Decorated Merge Trees for Persistent Topology

TL;DR

This paper introduces decorated merge trees (DMTs), a new invariant combining merge trees and persistent homology, with stability analysis and computational methods for applications in data analysis and visualization.

Contribution

The paper presents decorated merge trees as a novel invariant that unifies merge trees and persistent homology, along with stability results and efficient computational techniques.

Findings

DMTs distinguish filtrations that merge trees and persistent homology cannot.

Defined interleaving and bottleneck distances with stability guarantees.

Applied DMTs to real data for visualization and comparison tasks.

Abstract

This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent spaces. DMTs combine both $\pi_0$ and $H_n$ information into a single data structure that distinguishes filtrations that merge trees and persistent homology cannot distinguish alone. Three variants on DMTs, which emphasize category theory, representation theory and persistence barcodes, respectively, offer different advantages in terms of theory and computation. Two notions of distance -- an interleaving distance and bottleneck distance -- for DMTs are defined and a hierarchy of stability results that both refine and generalize existing stability results is proved here. To overcome some of the computational complexity inherent in these distances, we provide a novel use of Gromov-Wasserstein couplings to compute optimal merge tree alignments for a combinatorial version of our interleaving distance which can be tractably estimated. We introduce computational frameworks for generating, visualizing and comparing decorated merge trees derived from synthetic and real data. Example applications include comparison of point clouds, interpretation of persistent homology of sliding window embeddings of time series, visualization of topological features in segmented brain tumor images and topology-driven graph alignment.