# Distributional Shrinkage II: Higher-Order Scores Encode Brenier Map

arXiv: https://arxiv.org/abs/2512.09295 · HTML: https://mlsanity.com/paper/2512.09295/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 40 references that are in the map.
- First posted on arXiv 10 Dec 2025.

## Builds on (most cited first, 15 of 15)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [How Well Generative Adversarial Networks Learn Distributions](https://mlsanity.com/paper/1811.03179/) (2018)
- [Neural Empirical Bayes](https://mlsanity.com/paper/1903.02334/) (2019)
- [Density Estimation in Infinite Dimensional Exponential Families](https://mlsanity.com/paper/1312.3516/) (2013)
- [Deep Energy Estimator Networks](https://mlsanity.com/paper/1805.08306/) (2018)
- [Two Modeling Strategies for Empirical Bayes Estimation](https://mlsanity.com/paper/1409.2677/) (2014)
- [Optimal convex M-estimation via score matching](https://mlsanity.com/paper/2403.16688/) (2024)
- [Bounding the expectation of the supremum of empirical processes indexed by H"older classes](https://mlsanity.com/paper/2003.13530/) (2020)
- [Stein's unbiased risk estimate and Hyv"arinen's score matching](https://mlsanity.com/paper/2502.20123/) (2025)
- [Denoising Diffusions with Optimal Transport: Localization, Curvature, and Multi-Scale Complexity](https://mlsanity.com/paper/2411.01629/) (2024)
- [Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity](https://mlsanity.com/paper/2503.12966/) (2025)
- [No-Regret Generative Modeling via Parabolic Monge-Amp`ere PDE](https://mlsanity.com/paper/2504.09279/) (2025)
- [Constrained Denoising, Empirical Bayes, and Optimal Transport](https://mlsanity.com/paper/2506.09986/) (2025)
- [A New Perspective On Denoising Based On Optimal Transport](https://mlsanity.com/paper/2312.08135/) (2023)

## Abstract (arXiv)

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.