Moment varieties of measures on polytopes

Abstract

The uniform probability measure on a convex polytope induces piecewise polynomial densities on its projections. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex coordinates. We study projective varieties that are parametrized by finite collections of such rational functions. Our focus lies on determining the prime ideals of these moment varieties. Special cases include Hankel determinantal ideals for polytopal splines on line segments, and the relations among multisymmetric functions given by the cumulants of a simplex. In general, our moment varieties are more complicated than in these two special cases. They offer challenges for both numerical and symbolic computing in algebraic geometry.

Combinatorial Methods for Minkowski Tensors of Polytopes

“…There is also a close connection between these tensors and moments of the uniform distribution on convex bodies [7]. As shown in [6], moments on polytopes can be described via rational generating functions and are parametrized by certain algebraic varieties. In this paper we show that the methods employed in [6] for describing moments, and hence volume tensors, extend to give a description of surface tensors.…”

“…As shown in [6], moments on polytopes can be described via rational generating functions and are parametrized by certain algebraic varieties. In this paper we show that the methods employed in [6] for describing moments, and hence volume tensors, extend to give a description of surface tensors. We compute explicit generating functions for these tensors and define an analog of the adjoint polynomial of [15] for the boundary complex of a polytope.…”

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In this paper we use a generating function approach to record and calculate entries of the Minkowski tensors of a polytope. We focus on "surface tensors", extending the methods used in [6] for moments of the uniform distribution which correspond to volume tensors. In this context we also extend the definition of the adjoint polynomial to the boundary complex of a polytope with simplicial facets.

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PLMP - Point-Line Minimal Problems in Complete Multi-View Visibility

“…Specifically, we follow the monodromy solver framework outlined in [9] carrying out computation via a Macaulay2 package MonodromySolver 9 . Similar techniques have been successfully employed in a number of studies in applied algebraic geometry [17,22,8].…”

Moment Identifiability of Homoscedastic Gaussian Mixtures

“…In the context of algebraic statistics [19], moments of probability distributions have recently been explored from an algebraic and geometric point of view [1,4,11,13]. The key point for this connection is that in many cases the sets of moments define algebraic varieties, hence called moment varieties.…”