Optimal Transport Denoisers, Using Higher-Order Scores, Achieve Improved Signal Recovery with Wasserstein Metric
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The challenge of recovering clear signals from noisy data receives fresh attention from Tengyuan Liang of The University of Chicago, and colleagues. They introduce a new approach to signal denoising, framing it as a problem of optimal transport, where the goal is to reconstruct an unknown signal distribution from observations corrupted by Gaussian noise. This work establishes a hierarchy of denoisers that cleverly utilise higher-order score functions, progressively refining the denoising process and achieving improved accuracy, as measured by the Wasserstein metric. Crucially, this method operates independently of the specific signal distribution, and the team’s complete characterization of the underlying mathematical structure, using Bell polynomial recursions, reveals how these higher-order scores effectively encode the optimal transport map needed for signal recovery, opening new avenues in signal processing and statistical inference. Gaussian Smoothing, Bias, and Density Estimation This research presents a comprehensive collection of mathematical and statistical results focused on density estimation and empirical process theory. The study investigates methods for accurately estimating probability densities from data, with a particular emphasis on Gaussian smoothing techniques. Scientists rigorously analyze the bias and variance of these estimators, exploring how they perform with limited data and complex distributions. The work delves into higher-order asymptotic analysis to understand the subtle behavior of these estimators as the amount of data increases, utilizing tools from empirical process theory to assess their reliability. The findings have implications for a wide range of applications, including machine learning and statistical modeling. The study establishes a clear connection between the bandwidth of the Gaussian kernel and the accuracy of the density estimate, demonstrating how a wider kernel smooths the data but can also obscure important features. Scientists derive quantitative bounds on the bias and variance of the estimator, providing a framework for selecting the optimal bandwidth for a given dataset. This work builds upon established results in empirical process theory, providing a solid foundation for developing and analyzing more efficient density estimation techniques, and delivers a rigorous mathematical framework for understanding the trade-offs involved in density estimation. Higher-Order Score Functions for Optimal Denoising Scientists have developed a novel hierarchy of denoisers based on optimal transport theory to address the problem of signal denoising. The research aims to recover an unknown signal distribution from noisy observations without making assumptions about the underlying signal. Researchers constructed a series of denoisers that achieve increasingly accurate denoising as measured by the Wasserstein metric, a measure of distance between probability distributions. These denoisers converge towards the optimal transport map, which effectively transforms the noisy distribution into the true signal distribution, offering a significant improvement over traditional denoising methods.
The team demonstrated that the optimal transport map can be expressed as an infinite expansion, revealing its relationship to higher-order score functions of the signal distribution. This expansion forms the basis for constructing the hierarchy of denoisers, where each successive denoiser incorporates more refined score functions to improve denoising accuracy. Scientists established a uniform error bound for approximating the optimal transport map using these denoisers, resulting in denoisers independent of the signal distribution itself, relying solely on the higher-order score functions of the noisy data, making them versatile and applicable to a wide range of signal types and noise models.
Optimal Transport Maps for Signal Denoising Scientists have created a hierarchy of signal denoisers that function independently of the underlying signal distribution, achieving improved denoising quality as measured by the Wasserstein metric. This work introduces a series of agnostic denoisers, refined by progressively incorporating higher-order score functions at increasing noise resolution. Experiments reveal that the limiting denoiser precisely identifies the optimal transport map, effectively reconstructing the original signal from noisy observations.
The team characterized the combinatorial structure of this hierarchy using Bell polynomial recursions, demonstrating how higher-order score functions encode the optimal transport map for signal denoising, offering a significant advantage over traditional denoising methods. Researchers studied two estimation strategies for these higher-order scores, utilizing independent and identically distributed samples. The first approach employs plug-in estimation with Gaussian kernel smoothing, achieving a rate of convergence for estimating the score functions locally. The second strategy involves direct estimation via higher-order score matching, enabling global estimation of the functions and establishing a corresponding rate of convergence. Measurements confirm that these denoisers are nonparametric and agnostic to the specific form of the signal distribution, relying solely on higher-order score functions of the observed noise, delivering a fundamentally different approach to signal denoising by constructing denoisers directly on the observational space via higher-order score functions without estimating the prior signal distribution.
Optimal Transport Denoising via Polynomial Recursions Researchers have developed a new hierarchy of signal denoising methods grounded in optimal transport theory, offering improvements over traditional techniques that often over-shrink data distributions. This work establishes a series of denoisers, progressing from simple methods to those approaching the ideal optimal transport map, which accurately transforms noisy data into a clean signal distribution. The core of this achievement lies in utilizing higher-order score functions, refined through polynomial recursions, to progressively improve denoising accuracy as measured by the Wasserstein metric, offering a significant advantage over traditional denoising methods.
The team demonstrated that these denoisers can be constructed using increasingly complex polynomials, each building upon the previous one to achieve finer levels of accuracy in signal recovery. These polynomial formulas are linked to a fundamental mathematical relationship involving Bell polynomials, revealing an underlying structure to the denoising process. Researchers acknowledge that the performance of these methods relies on accurate estimation of the score functions from limited data, which presents a practical challenge, and future research will likely focus on improving these estimation techniques and exploring the application of this hierarchy to more complex signal types and noise models, potentially extending its use in fields like image processing and data analysis. 👉 More information 🗞 Distributional Shrinkage II: Optimal Transport Denoisers with Higher-Order Scores 🧠 ArXiv: https://arxiv.org/abs/2512.09295 Tags: