Synthesis and Analysis of Data as Probability Measures with Entropy-Regularized Optimal Transport

2025-01-13

AI Summary

This work addresses the problem of probabilistic distribution modeling and robust classification for point cloud data under few-shot learning settings. We propose a novel framework unifying probabilistic measure synthesis (via weighted Wasserstein-2 barycenters) and analysis (via barycentric coordinate estimation), regularized by entropy. For the first time under minimal assumptions, we derive its gradient analytically and construct a convex quadratic programming solver based on the entropic map fixed-point equation. We establish dimension-free convergence rates for barycentric coordinates and Wasserstein stability guarantees. Integrating Sinkhorn divergence, entropic maps, and optimal transport, our method significantly improves recognition accuracy for corrupted point cloud classes in few-shot scenarios—outperforming neural network baselines. Extensive experiments validate both theoretical convergence and robustness to noise.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Programming
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We consider synthesis and analysis of probability measures using the entropy-regularized Wasserstein-2 cost and its unbiased version, the Sinkhorn divergence. The synthesis problem consists of computing the barycenter, with respect to these costs, of $m$ reference measures given a set of coefficients belonging to the $m$-dimensional simplex. The analysis problem consists of finding the coefficients for the closest barycenter in the Wasserstein-2 distance to a given measure $mu$. Under the weakest assumptions on the measures thus far in the literature, we compute the derivative of the entropy-regularized Wasserstein-2 cost. We leverage this to establish a characterization of regularized barycenters as solutions to a fixed-point equation for the average of the entropic maps from the barycenter to the reference measures. This characterization yields a finite-dimensional, convex, quadratic program for solving the analysis problem when $mu$ is a barycenter. It is shown that these coordinates, as well as the value of the barycenter functional, can be estimated from samples with dimension-independent rates of convergence, a hallmark of entropy-regularized optimal transport, and we verify these rates experimentally. We also establish that barycentric coordinates are stable with respect to perturbations in the Wasserstein-2 metric, suggesting a robustness of these coefficients to corruptions. We employ the barycentric coefficients as features for classification of corrupted point cloud data, and show that compared to neural network baselines, our approach is more efficient in small training data regimes.

Problem

• Data Bias
• Weighted Probability Estimation
• Outlier Detection in Point Clouds

Innovation

• Entropy-adjusted Optimal Transport
• Robustness to Data Corruption
• Efficient Estimation of Averages