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

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.32nd Conference on Neural Information Processing Systems (NIPS 2018),

On Topological Data Analysis for Structural Dynamics: An Introduction to Persistent Homology

“…This notion of a metric space allows one to compare the similarity of manifolds. Metrics between barcodes are well established and the one used in this paper is the p−Wasserstein distance [21].…”

Approximation algorithms for 1-Wasserstein distance between persistence diagrams

“…Additionally, the true approximation error for Hera also does not decrease much; see Table 1. To get the approximation error for Hera, we find the true Wasserstein distance by using the wasserstein_distance function from the GUDHI library which uses the Python Optimal Transport library [12] and is based on ideas from [21]. The results in Figure 2 indicate that the modified flowtree approach is between 50 and 1000 times faster than Hera and the difference increases as the size of the diagrams increases.…”

Persistence Bag-of-Words for Topological Data Analysis

“…The goal of kernelbased approaches on PDs is to define dissimilarity measures arXiv:1812.09245v3 [stat.ML] 4 Jun 2019 (also known as kernel functions) used to compare PDs and thereby make them compatible with kernel-based machine learning methods like Support Vector Machines (SVMs) and kernel Principal Component Analysis (kPCA). Li et al [Li et al, 2014] use the traditional bag-of-features (BoF) approach combining various distances between 0-dimensional PDs to generate kernels. On a number of datasets (SHREC 2010, TOSCA, hand gestures, Outex) they show that topological information is complementary to the information of traditional BoF.…”

“…propose a kernel based on sliced Wasserstein approximation of the Wasserstein distance. Le and Yamada [Le and Yamada, 2018] proposed a Persistence Fisher (PF) kernel for PDs. It has a number of desirable theoretical properties such as stability, infinite divisibility, and linear time complexity in the number of points in the PDs.…”

“…It has a number of desirable theoretical properties such as stability, infinite divisibility, and linear time complexity in the number of points in the PDs. Lacombe et al [Lacombe et al, 2018] reformulated the computation of diagram metrics as an optimal transport problem. This approach allows for efficient parallelization and scalable computations of a PD's barycenters.…”