Topological Regularization via Persistence-Sensitive Optimization

• URL: http://arxiv.org/abs/2011.05290v1
• Date: Tue, 10 Nov 2020 18:19:43 GMT
• Title: Topological Regularization via Persistence-Sensitive Optimization
• Authors: Arnur Nigmetov, Aditi S. Krishnapriyan, Nicole Sanderson, Dmitriy Morozov
• Abstract summary: A key tool in machine learning and statistics, regularization relies on regularization to reduce overfitting. We propose a method that builds on persistence-sensitive simplification and translates required changes to the persistence diagram into changes on large subsets of the domain. This approach enables a faster and more precise topological regularization, the benefits of which we illustrate.
• Score: 10.29838087001588
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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