Semi-definite programs represent a frontier of efficient computation. While there has been much progress on semi-definite optimization, with moderate-sized instances currently solvable in practice by the interior-point method, the basic problem of sampling semi-definite solutions remains a formidable challenge. The direct application of known polynomial-time algorithms for sampling general convex bodies to semi-definite sampling leads to a prohibitively high running time. In addition, known general methods require an expensive rounding phase as pre-processing. Here we analyze the Dikin walk, by first adapting it to general metrics, then devising suitable metrics for the PSD cone with affine constraints. The resulting mixing time and per-step complexity are considerably smaller, and by an appropriate choice of the metric, the dependence on the number of constraints can be made polylogarithmic. We introduce a refined notion of self-concordant matrix functions and give rules for combining different metrics. Along the way, we further develop the theory of interior-point methods for sampling.

翻译：半定规划代表了高效计算的前沿。尽管半定优化已取得显著进展，中等规模的问题目前可通过内点法在实践中求解，但半定解的基本采样问题仍然充满挑战。将已知的通用凸体采样多项式时间算法直接应用于半定采样会导致过高的运行时间。此外，已知的通用方法需要昂贵的圆整阶段作为预处理。本文通过首先将Dikin游走推广到一般度量，然后为带仿射约束的PSD锥设计合适的度量，分析了该游走。其混合时间和每步计算复杂度显著降低，通过合理选择度量，对约束数量的依赖可达到多对数级别。我们引入了自和谐矩阵函数的精细概念，并给出了不同度量的组合规则。在此过程中，我们进一步发展了面向采样的内点方法理论。