Topological mixture estimation

TL;DR

This paper introduces a nonparametric, efficient method for estimating one-dimensional mixture models by leveraging topological and information-theoretic principles to identify unimodal components without restrictive assumptions.

Contribution

It presents a novel topological mixture estimation technique that is fully nonparametric, computationally efficient, and capable of determining unimodal mixture components directly from data.

Findings

Effective in estimating unimodal mixture components

Handles sample data directly with topological persistence

Avoids restrictive parametric assumptions

Abstract

Density functions that represent sample data are often multimodal, i.e. they exhibit more than one maximum. Typically this behavior is taken to indicate that the underlying data deserves a more detailed representation as a mixture of densities with individually simpler structure. The usual specification of a component density is quite restrictive, with log-concave the most general case considered in the literature, and Gaussian the overwhelmingly typical case. It is also necessary to determine the number of mixture components \emph{a priori}, and much art is devoted to this. Here, we introduce \emph{topological mixture estimation}, a completely nonparametric and computationally efficient solution to the one-dimensional problem where mixture components need only be unimodal. We repeatedly perturb the unimodal decomposition of Baryshnikov and Ghrist to produce a topologically and information-theoretically optimal unimodal mixture. We also detail a smoothing process that optimally exploits topological persistence of the unimodal category in a natural way when working directly with sample data. Finally, we illustrate these techniques through examples.