Distributional Shrinkage II: Higher-Order Scores Encode Brenier Map

Consider the additive Gaussian model Y = X + σ Z, where X ~ P is an unknown signal, Z ~ 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 T0, T1, ldots, T_∞ colon ℝ → ℝ that depend only on higher-order score functions q(m)/q, m ≥ 1, of Q and require no knowledge of the law P. The K-th order denoiser TK involves scores up to order 2K-1 and satisfies Wr(TK sharp Q, P) = O(σ2(K+1)) for every r ≥ 1; in the limit, T_∞ 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.