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

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 (OPTA) 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.

Sequential Empirical Coordination Under an Output Entropy Constraint

“…Then X and U are conditionally independent given U . Using this fact and the uniform Lipschitz property (20), we have for…”

“…The problem becomes more intricate when causality constraints are imposed, and decisions must be made in real time. There are many recent works on this topic dealing with various types of information constraints and structural assumptions about the source (see, e.g., [13]- [20]).…”

Zero-Delay Multiple Descriptions of Stationary Scalar Gauss-Markov Sources

“…All these scenarios may operate under strict requirements on latency and reliability. Particularly, delays play a critical role in the performance or stability of these systems [ 8 ].…”

“…Recently, the authors of [ 12 ] showed in the symmetric case, i.e., equal rates and distortions for each individual description, that the MD region for a colored Gaussian source subject to MSE distortion constraints can be achieved by predictive coding using filtering. However, similar to single-description source coding [ 8 ], the MD source coders whose performance is close to the fundamental rate-distortion bounds impose long delays on the end-to-end processing of information, i.e., the total delay only due to source coding [ 15 ].…”

“…In the single-description case, ZD rate-distortion theory has been increasingly more popular in recent decades, due to its significance in real-time communication systems and especially feedback systems. Some indicative results on ZD source coding for networked control systems and systems with and without feedback may be found in [ 5 , 6 , 7 , 8 , 17 , 19 , 20 , 21 ]. The results of [ 5 ] establish a novel information-theoretic lower bound on the average data-rate for a source coding scheme within a feedback loop by the directed information rate across the channel.…”