A FRAMEWORK TO DIFFERENTIATE PERSISTENT HOMOLOGY WITH APPLICATIONS IN MACHINE LEARNING AND STATISTICS - FRÉDÉRIC CHAZAL - INRIA SACLAY ILE-DE-FRANCE
Understanding the differentiable structure of persistent homology and 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-convex optimization, statistics and machine learning. However, the approaches proposed in the literature are usually anchored to a specific application and/or topological construction, and do not come with theoretical guarantees. In this talk, we will study the differentiability of a general map associated with the most common topological construction, that is, the persistence map. Building on real analytic geometry arguments, we propose a general framework that allows to define and compute gradients for persistence-based functions in a very simple way. As an application, we also provide a simple, explicit and sufficient condition for convergence of stochastic subgradient methods for such functions. If time permits, as another application, we will also show how this framework combined with standard geometric measure theory arguments leads to results on the statistical behavior of persistence diagrams of filtrations built on top of random point clouds.
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