Fast Comparative Analysis of Merge Trees Using Locality Sensitive Hashing

TL;DR

This paper introduces a fast, scalable framework for comparing merge trees in scalar field analysis using locality sensitive hashing, enabling efficient shape matching, clustering, and ensemble analysis.

Contribution

It proposes two novel LSH-based similarity measures for merge trees, significantly improving scalability and efficiency over existing methods.

Findings

The new measures closely match existing similarity metrics.

The framework enables efficient shape matching and clustering.

Applications include key event detection and ensemble summarization.

Abstract

Scalar field comparison is a fundamental task in scientific visualization. In topological data analysis, we compare topological descriptors of scalar fields -- such as persistence diagrams and merge trees -- because they provide succinct and robust abstract representations. Several similarity measures for topological descriptors seem to be both asymptotically and practically efficient with polynomial time algorithms, but they do not scale well when handling large-scale, time-varying scientific data and ensembles. In this paper, we propose a new framework to facilitate the comparative analysis of merge trees, inspired by tools from locality sensitive hashing (LSH). LSH hashes similar objects into the same hash buckets with high probability. We propose two new similarity measures for merge trees that can be computed via LSH, using new extensions to Recursive MinHash and subpath signature, respectively. Our similarity measures are extremely efficient to compute and closely resemble the results of existing measures such as merge tree edit distance or geometric interleaving distance. Our experiments demonstrate the utility of our LSH framework in applications such as shape matching, clustering, key event detection, and ensemble summarization.