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.Comment: 7 pages, 6 figures, accepted for publication in IEEE Information Theory Workshop (ITW

Information Nonanticipative Rate Distortion Function and Its Applications

This paper investigates applications of nonanticipative Rate Distortion Function (RDF) in a) zero-delay Joint Source-Channel Coding (JSCC) design based on average and excess distortion probability, b) in bounding the Optimal Performance Theoretically Attainable (OPTA) by noncausal and causal codes, and computing the Rate Loss (RL) of zero-delay and causal codes with respect to noncausal codes. These applications are described using two running examples, the Binary Symmetric Markov Source with parameter p, (BSMS(p)) and the multidimensional partially observed Gaussian-Markov source. For the multidimensional Gaussian-Markov source with square error distortion, the solution of the nonanticipative RDF is derived, its operational meaning using JSCC design via a noisy coding theorem is shown by providing the optimal encoding-decoding scheme over a vector Gaussian channel, and the RL of causal and zero-delay codes with respect to noncausal codes is computed. For the BSMS(p) with Hamming distortion, the solution of the nonanticipative RDF is derived, the RL of causal codes with respect to noncausal codes is computed, and an uncoded noisy coding theorem based on excess distortion probability is shown. The information nonanticipative RDF is shown to be equivalent to the nonanticipatory epsilon-entropy, which corresponds to the classical RDF with an additional causality or nonanticipative condition imposed on the optimal reproduction conditional distribution.Comment: 34 pages, 12 figures, part of this paper was accepted for publication in IEEE International Symposium on Information Theory (ISIT), 2014 and in book Coordination Control of Distributed Systems of series Lecture Notes in Control and Information Sciences, 201

A zero-delay sequential scheme for lossy coding of individual sequences

We consider adaptive sequential lossy coding of bounded individual sequences when the performance is measured by the sequentially accumulated mean squared distortion. The encoder and the decoder are connected via a noiseless channel of capacity $R$ and both are assumed to have zero delay. No probabilistic assumptions are made on how the sequence to be encoded is generated. For any bounded sequence of length $n$, the distortion redundancy is defined as the normalized cumulative distortion of the sequential scheme minus the normalized cumulative distortion of the best scalar quantizer of rate $R$ which is matched to this particular sequence. We demonstrate the existence of a zero-delay sequential scheme which uses common randomization in the encoder and the decoder such that the normalized maximum distortion redundancy converges to zero at a rate $n^{-1/5}\log n$ as the length of the encoded sequence $n$ increases without bound.Lossy source coding, scalar quantization, sequential prediction, individual sequences