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== Citation == == Online version == [http://arxiv.org/pdf/0910.0902v3] == Summary == This [[Category::paper]] introduces Reduced-Rank Hidden Markov Models (RR-HMMs). A RR-HMM is similar to standard [[UsesMethod::HMM]], except the rank of the transition matrix is less than the number of hidden states. The dynamics evolve in a subspace of the hidden state probability space. == Method == Let <math>x_t</math> be the observed output of the RR-HMM and let: [[File:Siddiqi et al 2009 Definition of P.png]] The learning algorithm uses a singular value decomposition (SVD) of the correlation matrix between past and future observations. The algorithm is borrowed from Hsu et al 2009, with no change for the reduced-rank case. [[File:Siddiqi et al 2009 Algorithm.png]] <math>\hat{b}_1</math> is the initial state distribution, <math>\hat{b}_\infty</math> is the final state distribution, and <math>\hat{B}_x</math> is the transition matrix when x is observed. <math>m</math> is the rank of the reduced state space. Note that <math>X^+</math> denotes the Moore-Penrose pseudo-inverse of the matrix <math>X</math>. Inference can be performed using the model parameters: [[File:Siddiqi et al 2009 Inference.png]] == Experimental Result == [[File:Siddiqi et al 2009 Results.png]] == Related Papers == In progress by [[User:Jmflanig]] == Comment == This stuff is super cool but a little tricky to wrap one's head around. I wrote up these notes about the older Hsu paper, and Siddiqi too, a while ago and they may not be totally correct but here we go: http://brenocon.com/matrix_hmm.pdf --[[User:Brendan|Brendan]] 23:21, 13 October 2011 (UTC)

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