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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.09500 |
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| _version_ | 1866917363360727040 |
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| author | Liang, Tengyuan |
| author_facet | Liang, Tengyuan |
| contents | 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, $$\mathbf{T}_1(y) = y + \frac{σ^2}{2} \nabla \log q(y),$$ $$\mathbf{T}_2(y) = y + \frac{σ^2}{2} \nabla \log q(y) - \frac{σ^4}{8} \nabla \!\left( \frac{1}{2} \| \nabla \log q(y) \|^2 + \nabla \cdot \nabla \log q(y) \right)\!.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distributional Shrinkage I: Universal Denoiser Beyond Tweedie's Formula Liang, Tengyuan Machine Learning Statistics Theory Methodology 62G05, 62C12, 49Q22, 60E15 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, $$\mathbf{T}_1(y) = y + \frac{σ^2}{2} \nabla \log q(y),$$ $$\mathbf{T}_2(y) = y + \frac{σ^2}{2} \nabla \log q(y) - \frac{σ^4}{8} \nabla \!\left( \frac{1}{2} \| \nabla \log q(y) \|^2 + \nabla \cdot \nabla \log q(y) \right)\!.$$ |
| title | Distributional Shrinkage I: Universal Denoiser Beyond Tweedie's Formula |
| topic | Machine Learning Statistics Theory Methodology 62G05, 62C12, 49Q22, 60E15 |
| url | https://arxiv.org/abs/2511.09500 |