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Bayesian Signal Component Decomposition via Diffusion-within-Gibbs Sampling

Yi Zhang, Rui Guo, Yonina C. Eldar

TL;DR

This work tackles multi-component signal decomposition under a linear mixture model by introducing Diffusion-within-Gibbs (DiG), a Bayesian framework that combines per-component diffusion priors with Gibbs updates to sample from the posterior over components. It unifies model-driven and data-driven priors through diffusion priors, enabling modular learning of component priors without retraining and providing asymptotic consistency under perfect priors. The authors develop conditional diffusion-based samplers for Gibbs updates, extend to general sensing operators via relaxation, and employ parameter annealing to ensure robust mixing. Theoretical results establish consistency with the relaxed posterior and position DiG relative to existing diffusion samplers, while experiments on image decomposition and heartbeat extraction demonstrate improved decomposition quality, particularly with hybrid priors and limited training data. Overall, DiG offers a scalable, principled approach for Bayesian component separation that leverages both physics-based priors and learned statistics in a cohesive diffusion framework.

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.

Bayesian Signal Component Decomposition via Diffusion-within-Gibbs Sampling

TL;DR

This work tackles multi-component signal decomposition under a linear mixture model by introducing Diffusion-within-Gibbs (DiG), a Bayesian framework that combines per-component diffusion priors with Gibbs updates to sample from the posterior over components. It unifies model-driven and data-driven priors through diffusion priors, enabling modular learning of component priors without retraining and providing asymptotic consistency under perfect priors. The authors develop conditional diffusion-based samplers for Gibbs updates, extend to general sensing operators via relaxation, and employ parameter annealing to ensure robust mixing. Theoretical results establish consistency with the relaxed posterior and position DiG relative to existing diffusion samplers, while experiments on image decomposition and heartbeat extraction demonstrate improved decomposition quality, particularly with hybrid priors and limited training data. Overall, DiG offers a scalable, principled approach for Bayesian component separation that leverages both physics-based priors and learned statistics in a cohesive diffusion framework.

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.
Paper Structure (24 sections, 5 theorems, 54 equations, 2 figures, 2 tables, 3 algorithms)

This paper contains 24 sections, 5 theorems, 54 equations, 2 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Proposed Method
  4. Preliminaries on Diffusion Models
  5. Forward and Reverse-Time Diffusion Processes
  6. Implementation of Diffusion Models
  7. Prior Modeling
  8. Problem Formulation and Assumptions
  9. Incorporating Component Priors into a Diffusion Model
  10. Diffusion-within-Gibbs Sampling
  11. Conditional Sampling via Diffusion When $\boldsymbol{H}_k=\boldsymbol{I}$
  12. Reducing General $\boldsymbol{H}_k$ to the Identity Case via Relaxation
  13. Parameter Annealing Technique
  14. Theoretical Analysis and Discussion
  15. Consistency of the DiG Sampler
  16. ...and 9 more sections

Key Result

Lemma 1

Let $\left(\bar{\boldsymbol{x}}_{k,t}\right)_{t=0}^T$ be defined as in Assumption assump:diff. Then given $\eta>0$, for any $\boldsymbol{z}\in\mathbb{R}^{d_k}$, we have where $\sigma^{-1}$ denotes the inverse of the noise-scale function $\sigma(\cdot)$ in Assumption assump:diff.

Figures (2)

  • Figure 1: Estimates given by all algorithms in a representative problem instance. First row: the observation and ground-truth of all component signals. Other rows: estimates given by each algorithm. The red boxes highlight representative artifact regions in the estimates produced by each method.
  • Figure 2: Estimates of heartbeat signals given by all algorithms in representative problem instances with different SIR--SNR settings. GT: ground-truth.

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Corollary 5