Baby Morse Theory in Data Analysis

TL;DR

This paper introduces a Morse Theory-based method for inferring the topology of 3D point cloud data, providing both topological estimates and their uncertainty in a probabilistic framework.

Contribution

It presents a novel approach that combines Morse Theory with probabilistic sampling to estimate and assess the topology of point cloud data.

Findings

Method accurately estimates topological features like genus.

Provides probabilistic interval estimates for topology.

Demonstrated on multiple 3D point cloud examples.

Abstract

A methodology is proposed for inferring the topology underlying point cloud data. The approach employs basic elements of Morse Theory, and is capable of producing not only a point estimate of various topological quantities (e.g., genus), but it can also assess their sampling uncertainty in a probabilistic fashion. Several examples of point cloud data in three dimensions are utilized to demonstrate how the method yields interval estimates for the topology of the data as a 2-dimensional surface embedded in R^3.