We study the problem of optimally projecting the transition matrix of a finite ergodic multivariate Markov chain onto a lower-dimensional state space, as well as the problem of finding an optimal partition of coordinates such that the factorized Markov chain gives minimal information loss compared to the original multivariate chain. Specifically, we seek to construct a 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 or partition selection problems over multivariate Markov chains and leverage the (k-)submodular (or (k-)supermodular) structures of the objective functions to develop efficient greedy-based algorithms with theoretical guarantees. Along the way, we 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 models.

翻译：我们研究将有限遍历多元马尔可夫链的转移矩阵最优投影到低维状态空间的问题，以及寻找坐标最优划分使得因子化马尔可夫链相较于原始多元链信息损失最小的问题。具体而言，我们旨在构建在基数约束下优化多种信息论准则的马尔可夫链。这些准则包括熵率、与可因子化性的信息论距离、独立性及平稳性。我们将这些任务形式化为多元马尔可夫链上的最优子集或划分选择问题，并利用目标函数的（k-）子模（或（k-）超模）结构，开发具有理论保证的高效贪心算法。在此过程中，我们引入了一种广义版本的扭曲贪心算法，其本身可能具有独立的研究价值。最后，我们通过在Bernoulli–Laplace模型和Curie–Weiss模型相关的多元马尔可夫链上进行大量数值实验，并辅以公开代码，对理论与算法进行了阐释。