This work presents a low-rank tensor model for multi-dimensional Markov chains. A common approach to simplify the dynamical behavior of a Markov chain is to impose low-rankness on the transition probability matrix. Inspired by the success of these matrix techniques, we present low-rank tensors for representing transition probabilities on multi-dimensional state spaces. Through tensor decomposition, we provide a connection between our method and classical probabilistic models. Moreover, our proposed model yields a parsimonious representation with fewer parameters than matrix-based approaches. Unlike these methods, which impose low-rankness uniformly across all states, our tensor method accounts for the multi-dimensionality of the state space. We also propose an optimization-based approach to estimate a Markov model as a low-rank tensor. Our optimization problem can be solved by the alternating direction method of multipliers (ADMM), which enjoys convergence to a stationary solution. We empirically demonstrate that our tensor model estimates Markov chains more efficiently than conventional techniques, requiring both fewer samples and parameters. We perform numerical simulations for both a synthetic low-rank Markov chain and a real-world example with New York City taxi data, showcasing the advantages of multi-dimensionality for modeling state spaces.

翻译：本研究提出了一种用于多维马尔可夫链的低秩张量模型。简化马尔可夫链动态行为的常用方法是在转移概率矩阵上施加低秩约束。受这些矩阵方法成功的启发，我们提出了低秩张量来表示多维状态空间上的转移概率。通过张量分解，我们建立了该方法与经典概率模型之间的联系。此外，所提出的模型能够产生比基于矩阵的方法参数更少的简约表示。与这些在所有状态上统一施加低秩约束的方法不同，我们的张量方法考虑了状态空间的多维特性。我们还提出了一种基于优化的方法，将马尔可夫模型估计为低秩张量。我们的优化问题可以通过交替方向乘子法（ADMM）求解，该方法能够收敛至平稳解。我们通过实验证明，与常规技术相比，我们的张量模型能够更高效地估计马尔可夫链，同时需要更少的样本和参数。我们分别对合成低秩马尔可夫链和纽约市出租车数据的真实案例进行了数值模拟，展示了多维特性在状态空间建模中的优势。