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：我们研究了有限遍历多元马尔可夫链转移矩阵向低维状态空间的最优投影问题，以及寻找最优坐标划分以使因子化马尔可夫链相较原始多元链信息损失最小化的问题。具体而言，我们旨在构建满足基数约束下多种信息论准则最优化的马尔可夫链，这些准则包括熵率、与可因子化性的信息论距离、独立性和平稳性。我们将这些任务建模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，开发了具有理论保证的高效贪心算法。在此过程中，我们引入了扭曲贪心算法的一个广义版本，该版本可能具有独立的研究价值。最后，我们通过公开代码，在伯努利-拉普拉斯和居里-外斯模型相关的多元马尔可夫链上进行了大量数值实验，展示了相关理论与算法的有效性。