Solving Inverse Problems with Score-Based Generative Priors learned from Noisy Data

TL;DR

SURE-Score introduces a method to learn score-based generative models from noisy data, enabling effective inverse problem solving without requiring large clean datasets, demonstrated on wireless channel estimation and MRI reconstruction.

Contribution

The paper proposes a novel loss function based on Stein's unbiased risk estimate to learn score models directly from noisy data, extending the applicability of diffusion models to noisier training conditions.

Findings

Effective learning from noisy data at 0 and 10 dB SNR.

Competitive performance in wireless channel estimation.

Successful MRI reconstruction with noisy training samples.

Abstract

We present SURE-Score: an approach for learning score-based generative models using training samples corrupted by additive Gaussian noise. When a large training set of clean samples is available, solving inverse problems via score-based (diffusion) generative models trained on the underlying fully-sampled data distribution has recently been shown to outperform end-to-end supervised deep learning. In practice, such a large collection of training data may be prohibitively expensive to acquire in the first place. In this work, we present an approach for approximately learning a score-based generative model of the clean distribution, from noisy training data. We formulate and justify a novel loss function that leverages Stein's unbiased risk estimate to jointly denoise the data and learn the score function via denoising score matching, while using only the noisy samples. We demonstrate the generality of SURE-Score by learning priors and applying posterior sampling to ill-posed inverse problems in two practical applications from different domains: compressive wireless multiple-input multiple-output channel estimation and accelerated 2D multi-coil magnetic resonance imaging reconstruction, where we demonstrate competitive reconstruction performance when learning at signal-to-noise ratio values of 0 and 10 dB, respectively.