We consider sources and channels with memory observed through erasure channels. In particular, we examine the impact of sporadic erasures on the fundamental limits of lossless data compression, lossy data compression, channel coding, and denoising. We define the erasure entropy of a collection of random variables as the sum of entropies of the individual variables conditioned on all the rest. The erasure entropy measures the information content carried by each symbol knowing its context. The erasure entropy rate is shown to be the minimal amount of bits per erasure required to recover the lost information in the limit of small erasure probability. When we allow recovery of the erased symbols within a prescribed degree of distortion, the fundamental tradeoff is described by the erasure rate–distortion function which we characterize. We show that in the regime of sporadic erasures, knowledge at the encoder of the erasure locations does not lower the rate required to achieve a given distortion. When no additional encoded information is available, the erased information is reconstructed solely on the basis of its context by a denoiser. Connections between erasure entropy and discrete denoising are developed. The decrease of the capacity of channels with memory due to sporadic memoryless erasures is also characterized in wide generality.

Abstract—In many applications, an uncompressed source stream is systematically encoded by a channel code (which ignores the source redundancy) for transmission over a discrete memoryless channel. The decoder knows the channel and the code but does not know the source statistics. This paper proposes several universal channel decoders that take advantage of the source redundancy without requiring prior knowledge of its statistics. Index Terms—Belief propagation, channel decoding, denoising, discrete memoryless channels, joint source–channel decoding, lossless compression, soft decoding, universal algorithms. I.

Abstract — Erasure entropy rate (introduced recently by Verdú and Weissman) differs from Shannon’s entropy rate in that the conditioning occurs with respect to both the past and the future, as opposed to only the past (or the future). In this paper, universal algorithms for estimating erasure entropy rate are proposed based on the basic and extended context-tree weighting (CTW) algorithms. Consistency results are shown for those CTW based algorithms. Simulation results for those algorithms applied to Markov sources, tree sources and English texts are compared to those obtained by fixed-order plug-in estimators with different orders. An estimate of the erasure entropy of English texts based on the proposed algorithms is about 0.22 bits per letter, which can be compared to an estimate of about 1.3 bits per letter for the entropy rate of English texts by a similar CTW based algorithm.

Abstract—A relationship between information theory and estimation theory was recently shown for the Gaussian channel, relating the derivative of mutual information with the minimum mean-square error. This paper generalizes the link between information theory and estimation theory to arbitrary channels, giving representations of the derivative of mutual information as a function of the conditional marginal input distributions given the outputs. We illustrate the use of this representation in the efficient numerical computation of the mutual information achieved by inputs such as specific codes or natural language. Index Terms—Computation of mutual information, extrinsic information, input estimation, low-density parity-check (LDPC) codes, minimum mean square error (MMSE), mutual information, soft channel decoding. I.

Abstract—Erasure entropy rate differs from Shannon’s entropy rate in that the conditioning occurs with respect to both the past and the future, as opposed to only the past (or the future). In this paper, consistent universal algorithms for estimating erasure entropy rate are proposed based on the basic and extended context-tree weighting (CTW) algorithms. Simulation results for those algorithms applied to Markov sources, tree sources, and English texts are compared to those obtained by fixed-order plug-in estimators with different orders. Index Terms—Bidirectional context tree, context-tree weighting, data compression, entropy rate, universal algorithms, universal modeling. I.

A fundamental relationship between information theory and estimation theory was recently unveiled for the Gaussian channel, relating the derivative of mutual information with the minimum mean-square error. This paper generalizes this fundamental link between information theory and estimation theory to arbitrary channels and in particular encompasses the discrete memoryless channel (DMC). In addition to the intrinsic theoretical interest of such a result, it naturally leads to an efficient numerical computation of mutual information for cases in which it was previously infeasible such as with LDPC codes. 1

We consider the Wyner–Ziv (WZ) problem of lossy compression where the decompressor observes a noisy version of the source, whose statistics are unknown. A new family of WZ coding algorithms is proposed and their universal optimality is proven. Compression consists of sliding-window processing followed by Lempel–Ziv (LZ) compression, while the decompressor is based on a modification of the discrete universal denoiser (DUDE) algorithm to take advantage of side information. The new algorithms not only universally attain the fundamental limits, but also suggest a paradigm for practical WZ coding. The effectiveness of our approach is illustrated with experiments on binary images, and English text using a low complexity algorithm motivated by our class of universally optimal WZ codes.