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Register a new userOn the Cover and Pombra Gaussian Feedback Capacity: Complete Sequential Characterizations via a Sufficient Statistic
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Added by
Stelios Louka
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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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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.
We consider the problem of decentralized sequential active hypothesis testing (DSAHT), where two transmitting agents, each possessing a private message, are actively helping a third agent--and each other--to learn the message pair over a discrete memoryless multiple access channel (DM-MAC). The third agent (receiver) observes the noisy channel output, which is also available to the transmitting agents via noiseless feedback. We formulate this problem as a decentralized dynamic team, show that optimal transmission policies have a time-invariant domain, and characterize the solution through a dynamic program. Several alternative formulations are discussed involving time-homogenous cost functions and/or variable-length codes, resulting in solutions described through fixed-point, Bellman-type equations. Subsequently, we make connections with the problem of simplifying the multi-letter capacity expressions for the noiseless feedback capacity of the DM-MAC. We show that restricting attention to distributions induced by optimal transmission schemes for the DSAHT problem, without loss of optimality, transforms the capacity expression, so that it can be thought of as the average reward received by an appropriately defined stochastic dynamical system with time-invariant state space.
The feedback sum-rate capacity is established for the symmetric $J$-user Gaussian multiple-access channel (GMAC). The main contribution is a converse bound that combines the dependence-balance argument of Hekstra and Willems (1989) with a variant of the factorization of a convex envelope of Geng and Nair (2014). The converse bound matches the achievable sum-rate of the Fourier-Modulated Estimate Correction strategy of Kramer (2002).
This paper studies the capacity of the peak-and-average-power-limited Gaussian channel when its output is quantized using a dithered, infinite-level, uniform quantizer of step size $Delta$. It is shown that the capacity of this channel tends to that of the unquantized Gaussian channel when $Delta$ tends to zero, and it tends to zero when $Delta$ tends to infinity. In the low signal-to-noise ratio (SNR) regime, it is shown that, when the peak-power constraint is absent, the low-SNR asymptotic capacity is equal to that of the unquantized channel irrespective of $Delta$. Furthermore, an expression for the low-SNR asymptotic capacity for finite peak-to-average-power ratios is given and evaluated in the low- and high-resolution limit. It is demonstrated that, in this case, the low-SNR asymptotic capacity converges to that of the unquantized channel when $Delta$ tends to zero, and it tends to zero when $Delta$ tends to infinity. Comparing these results with achievability results for (undithered) 1-bit quantization, it is observed that the dither reduces capacity in the low-precision limit, and it reduces the low-SNR asymptotic capacity unless the peak-to-average-power ratio is unbounded.
Inference on vertex-aligned graphs is of wide theoretical and practical importance.There are, however, few flexible and tractable statistical models for correlated graphs, and even fewer comprehensive approaches to parametric inference on data arising from such graphs. In this paper, we consider the correlated Bernoulli random graph model (allowing different Bernoulli coefficients and edge correlations for different pairs of vertices), and we introduce a new variance-reducing technique -- called emph{balancing} -- that can refine estimators for model parameters. Specifically, we construct a disagreement statistic and show that it is complete and sufficient; balancing can be interpreted as Rao-Blackwellization with this disagreement statistic. We show that for unbiased estimators of functions of model parameters, balancing generates uniformly minimum variance unbiased estimators (UMVUEs). However, even when unbiased estimators for model parameters do {em not} exist -- which, as we prove, is the case with both the heterogeneity correlation and the total correlation parameters -- balancing is still useful, and lowers mean squared error. In particular, we demonstrate how balancing can improve the efficiency of the alignment strength estimator for the total correlation, a parameter that plays a critical role in graph matchability and graph matching runtime complexity.
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