We study the problem of optimally projecting the transition matrix of a finite ergodic multivariate Markov chain onto a lower-dimensional state space. Specifically, we seek to construct a projected Markov chain that optimizes various information-theoretic criteria under cardinality constraints. These criteria include entropy rate, information-theoretic distance to factorizability, independence, and stationarity. We formulate these tasks as best subset selection problems over multivariate Markov chains and leverage the submodular (or supermodular) structure of the objective functions to develop efficient greedy-based algorithms with theoretical guarantees. We extend our analysis to $k$-submodular settings and introduce a generalized version of the distorted greedy algorithm, which may be of independent interest. Finally, we illustrate the theory and algorithms through extensive numerical experiments with publicly available code on multivariate Markov chains associated with the Bernoulli-Laplace and Curie-Weiss model.

翻译：本文研究有限遍历多元马尔可夫链转移矩阵在低维状态空间上的最优投影问题。具体而言，我们旨在构建一个投影马尔可夫链，使其在基数约束下优化多种信息论准则，包括熵率、与可分解性的信息论距离、独立性与平稳性。我们将这些任务形式化为多元马尔可夫链上的最优子集选择问题，并利用目标函数的子模（或超模）结构，开发具有理论保证的高效贪心算法。我们将分析扩展至$k$-子模场景，并提出广义版本的扭曲贪心算法，该算法本身可能具有独立的研究价值。最后，我们通过基于Bernoulli-Laplace模型和Curie-Weiss模型的多元马尔可夫链进行大量数值实验（代码已公开），对理论与算法进行了验证。