This paper develops a low-nonnegative-rank approximation method to identify the state aggregation structure of a finite-state Markov chain under an assumption that the state space can be mapped into a handful of meta-states. The number of meta-states is characterized by the nonnegative rank of the Markov transition matrix. Motivated by the success of the nuclear norm relaxation in low rank minimization problems, we propose an atomic regularizer as a convex surrogate for the nonnegative rank and formulate a convex optimization problem. Because the atomic regularizer itself is not computationally tractable, we instead solve a sequence of problems involving a nonnegative factorization of the Markov transition matrices by using the proximal alternating linearized minimization method. Two methods for adjusting the rank of factorization are developed so that local minima are escaped. One is to append an additional column to the factorized matrices, which can be interpreted as an approximation of a negative subgradient step. The other is to reduce redundant dimensions by means of linear combinations. Overall, the proposed algorithm very likely converges to the global solution. The efficiency and statistical properties of our approach are illustrated on synthetic data. We also apply our state aggregation algorithm on a Manhattan transportation data set and make extensive comparisons with an existing method.

中文翻译：

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马尔可夫链状态聚合的自适应低非负秩近似

本文开发了一种低非负秩近似方法，以在状态空间可以映射到少数元状态的假设下识别有限状态马尔可夫链的状态聚合结构。元状态的数量以马尔可夫转移矩阵的非负秩为特征。受低秩最小化问题中核范数松弛成功的启发，我们提出了一个原子正则化器作为非负秩的凸代理，并制定了一个凸优化问题。由于原子正则化器本身在计算上不易于处理，因此我们通过使用近端交替线性化最小化方法来解决一系列涉及马尔可夫转移矩阵的非负因式分解的问题。开发了两种调整因式分解等级的方法，以便避开局部最小值。一种是在分解矩阵后附加一列，这可以解释为负次梯度步骤的近似值。另一种是通过线性组合的方式减少冗余维度。总的来说，所提出的算法很可能收敛到全局解。我们的方法的效率和统计特性在合成数据上进行了说明。我们还将我们的状态聚合算法应用于曼哈顿交通数据集，并与现有方法进行了广泛的比较。另一种是通过线性组合的方式减少冗余维度。总的来说，所提出的算法很可能收敛到全局解。我们的方法的效率和统计特性在合成数据上进行了说明。我们还将我们的状态聚合算法应用于曼哈顿交通数据集，并与现有方法进行了广泛的比较。另一种是通过线性组合的方式减少冗余维度。总的来说，所提出的算法很可能收敛到全局解。我们的方法的效率和统计特性在合成数据上进行了说明。我们还将我们的状态聚合算法应用于曼哈顿交通数据集，并与现有方法进行了广泛的比较。