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Optimal Kullback–Leibler approximation of Markov chains via nuclear norm regularisation

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  • Kun Deng
  • Dayu Huang

Abstract

This paper is concerned with model reduction for Markov chain models. The goal is to obtain a low-rank approximation to the original Markov chain. The Kullback–Leibler divergence rate is used to measure the similarity between two Markov chains; the nuclear norm is used to approximate the rank function. A nuclear-norm regularised optimisation problem is formulated to approximately find the optimal low-rank approximation. The proposed regularised problem is analysed and performance bounds are obtained through the convex analysis. An iterative fixed point algorithm is developed based on the proximal splitting technique to compute the optimal solutions. The effectiveness of this approach is illustrated via numerical examples.

Suggested Citation

  • Kun Deng & Dayu Huang, 2015. "Optimal Kullback–Leibler approximation of Markov chains via nuclear norm regularisation," International Journal of Systems Science, Taylor & Francis Journals, vol. 46(11), pages 2029-2047, August.
    • Handle: RePEc:taf:tsysxx:v:46:y:2015:i:11:p:2029-2047
    DOI: 10.1080/00207721.2013.844284
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    References listed on IDEAS

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    1. Hua Han & Yongsheng Ding & Kuangrong Hao & Liangjian Hu, 2013. "Particle filter for state estimation of jump Markov nonlinear system with application to multi-targets tracking," International Journal of Systems Science, Taylor & Francis Journals, vol. 44(7), pages 1333-1343.
    2. Carl Eckart & Gale Young, 1936. "The approximation of one matrix by another of lower rank," Psychometrika, Springer;The Psychometric Society, vol. 1(3), pages 211-218, September.
    3. Guoying Miao & Shengyuan Xu & Yun Zou, 2013. "Necessary and sufficient conditions for mean square consensus under Markov switching topologies," International Journal of Systems Science, Taylor & Francis Journals, vol. 44(1), pages 178-186.
    4. Alexander Brownlee & Olivier Regnier-Coudert & John McCall & Stewart Massie & Stefan Stulajter, 2013. "An application of a GA with Markov network surrogate to feature selection," International Journal of Systems Science, Taylor & Francis Journals, vol. 44(11), pages 2039-2056.
    5. Z. Kowalczuk & M. Domżalski, 2012. "Optimal asynchronous estimation of 2D Gaussian–Markov processes," International Journal of Systems Science, Taylor & Francis Journals, vol. 43(8), pages 1431-1440.
    6. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
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