Topological Regularization via Persistence-Sensitive Optimization

TL;DR

This paper introduces a novel topological regularization method that improves upon existing techniques by enabling faster and more precise control over the solution's topological features through persistence-sensitive simplification.

Contribution

It proposes a new optimization-based topological regularization approach that considers both critical and regular points, enhancing efficiency and accuracy over previous gradient-based methods.

Findings

Faster topological regularization achieved compared to existing methods.

More precise control over topological features demonstrated.

Experimental results validate the effectiveness of the proposed approach.

Abstract

Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have emerged as a way to provide a more precise and expressive control over the solution, relying on persistent homology to quantify and reduce its roughness. All such existing techniques back-propagate gradients through the persistence diagram, which is a summary of the topological features of a function. Their downside is that they provide information only at the critical points of the function. We propose a method that instead builds on persistence-sensitive simplification and translates the required changes to the persistence diagram into changes on large subsets of the domain, including both critical and regular points. This approach enables a faster and more precise topological regularization, the benefits of which we illustrate with experimental evidence.