Statistical Inference using the Morse-Smale Complex

TL;DR

This paper explores statistical methods for estimating Morse-Smale complexes, enabling improved visualization, clustering, and hypothesis testing for multivariate functions in nonparametric settings.

Contribution

It provides new statistical results for Morse-Smale complex estimation and introduces two novel methods: a visualization technique and a two-sample hypothesis test.

Findings

Enhanced understanding of Morse-Smale complex estimation

New visualization method for multivariate functions

A two-sample hypothesis test based on Morse-Smale complexes

Abstract

The Morse-Smale complex of a function $f$ decomposes the sample space into cells where $f$ is increasing or decreasing. When applied to nonparametric density estimation and regression, it provides a way to represent, visualize, and compare multivariate functions. In this paper, we present some statistical results on estimating Morse-Smale complexes. This allows us to derive new results for two existing methods: mode clustering and Morse-Smale regression. We also develop two new methods based on the Morse-Smale complex: a visualization technique for multivariate functions and a two-sample, multivariate hypothesis test.