用高阶分数函数构建渐进最优去噪器，直达最优传输映射
Distributional Shrinkage II: Higher-Order Scores Encode Brenier Map
- 基于高阶分数函数构造分层去噪器，无需信号分布先验
- 第K阶去噪器误差为O(σ^{2(K+1)})，极限下精确恢复Brenier映射
- 揭示分数统计、最优传输与整数分拆的深层数学关联
考虑加性高斯模型 $Y = X + σZ$，其中 $X \sim P$ 为未知信号，$Z \sim N(0,1)$ 独立于 $X$，$σ>0$ 已知。令 $Q$ 表示 $Y$ 的分布。我们构建了一族仅依赖于 $Q$ 的高阶分数函数 $q^{(m)}/q$（$m \geq 1$）的去噪器 $T_0, T_1, \ldots, T_\infty \colon \mathbb{R} \to \mathbb{R}$，无需知晓 $P$ 的分布。第 $K$ 阶去噪器 $T_K$ 涉及至 $2K-1$ 阶分数函数，满足对任意 $r \geq 1$ 有 $W_r(T_K \sharp Q, P) = O(σ^{2(K+1)})$；在极限情况下，$T_\infty$ 精确恢复将 $Q$ 推至 $P$ 的单调最优传输映射（Brenier 映射）。我们通过部分贝尔多项式递推完整刻画了该层级结构的组合性质，精确揭示了高阶分数函数如何编码 Brenier 映射。进一步在两种互补策略下建立了从 $n$ 个独立同分布样本估计这些分数函数的收敛速率：(i) 插值核密度估计，(ii) 高阶分数匹配。该构造揭示了高阶 Fisher 型信息、最优传输与整数分拆之间的精确互动。
原文摘要 · Abstract (English)
Consider the additive Gaussian model $Y = X + σZ$, where $X \sim P$ is an unknown signal, $Z \sim N(0,1)$ is independent of $X$, and $σ> 0$ is known. Let $Q$ denote the law of $Y$. We construct a hierarchy of denoisers $T_0, T_1, \ldots, T_\infty \colon \mathbb{R} \to \mathbb{R}$ that depend only on higher-order score functions $q^{(m)}/q$, $m \geq 1$, of $Q$ and require no knowledge of the law $P$. The $K$-th order denoiser $T_K$ involves scores up to order $2K{-}1$ and satisfies $W_r(T_K \sharp Q, P) = O(σ^{2(K+1)})$ for every $r \geq 1$; in the limit, $T_\infty$ recovers the monotone optimal transport map (Brenier map) pushing $Q$ onto $P$. We provide a complete characterization of the combinatorial structure governing this hierarchy through partial Bell polynomial recursions, making precise how higher-order score functions encode the Brenier map. We further establish rates of convergence for estimating these scores from $n$ i.i.d.\ draws from $Q$ under two complementary strategies: (i) plug-in kernel density estimation, and (ii) higher-order score matching. The construction reveals a precise interplay among higher-order Fisher-type information, optimal transport, and the combinatorics of integer partitions.
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