Provable Mixed-Noise Learning with Flow-Matching

2025-08-25

AI Summary

This work addresses the joint inference of unknown, spatially heterogeneous additive and multiplicative Gaussian mixture noise in Bayesian inverse problems. We propose a novel framework integrating conditional flow matching with the Expectation-Maximization (EM) algorithm. Crucially, we embed an ODE-based flow matching model within the EM loop: the E-step leverages the flow model to efficiently approximate the high-dimensional posterior distribution, while the M-step updates the unknown noise parameters. We provide the first theoretical guarantee that this method converges to the true noise parameters in the infinite-data limit. Experiments demonstrate that our approach simultaneously and accurately recovers both the heterogeneous noise structure and the posterior distribution. It exhibits scalability and robustness in high-dimensional settings and significantly outperforms conventional inference methods that assume fixed, homogeneous noise.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Estimating unknown mixed noise parameters
• Solving Bayesian inverse problems with heterogeneous noise
• Jointly learning posterior samplers and noise characteristics

Innovation

• Flow matching with EM algorithm
• Simulation-free ODE-based generative model
• Joint posterior and noise estimation