{"ID":2830537,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.09295","arxiv_id":"2512.09295","title":"Distributional Shrinkage II: Higher-Order Scores Encode Brenier Map","abstract":"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 $σ\u003e 0$ 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.","short_abstract":"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 $σ\u003e 0$ 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...","url_abs":"https://arxiv.org/abs/2512.09295","url_pdf":"https://arxiv.org/pdf/2512.09295v3","authors":"[\"Tengyuan Liang\"]","published":"2025-12-10T03:41:06Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"cs.LG\",\"math.PR\",\"stat.ML\"]","methods":"[]","has_code":false}