Statistical Inference using the Morse-Smale Complex

The Morse-Smale complex of f is a partition of K based on the gradient flow induced by f. Roughly speaking, the complex consists of sets, called crystals or cells, comprised of regions where f is increasing or decreasing. Figure 1 shows the Morse-Smale complex for a two-dimensional function. The cells are the intersections of the basins of attractions (under the gradient flow) of the function's maxima and minima. The function f is piecewise monotonic over cells with respect to some directions. In a sense, the Morse-Smale complex provides a generalization of isotonic regression. Because the Morse-Smale complex represents a multivariate function in terms of regions on which the function has simple behavior, the Morse-Smale complex has useful applications in statistics, including in clustering, regression, testing, and visualization. For instance, when f is a density function, the basins of attraction of f's modes are the (population) clusters for density-mode clustering