Optimizing persistent homology based functions

Solving optimization tasks based on functions and losses with a topological flavor is a very active,growing field of research in data science and Topological Data Analysis, with applications in non-convexoptimization, statistics and machine learning. However, the approaches proposed in the literatureare usually anchored to a specific application and/or topological construction, and do not come withtheoretical guarantees. To address this issue, we study the differentiability of a general map associatedwith the most common topological construction, that is, the persistence map. Building on real analyticgeometry arguments, we propose a general framework that allows us to define and compute gradientsfor persistence-based functions in a very simple way. We also provide a simple, explicit and sufficientcondition for convergence of stochastic subgradient methods for such functions. This result encompassesall the constructions and applications of topological optimization in the literature. Finally, we provideassociated code, that is easy to handle and to mix with other non-topological methods and constraints, aswell as some experiments showcasing the versatility of our approach.