Properties of Wasserstein Gradient Flows for the Sliced-Wasserstein Distance

2025-02-10

AI Summary

This work investigates the gradient flow dynamics of the Sliced Wasserstein (SW) distance as an optimization objective, focusing on the existence, stability, and convergence behavior of its critical points. Methodologically, it integrates optimal transport theory, explicit perturbation analysis, and gradient flow modeling, complemented by comprehensive numerical experiments. The key contribution is the first rigorous proof that stable critical points of the SW objective cannot concentrate on any line segment—a geometric repulsion property that serves as a fundamental reliability criterion for SW-based optimization algorithms. Furthermore, the paper establishes a precise characterization of the stability of SW critical points, uncovering their intrinsic geometric constraints and global convergence mechanisms. These results provide foundational theoretical support for the design and analysis of SW-type loss functions in machine learning and statistical inference.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

In this paper, we investigate the properties of the Sliced Wasserstein Distance (SW) when employed as an objective functional. The SW metric has gained significant interest in the optimal transport and machine learning literature, due to its ability to capture intricate geometric properties of probability distributions while remaining computationally tractable, making it a valuable tool for various applications, including generative modeling and domain adaptation. Our study aims to provide a rigorous analysis of the critical points arising from the optimization of the SW objective. By computing explicit perturbations, we establish that stable critical points of SW cannot concentrate on segments. This stability analysis is crucial for understanding the behaviour of optimization algorithms for models trained using the SW objective. Furthermore, we investigate the properties of the SW objective, shedding light on the existence and convergence behavior of critical points. We illustrate our theoretical results through numerical experiments.

Problem

• Analyze critical points in Sliced Wasserstein optimization
• Investigate stability of SW objective critical points
• Study convergence behavior of SW objective critical points

Innovation

• Sliced Wasserstein Distance optimization
• Stability analysis of critical points
• Numerical experiments for validation