Statistical Inference using the Morse-Smale Complex
This work addresses statistical inference challenges for researchers in multivariate data analysis, though it appears incremental as it builds on existing Morse-Smale complex applications.
The paper tackled the problem of estimating Morse-Smale complexes for nonparametric density estimation and regression, resulting in new statistical results that improved existing methods like mode clustering and Morse-Smale regression, and introduced new methods including a visualization technique and a two-sample hypothesis test.
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.