A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis

TL;DR

This paper presents a new gradient sampling algorithm tailored for stratifiably smooth functions, with applications in Topological Data Analysis, achieving sub-linear convergence and demonstrating effectiveness on topological optimization problems.

Contribution

The paper extends gradient sampling to stratifiably smooth functions and applies it to topological data analysis, introducing efficient stratification exploration methods.

Findings

Achieves sub-linear convergence rate for stratifiably smooth functions.

Effectively applies to topological optimization problems.

Demonstrates utility through benchmark and novel problems.

Abstract

We introduce a novel gradient descent algorithm extending the well-known Gradient Sampling methodology to the class of stratifiably smooth objective functions, which are defined as locally Lipschitz functions that are smooth on some regular pieces-called the strata-of the ambient Euclidean space. For this class of functions, our algorithm achieves a sub-linear convergence rate. We then apply our method to objective functions based on the (extended) persistent homology map computed over lower-star filters, which is a central tool of Topological Data Analysis. For this, we propose an efficient exploration of the corresponding stratification by using the Cayley graph of the permutation group. Finally, we provide benchmark and novel topological optimization problems, in order to demonstrate the utility and applicability of our framework.