A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis
Résumé
We introduce a novel gradient descent algorithm refining the well-known Gradient Sampling algorithm
on 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. On 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
benchmarks and novel topological optimization problems that demonstrate the utility and applicability
of our framework.
Origine : Fichiers produits par l'(les) auteur(s)
