Simplicial Regularizability of the Pseudo-Moment Cone and Carath\'eodory-Type Atomic Decomposition of Moment Matrices

TL;DR

This paper explores the geometry of the pseudo-moment cone and introduces an efficient atomic decomposition algorithm for moment matrices, with strong recovery performance demonstrated through numerical experiments.

Contribution

It establishes the simplicial structure of minimal faces in the pseudo-moment cone and develops a Carathéodory-type decomposition algorithm tailored for spectrahedral cones.

Findings

Minimal face of a moment matrix with O(n^d) atoms is simplicial and generated by rank-one atoms.

The proposed algorithm provides an efficient atomic decomposition for generically generated moment matrices.

Numerical experiments show strong recovery performance and potential as a practical high-rank extreme ray sampler.

Abstract

We study the facial geometry of the homogeneous pseudo-moment cone \(\Sigma_{n,2d}^*\) and its implications for atomic decomposition of moment matrices. For fixed \(d \ge 2\), we show that if a moment matrix is formed by \(O(n^d)\) generically chosen weighted atoms, then its minimal face in the matrix realization of the pseudo-moment cone is \emph{simplicial} and generated by the planted rank-one atoms. Based on this geometric result, we develop a Carath\'eodory-type extreme-ray decomposition algorithm for spectrahedral cones and show that, when specialized to the pseudo-moment cone, it yields an efficient atomic decomposition method for generically generated moment matrices in the same regime. A stabilized numerical implementation demonstrates strong recovery performance and suggests that, outside the guaranteed regime, the algorithm may serve as a practical sampler of high-rank extreme rays.