{"ID":2842936,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.09500","arxiv_id":"2511.09500","title":"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 $Y = X + σZ$ with known $σ\\in (0,1)$. 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 $O(σ^2)$ accuracy. They shrink $P_Y$ toward $P_X$ with $O(σ^4)$ and $O(σ^6)$ accuracy in matching generalized moments and densities. Drawing on optimal transport theory, our denoisers approximate the Monge--Ampère 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, $$\\mathbf{T}^*(y) = y + σ^2 \\nabla \\log q(y),$$ with denoisers exhibiting less-aggressive distributional shrinkage, $$\\mathbf{T}_1(y) = y + \\frac{σ^2}{2} \\nabla \\log q(y),$$ $$\\mathbf{T}_2(y) = y + \\frac{σ^2}{2} \\nabla \\log q(y) - \\frac{σ^4}{8} \\nabla \\!\\left( \\frac{1}{2} \\| \\nabla \\log q(y) \\|^2 + \\nabla \\cdot \\nabla \\log q(y) \\right)\\!.$$","short_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 $Y = X + σZ$ with known $σ\\in (0,1)$. We propose \\emph{universal} denoisers, agnostic to both signal and noise distributions, that recover the signal di...","url_abs":"https://arxiv.org/abs/2511.09500","url_pdf":"https://arxiv.org/pdf/2511.09500v4","authors":"[\"Tengyuan Liang\"]","published":"2025-11-12T17:20:42Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\",\"math.ST\",\"stat.ME\"]","methods":"[]","has_code":false}