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
abla \log q(y), with denoisers exhibiting less-aggressive distributional shrinkage,
\mathbf{T}_1(y) = y + \frac{\sigma^2}{2}
abla \log q(y), \mathbf{T}_2(y) = y + \frac{\sigma^2}{2}
abla \log q(y) - \frac{\sigma^4}{8}
abla \!\left( \frac{1}{2} \|
abla \log q(y) \|^2 +
abla \cdot
abla \log q(y) \right)\!.
