Asymptotic Reverse Waterfilling Algorithm of NRDF for Certain Classes of Vector Gauss–Markov Processes

Photios A Stavrou,Mikael Skoglund

doi:10.1109/tac.2021.3099444

Abstract

The existence of an optimal reverse-waterfilling algorithm to compute the nonanticipative rate distortion function (NRDF) for time-invariant vector-valued Gauss–Markov processes with a mean-squared-error distortion has been an open question since the pioneering work of Tatikonda <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> on stochastic linear control over a communication channel, in 2004. In this article, we derive <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">strong structural properties</i> on the time-invariant multidimensional Gauss–Markov processes that allow for an optimization problem that can be computed optimally via a reverse-waterfilling algorithm. Moreover, we propose an elegant optimal iterative scheme that computes this reverse-waterfilling algorithm. We show that the specific scheme operates much faster than any existing algorithmic approach that solves the same problem optimally and is also scalable. Finally, using our new results, we derive for the first time a nontrivial analytical solution of the asymptotic NRDF using a correlated time-invariant 2-D Gauss–Markov process.

Highlights

Tatikonda et al in [2] studied the fundamental limitations of a multidimensional closed-loop control system when a communication link intervenes between a stochastic linear fully observable time-invariant plant driven by a Gaussian noise process and a controller whereas the performance criterion is the classical linear quadratic cost
To compute the communication cost, they introduced a Gaussian sequential rate distortion function (SRDF) defined by minimizing a variant of directed information [3] subject to a pointwise mean squared error (MSE) distortion function, a definition that is attributed to Gorbunov and Pinsker in [4]
(1) We show that for certain classes of time-invariant vectorvalued Gauss-Markov processes, [1, Theorem 3] simplifies to a reverse-waterfilling algorithm obtained in matrix form

Summary

Introduction

Tatikonda et al in [2] studied the fundamental limitations of a multidimensional closed-loop control system when a communication link intervenes between a stochastic linear fully observable time-invariant plant driven by a Gaussian noise process and a controller whereas the performance criterion is the classical linear quadratic cost. To compute the communication cost, they introduced a Gaussian sequential rate distortion function (SRDF) defined by minimizing a variant of directed information [3] subject to a pointwise mean squared error (MSE) distortion function, a definition that is attributed to Gorbunov and Pinsker in [4]. This information measure when solved, it ensures a lower bound on the communication cost irrespectively of whether the channel is noiseless or noisy.

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