Persistent homology has emerged as a useful theoretical and practical tool to infer and compute robust topological invariants of data sets, the so-called persistence diagrams, that have found various applications in data analysis and machine learning. Understanding the differentiable structure of persistent homology to solve optimization tasks based on functions with a topological flavor is an active and 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, after a brief introduction to persistent homology (from a data analysis perspective) and its stability properties, building on basic real analytic geometry arguments, we will propose a general framework that allows to define and compute gradients for persistence-based functions in a very simple way. As an application, we will 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. 

Diffusion sur: https://webconf.imo.universite-paris-saclay.fr/b/jer-7cp-7mk