Geiger, Bernhard; Petrov, TatjanaIST Austria; Kubin, Gernot; Koeppl, Heinz
In this paper, we present a method for reducing a regular, discrete-time Markov chain (DTMC) to another DTMC with a given, typically much smaller number of states. The cost of reduction is defined as the Kullback-Leibler divergence rate between a projection of the original process through a partition function and a DTMC on the correspondingly partitioned state space. Finding the reduced model with minimal cost is computationally expensive, as it requires an exhaustive search among all state space partitions, and an exact evaluation of the reduction cost for each candidate partition. Our approach deals with the latter problem by minimizing an upper bound on the reduction cost instead of minimizing the exact cost. The proposed upper bound is easy to compute and it is tight if the original chain is lumpable with respect to the partition. Then, we express the problem in the form of information bottleneck optimization, and propose using the agglomerative information bottleneck algorithm for searching a suboptimal partition greedily, rather than exhaustively. The theory is illustrated with examples and one application scenario in the context of modeling bio-molecular interactions.
IEEE Transactions on Automatic Control
This work was supported by the Austrian Research Association under Project 06/12684, by the Swiss National Science Foundation (SNSF) under Grant PP00P2 128503/1, by the SystemsX.ch (the Swiss Inititative for Systems Biology), and by a SNSF Early Postdoc.Mobility Fellowship grant P2EZP2_148797.
1010 - 1022
Geiger B, Petrov T, Kubin G, Koeppl H. Optimal Kullback-Leibler aggregation via information bottleneck. IEEE Transactions on Automatic Control. 2015;60(4):1010-1022. doi:10.1109/TAC.2014.2364971
Geiger, B., Petrov, T., Kubin, G., & Koeppl, H. (2015). Optimal Kullback-Leibler aggregation via information bottleneck. IEEE Transactions on Automatic Control, 60(4), 1010–1022. https://doi.org/10.1109/TAC.2014.2364971
Geiger, Bernhard, Tatjana Petrov, Gernot Kubin, and Heinz Koeppl. “Optimal Kullback-Leibler Aggregation via Information Bottleneck.” IEEE Transactions on Automatic Control 60, no. 4 (2015): 1010–22. https://doi.org/10.1109/TAC.2014.2364971.
B. Geiger, T. Petrov, G. Kubin, and H. Koeppl, “Optimal Kullback-Leibler aggregation via information bottleneck,” IEEE Transactions on Automatic Control, vol. 60, no. 4, pp. 1010–1022, 2015.
Geiger B, Petrov T, Kubin G, Koeppl H. 2015. Optimal Kullback-Leibler aggregation via information bottleneck. IEEE Transactions on Automatic Control. 60(4), 1010–1022.
Geiger, Bernhard, et al. “Optimal Kullback-Leibler Aggregation via Information Bottleneck.” IEEE Transactions on Automatic Control, vol. 60, no. 4, IEEE, 2015, pp. 1010–22, doi:10.1109/TAC.2014.2364971.