有效可数广义矩问题
The Effective Countable Generalized Moment Problem
AI总结：
本文研究可数约束广义矩问题的 Moment-SoS 松弛，在对偶可达、$S$-满性和阿基米德条件下给出几何自适应的最优值、可行集与最优测度收敛速率，并应用于对称张量分解。
摘要：
我们在对偶最优解可达、$S$-满性和阿基米德条件下，针对测度向量上具有可数矩约束的广义矩问题（GMP），建立了矩-平方和（Moment-SoS）松弛的新收敛速率。这些界能够适应底层半代数集的几何结构，既适用于最优值的收敛，也适用于松弛可行集与 GMP 可行集之间 Hausdorff 距离意义下的收敛。我们表明，在上述条件下，只要最优测度唯一，松弛的最优解序列就在 weak$^*$ 拓扑下收敛到 GMP 的最优解。本研究为表述为测度上线性规划的 GMP 提供了定量的、几何自适应的速率。它补充了早期对特定 GMP 实例（例如多项式优化）的分析，也补充了近期已应用于体积计算和最优控制的方法框架。我们将该收敛速率分析应用于对称张量分解问题，为张量分解的 Moment-SoS 层次收敛提供了新的有效误差界。
英文摘要：
We establish new convergence rates for the Moment-Sum-of-Squares (Moment-SoS) relaxations for the Generalized Moment Problem (GMP) with countable moment constraints on vectors of measures, under dual optimum attainment, $S$-fullness and Archimedean conditions. These bounds, which adapt to the geometry of the underlying semi-algebraic set, apply to both the convergence of optima, and to the convergence in Hausdorff distance between the relaxation feasibility set and the GMP feasibility set. We show that under the previous conditions, the sequence of optimizers of the relaxations converge to the optimizer of the GMP for the weak$^*$ topology, provided this optimal measure is unique. This research provides quantitative geometry-adaptive rates for GMPs cast as linear programs on measures. It complements earlier analyses of specific GMP instances (e.g., polynomial optimization) as well as recent methodological frameworks that have been applied to volume computation and optimal control. We apply the convergence rate analysis to symmetric tensor decomposition problems, providing new effective error bounds for the convergence of the Moment-SoS hierarchies for tensor decomposition.
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