  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.
