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Charalambos D. Charalambous

Publications and source records attributed to Charalambos D. Charalambous.

At least 37 records · Page 2Linked to original sources

arXiv2021

On the Cover and Pombra Gaussian Feedback Capacity: Complete Sequential Characterizations via a Sufficient Statistic

The main objective of this paper is to derive a new sequential characterization of the Cover and Pombra \cite{cover-pombra1989} characterization of the $n-$finite block or transmission feedback information ($n$-FTFI) capacity, which clarifies several issues of confusion and incorrect interpretation of results in literature. The optimal channel input processes of the new equivalent sequential characterizations are expressed as functionals of a sufficient statistic and a Gaussian orthogonal innovations process. From the new representations follows that the Cover and Pombra characterization of the $n-$FTFI capacity is expressed as a functional of two generalized matrix difference Riccati equations (DRE) of filtering theory of Gaussian systems. This contradicts results which are redundant in the literature, and illustrates the fundamental complexity of the feedback capacity formula.

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arXiv2021

Joint Rate Distortion Function of a Tuple of Correlated Multivariate Gaussian Sources with Individual Fidelity Criteria

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, $X_1 : Ω\rightarrow {\mathbb R}^{p_1}, X_2 : Ω\rightarrow {\mathbb R}^{p_2}$ 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.

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arXiv2021

Joint Nonanticipative Rate Distortion Function for a Tuple of Random Processes with Individual Fidelity Criteria

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. Further, for the application example of the joint NRDF of a tuple of jointly multivariate Gaussian Markov processes 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.

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arXiv2021

A New Approach to Lossy Network Compression of a Tuple of Correlated Multivariate Gaussian RVs

The classical Gray and Wyner source coding for a simple network for sources that generate a tuple of multivariate, correlated Gaussian random variables $(Y_1,Y_2)$ is re-examined using the geometric approach of Gaussian random variables, and the weak stochastic realization of correlated Gaussian random variables. New results are: (1) The formulation, methods and algorithms to parametrize all random variables $W : Ω\rightarrow {\mathbb R}^n $ which make the two components of the tuple $(Y_1,Y_2)$ conditionally independent, according to the weak stochastic realization of $(Y_1, Y_2)$. (2) The transformation of random variables $(Y_1,Y_2)$ via non-singular transformations $(S_1,S_2)$, into their canonical variable form. (3) A formula for Wyner's lossy common information for joint decoding with mean-square error distortions. (4) The methods are shown to be of fundamental importance to the parametrization of the lossy rate region of the Gray and Wyner source coding problem, and the calculation of the smallest common message rate $R_0$ on the Gray and Wyner source problem, when the sum rate $R_0+R_1+R_2$ is arbitrary close to the joint rate distortion function $R_{Y_1, Y_2}(Δ_1, Δ_2)$ of joint decoding. The methods and algorithms may be applicable to other problems of multi-user communication, such as, the multiple access channel, etc. The discussion is largely self-contained and proceeds from first principles; basic concepts of weak stochastic realization theory of multivariate correlated Gaussian random variables are reviewed, while certain results are developed to meet the requirement of results (1)-(4).

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arXiv2020

Structural Properties of Optimal Test Channels for Distributed Source Coding with Decoder Side Information for Multivariate Gaussian Sources with Square-Error Fidelity

This paper focuses on the structural properties of test channels, of Wyner's operational information rate distortion function (RDF), $\overline{R}(Δ_X)$, of a tuple of multivariate correlated, jointly independent and identically distributed Gaussian random variables (RVs), $\{X_t, Y_t\}_{t=1}^\infty$, $X_t: Ω\rightarrow {\mathbb R}^{n_x}$, $Y_t: Ω\rightarrow {\mathbb R}^{n_y}$, with average mean-square error at the decoder, $\frac{1}{n} {\bf E}\sum_{t=1}^n||X_t - \widehat{X}_t||^2\leq Δ_X$, when $\{Y_t\}_{t=1}^\infty$ is the side information available to the decoder only. We construct optimal test channel realizations, which achieve the informational RDF, $\overline{R}(Δ_X) \triangleq\inf_{{\cal M}(Δ_X)} I(X;Z|Y)$, where ${\cal M}(Δ_X)$ is the set of auxiliary RVs $Z$ such that, ${\bf P}_{Z|X,Y}={\bf P}_{Z|X}$, $\widehat{X}=f(Y,Z)$, and ${\bf E}\{||X-\widehat{X}||^2\}\leq Δ_X$. We show the fundamental structural properties: (1) Optimal test channel realizations that achieve the RDF, $\overline{R}(Δ_X)$, satisfy conditional independence, $ {\bf P}_{X|\widehat{X}, Y, Z}={\bf P}_{X|\widehat{X},Y}={\bf P}_{X|\widehat{X}}, \hspace{.2in} {\bf E}\Big\{X\Big|\widehat{X}, Y, Z\Big\}={\bf E}\Big\{X\Big|\widehat{X}\Big\}=\widehat{X} $ and (2) similarly for the conditional RDF, ${R}_{X|Y}(Δ_X) \triangleq \inf_{{\bf P}_{\widehat{X}|X,Y}:{\bf E}\{||X-\widehat{X}||^2\} \leq Δ_X} I(X; \widehat{X}|Y)$, when $\{Y_t\}_{t=1}^\infty$ is available to both the encoder and decoder, and the equality $\overline{R}(Δ_X)={R}_{X|Y}(Δ_X)$.

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arXiv2020

New Formulas of Feedback Capacity for AGN Channels with Memory: A Time-Domain Sufficient Statistic Approach

In the recent paper [1] it is shown, via an application example, that the Cover and Pombra [2] "characterization of the $n-$block or transmission" feedback capacity formula, of additive Gaussian noise (AGN) channels, is the subject of much confusion in the literature, with redundant incorrect results. The main objective of this paper is to clarify the main points of confusion and remove any further ambiguity. The first part of the paper applies time-domain methods, to derive for a first time, equivalent sequential characterizations of the Cover and Pombra characterization of feedback capacity of AGN channels driven by nonstationary and nonergodic Gaussian noise. The optimal channel input processes of the new equivalent sequential characterizations are expressed as functionals of a sufficient statistic and a Gaussian orthogonal innovations process. From the new representations follows that the Cover and Pombra $n-$block capacity formula is expressed as a functional of two generalized matrix difference Riccati equations (DRE) of filtering theory of Gaussian systems, contrary to results that appeared in the literature. In the second part of the paper the existence of the asymptotic limit of the $n-$block feedback capacity formula is shown to be equivalent to the convergence properties of solutions of the two generalized DREs. Further, necessary and or sufficient conditions are identified for existence of the asymptotic limits, for stable and unstable Gaussian noise, when the optimal input distributions are time-invariant, but not necessarily stationary. The paper contains an in depth analysis, with examples, of the specific technical issues, which are overlooked in past literature [3-7], that studied the AGN channel of [2], for stationary noises.

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arXiv2020

Numerical Evaluation of Exact Person-by-Person Optimal Nonlinear Control Strategies of the Witsenhausen Counterexample

Witsenhausen's 1968 counterexmaple is a simple two-stage decentralized stochastic control problem that highlighted the difficulties of sequential decision problems with non-classical information structures. Despite extensive prior efforts, what is known currently, is the exact Person-by-Person (PbP) optimal nonlinear strategies, which satisfy two nonlinear integral equations, announced in 2014, and obtained using Girsanov's change of measure transformations. In this paper, we provide numerical solutions to the two exact nonlinear PbP optimal control strategies, using the Gauss Hermite Quadrature to approximate the integrals and then solve a system of non-linear equations to compute the signaling levels. Further, we analyse and compare our numerical results to existing results previously reported in the literature.

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arXiv2020

Characterization of Conditional Independence and Weak Realizations of Multivariate Gaussian Random Variables: Applications to Networks

The Gray and Wyner lossy source coding for a simple network for sources that generate a tuple of jointly Gaussian random variables (RVs) $X_1 : Ω\rightarrow {\mathbb R}^{p_1}$ and $X_2 : Ω\rightarrow {\mathbb R}^{p_2}$, with respect to square-error distortion at the two decoders is re-examined 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 ${\bf P}_{X_1, X_2, W}$ by Gaussian RVs $W : Ω\rightarrow {\mathbb 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 ${\bf E}\big\{||X_i-\hat{X}_i||_{{\mathbb R}^{p_i}}^2 \big\}\leq Δ_i \in [0,\infty], i=1,2$, 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 $C_{WL}(X_1,X_2)=C_W(X_1,X_2) = \frac{1}{2} \sum_{j=1}^n \ln \left( \frac{1+d_j}{1-d_j} \right),$ for the distortion region $ 0\leq Δ_1 \leq \sum_{j=1}^n(1-d_j)$, $0\leq Δ_2 \leq \sum_{j=1}^n(1-d_j)$, and where $1 > d_1 \geq d_2 \geq \ldots \geq d_n>0$ in $(0,1)$ are {\em 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.

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arXiv2018

Zero-Delay Rate Distortion via Filtering for Vector-Valued Gaussian Sources

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 (OPTA) function 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. 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. 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.

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arXiv2017

An Upper Bound to Zero-Delay Rate Distortion via Kalman Filtering for Vector Gaussian Sources

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.

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arXiv2017

Optimal Estimation via Nonanticipative Rate Distortion Function and Applications to Time-Varying Gauss-Markov Processes

In this paper, we develop {finite-time horizon} causal filters using the nonanticipative rate distortion theory. We apply the {developed} theory to {design optimal filters for} time-varying multidimensional Gauss-Markov processes, subject to a mean square error fidelity constraint. We show that such filters are equivalent to the design of an optimal \texttt{\{encoder, channel, decoder\}}, which ensures that the error satisfies {a} fidelity constraint. Moreover, we derive a universal lower bound on the mean square error of any estimator of time-varying multidimensional Gauss-Markov processes in terms of conditional mutual information. Unlike classical Kalman filters, the filter developed is characterized by a reverse-waterfilling algorithm, which ensures {that} the fidelity constraint is satisfied. The theoretical results are demonstrated via illustrative examples.

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arXiv2017

Single Letter Expression of Capacity for a Class of Channels with Memory

We study finite alphabet channels with Unit Memory on the previous Channel Outputs called UMCO channels. 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 single letter, expressions, 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. Moreover, for UMCO channels, we give a simple expression for the ML error exponent, and we identify sufficient conditions to test whether feedback does not increase capacity. We derive similar results, when transmission cost constraints are imposed. We apply the results to a special class of the UMCO channels, the Binary State Symmetric Channel (BSSC) with and without transmission cost constraints, to show that the optimization problem of feedback capacity is non-nested, the capacity achieving channel input distribution and the corresponding channel output transition probability distribution are time-invariant, and feedback capacity is characterized by a single letter formulae, precisely as Shannon's single letter characterization of capacity of memoryless channels. Then we derive closed form expressions for the capacity achieving channel input distribution and feedback capacity. We use the closed form expressions to evaluate an error exponent for ML decoding.

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arXiv2016

Feedback Does Not Increase the Capacity of Compound Channels with Additive Noise

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 single-state channels with full transmitter (Tx) channel state information (CSI) to the compound setting. 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. This can also be expressed as a saddle-point in the information-theoretic game between the transmitter (who selects the input distribution) and the nature (who selects the channel state), even though the objective function (the inf-information rate) is not convex/concave in the right way. Cases where the Tx CSI does increase the capacity are identified. Conditions under which the strong converse holds for this channel are studied. The ergodic behaviour of the worst-case noise in otherwise information-unstable channel is shown to be both sufficient and necessary for the strong converse to hold, including feedback and no feedback cases.

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arXiv2016

Optimal Signaling for Secure Communications over Gaussian MIMO Wiretap Channels

Optimal signalling over the Gaussian MIMO 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.

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arXiv2016

Information Structures of Maximizing Distributions of Feedback Capacity for General Channels with Memory & Applications

For any class of channel conditional distributions, with finite memory dependence on channel input RVs $A^n {\stackrel{\triangle}{=}} \{A_i: i=0, \ldots, n\}$ or channel output RVs $B^n {\stackrel{\triangle}{=}} \{B_i: i=0, \ldots, n\}$ or both, we characterize the sets of channel input distributions, which maximize directed information defined by $ I(A^n \rightarrow B^n) {\stackrel{\triangle}{=}} \sum_{i=0}^n I(A^i;B_i|B^{i-1}) $ and we derive the corresponding expressions, called "characterizations of Finite Transmission Feedback Information (FTFI) capacity". The main theorems state that optimal channel input distributions occur in subsets ${\cal P}_{[0,n]}^{CI}\subseteq {\cal P}_{[0,n]} {\stackrel{\triangle}{=}}\big\{ {\bf P}_{A_i|A^{i-1}, B^{i-1}}: i=0, \ldots, n\big\}$, which satisfy conditional independence on past information. We derive similar characterizations, when general transmission cost constraints are imposed. Moreover, we also show that the structural properties apply to general nonlinear and linear autoregressive channel models defined by discrete-time recursions on general alphabet spaces, and driven by arbitrary distributed noise processes. We derive these structural properties by invoking stochastic optimal control theory and variational equalities of directed information, to identify tight upper bounds on $I(A^n \rightarrow B^n)$, which are achievable over subsets of conditional distributions ${\cal P}_{[0,n]}^{CI} \subseteq {\cal P}_{[0,n]}$, which satisfy conditional independence and they are specified by the dependence of channel distributions and transmission cost functions on inputs and output symbols. We apply the characterizations to recursive Multiple Input Multiple Output Gaussian Linear Channel Models with limited memory and we show a separation principle between the computation of the elements of the optimal strategies.

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arXiv2016

Information Structures for Feedback Capacity of Channels with Memory and Transmission Cost: Stochastic Optimal Control & Variational Equalities-Part I

The Finite Transmission Feedback Information (FTFI) capacity is characterized for any class of channel conditional distributions $\big\{{\bf P}_{B_i|B^{i-1}, A_i} :i=0, 1, \ldots, n\big\}$ and $\big\{ {\bf P}_{B_i|B_{i-M}^{i-1}, A_i} :i=0, 1, \ldots, n\big\}$, where $M$ is the memory of the channel, $B^n {\stackrel{\triangle}{=}} \{B_j: j=\ldots, 0,1, \ldots, n\}$ are the channel outputs and $A^n{\stackrel{\triangle}{=}} \{A_j: j=\ldots, 0,1, \ldots, n\}$ are the channel inputs. The characterizations of FTFI capacity, are obtained by first identifying the information structures of the optimal channel input conditional distributions ${\cal P}_{[0,n]} {\stackrel{\triangle}{=}} \big\{ {\bf P}_{A_i|A^{i-1}, B^{i-1}}: i=0, \ldots, n\big\}$, which maximize directed information. The main theorem states, for any channel with memory $M$, the optimal channel input conditional distributions occur in the subset satisfying conditional independence $\stackrel{\circ}{\cal P}_{[0,n]}{\stackrel{\triangle}{=}} \big\{ {\bf P}_{A_i|A^{i-1}, B^{i-1}}= {\bf P}_{A_i|B_{i-M}^{i-1}}: i=1, \ldots, n\big\}$, and the characterization of FTFI capacity is given by $C_{A^n \rightarrow B^n}^{FB, M} {\stackrel{\triangle}{=}} \sup_{ \stackrel{\circ}{\cal P}_{[0,n]} } \sum_{i=0}^n I(A_i; B_i|B_{i-M}^{i-1}) $. The methodology utilizes stochastic optimal control theory and a variational equality of directed information, to derive upper bounds on $I(A^n \rightarrow B^n)$, which are achievable over specific subsets of channel input conditional distributions ${\cal P}_{[0,n]}$, which are characterized by conditional independence. For any of the above classes of channel distributions and transmission cost functions, a direct analogy, in terms of conditional independence, of the characterizations of FTFI capacity and Shannon's capacity formulae of Memoryless Channels is identified.

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arXiv2016

Rank-Deficient Solutions for Optimal Signaling over Wiretap MIMO Channels

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 a weak eavesdropper is considered in detail and the optimal covariance is established in an explicit, closed form with no extra assumptions. This provides lower and upper bounds to the secrecy capacity in the general case with a bounded gap, which are tight for a weak eavesdropper or/and low SNR. Closed form solutions are also obtained for isotropic and omnidirectional eavesdroppers, based on which lower and upper bounds to the secrecy capacity are established in the general case. Sufficient and necessary conditions for optimality of 3 popular transmission techniques, namely the zero-forcing (ZF), the standard water-filling (WF) over the channel eigenmodes and the isotropic signaling (IS), are established for the MIMO wiretap channel. These solutions are appealing due to their lower complexity. In particular, no wiretap codes are needed for the ZF transmission, and no precoding or feedback is needed for the isotropic signaling.

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arXiv2016

Sequential Necessary and Sufficient Conditions for Capacity Achieving Distributions of Channels with Memory and Feedback

We derive sequential necessary and sufficient conditions for any channel input conditional distribution ${\cal P}_{0,n}\triangleq\{P_{X_t|X^{t-1},Y^{t-1}}:~t=0,\ldots,n\}$ to maximize the finite-time horizon directed information defined by $$C^{FB}_{X^n \rightarrow Y^n} \triangleq \sup_{{\cal P}_{0,n}} I(X^n\rightarrow{Y^n}),~~~ I(X^n \rightarrow Y^n) =\sum_{t=0}^n{I}(X^t;Y_t|Y^{t-1})$$ for channel distributions $\{P_{Y_t|Y^{t-1},X_t}:~t=0,\ldots,n\}$ and $\{P_{Y_t|Y_{t-M}^{t-1},X_t}:~t=0,\ldots,n\}$, where $Y^t\triangleq\{Y_0,\ldots,Y_t\}$ and $X^t\triangleq\{X_0,\ldots,X_t\}$ are the channel input and output random processes, and $M$ is a finite nonnegative integer. \noi We apply the necessary and sufficient conditions to application examples of time-varying channels with memory and we derive recursive closed form expressions of the optimal distributions, which maximize the finite-time horizon directed information. Further, we derive the feedback capacity from the asymptotic properties of the optimal distributions by investigating the limit $$C_{X^\infty \rightarrow Y^\infty}^{FB} \triangleq \lim_{n \longrightarrow \infty} \frac{1}{n+1} C_{X^n \rightarrow Y^n}^{FB}$$ without any á priori assumptions, such as, stationarity, ergodicity or irreducibility of the channel distribution. The necessary and sufficient conditions can be easily extended to a variety of channels with memory, beyond the ones considered in this paper.

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