Bayesian Signal Component Decomposition via Diffusion-within-Gibbs Sampling

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 component-wise model-driven and data-driven priors into diffusion models in a unified manner. Moreover, the proposed posterior sampler allows component priors to be learned separately and flexibly combined for different decomposition tasks at inference time. Under suitable assumptions, the proposed Diffusion-within-Gibbs (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.