Whitened Score Diffusion: A Structured Prior for Imaging Inverse Problems

Yu Sun (California Polytechnic State University, Pomona) · Jeffrey Alido (Boston University) · Tongyu Li (Boston University, Boston University) · Lei Tian (Boston University)

anisotropic gaussian diffusionbayesian priorsceleba datasetscifarcomputational imaging taskscovariance inversioncovariance matricesdenoising score matchingflow matchinggaussian forward noising processesimaging inverse problems score-based diffusion modelsspectral inductive biasesstochastic differential equationsstructured noisewhitened score function

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.