Simplicial Regularizability of the Pseudo-Moment Cone and Carathéodory-Type Atomic Decomposition of Moment Matrices

We study the facial geometry of the homogeneous pseudo-moment cone ${\mathrm{\Sigma }}_{n,2d}^{\ast }$ 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 Carathéodory-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.