从矩恢复测度的低维半代数支集
Recovering Lower-Dimensional Semialgebraic Support of a Measure from its Moments
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中文总结 AI 辅助
本文针对余维数至少为一的紧致半代数支集，结合矩矩阵核与Christoffel--Darboux核提出离散近似方法，并在解析矩和伪矩上验证，补充了代数支集和全维支集恢复的近期工作。
中文摘要 AI 辅助
从矩恢复概率测度在许多应用中具有广泛用途，特别是在统计学和优化中的矩方法相关领域。在测度不必是有限原子的，但其支集已知为紧致且半代数、余维数至少为一的情况下，该问题仍然是开放的。我们将矩矩阵核信息与Christoffel--Darboux核相结合，提供了支集的离散近似。为验证所提出的方法，我们在解析计算的矩以及由不具有唯一全局最小值的多项式优化问题产生的伪矩上测试了我们的算法。这补充了近期关于恢复具有代数支集的测度的知名工作（其中矩矩阵的核可以揭示在支集上消失的多项式），以及恢复足够正则的全维支集的工作（其中通过阈值化Christoffel--Darboux核构建的估计量已知渐近收敛到支集）。
英文摘要
Recovering probability measures from their moments has numerous applications, esp. in connection with the method of moments in statistics and optimization. In the setting where measure need not be finitely atomic, but its support is known to be compact and semialgebraic with codimension at least one, the problem is still open. We combine moment-matrix kernel information with the Christoffel--Darboux kernel to provide a discrete approximation of the support. To validate the proposed approach, we test our algorithm on analytically computed moments and pseudo-moments arising from polynomial optimization problems without unique global minimizers. This complements well-known recent work on recovery of measures with algebraic support, where the kernel of a moment matrix can reveal polynomials vanishing on the support, and on recovery of sufficiently regular full-dimensional supports, where estimators constructed by thresholding the Christoffel--Darboux kernel are known to converge asymptotically to the support.
发表机构
- Czech Technical University in Prague（布拉格捷克理工大学）
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