Topological Regularization via Persistence-Sensitive Optimization

DOI10.1016/J.COMGEO.2024.102086arXiv2011.05290MaRDI QIDQ6353450

Nicole F. Sanderson, Dmitriy Morozov, Arnur Nigmetov, Aditi Krishnapriyan

Publication date: 10 November 2020

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.

Mathematics Subject Classification ID

Persistent homology and applications, topological data analysis (55N31) Mathematical programming (90Cxx) Computing methodologies and applications (68Uxx) Computational aspects of data analysis and big data (68T09) Topological data analysis (62R40)

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