arXiv subjects

Charalambos D. Charalambous

Publications and source records attributed to Charalambos D. Charalambous.

At least 55 records · Page 3Linked to original sources

arXiv2016

A General Formula for Compound Channel Capacity

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 information-unstable and information-stable regimes of a compound channel is established. The results are extended to include $\varepsilon$-capacity of compound channels. Sufficient and necessary conditions for the strong converse to hold are given.

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arXiv2016

Capacity Achieving Distributions & Information Lossless Randomized Strategies for Feedback Channels with Memory: The LQG Theory of Directed Information-Part II

A methodology is developed to realized optimal channel input conditional distributions, which maximize the finite-time horizon directed information, for channels with memory and feedback, by information lossless randomized strategies. The methodology is applied to general Time-Varying Multiple Input Multiple Output (MIMO) Gaussian Linear Channel Models (G-LCMs) with memory, subject to average transmission cost constraints of quadratic form. The realizations of optimal distributions by randomized strategies are shown to exhibit a decomposion into a deterministic part and a random part. The decomposition reveals the dual role of randomized strategies, to control the channel output process and to transmit new information over the channels. Moreover, a separation principle is shown between the computation of the optimal deterministic part and the random part of the randomized strategies. The dual role of randomized strategies generalizes the Linear-Quadratic-Gaussian (LQG) stochastic optimal control theory to directed information pay-offs. The characterizations of feedback capacity are obtained from the per unit time limits of finite-time horizon directed information, without imposing á priori assumptions, such as, stability of channel models or ergodicity of channel input and output processes. For time-invariant MIMO G-LCMs with memory, it is shown that whether feedback increases capacity, is directly related to the channel parameters and the transmission cost function, through the solutions of Riccati matrix equations, and moreover for unstable channels, feedback capacity is non-zero, provided the power exceeds a critical level.

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arXiv2015

Directed Information on Abstract Spaces: Properties and Variational Equalities

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 for general abstract alphabets (complete separable metric spaces) using the topology of weak convergence of probability measures. These include 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 these 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 of directed information. Finally, we derive variational equalities for directed information, including sequential versions. These may be viewed as the analogue of the variational equalities of mutual information (utilized in Blahut-Arimoto algorithm). In summary, we extend the basic functional and topological properties of mutual information to directed information. These properties are discussed in the context of extremum problems of directed information.

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arXiv2015

Infinite Horizon Average Cost Dynamic Programming Subject to Total Variation Distance Ambiguity

We analyze the infinite horizon minimax average 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 $R\in [0,2]$, in which the minimization is over the control strategies and the maximization is over conditional distributions. Upon performing the maximization, a dynamic programming equation is obtained which includes, in addition to the standard terms, the oscillator semi-norm of the cost-to-go. First, the dynamic programming equation is analyzed for finite state and control spaces. We show that if the nominal controlled process distribution is irreducible, then for every stationary Markov control policy the maximizing conditional distribution of the controlled process is also irreducible for $R \in [0,R_{max}]$. Second, the generalized dynamic programming is analyzed for Borel spaces. We derive necessary and sufficient conditions for any control strategy to be optimal. Through our analysis, new dynamic programming equations and new policy iteration algorithms are derived. The main feature of the new policy iteration algorithms (which are applied for finite alphabet spaces) is that the policy evaluation and policy improvement steps are performed by using the maximizing conditional distribution, which is obtained via a water filling solution. Finally, the application of the new dynamic programming equations and the corresponding policy iteration algorithms are shown via illustrative examples.

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arXiv2015

An Algorithm for Global Maximization of Secrecy Rates in Gaussian MIMO Wiretap Channels

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 \textit{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.

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arXiv2015

Capacity of Binary State Symmetric Channel with and without Feedback and Transmission Cost

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, of this BSSC with and without feedback and transmission cost. We also show that the capacity of the BSSC is not increased by feedback, and it is achieved by a first order symmetric Markov process.

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arXiv2014

Approximation of Markov Processes by Lower Dimensional Processes via Total Variation Metrics

The aim of this paper is to approximate a finite-state Markov process by another process with fewer states, called herein the approximating process. The approximation problem is formulated using two different methods. The first method, utilizes the total variation distance to discriminate the transition probabilities of a high dimensional Markov process and a reduced order Markov process. The approximation is obtained by optimizing a linear functional defined in terms of transition probabilities of the reduced order Markov process over a total variation distance constraint. The transition probabilities of the approximated Markov process are given by a water-filling solution. The second method, utilizes total variation distance to discriminate the invariant probability of a Markov process and that of the approximating process. The approximation is obtained via two alternative formulations: (a) maximizing a functional of the occupancy distribution of the Markov process, and (b) maximizing the entropy of the approximating process invariant probability. For both formulations, once the reduced invariant probability is obtained, which does not correspond to a Markov process, a further approximation by a Markov process is proposed which minimizes the Kullback-Leibler divergence. These approximations are given by water-filling solutions. Finally, the theoretical results of both methods are applied to specific examples to illustrate the methodology, and the water-filling behavior of the approximations.

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arXiv2014

Applications of Information Nonanticipative Rate Distortion Function

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.

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arXiv2014

Source-Channel Matching for Sources with Memory

In this paper we analyze the probabilistic matching of sources with memory to channels with memory so that symbol-by-symbol code with memory without anticipation are optimal, with respect to an average distortion and excess distortion probability. We show achievability of such a symbolby- symbol code with memory without anticipation, and we show matching for the Binary Symmetric Markov source (BSMS(p)) over a first-order symmetric channel with a cost constraint.

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arXiv2014

Dynamic Programming Subject to Total Variation Distance Ambiguity

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 pay-off 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; 3) 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.

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arXiv2014

Nonanticipative Rate Distortion Function and Filtering Theory: A weak Convergence Approach

In this paper the relation between nonanticipative rate distortion function (RDF) and Bayesian filtering theory is further investigated on general 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 using the topology of weak convergence of probability measures. Subsequently, we use the solution of the nonanticipative RDF to present the realization of a multidimensional partially observable source over a scalar Gaussian channel. We show that linear encoders are optimal, establishing joint source-channel coding in real-time.

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arXiv2013

Dynamic Team Theory of Stochastic Differential Decision Systems with Decentralized Noisy Information Structures via Girsanov's Measure Transformation

In this paper, we present two methods which generalize static team theory to dynamic team theory, in the context of continuous-time stochastic nonlinear differential decentralized decision systems, with relaxed strategies, which are measurable to different noisy information structures. For both methods we apply Girsanov's measure transformation to obtain an equivalent dynamic team problem under a reference probability measure, so that the observations and information structures available for decisions, are not affected by any of the team decisions. The first method is based on function space integration with respect to products of Wiener measures, and generalizes Witsenhausen's [1] definition of equivalence between discrete-time static and dynamic team problems. The second method is based on stochastic Pontryagin's maximum principle. The team optimality conditions are given by a "Hamiltonian System" consisting of forward and backward stochastic differential equations, and a conditional variational Hamiltonian with respect to the information structure of each team member, expressed under the initial and a reference probability space via Girsanov's measure transformation. Under global convexity conditions, we show that that PbP optimality implies team optimality. In addition, we also show existence of team and PbP optimal relaxed decentralized strategies (conditional distributions), in the weak$^*$ sense, without imposing convexity on the action spaces of the team members. Moreover, using the embedding of regular strategies into relaxed strategies, we also obtain team and PbP optimality conditions for regular team strategies, which are measurable functions of decentralized information structures, and we use the Krein-Millman theorem to show realizability of relaxed strategies by regular strategies.

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arXiv2013

Dynamic Team Theory of Stochastic Differential Decision Systems with Decentralized Noiseless Feedback Information Structures via Girsanov's Measure Transformation

In this paper we generalized static team theory to dynamic team theory, in the context of stochastic differential decision system with decentralized noiseless feedback information structures. We apply Girsanov's theorem to transformed the initial stochastic dynamic team problem to an equivalent team problem, under a reference probability space, with state process and information structures independent of any of the team decisions. Subsequently, we show, under certain conditions, that continuous-time and discrete-time stochastic dynamic team problems, can be transformed to equivalent static team problems, although computing the optimal team strategies using this method might be computational intensive. Therefore, we propose an alternative method, by deriving team and Person-by-Person (PbP) optimality conditions, via the stochastic Pontryagin's maximum 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 equation to the value process of the stochastic team problem.

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arXiv2013

Nonanticipative Rate Distortion Function and Relations to Filtering Theory

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.

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arXiv2013

Team and Person-by-Person Optimality Conditions of Differential Decision Systems

In this paper, we derive team and person-by-person optimality conditions for distributed differential decision systems with different or decentralized 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 decentralized information structures. Under certain global convexity conditions it is shown that the optimality conitions are also sufficient.

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arXiv2013

Variational Equalities of Directed Information and Applications

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) ${R}^{na}_{0,n}(D)$ defined via directed information introduced in [1], and we establish its equivalence to Gorbunov-Pinsker's nonanticipatory $ε$-entropy $R^{\varepsilon}_{0,n}(D)$. By invoking certain results we first establish existence of the infimizing reproduction distribution for ${R}^{na}_{0,n}(D)$, 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 ${R}^{na}_{0,n}(D)$.

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arXiv2013

Rate Distortion Function for a Class of Relative Entropy Sources

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.

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arXiv2013

On the relation of nonanticipative rate distortion function and filtering theory

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.

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