criteria. First, we derive structural properties of the realizations of the reproduction Random Variables (RVs), which induce the corresponding optimal test channel distributions of the joint RDF. Second, we consider a tuple of correlated multivariate jointly Gaussian RVs, X1:Ω→Rp1,X2:Ω→Rp2 with two square-error fidelity criteria, and we derive additional structural properties of the optimal realizations, and use these to characterize the RDF as a convex optimization problem with respect to the parameters of the realizations. We show that the computation of the joint RDF can be performed by semidefinite programming. Further, we derive closed-form expressions of the joint RDF, such that Gray's [1] lower bounds hold with equality, and verify their consistency with the semidefinite programming computations.
M. Gkagkos and C. D. Charalambous, "Structural Properties of Test Channels of the RDF for Gaussian Multivariate Distributed Sources," 2021 IEEE International Symposium on Information Theory (ISIT), 2021, pp. 2631-2636
Abstract
In this paper, we consider Wyner's [1] lossy distributed source coding problem of Fig. 1, for multivariate Gaussian distributed sources with square-error fidelity, when i) the side information, random variable (RV) Y , is available at the decoder only, and ii) Y is available at both the encoder and the decoder. We derive structural properties of optimal test channels that achieve the operational information rate distortion functions (RDFs) for i) and ii), i.e., R¯¯¯¯(d) and RX|Y(d) , respectively. We also show there is no loss of compression if the side information Y is only available at the decoder, i.e., R¯¯¯¯(d)=RX|Y(d) . This paper also points out that the two test channels of the RDF of the distributed remote source coding problem given in [2] and [3], do not generate Wyner's value of the RDF of scalar RVs, as one would expect; this observation implies that the results in [2, Theorem 4] and [3, Theorem 3A] should be read with caution.
B. Telsang, S. Djouadi and C. Charalambous, "Numerical Evaluation of Exact Person-by-Person Optimal Nonlinear Control Strategies of the Witsenhausen Counterexample," 2021 American Control Conference (ACC), 2021, pp. 1250-1255
Abstract
The Witsenhausen's counterexmaple formulated in 1968 is a decentralized stochastic control problem that highlights the importance of information structure in a seemingly simple two player team problem. Despite immense attention towards the problem, the non-classical information structure therein had remained the key challenge to obtain the optimal solution until recently (2014), when the Person-by-Person nonlinear optimal laws that satisfy integral equations were derived using Girsanov's transformation. In this paper, we compute and implement the nonlinear optimal laws using the Gauss Hermite Quadrature to approximate the integrals and then solving a system of non-linear equations to compute the signaling levels. We then analyse and compare our results with costs previously reported in the literature.
C. D. Charalambous and J. H. van Schuppen, "Characterization of Conditional Independence and Weak Realizations of Multivariate Gaussian Random Variables: Applications to Networks," in Proceedings of the 2020 IEEE International Symposium on Information Theory, pages-6, Los Angeles, USA, July 21-26, 2020
Abstract
The Gray and Wyner lossy source coding for a simple network for sources that generate a tuple of jointly Gaussian random variables (RVs) X 1 : Ω → R p1 and X 2 : Ω → R p2 , with respect to square-error distortion at the two decoders is reexamined using (1) Hotelling's geometric approach of Gaussian RVs-the canonical variable form, and (2) van Putten's and van Schuppen's parametrization of joint distributions P X1,X2,W by Gaussian RVs W : Ω → R n which make (X 1 ,X 2 ) conditionally independent, and the weak stochastic realization of (X 1 ,X 2 ). Item (2) is used to parametrize the lossy rate region of the Gray and Wyner source coding problem for joint decoding with mean-square error distortions E{||X i - X i || R p i 2 } ≤ Δ i ∈ [0,∞],i = 1,2, by the covariance matrix of RV W. From this then follows Wyner's common information C W (X 1 ,X 2 ) (information definition) is achieved by W with identity covariance matrix, while a formula for Wyner's lossy common information (operational definition) is derived, given by C WL (X 1 ,X 2 ) =CW (X 1 ,X 2 ) = 1/2 Σ j=1 n In (1+d j /1-d j ), for the distortion region 0 ≤ Δ 1 ≤ n(1-d 1 ), 0 ≤ Δ 2 ≤ n(1 - d 1 ), and where 1 > d 1 ≥ d 2 ≥ ... ≥ d n > 0 in (0,1) are the canonical correlation coefficients computed from the canonical variable form of the tuple (X 1 ,X 2 ). The methods are of fundamental importance to other problems of multi-user communication, where conditional independence is imposed as a constraint.
C. D. Charalambous, T. Charalambous, C. Kourtellaris and J. H. van Schuppen, "Structural Properties of Nonanticipatory Epsilon Entropy of Multivariate Gaussian Sources," in Proceedings of the 2020 IEEE International Symposium on Information Theory, pages-6, Los Angeles, USA, July 21-26, 2020
Abstract
The complete characterization of the Gorbunov and Pinsker [1], [2] nonanticipatory epsilon entropy of multivariate Gauss-Markov sources with square-error fidelity is derived, which remained an open problem since 1974. Specifically, it is shown that the optimal matrices of the stochastic realization of the optimal test channel or reproduction distribution, admit spectral representations with respect to the same unitary matrices, and that the optimal reproduction process is generated, subject to pre-processing and post-processing by memoryless parallel additive Gaussian noise channels. The derivations and analyses are new and bring out several properties of such optimization problems over the space of conditional distributions and their realizations.
C. Kourtellaris and C.D. Charalambous and S. Loyka, “New Formulas for Ergodic Feedback Capacity of AGN Channels Driven by Stable and Unstable Autoregressive Noise,” in Proceedings of the 2020 IEEE International Symposium on Information Theory, pages-6, Los Angeles, USA, July 21-26, 2020
Abstract
In this paper we characterize the feedback capacity of Additive Gaussian Noise (AGN) channels driven by stable and unstable autoregressive noise, for time-invariant feedback codes (channel input distributions). For stable (resp. unstable) channel noise we identify necessary and sufficient conditions for the optimal input process to induce asymptotic stationarity and ergodicity of the channel output (resp. innovations) process. We call this the {\emph{ergodic feedback capacity}}. From our characterization follows the surprising result: for a time-invariant unit memory Gaussian autoregressive noise AR($c$), $c\in (-\infty,\infty)$, (i) feedback does not increase capacity for the region with $c\in (-1,1)$ and certain unstable $c$, and total transmit power $\kappa \in [0,\infty)$, and (ii) feedback increases capacity for the compliment of the region of values of $(c, \kappa)$, not covered in (i).
C. Kourtellaris and C.D. Charalambous and S. Loyka, “From Feedback Capacity to Tight Achievable Bounds without Feedback for AGN Channels with Stable and Unstable Autoregressive Noise,” in Proceedings of the 2020 IEEE International Symposium on Information Theory, pages-6, Los Angeles, USA, July 21-26, 2020
Abstract
In this paper we employ the information structures of optimal channel inputs of feedback capacity of additive Gaussian noise (AGN) channels, driven by stable and unstable autoregressive noise, to derive achievable rates for nofeedback capacity, and the corresponding channel input processes. The expressions of rates are derived using time-domain methods, they hold for stable and unstable noise, for all values of transmit power. The method avoids power spectral techniques and their limitations. The lower bounds are evaluated against the classical nonfeedback capacity obtained via water-filling frequency-domain techniques.
I. Tzortzis and C.D. Charalambous and T. Charalambous, “Canonical Dynamic Programming Equations Subject to Ambiguity,” in Proceedings of the 21st IFAC Wold Congress, Berlin, Germany, July 12-17, 2020
Jan H. van Schuppen and C.D. Charalambous, “Concepts, Decompositions, and Optimal Laws for a Gaussian Team Problem,” in Proceedings of the 59th IEEE Conference on Decision Control, Nice, France, December 11-13, 2019
Abstract
It is proven that the Gaussian team problem with two controllers and without dynamics has optimal control laws which depend only on the common and correlated information of the two controllers. The private information of both controllers is not used at all. Explicit control laws are derived.
C.D. Charalambous, C. Kourtellaris, S. Loyka, T. Charalambous, “Optimal Encoders and MSE Decoders for Communicating Unstable Markov Processes over Unstable Gaussian Recusrive Models with Feedback: A Nonanticipative RDF Approach,” in Proceedings of the 2019 IEEE International Symposium on Information Theory, pages-5, July 7-12, Paris, France, 2019
Abstract
Shannon's coding capacity of memoryless additive Gaussian noise (AGN) channels with noiseless feedback, is known to be achieved by the Elias [1] coding scheme of communicating the mean square-error (MSE) of a Gaussian RV X ~ N(0, σ'), from past channel outputs, and decoding it using a MSE decoder. Further, it is known that among all encoders, and decoders that minimize the MSE, then the Elias encoder and decoder are globally optimal. In this paper we derive analogous results, for communicating unstable Gaussian Markov processes over unstable multiple-input multiple-output (MIMO) Gaussian recursive models (GRM) with memory, often called infinite impulse response (IIR) models, subject to an average cost of quadratic form. However, unlike memoryless AGNs, certain conditions are required, for such generalizations. Further, to show global optimality, we need to invoke Gorbunov and Pinsker [2] nonanticipatory € entropy, instead of the classical rate distortion function of the source. Another important observation is that we need a two-parameter coding scheme, instead of the one-parameter coding scheme of memoryless AGN channels.
C.D. Charalambous and T. Charalambous and C. Kourtellaris and J.H. van Schuppen, “Generalizations of Nonanticipative Rate Distortion Function to Nonstationary Gaussian Autoregressive Processes,” in Proceedings of the 59th IEEE Conference on Decision Control, Nice, France, December 11-13, 2019
Abstract
The characterizations of nonanticipative rate distortion function (NRDF) on a finite horizon are generalized to nonstationary multivariate Gaussian order L autoregressive, AR(L), source processes, with respect to mean square error (MSE) distortion functions. It is shown that the optimal reproduction distributions are induced by a reproduction process, which is a linear function of the state of the source, its best mean-square error estimate, and a Gaussian random process.
I. Tzortzis and C.D. Charalambous and C. Hadjicostis, “Robust Linear Quadratic Regulators for Markov Jump Systems,” in Proceedings of the 59th IEEE Conference on Decision Control, Nice, France, December 11-13, 2019
Abstract
This paper develops a robust LQG approach applicable to non-homogeneous Markov jump linear systems with uncertain transition probability distributions. The stochastic control problem is formulated using (i) minimax optimization theory, and (ii) a total variation distance metric as a tool for codifying the level of uncertainty of the jump process. By following a dynamic programming approach, a robust optimal controller is derived, which in addition to minimizing the quadratic cost, it also restricts the influence of uncertainty. A solution procedure for the LQG problem is also proposed, and an illustrative example is presented. Numerical results indicate the applicability and effectiveness of the proposed approach.
C.D. Charalambous, C. Kourtellaris and S. Loyka, “Control-Coding Capacity of Decision Models Driven by Correlated Noise and Gaussian Application Examples” in Proceedings of the 57th IEEE Conference on Decision Control, Miami, FL, USA, December, 2018
Abstract
We characterize the n -finite time feedback information (FTFI) control-coding capacity of decision models (DMs) driven by correlated noise. Under information stability the per unit limit, called control-coding (CC) capacity of the DM is operational, and analogous to Shannon's coding capacity of noisy communication channels, with the encoder replaced by a controller-encoder. We also analyze application examples of recursive linear DMs driven by correlated Gaussian noise, subject to an average cost constraint of quadratic form, called linear-quadratic Gaussian DMs (LQG-DMs). In one of the main theorems we show that the optimal randomized control strategies that achieve the n -FTFI CC capacity of the LQG-DMs, consist of multiple parts, that include control, estimation, and information transmission/signalling strategies, and that these strategies are determined using decentralized optimization techniques.
C.D. Charalambous, C. Kourtellaris and S. Loyka, “Control-Coding Capacity of Decision Models Driven by Correlated Noise and Gaussian Application Examples” in Proceedings of the 57th IEEE Conference on Decision Control, Miami, FL, USA, December, 2018
Abstract
We characterize the n -finite time feedback information (FTFI) control-coding capacity of decision models (DMs) driven by correlated noise. Under information stability the per unit limit, called control-coding (CC) capacity of the DM is operational, and analogous to Shannon's coding capacity of noisy communication channels, with the encoder replaced by a controller-encoder. We also analyze application examples of recursive linear DMs driven by correlated Gaussian noise, subject to an average cost constraint of quadratic form, called linear-quadratic Gaussian DMs (LQG-DMs). In one of the main theorems we show that the optimal randomized control strategies that achieve the n -FTFI CC capacity of the LQG-DMs, consist of multiple parts, that include control, estimation, and information transmission/signalling strategies, and that these strategies are determined using decentralized optimization techniques.
P. Stavrou, T. Charalambous, C.D. Charalambous and M. Skoglund, “Asymptotic Reverse WaterFilling Solution of Nonanticipative Rate Distortion Function for Vector-Valued Gauss-Markov Sources,” in Proceedings of the 57th IEEE Conference on Decision Control, Miami, FL, USA, December, 2018
Abstract
We analyze the asymptotic nonanticipative rate distortion function (NRDF) of vector-valued Gauss-Markov processes subject to a mean-squared error (MSE) distortion function. We derive a parametric characterization in terms of a reverse-waterfilling algorithm, that requires the solution of a matrix Riccati algebraic equation (RAE). Further, we develop an algorithm reminiscent of the classical reverse-waterfilling algorithm that provides an upper bound to the optimal solution of the reverse-waterfilling optimization problem, and under certain cases, it operates at the NRDF. Moreover, using the characterization of the reverse-waterfilling algorithm, we derive the analytical solution of the NRDF, for a simple two-dimensional parallel Gauss-Markov process. The efficacy of our proposed algorithm is demonstrated via an example.
S. Loyka and C.D. Charalambous, “The Operational Capacity of Compound Uniformy-Ergodic Fading Channels,” in 2018 International Zurich Seminar on Information and Communication, Zurich, Switzerland, February 21-23, 2018
Abstract
The impact of distribution uncertainty on the per- formance of compound fading channels is studied. To this end, a new class of fading channels, termed ”uniformly-ergodic”, is introduced and several of its equivalent (easy-to-use) characteri- zations and examples are presented. A single-letter expression for the operational capacity of this class of channels is obtained under the full Rx CSI using the recent general formula for compound channel capacity and the information spectrum approach. The saddle-point property is established, whereby the compound channel capacity is the same as the worst-case capacity so that the full knowledge of the fading distribution at the transmitter does not increase the capacity of this class of channels.
C.D. Charalambous, C. Kourtellaris and S. Loyka, “Control-Coding Capacity of Decision Models Driven by Correlated Noise and Gaussian Application Examples” in Proceedings of the 57th IEEE Conference on Decision Control, Miami, FL, USA, December, 2018
Abstract
We characterize the n -finite time feedback information (FTFI) control-coding capacity of decision models (DMs) driven by correlated noise. Under information stability the per unit limit, called control-coding (CC) capacity of the DM is operational, and analogous to Shannon's coding capacity of noisy communication channels, with the encoder replaced by a controller-encoder. We also analyze application examples of recursive linear DMs driven by correlated Gaussian noise, subject to an average cost constraint of quadratic form, called linear-quadratic Gaussian DMs (LQG-DMs). In one of the main theorems we show that the optimal randomized control strategies that achieve the n -FTFI CC capacity of the LQG-DMs, consist of multiple parts, that include control, estimation, and information transmission/signalling strategies, and that these strategies are determined using decentralized optimization techniques.
C.D. Charalambous, “A General Sufficient Conditions for Decentralized Cooperative Stochastic Differential Games,” in Proceedings of the 58th IEEE Conference on Decision Control, Miami, FL, USA, December, 2018
Abstract
Martingale techniques are applied to derive sufficient decentralized optimality conditions for general stochastic differential games, with multiple Decision Makers (DMs), who aim at optimizing a common pay-off, based on the notion of decentralized Person-by-Person (PbP) optimality. The methodology utilizes the value processes of each one of the DMs of the game, and relates them to solutions of backward stochastic differential equations (SDEs). The sufficient conditions for decentralized PbP optimality are expressed in terms of conditional Hamiltonians, conditioned on the information structures of the DMs. The mathematical models are generalizations of the ones considered in [1], while the decentralized PbP optimality conditions of this paper degenerate to the ones derived in [1] via the stochastic maximum principle.
C.D. Charalambous, C. Kourtellaris and T. Charalambous, “A General Coding Scheme of Signalling Gaussian Processes over Gaussian Decision Models” in Proceedings of the 19th IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Kalamata, Greece, June 25-28, 2018
Abstract
In this paper, we transform the n-finite transmission feedback information (FTFI) capacity of unstable Gaussian decision models with memory on past outputs, subject to an average cost constraint of quadratic form derived in [1], into controllers-encoders-decoders that control the output process, encode a Gaussian process, reconstruct the Gaussian process via a mean-square error (MSE) decoder, and achieve the n-FTFI capacity. For a Gaussian RV message X N(0,σ 2 X) it is shown that the MSE decays according to E X-X' n n 2 = -2C 0, n (k)σ X 2 , K ∈ (k min ,∞), where C 0, n (k) is the n-FTFI capacity, and k min is the threshold on the power to ensure convergence.
C.D. Charalambous and C. Kourtellaris and I. Tzortzis and S. Loyka, “The Decentralized Structures of Capacity Achieving Distributions of Channels with Memory and Feedback,” in 2018 International Zurich Seminar on Information and Communication, Zurich, Switzerland, February 21-23, 2018
Abstract
We consider extremum problems of feedback capacity for models with memory, subject to average cost constraints. We show the optimal input process that maximizes directed information consists of two parts, one responsible to control the output process, and one responsible to transmit new information that interact. Unlike [1], the decentralized structure of the optimal input process is demonstrated for Gaussian models with memory on past inputs and outputs. A semi-separation principle is shown that states, the optimal input process is generated from multiple strategies of a decentralized optimization problem, of control and information transmission. Further, it is shown that the derivation of directed information stability is semi-separable, in the sense that it separates into a statement about the ergodic properties of the stochastic optimal control problem with partial information, and a statement related to an information transmission problem.
P. Stavrou, J. Ostegaard, C.D. Charalambous, “An Upper Bound to Zero-Delay Rate Distortion via Kalman Filtering for Vector Gaussian Sources,” in Proceedings of the 2017 IEEE Information Theory Workshop, pages-6, November 6-10, Kaohsiung, Taiwan, 2017
Abstract
We deal with zero-delay source coding of a vector Gaussian autoregressive (AR) source subject to an average mean squared error (MSE) fidelity criterion. Toward this end, we consider the nonanticipative rate distortion function (NRDF) which is a lower bound to the causal and zero-delay rate distortion function (RDF). We use the realization scheme with feedback proposed in [1] to model the corresponding optimal “test-channel” of the NRDF, when considering vector Gaussian AR(1) sources subject to an average MSE distortion. We give conditions on the vector Gaussian AR(1) source to ensure asymptotic stationarity of the realization scheme (bounded performance). Then, we encode the vector innovations due to Kalman filtering via lattice quantization with subtractive dither and memoryless entropy coding. This coding scheme provides a tight upper bound to the zero-delay Gaussian RDF. We extend this result to vector Gaussian AR sources of any finite order. Further, we show that for infinite dimensional vector Gaussian AR sources of any finite order, the NRDF coincides with the zero-delay RDF. Our theoretical framework is corroborated with a simulation example.
C.D. Charalambous, C. Kourtellaris, I. Tzortzis, S. Loyka, “The Capacity of Unstable Dynamical Systems-Interaction of Control and Information Transmission,” in Proceedings of the 2017 IEEE International Symposium on Information Theory, pages-6, June 25-30, Aachen, Germany, 2017
Abstract
Feedback capacity is extended beyond classical communication channels, to stochastic dynamical systems, which may correspond to unstable control systems or unstable communication channels, subject to average cost constraints of total power κ ∈ [0, ∞). It is shown that optimal conditional distributions or randomized strategies, have a dual role, to simultaneously control the output process and to encode information. The dual role is due to the interaction of control and information transmission; it states that encoders in communication channels operate as encoders-controllers, while controllers in control systems operate as controllers-encoders. The concepts are illustrated through the analysis of Gaussian control systems with randomized strategies, which are equivalent to Additive Gaussian Noise channels, Stable or Unstable, with arbitrary memory on past outputs, with an average constraint of quadratic form. It is shown that such unstable dynamical systems have Control-Coding Capacity which is operational, precisely as in Shannon's operational definition. However, the control-coding capacity is zero, unless the power κ allocated to the system, exceeds a threshold K min , where K min is the minimum cost of ensuring asymptotic stability and ergodicity. The excess power κ - K min is turned into an achievable rate of information transmission over the dynamical system.
C.D. Charalambous and C. Kourtellaris and I. Tzortzis, “Signalling of Information in Networked Stochastic Dynamical Control Systems,” in Proceedings of the 56th IEEE Conference on Decision Control, Melbourne, Australia, December, 2017
Abstract
Signalling of information is made feasible from one controller to another controller, in a networked control system consisting of two interconnected control systems. In this application, controller 2 of control system 2 (CS-2) has access to feedback information from its output, while controller 1 of control system 1 (CS-1) does not have access to feedback information from its output. The methodology is based on computing the Control-Coding (CC) Capacity of CS-2, and then transforming the randomized control strategy which achieves the CC Capacity of CS-2, into a controller-encoder, which simultaneously controls the output of CS-2, and encodes the output of CS-1, then decodes it and applies it to the controller of CS-1, to minimize the pay-off of CS-1. The paper demonstrates that randomized strategies of control systems simultaneously control and encode information, and signal information from one processor to another processor, through the control system, provided the rate at which information is signalled through the control system is below the CC Capacity of the control system.
C. Kourtellaris, C.D. Charalambous, “Two-Letter Capacity Formula for Channels with Memory and Feedback,” in Proceedings of the 2017 IEEE Information Theory Workshop, pages-6, November 6-10, Kaohsiung, Taiwan, 2017
Abstract
For a class of channels with unit memory on previous channels outputs, we identify necessary and sufficient conditions, to test whether the capacity achieving channel input distributions with feedback are time-invariant, and whether feedback capacity is characterized by a two-letter expression, similar to that of memoryless channels. The method is based on showing that a certain dynamic programming equation, which in general, is a nested optimization problem over the sequence of channel input distributions, reduces to a non-nested optimization problem. We then apply these conditions to derive a two-letter expression for the feedback capacity of the Binary State Symmetric Channel, which is evaluated explicitly. Further, we derive computationally efficient upper bounds on the probability of maximum likelihood decoding error in the finite-blocklength regime.
C.D. Charalambous and C. Kourtellaris and I. Tzortzis, “Hierarchical Optimality of Linear Controllers-Encoders-Decoders Operating at Control-Coding Capacity of LQG Systems,” in Proceedings of the 56th IEEE Conference on Decision Control, Melbourne, Australia, December, 2017
Abstract
Randomized control strategies which achieve the Control-Coding Capacity of Linear Quadratic Gaussian Control systems with complete information, are transformed, hierarchically into controller-encoder strategies, which stabilize the control system, encode information signals, and operate at the control-coding capacity. Further, it is shown that among all controllers, encoders and decoders which minimize Mean Square Error (MSE), the conditional mean decoder is optimal and linear controller-encoder-decoders are optimal.
C.D. Charalambous, C. Kourtellaris, I. Tzortzis, S. Loyka, “The Capacity of Unstable Dynamical Systems-Interaction of Control and Information Transmission,” in Proceedings of the 2017 IEEE International Symposium on Information Theory, pages-6, June 25-30, Aachen, Germany, 2017
Abstract
Feedback capacity is extended beyond classical communication channels, to stochastic dynamical systems, which may correspond to unstable control systems or unstable communication channels, subject to average cost constraints of total power κ ∈ [0, ∞). It is shown that optimal conditional distributions or randomized strategies, have a dual role, to simultaneously control the output process and to encode information. The dual role is due to the interaction of control and information transmission; it states that encoders in communication channels operate as encoders-controllers, while controllers in control systems operate as controllers-encoders. The concepts are illustrated through the analysis of Gaussian control systems with randomized strategies, which are equivalent to Additive Gaussian Noise channels, Stable or Unstable, with arbitrary memory on past outputs, with an average constraint of quadratic form. It is shown that such unstable dynamical systems have Control-Coding Capacity which is operational, precisely as in Shannon's operational definition. However, the control-coding capacity is zero, unless the power κ allocated to the system, exceeds a threshold K min , where K min is the minimum cost of ensuring asymptotic stability and ergodicity. The excess power κ - K min is turned into an achievable rate of information transmission over the dynamical system.
C.D. Charalambous and C. Kourtellaris and I. Tzortzis, “Hierarchical Decomposition of Optimal Control and Information Strategies in Control-Coding Capacity of Stochastic Systems,” in Proceedings of the 56th IEEE Conference on Decision Control, Melbourne, Australia, December, 2017
Abstract
The Control-Coding Capacity (CC Capacity) is defined as the maximum amount of information in bits/second, which can be encoded into randomized control strategies, transmitted over the control system, and decoded at its outputs, with arbitrary asymptotic small probability of error. This paper shows that optimal randomized control strategies which achieve the CC Capacity impose a natural hierarchical decomposition into two simpler sub-optimization problems, one related to the control objectives and one related to the information transmission objectives. The hierarchical decomposition states that control signals and communication signals interact in a specific order, optimal strategies are decentralized, and the information transmission rate is zero, unless the power allocated to the overall system is above a certain threshold, which is the minimum cost to achieve the control objectives.
S. Loyka and C.D. Charalambous, “Feedback Does not Increase Capacity of Compound Channels with Additive,” in Proceedings of the 2016 IEEE International Symposium on Information Theory, pages-6, July 2016
Abstract
A discrete compound channel with memory is considered, where no stationarity, ergodicity or information stability is required, and where the uncertainty set can be arbitrary. When the discrete noise is additive but otherwise arbitrary and there is no cost constraint on the input, it is shown that the causal feedback does not increase the capacity. This extends the earlier result obtained for general channels with full transmitter (Tx) channel state information (CSI). It is further shown that, for this compound setting and under a mild technical condition on the additive noise, the addition of the full Tx CSI does not increase the capacity either, so that the worst-case and compound channel capacities are the same, thus revealing a saddle-point property.
P.A. Stavrou and T. Charalambous and C.D. Charalambous, “Filtering with Fidelity for Time-Varying Gauss-Markov Processes,” in Proceedings of the 55th IEEE Conference on Decision Control, Las Vegas, USA, December, 2016
Abstract
In this paper, we revisit the relation between Nonanticipative Rate Distortion (NRD) theory and real-time realizable filtering theory. Specifically, we give the closed form expression for the optimal nonstationary (time-varying) reproduction distribution of the Finite Time Horizon (FTH) Nonanticipative Rate Distortion Function (NRDF) and we establish its connection to real-time realizable filtering theory via a realization scheme utilizing time-varying fully observable multidimensional Gauss-Markov processes. As an application we provide the optimal filter with respect to a mean square error constraint. Unlike classical filtering theory, our filtering approach based on FTH NRDF is performed with waterfilling. We also derive a universal lower bound to the mean square error of any causal estimator to Gaussian processes based on the closed form expression of FTH NRDF. Our theoretical results are demonstrated via an illustrative example.
S. Loyka and C.D. Charalambous, “Strong Converse for General Compound Channels,” in 2016 International Zurich Seminar on Communications, Zurich, Switzerland, March 2-4, 2016
Abstract
A general compound channel is considered, where no stationarity, ergodicity or information stability is required. Following the recent result on the capacity of this channel under the full Rx CSI, sufficient and necessary conditions are obtained for the strong converse to hold. In a nutshell, even though no information satiability is required upfront, the conditions imply that there exists a sub-sequence of (bad) channel states (indexed by the blocklength) for which the respective information density rates converge in probability to the compound channel capacity, i.e. this sub-sequence is information stable.
C. Kourtellaris and C.D. Charalambous, “Information Structures of Capacity Capacity Achieving Distributions for Channels with Memory and Feedback,” in Proceedings of the 2016 IEEE International Symposium on Information Theory, pages-6, July 2016
Abstract
The information structures of the optimal channel input distributions P [0,n] = Δ {P Ai|Ai-1, Bi-1 : i = 0,1,..., n}, which correspond to the extremum problem of feedback capacity C An→Bn FB = Δ sup P [0,n] Σ i=0 n I(A i ;B i |B i-1 ) are identified, for any class of channel distributions {P Bi|Bi-1,Ai : i = 0,1,...,n} and {P Bi|B i-M i-1 ,Ai : i = 0,1,...,n}, where B n = Δ {B j : j = 0,1,...,n} are the channel output RVs, A n = Δ {A j : j = 0,1,...,n} are the channel inputs RVs, and M is a finite nonnegative integer. The methodology utilizes stochastic optimal control theory, to identify the control process, the controlled process, and a variational equality of directed information, to derive upper bounds on I(A n → B n )= Δ Σ i=0 n I(A i ;B i |B i-1 ), which are achievable over specific subsets of P [0,n] , which satisfy conditional independence. The main theorem states, that for any channel with memory M, the optimal channel input conditional distribution occur in the subset P [0,n] = Δ {P Ai|B i-M i-1 : i = 1,...,n} ⊂ P [0,n] , and the corresponding extremum problem simplifies to the following characterization C An→Bn FB,M = Δ sup P [0,n] Σ i=0 n I(A i ;B i |B i-M i-1 ).
I. Tzortzis and C.D. Charalambous and T. Charalambous and C. Kourtellaris and C Hadjicostis, “Robust Linear Quadratic Regulator for Uncertain Systems,” in Proceedings of the 55th IEEE Conference on Decision Control, Las Vegas, USA, December, 2016
Abstract
This paper develops a Linear Quadratic Regulator (LQR), which is robust to disturbance variability, by using the total variation distance as a metric. The robust LQR problem is formulated as a minimax optimization problem, resulting in a robust optimal controller which in addition to minimizing the quadratic cost it also minimizes the level of disturbance variability. A procedure for solving the LQR problem is also proposed and an example is presented which clearly illustrates the effectiveness of our developed methodology.
C.D. Charalambous and C. Kourtellaris and I. Tzortzis, “Information Transfer in Stochastic Optimal Control Theory with Randomized Strategies and Directed Information,” in Proceedings of the 55th IEEE Conference on Decision Control, Las Vegas, USA, December, 2016
Abstract
We show that stochastic dynamical control systems are capable of information transfer from control processes to output processes, with operational meaning as defined by Shannon. Moreover, we show that optimal control strategies have a dual role, specifically, (i) to transfer information from the control process to the output process, and (ii) to stabilize the output process. We illustrate that information transfer is feasible by considering general Gaussian Linear Decision Models, and relate it to the well-known Linear-Quadratic-Gaussian (LQG) control theory.
I. Tzortzis and C.D. Charalambous and T. Charalambous, “Infinite Horizon Discounted Dynamic Programming Subject to Total Variation Ambiguity on Conditional Distribution,” in Proceedings of the 55th IEEE Conference on Decision Control, Las Vegas, USA, December, 2016
Abstract
We analyze the infinite horizon minimax discounted cost Markov Control Model (MCM), for a class of controlled process conditional distributions, which belong to a ball, with respect to total variation distance metric, centered at a known nominal controlled conditional distribution with radius R ∈ [0, 2], in which the minimization is over the control strategies and the maximization is over conditional distributions. Through our analysis (i) we derive a new discounted dynamic programming equation, (ii) we show the associated contraction property, and (iii) we develop a new policy iteration algorithm. Finally, the application of the new dynamic programming and the corresponding policy iteration algorithm are shown via an illustrative example.
C.D. Charalambous, “Decentralized Sufficient and Necessary Conditions Optimality Conditions for Cooperative Stochastic Differential Decision Problems,” in Proceedings of the 55th IEEE Conference on Decision Control, Las Vegas, USA, December, 2016
Abstract
In this paper we derive necessary and sufficient Person by Person and Global (team) decentralized optimality conditions, for stochastic differential decision problems with multiple Decision Makers (DMs) or Agents aimed at optimizing a common pay-off. These consist of forward and backward stochastic differential equations, and a set of conditional variational Hamiltonians with respect to the information structures of the DMs.
Photios Stavrou and C.D. Charalambous and C. Kourtellaris, “Sequential Necessary and Sufficient Conditions for Optimal Channel Input Distributions of Channels with Memory and Feedback,” in Proceedings of the 2016 IEEE International Symposium on Information Theory, pages-6, July 2016
Abstract
We derive Sequential Necessary and Sufficient Conditions (SNSC) for any channel input distribution P 0,n =̑{P((X t )|X t-1 ,Y t-1 ):t=0,1,...,n} to maximize directed information for channel distributions of the form {P(Y t |Y t-M t-1 ,x t ): t=0,1,...,n} where X n =̑{X 0 ...,X n } , and Y n =̑{Y 0 ,...,Y n } are the channel input and output random variables, and M is nonnegative and finite. The results are obtained using the information structures of the optimal channel input distributions and the corresponding Finite Transmission Feedback Information (FTFI) capacity, convexity properties of directed information, and dynamic programming recursions. The conditions are applied to a finite alphabet channel with M = 1 to derive recursive closed form expressions for the optimal (nonstationary) distributions, which achieve the FTFI capacity. Further, ergodic feedback capacity is obtained in closed form, using the asymptotic properties of the optimal distributions. A numerical example is presented to illustrate the convergence properties of the per unit time limiting version of the FTFI capacity.
C. Kourtellaris, and C.D. Charalambous, “Capacity of Binary State Symmetric Channel with and without Feedback and Transmission Cost,” in Proceedings of the 2015 IEEE Information Theory Workshop, Jerusalem, Israel, April 26-May 1, 2015
Abstract
We consider a unit memory channel, called Binary State Symmetric Channel (BSSC), in which the channel state is the modulo2 addition of the current channel input and the previous channel output. We derive closed form expressions for the capacity and corresponding channel input distribution for the BSSC with and without feedback and transmission cost. We also show that the capacity of the BSSC, with or without feedback, is achieved by a first order symmetric Markov process.
S. Loyka and C.D. Charalambous, “A General Formula for Compound Channel Capacity,” in Proceedings of the 2015 IEEE International Symposium on Information Theory, Hong Kong, China, June 14-19, 2015
Abstract
A general formula for the capacity of arbitrary compound channels, which are not necessarily ergodic, stationary or information-stable, is obtained using the information density approach. A direct (constructive) proof is given. To prove achievability, we generalize Feinstein Lemma to the compound channel setting, and to prove converse, we generalize Verdu-Han Lemma to the same compound setting. This extends the general formula for channel capacity in [8] to arbitrary compound channels (not necessarily finite-state or countable).
C. Kourtellaris, and C.D. Charalambous, “Capacity of Binary State Symmetric Channel with and without Feedback and Transmission Cost,” in Proceedings of the 2015 IEEE Information Theory Workshop, Jerusalem, Israel, April 26-May 1, 2015
Abstract
We consider a unit memory channel, called Binary State Symmetric Channel (BSSC), in which the channel state is the modulo2 addition of the current channel input and the previous channel output. We derive closed form expressions for the capacity and corresponding channel input distribution for the BSSC with and without feedback and transmission cost. We also show that the capacity of the BSSC, with or without feedback, is achieved by a first order symmetric Markov process.
I. Tziortzis and C.D. Charalambous, “Stochastic Markov Decision Subject to Ambiguity,” in Proceedings of the 2015 SIAM Conference on Control and its Applications, Paris, France, July 8-10, 2015
I. Tziortzis and C.D. Charalambous, “Infinite Horizon Average Cost Dynamic Programming Subject to Ambiguity on Conditional Distribution,” in Proceedings of the 54th IEEE Conference on Decision Control, Osaka, December 15-18, 2015 (accepted)
Abstract
We analyze the per unit-time infinite horizon average cost Markov control model, subject to a total variation distance ambiguity on the controlled process conditional distribution. This stochastic optimal control problem is formulated as a minimax optimization problem in which the minimization is over the admissible set of control strategies, while the maximization is over the set of conditional distributions which are in a ball, with respect to the total variation distance, centered at a nominal distribution. We derive two new equivalent dynamic programming equations, and a new policy iteration algorithm. The main feature of the new dynamic programming equations is that the optimal control strategies are insensitive to inaccuracies or ambiguities in the controlled process conditional distribution. The main feature of the new policy iteration algorithm is that the policy evaluation and policy improvement steps are performed using the maximizing conditional distribution, which is obtained via a water filling solution of aggregating states together to form new states. Throughout the paper, we illustrate the new dynamic programming equations and the corresponding policy iteration algorithm to various examples. Read More: https://epubs.siam.org/doi/abs/10.1137/18M1210514
I. Tzortzis and C.D. Charalambous, “Stochastic Markov Decision Subject to Ambiguity,” in Proceedings of the 2015 SIAM Conference on Control and its Applications, Paris, France, July 8-10, 2015
Abstract
This paper addresses the optimality of stochastic control strategies based on the infinite horizon average cost criterion, subject to total variation distance ambiguity on the conditional distribution of the controlled process. This stochastic optimal control problem is formulated using minimax theory, in which the minimization is over the control strategies and the maximization is over the conditional distributions. Under the assumption that, for every stationary Markov control law the maximizing conditional distribution of the controlled process is irreducible, we derive a new dynamic programming recursion which minimizes the future ambiguity, and we propose a new policy iteration algorithm. The new dynamic programming recursion includes, in addition to the standard terms, the oscillator semi-norm of the cost-to-go. The maximizing conditional distribution is found via a water-filling algorithm. The implications of our results are demonstrated through an example.
C. Kourtellaris, C.D. Charalambous, and J. Boutros, “Nonanticipative Transmission for Sources and Channels with Memory” in Proceedings of the 2015 IEEE International Symposium on Information, Hong Kong, China, June 14-19, 2015
Abstract
In this paper we analyze nonanticipative (delayless) transmission of source symbols with memory over channels with memory (with and without feedback). We employ duality of {source, channel} pairs with respect to {distortion function, transmission cost} pairs to show achievability of nonanticipative transmission in terms of excess distortion probability. We apply the method to the Binary Markov source with Hamming distortion function and the Binary Unit Memory channel with transmission cost, with the joint-design operating optimally and in real-time, with and without feedback encoding and decoding.
C.D. Charalambous, “A Sufficient Condition for Decentralized Non-Cooperative Stochastic Differential Games,” in Proceedings of the 54th IEEE Conference on Decision Control, Osaka, December 15-18, 2015
Abstract
In this paper we use martingale techniques to derive a sufficient condition for existence of decentralized strategies in non-cooperative stochastic differential games. The sufficient condition is given by decentralized conditional Isaacs conditions, linked to backward stochastic differential equations, whose solutions are related to the value processes of each one of the Decision Makers (DMs) or Players of the game. Our formulation and results include decentralized noncooperative stochastic games with mean field and conditional mean field components.
S. Loyka and C.D. Charalambous, “Ergodic Capacity under Channel Distribution Uncertainty,” in Proceedings of the 52nd Annual Allerton Conference on Control, Communication, and Computing, University of Illinois at Urbana Champaign, Illinois, USA, October 1-3, 2014
Abstract
The impact of channel distribution uncertainty on the performance of fading channels is studied. The compound capacity of a class of ergodic fading channels subject to channel distribution uncertainty is obtained, for arbitrary noise and nominal channel distribution. The saddle-point property is established, so that the compound capacity equals to the worst-case channel capacity, which is characterized as 1-D convex optimization problem. The properties of worst-case mutual information and channel distribution are studied. Closed-form solutions are obtained in the asymptotic regimes of small and large uncertainty, and an error floor effect is established in the latter case. The known results for the ergodic capacity of the Gaussian MIMO channel under i.i.d. Rayleigh fading are shown to hold under the channel distribution uncertainty as well.
P. Stavrou, C. Kourtellaris, and C.D. Charalambous, “Applications of Information Nonanticipative Rate Distortion Function,” in Proceedings of the 2014 IEEE International Symposium on Information Theory, Honolulu, Hawaii, MA, June 29-July 4, 2014
Abstract
The objective of this paper is to further investigate various applications of information Nonanticipative Rate Distortion Function (NRDF) by discussing two working examples, the Binary Symmetric Markov Source with parameter p (BSMS(p)) with Hamming distance distortion, and the multidimensional partially observed Gaussian-Markov source. For the BSMS(p), we give the solution to the NRDF, and we use it to compute the Rate Loss (RL) of causal codes with respect to noncausal codes. For the multidimensional Gaussian-Markov source, we give the solution to the NRDF, we show its operational meaning via joint source-channel matching over a vector of parallel Gaussian channels, and we compute the RL of causal and zero-delay codes with respect to noncausal codes.
C.D. Charalambous and P. A. Ahmed, “Optimization of Directed Information and Relations to Filtering Theory,” in Proceedings of the 2014 European Control Conference, Strasbourg, France, June 24-27, 2014
Abstract
In this paper we generalize the relation between nonanticipative Rate Distortion Function (RDF) and filtering theory, to processes which are affected by the reproduction process, by utilizing the topology of weak convergence of probability measures. Specifically, this generalization is established via an optimization on the space of conditional distributions of the so-called directed information, subject to a fidelity constraint. Existence of the optimal reproduction distribution of the general nonanticipative RDF is shown, while the closed form expression of the optimal reproduction distribution is obtained for nonstationary processes. The expression of the optimal reproduction conditional distribution is recursively computed, backward in time. Finally, the realization procedure of the general nonanticipative RDF which is equivalent to joint-source channel matching for symbol-by-symbol transmission is described.
C. D Charalambous, “Team Theory of Stochastic Decision Systems with Decentralized Noiseless Information Structures via Girsanov’s,” in Proceedings of the 21st International Symposium of Mathematical Theory of Networks and Systems, Groninger, Netherlands, July 7-11, December 2014
Abstract
In this paper we generalized static team theory to dynamic team theory, in the context of stochastic differential decision system with decentralized noiseless feedback informa- tion structures. We apply Girsanov’s theorem to transformed the initial stochastic dynamic team problem to an equivalent team prob- lem, under a reference probability space, with state process and information structures independent of any of the team decisions. Subsequently, we derive team and Person-by-Person (PbP) optimality conditions, via the stochastic Pontryagin’s maxi- mum principle, consisting of forward and backward stochastic differential equations, and a set of conditional variational Hamiltonians with respect to the information structures of the team members. Finally, we relate the backward stochastic differential equa- tion to the value process of the stochastic team problem.
C.D. Charalambous and P. A. Ahmed, “Optimization of Directed Information and Relations to Filtering Theory,” in Proceedings of the 2014 European Control Conference, Strasbourg, France, June 24-27, 2014
Abstract
In this paper we generalize the relation between nonanticipative Rate Distortion Function (RDF) and filtering theory, to processes which are affected by the reproduction process, by utilizing the topology of weak convergence of probability measures. Specifically, this generalization is established via an optimization on the space of conditional distributions of the so-called directed information, subject to a fidelity constraint. Existence of the optimal reproduction distribution of the general nonanticipative RDF is shown, while the closed form expression of the optimal reproduction distribution is obtained for nonstationary processes. The expression of the optimal reproduction conditional distribution is recursively computed, backward in time. Finally, the realization procedure of the general nonanticipative RDF which is equivalent to joint-source channel matching for symbol-by-symbol transmission is described.
C. D Charalambous, T. Charalambous, and C. Hadjicostics, “Team Optimality Conditions of Differential Decision Systems with Nonclassical Information Structures,” in Proceedings of the European Control Conference, Strasbourg France, June 24-27, 2014
Abstract
We derive team optimality conditions for differential decision systems with nonclassical information structures. The necessary conditions of optimality are given in terms of Hamiltonian system of equations consisting of a coupled backward and forward differential equations and a Hamiltonian projected onto the subspace generated by the information structures. Under certain global convexity conditions it is shown that person-by-person optimality implies team optimality.
C. D Charalambous and P. Stavrou, “Optimization of Directed Information and Relations to Filtering Theory,” in Proceedings of the European Control Conference, Strasbourg France, July 24-27, 2014
Abstract
In this paper we generalize the relation between nonanticipative Rate Distortion Function (RDF) and filtering theory, to processes which are affected by the reproduction process, by utilizing the topology of weak convergence of probability measures. Specifically, this generalization is established via an optimization on the space of conditional distributions of the so-called directed information, subject to a fidelity constraint. Existence of the optimal reproduction distribution of the general nonanticipative RDF is shown, while the closed form expression of the optimal reproduction distribution is obtained for nonstationary processes. The expression of the optimal reproduction conditional distribution is recursively computed, backward in time. Finally, the realization procedure of the general nonanticipative RDF which is equivalent to joint-source channel matching for symbol-by-symbol transmission is described.
C. D Charalambous and N.U. Ahmed, “Equivalence of Decentralized Stochastic Dynamic Decision Systems via Girsanov’s Theorem Measure Transformation,” in Proceedings of the 53rd Conference on Decision and Control, Los Angeles, Californian, USA, December 15-17, 2014
Abstract
In this paper we present two methods for decentralized optimization of continuous and discrete-time stochastic dynamic decision systems, with multiple decision makers having nonclassical information structures. For both methods we apply Girsanov's change of measure to transform such problems into equivalent optimization problems, under a reference probability measure, with corresponding observations and information structures available for decisions, which are not affected by any of the decisions. © 2014 IEEE.
C. D Charalambous and N.U. Ahmed, “Maximum Principle for Decentralized Stochastic Differential Decision Systems,” in Proceedings of the 53rd Conference on Decision and Control, Los Angeles, Californian, USA, December 15-17, 2014
Abstract
In this paper we derive team and Person-by-Person (PbP) optimality conditions for Itô SDEs with nonclassical information structures. The optimality conditions are given in terms of a Hamiltonian System described by coupled backward and forward SDEs and conditional Hamiltonians, conditioned on the information structures, for regular (measurable functions) and relaxed strategies (conditional distributions).
S. Loyka and C.D. Charalambous, “Rank Deficient Solutions for Optimal Signaling over Secure MIMO Channels,” in Proceedings of the 2014 IEEE International Symposium on Information Theory, Honolulu, Hawaii, MA, June 29-July 4, 2014
Abstract
Capacity-achieving signaling strategies for the Gaussian wiretap MIMO channel are investigated without the degradedness assumption. In addition to known solutions, a number of new rank-deficient solutions for the optimal transmit covariance matrix are obtained. The case of weak eavesdropper is considered in details and the optimal covariance is established in an explicit, closed-form with no extra assumptions. The conditions for optimality of zero-forcing signaling are established, and the standard water-filling is shown to be optimal under those conditions. No wiretap codes are needed in this case. The case of identical right singular vectors for the required and eavesdropper channels is studied and the optimal covariance is established in an explicit closed form. As a by-product of this analysis, we establish a generalization of celebrated Hadamard determinantal inequality using information-theoretic tools.
P.A. Stavrou and C.D. Charalambous, “On the Relation of Nonanticipative Rate Distortion Function and Filtering Theory,” in Proceedings of the 2013 European Control Conference, Zurich, Switzerland, July 17-19, 2013
Abstract
In this paper the relation between nonanticipative rate distortion function (RDF) and Bayesian filtering theory is investigated using the topology of weak convergence of probability measures on Polish spaces. The relation is established via an optimization on the space of conditional distributions of the so-called directed information subject to fidelity constraints. Existence of the optimal reproduction distribution of the nonanticipative RDF is shown, while the optimal nonanticipative reproduction conditional distribution for stationary processes is derived in closed form. The realization procedure of nonanticipative RDF which is equivalent to joint-source channel matching for symbol-by-symbol transmission is described, while an example is introduced to illustrate the concepts.
S. Loyka and C.D. Charalambous, “Further Results on Optimal Signalling Over Secure MIMO Channels,” in Proceedings of the 2012 IEEE International Symposium on Information Theory, Istanbul, Turkey, July 7-12, 2013
Abstract
Optimal signalling over the wire-tap MIMO Gaussian channel is studied under the total transmit power constraint. The recent results are extended in several directions, including a rank-deficient solution for the optimal covariance, lower and upper capacity bounds for the general case, and characterization of optimality of the isotropic signaling. An isotropic eavesdropper model is studied, which provides (tight) upper and lower capacity bounds for the non-isotropic case and also serves as the worst-case scenario. The optimal signaling for this model is obtained in an explicit form and its properties are studied, including the high and low-SNR behavior, the conditions for the eavesdropper to be negligible and the capacity saturation effect.
P.A. Stavrou and C.D. Charalambous, “Variational Equalities of Directed Information and Applications,” in Proceedings of the 2012 IEEE International Symposium on Information Theory, Istanbul, Turkey, July 7-12, 2013
Abstract
In this paper we introduce two variational equalities of directed information, which are analogous to those of mutual information employed in the Blahut-Arimoto Algorithm (BAA). Subsequently, we introduce nonanticipative Rate Distortion Function (RDF) R o, n na (D) defined via directed information introduced in, and we establish its equivalence to Gorbunov-Pinsker's nonanticipatory ε-entropy R o, n ε (D). By invoking certain results we first establish existence of the infimizing reproduction distribution for R o, n na (D), and then we give its implicit form for the stationary case. Finally, we utilize one of the variational equalities and the closed form expression of the optimal reproduction distribution to provide an algorithm for the computation of R o, n na (D).
C. D Charalambous, I. Tziortzis, Themistoklis Charalambous, and S. Loyka, “Extremum Problems with Total Variation Distance,” in Proceedings of the 52nd IEEE Conference on Decision and Control European Control, Firenze, Italy, December 2013
Abstract
The aim of this paper is to investigate extremum problems with pay-off being the total variation distance metric defined on the space of probability measures, subject to linear functional constraints on the space of probability measures, and vice-versa; that is, with the roles of total variation metric and linear functional interchanged. Utilizing concepts from signed measures, the extremum probability measures of such problems are obtained in closed form, by identifying the partition of the support set and the mass of these extremum measures on the partition. The results are derived for abstract spaces; specifically, complete separable metric spaces known as Polish spaces, while the high level ideas are also discussed for denumerable spaces endowed with the discrete topology. These extremum problems often arise in many areas, such as, approximating a family of probability distributions by a given probability distribution, maximizing or minimizing entropy subject to total variation distance metric constraints, quantifying uncertainty of probability distributions by total variation distance metric, stochastic minimax control, and in many problems of information, decision theory, and minimax theory.
P.A. Stavrou and C.D. Charalambous, “On the Relation of Nonanticipative Rate Distortion Function and Filtering Theory,” in Proceedings of the 2013 European Control Conference, Zurich, Switzerland, July 17-19, 2013
Abstract
In this paper the relation between nonanticipative rate distortion function (RDF) and Bayesian filtering theory is investigated using the topology of weak convergence of probability measures on Polish spaces. The relation is established via an optimization on the space of conditional distributions of the so-called directed information subject to fidelity constraints. Existence of the optimal reproduction distribution of the nonanticipative RDF is shown, while the optimal nonanticipative reproduction conditional distribution for stationary processes is derived in closed form. The realization procedure of nonanticipative RDF which is equivalent to joint-source channel matching for symbol-by-symbol transmission is described, while an example is introduced to illustrate the concepts.
C. D Charalambous, I. Tziortzis, Themistoklis Charalambous, and S. Loyka, “Extremum Problems with Total Variation Distance,” in Proceedings of the 52nd IEEE Conference on Decision and Control European Control, Firenze, Italy, December 2013
Abstract
The aim of this paper is to investigate extremum problems with pay-off the total variational distance metric subject to linear functional constraints both defined on the space of probability measures, as well as related problems. Utilizing concepts from signed measures, the extremum probability measures of such problems are obtained in closed form, by identifying the partition of the support set and the mass of these extremum measures on the partition. The results are derived for abstract spaces, specifically, complete separable metric spaces, while the high level ideas are also discussed for denumerable spaces endowed with the discrete topology.
C. D Charalambous and N.U. Ahmed, “Dynamic Team Optimality Conditions of Distributed Stochastic Differential Decision Systems with Decentralized Noisy Information Structures,” in Proceedings of the 52nd IEEE Conference on Decision and Control European Control, Firenze, Italy, December 2013
Abstract
We derive team and Person-by-Person (PbP) optimality conditions using the stochastic Pontryagin's maximum principle, for distributed stochastic differential decision systems with decentralized noisy information structures. The optimality conditions are given by a Hamiltonian system consisting of forward and backward stochastic differential equations, and conditional Hamiltonians. We also show that, under global convexity conditions, PbP optimality implies team optimality. Finally, we apply the stochastic maximum principle to examples from communications, filtering and control.
X. Ma, C.D. Charalambous, S.M. Djouadi, S. Denic, “Encoder, Controller and Decoder Design of Jump Linear Systems over Jump Communication Systems,” in Proceedings of the 2013 American Control Conference, Washington, DC., USA, June 27-29, 2013
Abstract
In this work, we focus on the encoder, decoder, and controller design for jump linear systems over jump communication channels. The plant state information (PSI) and the channel state information (CSI) are assumed to be known at the both transmitter and the receiver ends. Both the PSI and the CSI are governed by a finite state Markov chain. The control and communication system is designed for the general vector observations over two different channels. One is assumed to be a vector channel, and the other one is a scalar channel. We design the optimal encoder, controller and decoder for both channels and the design scheme achieves the desired distortion. The rate distortions for both cases are also computed.
C.D. Charalambous and P.A. Stavrou, “Directed Information on Abstract Spaces: Properties and Extremum Problems,” in Proceedings of the 2012 IEEE International Symposium on Information Theory, Boston, MA, USA, July 1-6, 2012
C.D. Charalambous, I. Tziortzis, T. Charalambous “Dynamic Programming with Total Variational Distance Uncertainty,” in Proceedings of the 51st IEEE Conference on Decision and Control, Maui, Hawaii, U.S.A., December 10-13, 2012
Abstract
The aim of this paper is to address optimality of stochastic control strategies via dynamic programming subject to total variational distance uncertainty on the conditional distribution of the controlled process. Utilizing concepts from signed measures, the maximization of a linear functional on the space of probability measures on abstract spaces is investigated, among those probability measures which are within a total variational distance from a nominal probability measure. The maximizing probability measure is found in closed form. These results are then applied to solve minimax stochastic control with deterministic control strategies, under a Markovian assumption on the conditional distributions of the controlled process. The results include: 1) Optimization subject to total variational distance constraints, 2) new dynamic programming recursions, which involve the oscillator seminorm of the value function.
S. Loyka and C.D. Charalambous, “Optimal Signalling Over Secure MIMO Channels,” in Proceedings of the 2012 IEEE International Symposium on Information Theory, Boston, MA, USA, July 1-6, 2012
Abstract
Optimal signalling over the wire-tap MIMO Gaussian channel is studied under the total transmit power constraint. A direct proof of the necessary condition of optimality (signaling on the positive directions of the difference channel) is given using the necessary KKT conditions. Based on it, an explicit, closed-form solution for the optimal transmit covariance matrix is given when the latter is of the full rank. The cases of weak eavesdropper and high SNR are considered. It is shown that the optimal covariance does not converge to a scaled identity in the latter regime. A refined estimate of the rank of an optimal covariance matrix is given for the general case.
X. Ma, M. M. Olama, S. Djouadi, C.D. Charalambous, “Estimation and Identification of Time-Varying Long Term Fading Channels via the Particle Filter and the EM Algorithm,” in 2011 Proceedings of the IEEE Conference on Radio and Wireless Symposium, Phoenix, Arizona, USA, New Orleans, January 16-19, 2011
Abstract
In this paper, we are concerned with the estimation and identification of time-varying wireless long-term fading channels. The dynamics of the fading channels are captured using a mean-reverting linear stochastic differential equation driven by a Brownian motion. Recursive estimation and identification algorithms solely from received signal strength data are developed. These algorithms are based on combining the particle filter (PF) with the expectation maximization (EM) algorithm that estimate and identify the power path-loss of the channel and its parameters, respectively. Numerical results are provided to evaluate the accuracy of the proposed algorithms.
X. Ma, M. M. Olama, S. Djouadi, C.D. Charalambous, “Estimation and Identification of Time-Varying Long Term Fading Channels via the Particle Filter and the EM Algorithm,” in 2011 Proceedings of the IEEE Conference on Radio and Wireless Symposium, Phoenix, Arizona, USA, New Orleans, January 16-19, 2011
Abstract
n this paper, we are concerned with the estimation and identification of time-varying wireless long-term fading channels. The dynamics of the fading channels are captured using a mean-reverting linear stochastic differential equation driven by a Brownian motion. Recursive estimation and identification algorithms solely from received signal strength data are developed. These algorithms are based on combining the particle filter (PF) with the expectation maximization (EM) algorithm that estimate and identify the power path-loss of the channel and its parameters, respectively. Numerical results are provided to evaluate the accuracy of the proposed algorithms.
C.D. Charalambous and I. Tziortzis “Stochastic Optimal Control Subject to Ambiguity,” in Proceeding of the 18th World Congress of the International Federation of Automatic Control, Milano, Italy, August 28-September 2, 2011 (Invited Session)
Abstract
The aim of this paper is to address optimality of control strategies for stochastic control systems subject to uncertainty and ambiguity. Uncertainty corresponds to the case when the true dynamics and the nominal dynamics are different but they are defined on the same state space. Ambiguity corresponds to the case when the true dynamics are defined on a higher dimensional state space than the nominal dynamics. The paper is motivated by a brief summary of existing methods dealing with optimality of stochastic systems subject to uncertainty, and a discussion on its shortcoming when stochastic systems are ambiguous. The issues which will be discussed are the following. 1) Modeling methods for ambiguous stochastic systems, 2) formulation of optimal stochastic control systems subject to ambiguity, 3) optimality criteria for ambiguous stochastic control systems.
C.D. Charalambous, C. Kourtellaris, C. Hadjicostis, “Optimal Encoders Maximizing Directed Information of Channels with Memory and Feedback: Control and Dynamic Programming,” in Proceeding of the 18th World Congress of the International Federation of Automatic Control, Milano, Italy, August 28-September 2, 2011 (Invited Session)
Abstract
This paper is concerned with the stochastic control formulation of the capacity of channel with memory and feedback. The pay-off is the directed information from the source sequence to the channel output sequence, while the objective is to find the optimal encoder which maximizes this quantity. By invoking a change of measure technique this problem is reformulated in terms of controlling the a posteriori probability via the choice of the encoder strategies to maximize the directed information. A principle of optimality is derived and a dynamic programming equation is obtained which describe how the optimal encoder strategy should be selected. Further, a generalization of posterior matching scheme is presented.
T. Charalambous, C.D. Charalambous and F. Rezaei, “Lossless Coding with Generalized Criteria,” in Proceeding of the IEEE International Symposium on Information Theory, Saint Petersburg, Russia, July 31-August 5, 2011
Abstract
This paper presents prefix codes which minimize various criteria constructed as a convex combination of maximum codeword length and average codeword length, or, a convex combination of the average of an exponential function of the codeword length and the average codeword length. This framework encompasses as a special case several criteria previously investigated in the literature, while relations to universal coding is discussed. The coding algorithm derived is parametric resulting in re-adjusting the initial source probabilities via a weighted probability vector according to a merging rule. An algorithm is presented to compute the weighting vector.
T. Charalambous, C.D. Charalambous and F. Rezaei, “Lossless Coding with Generalized Criteria,” in Proceeding of the IEEE International Symposium on Information Theory, Saint Petersburg, Russia, July 31-August 5, 2011
Abstract
This paper presents prefix codes which minimize various criteria constructed as a convex combination of maximum codeword length and average codeword length, or, a convex combination of the average of an exponential function of the codeword length and the average codeword length. This framework encompasses as a special case several criteria previously investigated in the literature, while relations to universal coding is discussed. The coding algorithm derived is parametric resulting in re-adjusting the initial source probabilities via a weighted probability vector according to a merging rule. An algorithm is presented to compute the weighting vector.
I. Krikidis, Z. Ding and C.D. Charalambous, “A Low-Cost Non-Coherent Transmission for Uncoded Two-Way Relay Channels,” in Proceedings of IEEE First Workshop on Distributed Antenna Systems for Broadband Mobile Communications in Conjunction with IEEE Globecom 2011, Huston, TX, USA, December 5, 2011
Abstract
In this paper, we investigate a two-phase two-way relay protocol for a scenario without channel coding or channel estimation. The considered configuration represents practical sensor networks, where the nodes wish to exchange some low rate messages (e.g. 1 bit) via a shared relay under a strict complexity or latency constraint (corresponding to our assumptions of no-coding and no-channel estimation). A new scheme which incorporates a single-threshold non-coherent energy detection and a digital network coding process at the relay node is proposed. The energy detection is optimized for both phases of the protocol and a threshold that exhibits no error floors is derived in a closed form. Numerical results validate the efficiency of the proposed scheme and introduce the two-way relay channel as an efficient structure also for non-coherent and uncoded systems.
C.D. Charalambous, I. Tziortzis, F. Rezaei “Stochastic Optimal Control of Discrete-Time Systems Subject to Conditional Distribution Uncertainty,” in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, Florida, U.S.A., December 12-15, 2011
Abstract
The aim of this paper is to address optimality of control strategies for stochastic discrete time control systems subject to conditional distribution uncertainty. This type of uncertainty is motivated from the fact that the value function involves expectation with respect to the conditional distribution. The issues which will be discussed are the following. 1) Optimal stochastic control systems subject to conditional distribution uncertainty, 2) optimality criteria for stochastic control systems with conditional distribution uncertainty, including principle of optimality and dynamic programming.
C.D. Charalambous, C. Kourtellaris, C. Hadjicostis, “Optimal Encoders Maximizing Directed Information of Channels with Memory and Feedback: Control and Dynamic Programming,” in Proceeding of the 18th World Congress of the International Federation of Automatic Control, Milano, Italy, August 28-September 2, 2011 (Invited Session)
C.D. Charalambous, I. Tziortzis, F. Rezaei “Stochastic Optimal Control of Discrete-Time Systems Subject to Conditional Distribution Uncertainty,” in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, Florida, U.S.A., December 12-15, 2011
Abstract
The aim of this paper is to address optimality of control strategies for stochastic discrete time control systems subject to conditional distribution uncertainty. This type of uncertainty is motivated from the fact that the value function involves expectation with respect to the conditional distribution. The issues which will be discussed are the following. 1) Optimal stochastic control systems subject to conditional distribution uncertainty, 2) optimality criteria for stochastic control systems with conditional distribution uncertainty, including principle of optimality and dynamic programming.
C.D. Charalambous, C. Kourtellaris, C. Hadjicostis, “Optimal Encoder and Controller Strategies in Stochastic Control Subject to Rate Constraints for Channels with Memory and Feedback: Dynamic Programming,” in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, Florida, U.S.A., December 12-15, pp. 4522-4527, 2011
Abstract
This paper is concerned with the joint design of encoder/control strategies which minimize a control payoff subject to a rate constraint on the feedback information provided by the system. The control system is described by a conditional distribution, while the rate is described by the directed information of a noisy communication channel. By generalizing the Posterior Matching Scheme (PMS), a scheme that achieves operational capacity, to channels with memory and feedback, the joint optimality of encoder/control strategies is analyzed. A principle of optimality is derived and a dynamic programming equation is obtained, which together describe how the optimal encoder/control strategies should be selected. It is shown that joint optimality separates into an optimal encoder strategy based on the PMS, and a control strategy obtained via the dynamic programming equation.
Ioanna Ioannou and C.D. Charalambous “Capacity of MIMO Systems at Low SNR,” in 4th International Symposium on Communication, Control and Signal Processing, Limassol, Cyprus, March 3-March 5, 2010
Ioanna Ioannou, C.D. Charalambous and S. Loyka “Outage Capacity and Probability for a Class of MIMO Channels,” in 4th International Symposium on Communication, Control and Signal Processing, Limassol, Cyprus, March 3-March 5, 2010
C.D. Charalambous, “Uncertain Stochastic Systems on Abstract Spaces: Applications to Control and Communication Systems,” in 4th International Symposium on Communication, Control and Signal Processing, Limassol, Cyprus, March 3-March 5, 2010
I. Tziortzis and C.D. Charalambous, “Optimal Immigration Policies Based on Linear Quadratic Theory,” in 4th International Symposium on Communication, Control and Signal Processing, Limassol, Cyprus, March 3-March 5, 2010
Abstract
In this paper, it is demonstrated that the Theory of Linear Quadratic is applicable in deriving optimum immigration policies, while maintaining population and immigration levels close to certain pre-specified reference trajectories. An already existed dynamic population model found in literature and statistical data obtained from Cyprus Statistics, are used for our simulation purposes. The numerical results presented illustrate that the applied technique results in optimum immigration policies that can be well formulated for fixed as well as for variable target sets.
F. Rezaei, C.D. Charalambous, and Photios A. Stavrou “Rate Distortion Function for a Class of Relative Entropy Sources,” ,” in 19th International Symposium on Network and Systems, Budapest, Hungary, July 5-July 9, 2010
Abstract
This paper deals with rate distortion or source coding with fidelity criterion, in measure spaces, for a class of source distributions. The class of source distributions is described by a relative entropy constraint set between the true and a nominal distribution. The rate distortion problem for the class is thus formulated and solved using minimax strategies, which result in robust source coding with fidelity criterion. It is shown that minimax and maxmin strategies can be computed explicitly, and they are generalizations of the classical solution. Finally, for discrete memoryless uncertain sources, the rate distortion theorem is stated for the class omitting the derivations while the converse is derived.
C.D. Charalambous and Christos Kourtellaris “On the Optimal Reconstruction Kernel of Causal Rate Distortion Function,” in 19th International Symposium on Network and Systems, Budapest, Hungary, July 5-July 9, 2010 [Invited Paper]
Abstract
This paper considers source coding of general sources with memory, when causal feedback is available at the decoder. The rate distortion function defined as the infimum of the directed information between source and reconstruction sequences over causal data compression channels, which satisfy a distortion fidelity constraint. The form of the optimal causal data compression Kernel is derived.
C.D. Charalambous and C. Kourtellaris and C. Hadjicostis “Capacity of Channels with Memory and Feedback: Encoder Properties and Dynamic Programming,” in Proceeding of the 48th Allerton Conference on Communication, Control, Computing, Urbana Champaign, Illinois, September 29-October 1, 2010
Abstract
This paper is concerned with capacity formulae for channels with memory and feedback, properties of the capacity achieving encoder, and dynamic programming for designing optimal encoders. The source is general and the techniques discussed include outputs of dynamic systems whose conditional probability distribution depends causally on the channel output and encoder law. First, encoder strategies are identified to maximize directed information, between the source and the channel output. Second, various definitions of information capacity are introduced via directed information, and converse coding theorems are derived. Encoder properties which lead to a tight upper bound on achievable rates are identified. Specifically, it is shown that channel inputs need to be independent of past channel outputs. Third, the form of the capacity achieving encoder is described. The encoder law is a functional of the a posteriori distribution of the source output given a sequence of observable channel outputs. Here a generalization of the Posterior Matching Scheme to channels with memory and feedback is shown to hold. Finally, dynamic programming is discussed, identifying analogies with optimal stochastic control under partial information.
C.D. Charalambous, F. Rezaei “Optimal Stochastic Control of Discrete Time Systems Subject to Total Variation Distance Uncertainty,” in Proceeding of the 49th IEEE Conference on Decision and Control, Atlanta, Georgia, December 15-17, 2010
Abstract
This paper presents another application of the results in, where existence of the maximizing measure over the total variation distance constraint is established, while the maximizing pay-off is shown to be equivalent to an optimization of a pay-off which is a linear combination of L 1 and L ∞ norms. Here emphasis is geared towards to uncertain discrete-time controlled stochastic dynamical system, in which the control seeks to minimize the pay-off while the measure seeks to maximize it over a class of measures described by a ball with respect to the total variation distance centered at a nominal measure. Two types of uncertain classes are considered; an uncertainty on the joint distribution, an uncertainty on the conditional distribution. The solution of the minimax problem is investigated via dynamic programming.
C.D. Charalambous and Christos Kourtellaris “Rate Distortion with Causal Decoding,” in Proceeding of the 49th IEEE Conference on Decision and Control, Atlanta, Georgia, December 15-17, 2010
Abstract
This paper considers source coding of general sources with memory, when causal feedback is available at the decoder. The rate distortion function is defined as the infimum of the so-called directed information over causal data compression channels which satisfy a distortion fidelity constraint. The expression of the optimal causal data compression is derived. Further a tight lower bound on the rate distortion bound is derived. These are variants of the classical non-causal rate distortion function, and associate Shannon lower bound. Generalization of the results to controlled sources is also discussed.
