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.

半定规划和优化中的采样过程，使用 Dikin 行走和内点方法结合，通过适当的指标选择，可以在多项式对数的复杂度下减少混合时间和步骤复杂度。