A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis
This provides a new optimization method for topological data analysis problems, though it appears to be an incremental extension of existing gradient sampling techniques.
The authors developed a gradient descent algorithm for stratifiably smooth functions that achieves sub-linear convergence, and applied it to topological optimization problems using persistent homology maps from Topological Data Analysis.
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.