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

Jacob Leygonie, Mathieu Carrière, Théo Lacombe, Steve Oudot

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