Abstract
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.
Highlights
Large-scale time- and space-discrete Markov chains are ubiquitous in many areas of quantitative science, where they arise as discretizations of continuous models [1,2,3,4], as formalization of network-based models [5,6,7], or as models of many other types of complex dynamics
Markov chains arising from discretized biomolecular systems often exhibit metastability, the phenomenon that on long time scales, the dynamics is determined by rare jumps between almost-invariant subsets of states [1,8,9]
We have demonstrated that in applications where the computation of the transition matrix is the computational bottleneck, this can lead to a speedup of factor 10 or more over conventional model reduction algorithms
Summary
Large-scale time- and space-discrete Markov chains are ubiquitous in many areas of quantitative science, where they arise as discretizations of continuous models [1,2,3,4], as formalization of network-based models [5,6,7], or as models of many other types of complex dynamics. Markov chains arising from discretized biomolecular systems often exhibit metastability, the phenomenon that on long time scales, the dynamics is determined by rare jumps between almost-invariant subsets of states [1,8,9] Another example are complex traffic networks, whose transition matrices often exhibit a low-rank structure, which can be explained by patterns in the large-scale traffic flow between neighborhoods of a city [10,11]. Due to its probabilistic nature, the algorithm is able to exploit the low-rank structure of the full transition matrix without detailed knowledge of it This gives our method a computational advantage of several orders of magnitude over methods that require the full transition matrix. The entry of the ith row and jth column of A is denoted by Aij
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