Simplicial Regularizability of the Pseudo-Moment Cone and Carathéodory-Type Atomic Decomposition of Moment Matrices
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914540758761472 |
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| author | Kang, Shucheng Yang, Heng |
| author_facet | Kang, Shucheng Yang, Heng |
| contents | We study the facial geometry of the homogeneous pseudo-moment cone \(Σ_{n,2d}^*\) 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06854 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Simplicial Regularizability of the Pseudo-Moment Cone and Carathéodory-Type Atomic Decomposition of Moment Matrices Kang, Shucheng Yang, Heng Optimization and Control We study the facial geometry of the homogeneous pseudo-moment cone \(Σ_{n,2d}^*\) 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 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. |
| title | Simplicial Regularizability of the Pseudo-Moment Cone and Carathéodory-Type Atomic Decomposition of Moment Matrices |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2605.06854 |