Sparse moments of univariate step functions and allele frequency spectra

TL;DR

This paper characterizes the univariate moment problem for piecewise-constant functions and applies it to population genetics, showing that minimal step functions can represent all moments and coalescence histories, with solutions formulated as semidefinite programs.

Contribution

It establishes tight bounds on the number of breakpoints needed for representing moments and coalescence manifolds, linking moment problems to spectrahedra and semidefinite programming.

Findings

Any collection of n moments can be realized by a step function with at most n-1 breakpoints.

Any point in the nth coalescence manifold can be achieved by a piecewise constant population history with at most n-2 changes.

Moment cones and coalescence manifolds are projected spectrahedra, enabling semidefinite programming approaches.

Abstract

We study the univariate moment problem of piecewise-constant density functions on the interval $[0,1]$ and its consequences for an inference problem in population genetics. We show that, up to closure, any collection of $n$ moments is achieved by a step function with at most $n-1$ breakpoints and that this bound is tight. We use this to show that any point in the $n$th coalescence manifold in population genetics can be attained by a piecewise constant population history with at most $n-2$ changes. Both the moment cones and the coalescence manifold are projected spectrahedra and we describe the problem of finding a nearest point on them as a semidefinite program.