Distributional Shrinkage I: Universal Denoisers in Multi-Dimensions

We revisit the problem of denoising from noisy measurements where only the noise level is known, not the noise distribution. In multi-dimensions, independent noise $Z$ corrupts the signal $X$, resulting in the noisy measurement $Y = X + σZ$, where $σ\in (0, 1)$ is a known noise level. Our goal is to recover the underlying signal distribution $P_X$ from denoising $P_Y$. We propose and analyze universal denoisers that are agnostic to a wide range of signal and noise distributions. Our distributional denoisers offer order-of-magnitude improvements over the Bayes-optimal denoiser derived from Tweedie's formula, if the focus is on the entire distribution $P_X$ rather than on individual realizations of $X$. Our denoisers shrink $P_Y$ toward $P_X$ optimally, achieving $O(σ^4)$ and $O(σ^6)$ accuracy in matching generalized moments and density functions. Inspired by optimal transport theory, the proposed denoisers are optimal in approximating the Monge-Ampère equation with higher-order accuracy, and can be implemented efficiently via score matching. Let $q$ represent the density of $P_Y$; for optimal distributional denoising, we recommend replacing the Bayes-optimal denoiser, \[ \mathbf{T}^*(y) = y + σ^2 \nabla \log q(y), \] with denoisers exhibiting less aggressive distributional shrinkage, \[ \mathbf{T}_1(y) = y + \frac{σ^2}{2} \nabla \log q(y), \] \[ \mathbf{T}_2(y) = y + \frac{σ^2}{2} \nabla \log q(y) - \frac{σ^4}{8} \nabla \left( \frac{1}{2} \| \nabla \log q(y) \|^2 + \nabla \cdot \nabla \log q(y) \right) . \]