Whitened Score Diffusion: A Structured Prior for Imaging Inverse Problems

TL;DR

This paper introduces Whitened Score diffusion models that improve imaging inverse problem solutions by avoiding covariance inversion, enabling stable training on arbitrary Gaussian noise, and outperforming traditional methods in various imaging tasks.

Contribution

The paper presents a novel framework for score-based diffusion models that learns the Whitened Score, allowing stable training on arbitrary Gaussian processes and enhancing performance in imaging inverse problems.

Findings

WS diffusion models outperform conventional diffusion priors on anisotropic Gaussian noise.

The approach enables stable training on arbitrary Gaussian forward noising processes.

Experiments on CIFAR and CelebA datasets demonstrate improved imaging results.

Abstract

Conventional score-based diffusion models (DMs) may struggle with anisotropic Gaussian diffusion processes due to the required inversion of covariance matrices in the denoising score matching training objective \cite{vincent_connection_2011}. We propose Whitened Score (WS) diffusion models, a novel framework based on stochastic differential equations that learns the Whitened Score function instead of the standard score. This approach circumvents covariance inversion, extending score-based DMs by enabling stable training of DMs on arbitrary Gaussian forward noising processes. WS DMs establish equivalence with flow matching for arbitrary Gaussian noise, allow for tailored spectral inductive biases, and provide strong Bayesian priors for imaging inverse problems with structured noise. We experiment with a variety of computational imaging tasks using the CIFAR, CelebA ($64\times64$), and CelebA-HQ ($256\times256$) datasets and demonstrate that WS diffusion priors trained on anisotropic Gaussian noising processes consistently outperform conventional diffusion priors based on isotropic Gaussian noise. Our code is open-sourced at \href{https://github.com/jeffreyalido/wsdiffusion}{\texttt{github.com/jeffreyalido/wsdiffusion}}.