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arXiv · 2605.06854

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

Shucheng Kang·Heng Yang
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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.

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Shucheng Kang, Heng Yang. 2026-05-07. Simplicial Regularizability of the Pseudo-Moment Cone and Carath\'eodory-Type Atomic Decomposition of Moment Matrices. https://arxiv.org/abs/2605.06854

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