Topological Regularization via Persistence-Sensitive Optimization
This work addresses the need for more effective regularization in machine learning, offering a novel approach that could enhance model robustness, though it appears incremental relative to prior topological methods.
The paper tackles the problem of overfitting in optimization by introducing a topological regularization method that improves upon existing techniques by applying changes across both critical and regular points, resulting in faster and more precise regularization as demonstrated experimentally.
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.