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Markov State Space Aggregation via the Information Bottleneck Method

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Języki publikacji
EN
Abstrakty
EN
Consider the problem of approximating a Markov chain by another Markov chain with a smaller state space that is obtained by partitioning the original state space. An information-theoretic cost function is proposed that is based on the relative entropy rate between the original Markov chain and a Markov chain defined by the partition. The state space aggregation problem can be sub-optimally solved by using the information bottleneck method.
Rocznik
Tom
Strony
45--56
Opis fizyczny
Bibliogr. 24 poz., rys.
Twórcy
autor
  • Institute for Communications Engineering TU Munich Arcisstraße 21, D-80333 Munich
Bibliografia
  • [1] Cover T.M., Thomas J.A., Elements of Information Theory, Wiley Interscience, Hoboken, NJ, 2nd edition, 2006.
  • [2] Deng K., Mehta P.G., Meyn S.P., Optimal Kullback-Leibler aggregation via spectral theory of Markov chains, IEEE Trans. Autom. Control 56 (12), Dec. 2011, pp. 2793–2808.
  • [3] Dhillon I., Mallela S., Modha D., Information-theoretic co-clusterings, Proc. ACM Int. Conf. on Knowledge Discovery and Data Mining (SIGKDD), Washington, D.C., Aug. 2003, pp. 89–98.
  • [4] Friedman A., Goldberger J., Information theoretic pairwise clustering, E. Hancock and M. Pelillo (Eds.), Proc. Similarity-Based Pattern Recognition,Springer Berlin LNCS 7953, 2013, pp. 106–119.
  • [5] Geiger B.C., Petrov T., Kubin G., Koeppl H., Optimal Kullback-Leibler aggregation via information bottleneck, Apr. 2013, Accepted for publication in IEEE Trans. Autom. Control; preprint available: arXiv:1304.6603 [cs.SY].
  • [6] Geiger B.C., Temmel C., Lumpings of Markov chains, entropy rate preservation, and higher-order lumpability, Dec. 2012. Accepted for publication in J. Appl. Prob., preprint available: arXiv:1212.4375 [cs.IT].
  • [7] Goldberger J., Erez K., Abeles M., A Markov clustering method for analyzing movement trajectories, Proc. IEEE Int. Workshop on Machine Learning for Signal Processing (MLSP), Thessaloniki, Aug. 2007, pp. 211–216.
  • [8] Gray R.M., Entropy and Information Theory, Springer, New York, NY, 1990.
  • [9] Gurvits L., Ledoux J., Markov property for a function of a Markov chain: A linear algebra approach, Linear Algebra Appl. 404, 2005, pp. 85–117.
  • [10] Hayes B., First links in the Markov chain, American Scientist 101, 2013, pp. 92–97.
  • [11] Katsoulakis M.A., Trashorras J., Information loss in coarse-graining of stochastic particle dynamics, J. Stat. Phys., 122 (1), 2006, pp. 115–135.
  • [12] Kemeny J.G., Snell J.L., Finite Markov Chains, Springer, 2nd edition, 1976.
  • [13] Manning C.D., Schütze H., Foundations of Statistical Natural Language Processing, MIT Press, Cambridge, MA, 2nd edition, 2000.
  • [14] Petrov T., Formal reductions of stochastic rule-based models of biochemical systems, PhD thesis, ETH Zürich, 2013.
  • [15] Rached Z., Alajaji F., Campbell L.L., The Kullback-Leibler divergence rate between Markov sources, IEEE Trans. Inf. Theory 50 (5), May 2004, pp. 917–921.
  • [16] Raj A., Wiggins C.H., An information-theoretic derivation of min-cut-based clustering, IEEE Trans. Pattern Anal. Mach. Intell. 32 (6), June 2010, pp. 988–995.
  • [17] Shannon C.E., A mathematical theory of communication, Bell Systems Technical Journal 27, Oct. 1948, pp. 379–423, 623–656.
  • [18] Slonim N., Tishby N., Agglomerative information bottleneck, Advances in Neural Information Processing Systems (NIPS), Denver, CO, Nov. 1999, pp. 617–623.
  • [19] Tishby N., Pereira F.C., Bialek W., The information bottleneck method, Proc. Allerton Conf. on Communication, Control, and Computing, Monticello, IL, Sept. 1999, pp. 368–377.
  • [20] Tishby N., Slonim N., Data clustering by Markovian relaxation and the information bottleneck method, Advances in Neural Information Processing Systems (NIPS), Denver, CO, Nov. 2000.
  • [21] Tzortzis I., Charalambous C.D., Charalambous T., Hadjicostis C.N., Johansson M., Approximation of Markov processes by lower dimensional processes via total variation metrics, Oct. 2014, arXiv:1410.3976 [math.OC].
  • [22] Vedaldi A., Fulkerson B. VLFeat: An open and portable library of computer vision algorithms, 2008, http://www.vlfeat.org/.
  • [23] Vidyasagar M., Reduced-order modeling of Markov and hidden Markov processes via aggregation, Proc. IEEE Conf. on Decision and Control (CDC), Atlanta, GA, Dec. 2010, pp. 1810–1815.
  • [24] White L.B., Mahony R., Brushe G.D., Lumpable hidden Markov models–model reduction and reduced complexity filtering, IEEE Trans. Autom. Control 45 (12), Dec. 2000, pp. 2297–2306.
Typ dokumentu
Bibliografia
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