Fast Comparative Analysis of Merge Trees Using Locality Sensitive Hashing

TL;DR

A new framework to facilitate the comparative analysis of merge trees, inspired by tools from locality sensitive hashing (LSH), and proposes two new similarity measures for merge trees that can be computed via LSH, using new extensions to Recursive MinHash and subpath signature, respectively.

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.

Figures (19)

• Figure 1: Illustration of merge trees. (a) A scalar field $f$ overlaid with critical points. (b) Merge tree. (c) Labeled merge tree with node labels $[3,4]$, $[2]$, and $[1]$ for nodes a, b, and c, respectively. (d) Merge tree of $-f$.
• Figure 2: (a) Generating MinHash: an $N \times M$ binary matrix is generated and then reduced to a dense $q \times M$ signature matrix using $q$-MinHash. Here, $q=4$. (b) LSH: a signature is divided into bands of length $r$. Here, $r=2$. LSH is applied to individual bands and candidate pairs are determined based on signature collision in any band (in red). The orange arrows are applications of hash functions.
• Figure 3: An illustration of RMH.
• Figure 4: A pipeline of LSH for comparing merge trees.
• Figure 5: Illustration of subpath signature for a labeled MT.