Statistical Inference using the Morse-Smale Complex

The Morse-Smale complex decomposes the sample space into cells where a given function $f$ is increasing or decreasing. When applied to nonparametric density estimation and regression, it provides a way to represent, visualize and compare functions, even in high dimensions. In this paper, we study the estimation of the Morse-Smale complex and we use our results for a variety of statistical problems including: nonparametric two-sample testing, density estimation, nonparametric regression and mode clustering.