Topological regularization via persistence-sensitive optimization

• DOI: 10.1016/j.comgeo.2024.102086
• Publication Date: 2024-02-28T10:01:04Z
• Authors: Arnur Nigmetov; Aditi Krishnapriyan; Nicole Sanderson; Dmitriy Morozov
• Keywords: FOS: Computer and information sciences; Computer Science - Machine Learning; math.AT; Numerical and Computational Mathematics; cs.LG; Mathematical sciences; Computation Theory and Mathematics; Geological & Geomatics Engineering; Pure Mathematics; 01 natural sciences; Machine Learning (cs.LG); Information and Computing Sciences; Augmented Reality and Games; Machine Learning and Artificial Intelligence; Information and computing sciences; FOS: Mathematics; Graphics; Algebraic Topology (math.AT); Mathematics - Algebraic Topology; 0101 mathematics

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.<br/>The first two authors contributed equally to this work<br/>