We study the problem of selecting optimal two-block partitions to accelerate the mixing of finite Markov chains under group-averaging transformations. The main objectives considered are the Kullback-Leibler (KL) divergence and the Frobenius distance to stationarity. We establish explicit connections between these objectives and the induced projection chain. In the case of the KL divergence, this reduction yields explicit decay rates in terms of the log-Sobolev constant. For the Frobenius distance, we identify a Cheeger-type functional that characterises optimal cuts. This formulation recasts two-block selection as a structured combinatorial optimisation problem admitting difference-of-submodular decompositions. We further propose several algorithmic approximations, including majorisation-minimisation and coordinate descent schemes, as computationally feasible alternatives to exhaustive combinatorial search. Our numerical experiments reveal that optimal cuts under the two objectives can substantially reduce total variation distance to stationarity and demonstrate the practical effectiveness of the proposed approximation algorithms.

翻译：本研究探讨了在群平均变换下，如何选择最优的双块划分以加速有限马尔科夫链的混合过程。主要考察的目标函数为平稳分布的Kullback-Leibler（KL）散度与Frobenius距离。我们建立了这些目标函数与诱导投影链之间的显式联系。对于KL散度，该约简过程通过log-Sobolev常数给出了显式衰减率。针对Frobenius距离，我们识别出一个刻画最优划分的Cheeger型泛函。此公式将双块选择问题重构为具有子模差分解的结构化组合优化问题。我们进一步提出了多种算法近似方案，包括主化最小化与坐标下降策略，作为穷举组合搜索的计算可行替代方案。数值实验表明，在两种目标函数下的最优划分能显著降低平稳分布的总变差距离，并验证了所提近似算法的实际有效性。