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, \[ \mathbf{T}_1(y) = y + \frac{σ^2}{2} \nabla \log q(y), \] \[ \mathbf{T}_2(y) = y + \frac{σ^2}{2} \nabla \log q(y) - \frac{σ^4}{8} \nabla \left( \frac{1}{2} \| \nabla \log q(y) \|^2 + \nabla \cdot \nabla \log q(y) \right) . \]

翻译：我们重新审视从含噪测量中恢复信号的问题，其中仅已知噪声水平而未知噪声分布。在多维情况下，独立噪声 $Z$ 污染信号 $X$，产生含噪测量 $Y = X + σZ$，其中 $σ∈ (0, 1)$ 为已知噪声水平。我们的目标是通过对 $P_Y$ 去噪来恢复潜在信号分布 $P_X$。我们提出并分析了适用于广泛信号与噪声分布的通用去噪器。若关注焦点在于整个分布 $P_X$ 而非 $X$ 的个体实现，我们的分布去噪器相比基于Tweedie公式推导的贝叶斯最优去噪器可实现数量级改进。我们的去噪器以最优方式将 $P_Y$ 收缩至 $P_X$，在匹配广义矩和密度函数方面分别达到 $O(σ^4)$ 和 $O(σ^6)$ 精度。受最优传输理论启发，所提去噪器能以更高阶精度逼近Monge-Ampère方程，并可通过分数匹配高效实现。令 $q$ 表示 $P_Y$ 的密度；对于最优分布去噪，我们建议将贝叶斯最优去噪器 \[ \mathbf{T}^*(y) = y + σ^2 \nabla \log q(y) \] 替换为具有更温和分布收缩特性的去噪器：\[ \mathbf{T}_1(y) = y + \frac{σ^2}{2} \nabla \log q(y) \]，\[ \mathbf{T}_2(y) = y + \frac{σ^2}{2} \nabla \log q(y) - \frac{σ^4}{8} \nabla \left( \frac{1}{2} \| \nabla \log q(y) \|^2 + \nabla \cdot \nabla \log q(y) \right) \]。