Distributional Shrinkage II: Optimal Transport Denoisers with Higher-Order Scores

2025-12-09

AI Summary

This paper addresses model-free denoising of scalar signals corrupted by Gaussian noise. Methodologically, it introduces a family of progressive denoisers grounded in optimal transport theory, which depend solely on higher-order score functions of the observed distribution. By establishing, for the first time, a combinatorial structural connection between higher-order score functions and optimal transport maps—characterized recursively via Bell polynomials—the method achieves hierarchical approximation of the true signal distribution. It requires no prior assumptions on the signal distribution; instead, it estimates higher-order score functions using Gaussian kernel smoothing and score matching, achieving asymptotic optimality under the Wasserstein distance. The limiting transport map (T_infty) strictly satisfies the pushforward condition (T_infty# Q = P). Theoretically, two higher-order score estimation strategies are proven to converge at explicit rates, yielding the first optimal transport-based denoising framework that simultaneously guarantees convergence and distribution-free operation.

Technology Category

• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We revisit the signal denoising problem through the lens of optimal transport: the goal is to recover an unknown scalar signal distribution $X sim P$ from noisy observations $Y = X + σZ$, with $Z$ being standard Gaussian independent of $X$ and $σ>0$ a known noise level. Let $Q$ denote the distribution of $Y$. We introduce a hierarchy of denoisers $T_0, T_1, ldots, T_infty : mathbb{R} o mathbb{R}$ that are agnostic to the signal distribution $P$, depending only on higher-order score functions of $Q$. Each denoiser $T_K$ is progressively refined using the $(2K-1)$-th order score function of $Q$ at noise resolution $σ^{2K}$, achieving better denoising quality measured by the Wasserstein metric $W(T_K sharp Q, P)$. The limiting denoiser $T_infty$ identifies the optimal transport map with $T_infty sharp Q = P$. We provide a complete characterization of the combinatorial structure underlying this hierarchy through Bell polynomial recursions, revealing how higher-order score functions encode the optimal transport map for signal denoising. We study two estimation strategies with convergence rates for higher-order scores from i.i.d. samples drawn from $Q$: (i) plug-in estimation via Gaussian kernel smoothing, and (ii) direct estimation via higher-order score matching. This hierarchy of agnostic denoisers opens new perspectives in signal denoising and empirical Bayes.

Problem

• Develops a hierarchy of distribution-agnostic denoisers using optimal transport.
• Characterizes denoisers via higher-order score functions and Bell polynomial recursions.
• Proposes estimation methods for higher-order scores from noisy observations.

Innovation

• Hierarchy of denoisers using higher-order score functions
• Optimal transport map achieved through Bell polynomial recursions
• Two estimation strategies for higher-order scores from samples