Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)

TL;DR

This paper introduces a computational framework for Principal Geodesic Analysis of merge trees using Wasserstein metrics, enabling efficient data reduction and visualization in topological data analysis.

Contribution

It formulates MT-PGA as a constrained optimization problem, develops an efficient iterative algorithm, and extends PCA applications to merge trees and persistence diagrams.

Findings

MT-PGA computations take minutes for large datasets

Effective data compression using first MT-PGA coordinates

Two-dimensional layouts facilitate visual analysis of feature variability

Abstract

This paper presents a computational framework for the Principal Geodesic Analysis of merge trees (MT-PGA), a novel adaptation of the celebrated Principal Component Analysis (PCA) framework [87] to the Wasserstein metric space of merge trees [92]. We formulate MT-PGA computation as a constrained optimization problem, aiming at adjusting a basis of orthogonal geodesic axes, while minimizing a fitting energy. We introduce an efficient, iterative algorithm which exploits shared-memory parallelism, as well as an analytic expression of the fitting energy gradient, to ensure fast iterations. Our approach also trivially extends to extremum persistence diagrams. Extensive experiments on public ensembles demonstrate the efficiency of our approach - with MT-PGA computations in the orders of minutes for the largest examples. We show the utility of our contributions by extending to merge trees two typical PCA applications. First, we apply MT-PGA to data reduction and reliably compress merge trees by concisely representing them by their first coordinates in the MT-PGA basis. Second, we present a dimensionality reduction framework exploiting the first two directions of the MT-PGA basis to generate two-dimensional layouts of the ensemble. We augment these layouts with persistence correlation views, enabling global and local visual inspections of the feature variability in the ensemble. In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used to reproduce our results.