Rate-Distortion via Markov Chain Monte Carlo

TL;DR

This paper introduces a novel MCMC-based method for lossy source coding that achieves optimal rate-distortion performance and demonstrates promising results in denoising applications.

Contribution

It presents a new MCMC algorithm for lossy source coding that leverages Gibbs sampling and annealing, achieving theoretical optimality and practical effectiveness.

Findings

Achieves near-optimal rate-distortion performance for stationary ergodic sources.

Complexity per iteration is independent of sequence length, depending only on a small context parameter.

Demonstrates competitive denoising performance compared to existing MCMC schemes and DUDE.

Abstract

We propose an approach to lossy source coding, utilizing ideas from Gibbs sampling, simulated annealing, and Markov Chain Monte Carlo (MCMC). The idea is to sample a reconstruction sequence from a Boltzmann distribution associated with an energy function that incorporates the distortion between the source and reconstruction, the compressibility of the reconstruction, and the point sought on the rate-distortion curve. To sample from this distribution, we use a `heat bath algorithm': Starting from an initial candidate reconstruction (say the original source sequence), at every iteration, an index i is chosen and the i-th sequence component is replaced by drawing from the conditional probability distribution for that component given all the rest. At the end of this process, the encoder conveys the reconstruction to the decoder using universal lossless compression. The complexity of each iteration is independent of the sequence length and only linearly dependent on a certain context parameter (which grows sub-logarithmically with the sequence length). We show that the proposed algorithms achieve optimum rate-distortion performance in the limits of large number of iterations, and sequence length, when employed on any stationary ergodic source. Experimentation shows promising initial results. Employing our lossy compressors on noisy data, with appropriately chosen distortion measure and level, followed by a simple de-randomization operation, results in a family of denoisers that compares favorably (both theoretically and in practice) with other MCMC-based schemes, and with the Discrete Universal Denoiser (DUDE).