Bayesian Signal Component Decomposition via Diffusion-within-Gibbs Sampling

TL;DR

This paper introduces a Bayesian signal decomposition method that combines Gibbs sampling with diffusion priors, enabling flexible incorporation of prior knowledge and improved posterior sampling for noisy superimposed signals.

Contribution

It develops a novel DiG sampler that unifies model-driven and data-driven priors, allowing flexible prior learning and superior signal component estimation.

Findings

DiG sampler provably samples from the posterior distribution.

The method outperforms existing approaches in numerical experiments.

DiG better exploits measurement model structure for certain sensing operators.

Abstract

In signal processing, the data collected from sensing devices is often a noisy linear superposition of multiple components, and the estimation of components of interest constitutes a crucial pre-processing step. In this work, we develop a Bayesian framework for signal component decomposition, which combines Gibbs sampling with plug-and-play (PnP) diffusion priors to draw component samples from the posterior distribution. Unlike many existing methods, our framework supports incorporating model-driven and data-driven prior knowledge into the diffusion prior in a unified manner. Moreover, the proposed posterior sampler allows component priors to be learned separately and flexibly combined without retraining. Under suitable assumptions, the proposed DiG sampler provably produces samples from the posterior distribution. We also show that DiG can be interpreted as an extension of a class of recently proposed diffusion-based samplers, and that, for suitable classes of sensing operators, DiG better exploits the structure of the measurement model. Numerical experiments demonstrate the superior performance of our method over existing approaches.