Model reduction of large Markov chains is an essential step in a wide array of techniques for understanding complex systems and for efficiently learning structures from high-dimensional data. We present a novel aggregation algorithm for compressing such chains that exploits a specific low-rank structure in the transition matrix which, e.g., is present in metastable systems, among others. It enables the recovery of the aggregates from a vastly undersampled transition matrix which in practical applications may gain a speedup of several orders of magnitude over methods that require the full transition matrix. Moreover, we show that the new technique is robust under perturbation of the transition matrix. The practical applicability of the new method is demonstrated by identifying a reduced model for the large-scale traffic flow patterns from real-world taxi trip data.

大型马尔可夫链的模型简化是用于理解复杂系统并从高维数据有效学习结构的众多技术中的重要步骤。我们提出了一种用于压缩这样的链的新颖的聚集算法，该算法利用了例如在亚稳系统中存在的过渡矩阵中的特定低秩结构。它使人们能够从一个采样率极低的过渡矩阵中恢复聚集体，在实际应用中，与需要完整过渡矩阵的方法相比，该矩阵可以将速度提高几个数量级。此外，我们证明了该新技术在过渡矩阵的扰动下具有鲁棒性。