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。尽管在半定优化方面取得了显著进展，中等规模的问题目前可通过内点法实际求解，但半定解的基本抽样问题仍是一个严峻挑战。将已知用于一般凸体抽样的多项式时间算法直接应用于半定抽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。本文分析了Dikin游走，首先将其推广到一般度量，然后为具有仿射约束的PSD锥设计合适的度量。由此得到的混合时间和每步复杂度显著降低，并且通过适当选择度量，对约束数量的依赖可达到多对数级别。我们引入了一个自和谐矩阵函数的精炼概念，并给出了组合不同度量的规则。在此过程中，我们进一步发展了用于抽样的内点方法理论。