A gradient sampling algorithm for stratified maps with applications to topological data analysis

• 成果类型: Article
• 署名作者: Leygonie, Jacob; Carriere, Mathieu; Lacombe, Theo; Oudot, Steve
• 署名单位: University of Oxford; Universite Cote d'Azur; Inria; Universite Gustave-Eiffel; ESIEE Paris; Institut Polytechnique de Paris; Ecole des Ponts ParisTech; Inria; Institut Polytechnique de Paris
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-01931-x
• 发表日期: 2023
• 页码: 199-239
• 关键词: persistent homology; descent methods; nonsmooth; CONVERGENCE; optimization

摘要

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.