Provable Mixed-Noise Learning with Flow-Matching

TL;DR

This paper introduces a new inference framework combining flow matching and EM algorithms to effectively solve high-dimensional Bayesian inverse problems with mixed, unknown Gaussian noise, demonstrating convergence and practical effectiveness.

Contribution

It proposes a novel EM-based approach with flow matching for joint noise parameter estimation and posterior sampling in mixed-noise inverse problems, scalable to high dimensions.

Findings

The method converges to true noise parameters under certain conditions.

Simulation-free ODE-based flow matching enhances scalability.

Numerical experiments validate the approach's effectiveness.

Abstract

We study Bayesian inverse problems with mixed noise, modeled as a combination of additive and multiplicative Gaussian components. While traditional inference methods often assume fixed or known noise characteristics, real-world applications, particularly in physics and chemistry, frequently involve noise with unknown and heterogeneous structure. Motivated by recent advances in flow-based generative modeling, we propose a novel inference framework based on conditional flow matching embedded within an Expectation-Maximization (EM) algorithm to jointly estimate posterior samplers and noise parameters. To enable high-dimensional inference and improve scalability, we use simulation-free ODE-based flow matching as the generative model in the E-step of the EM algorithm. We prove that, under suitable assumptions, the EM updates converge to the true noise parameters in the population limit of infinite observations. Our numerical results illustrate the effectiveness of combining EM inference with flow matching for mixed-noise Bayesian inverse problems.