Distributional Shrinkage II: Higher-Order Scores Encode Brenier Map

arXiv:2512.09295v3 Announce Type: replace-cross
Abstract: Consider the additive Gaussian model $Y = X + sigma Z$, where $X sim P$ is an unknown signal, $Z sim N(0,1)$ is independent of $X$, and $sigma > 0$ 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 $W_r(T_K sharp Q, P) = O(sigma^{2(K+1)})$ 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.