Reeb Graphs and Towers: Multiscale Skeletons for Data

Andrew J Steindl, Dhananjay Bhaskar, Smita Krishnaswamy

Track: Extended Abstract (non-archival, 4 pages)

Keywords: Reeb Graph, Multiscale topology, diffusion geometry, Manifold learning, single-cell

TL;DR: We introduce scReebTowers, a diffusion-refined, multiscale Reeb graph framework that extracts robust skeletons from noisy point clouds, enabling automated detection of clusters, branches, and cycles in single-cell and other high-dimensional data.

Abstract: Recently there has been a profusion of tools to analyze high dimensional point cloud data, popularized by the advent and popularity of single cell data in biology. As the number of datasets of this type grow, the need to automate analysis of the data and generate hypotheses grows. However, most of the techniques developed for single-cell data assume a particular type of structure in the data, i.e., typically cluster structure, tree structure, or trajectory structure, and provide specific heuristics to analyze such structure. For example a standard single cell pipeline consists of UMAP visualization followed by Louvain community detection, which assumes the need to visualize neighborhoods in the cellular data, and to separate them into clusters. But deciding which tool to use requires human intervention. Instead, here we provide a method for automating data shape detection from noisy single cell point clouds using the concept of a Reeb graph, called a scReebTower. A Reeb graph is a topological descriptor that uses a TDA-like filtration of the data to detect connected components of a level set of the data. Then each connected component is contracted to a single point resulting in a graph where nodes correspond to critical points and edges relationships between points. Here we propose to construct such graphs by first starting with a data diffusion operator and performing a Vietoris–Rips–like filtration over diffusion scales to reveal multiscale structure. scReebTower first creates a skeleton-Reeb graph of the data at each scale that reveals the fundamental structure of the data beneath the vast amount of noise in single cell data. Next, scReebTower computes the persistence homology of each floor as a quantitative signature of the shape at each scale. This set of signatures then allows classification methods to distinguish between datasets that have cluster structure, tree structure, trajectory structure and undifferentiated (blob--like) structure at a particular scale which naturally suggests downstream analysis tools. Preliminary results show single Reeb graphs and scReebTowers atop single cell embryoid body\cite{ebd} data and constructed toy data sets.

Submission Number: 33