Distributional Shrinkage I: Universal Denoiser Beyond Tweedie's Formula

Abstract

We study the problem of denoising when only the noise level is known, not the noise distribution. Independent noise $Z$ corrupts a signal $X$, yielding the observation $Y = X + \sigma Z$ with known $\sigma \in (0,1)$. We propose \emph{universal} denoisers, agnostic to both signal and noise distributions, that recover the signal distribution $P_X$ from $P_Y$. When the focus is on distributional recovery of $P_X$ rather than on individual realizations of $X$, our denoisers achieve order-of-magnitude improvements over the Bayes-optimal denoiser derived from Tweedie's formula, which achieves $O(\sigma^2)$ accuracy. They shrink $P_Y$ toward $P_X$ with $O(\sigma^4)$ and $O(\sigma^6)$ accuracy in matching generalized moments and densities. Drawing on optimal transport theory, our denoisers approximate the Monge--Amp\`ere equation with higher-order accuracy and can be implemented efficiently via score matching. Let $q$ denote the density of $P_Y$. For distributional denoising, we propose replacing the Bayes-optimal denoiser, $$\mathbf{T}^*(y) = y + \sigma^2 \nabla \log q(y),$$ with denoisers exhibiting less-aggressive distributional shrinkage, $$\mathbf{T}_1(y) = y + \frac{\sigma^2}{2} \nabla \log q(y),$$ $$\mathbf{T}_2(y) = y + \frac{\sigma^2}{2} \nabla \log q(y) - \frac{\sigma^4}{8} \nabla \!\left( \frac{1}{2} \| \nabla \log q(y) \|^2 + \nabla \cdot \nabla \log q(y) \right)\!.$$