Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport

TL;DR

This paper introduces a scalable, GPU-accelerated framework for computing means and clusters of persistence diagrams using optimal transport reformulations, enabling large-scale topological data analysis.

Contribution

It reformulates persistence diagram metrics as optimal transport problems and leverages Sinkhorn algorithm for efficient, large-scale computations of barycenters and clustering.

Findings

Efficient computation of PD distances and barycenters at scale

Clustering of thousands of PDs demonstrated on large datasets

GPU-accelerated implementation enables real-time analysis

Abstract

Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data. Despite several appealing properties, incorporating PDs in learning pipelines can be challenging because their natural geometry is not Hilbertian. Indeed, this was recently exemplified in a string of papers which show that the simple task of averaging a few PDs can be computationally prohibitive. We propose in this article a tractable framework to carry out standard tasks on PDs at scale, notably evaluating distances, estimating barycenters and performing clustering. This framework builds upon a reformulation of PD metrics as optimal transport (OT) problems. Doing so, we can exploit recent computational advances: the OT problem on a planar grid, when regularized with entropy, is convex can be solved in linear time using the Sinkhorn algorithm and convolutions. This results in scalable computations that can stream on GPUs. We demonstrate the efficiency of our approach by carrying out clustering with diagrams metrics on several thousands of PDs, a scale never seen before in the literature.