Minimal realization problem for Hidden Markov Models

In this paper, we study the problem of finding a minimal order (quasi-) Hidden Markov Model for a random process, which is the output process of an unknown stationary HMM of finite order. In the main theorem, we show that excluding a measure zero set in the parameter space of transition and observation probability matrices, both the minimal quasi-HMM realization and the minimal HMM realization can be efficiently constructed based on the joint probabilities of length N output strings, for N > max(4 logd(k) + 1,3), where d is the size of the output alphabet size, and k is the minimal order of the realization. We also aim to connect the two lines of literature: realization theory of HMMs/automatas, and the recent development in learning latent variable models with tensor techniques.

[1]  Lieven De Lathauwer,et al.  Fourth-Order Cumulant-Based Blind Identification of Underdetermined Mixtures , 2007, IEEE Transactions on Signal Processing.

[2]  Raphaël Bailly Quadratic Weighted Automata: Spectral Algorithm and Likelihood Maximization , 2011, ACML 2011.

[3]  Elchanan Mossel,et al.  Learning nonsingular phylogenies and hidden Markov models , 2005, STOC '05.

[4]  Ariadna Quattoni,et al.  Spectral learning of weighted automata , 2014, Machine Learning.

[5]  Aditya Bhaskara,et al.  Uniqueness of Tensor Decompositions with Applications to Polynomial Identifiability , 2013, COLT.

[6]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[7]  Mathukumalli Vidyasagar,et al.  The complete realization problem for hidden Markov models: a survey and some new results , 2011, Math. Control. Signals Syst..

[8]  Dean Alderucci A SPECTRAL ALGORITHM FOR LEARNING HIDDEN MARKOV MODELS THAT HAVE SILENT STATES , 2015 .

[9]  Anima Anandkumar,et al.  Tensor decompositions for learning latent variable models , 2012, J. Mach. Learn. Res..

[10]  Nikos D. Sidiropoulos,et al.  Kruskal's permutation lemma and the identification of CANDECOMP/PARAFAC and bilinear models with constant modulus constraints , 2004, IEEE Transactions on Signal Processing.

[11]  Lieven De Lathauwer,et al.  A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization , 2006, SIAM J. Matrix Anal. Appl..

[12]  N. Sidiropoulos,et al.  On the uniqueness of multilinear decomposition of N‐way arrays , 2000 .

[13]  Bart De Moor,et al.  Equivalence of state representations for hidden Markov models , 2007, 2007 European Control Conference (ECC).

[14]  C. Matias,et al.  Identifiability of parameters in latent structure models with many observed variables , 2008, 0809.5032.

[15]  Brian D. O. Anderson,et al.  The Realization Problem for Hidden Markov Models , 1999, Math. Control. Signals Syst..

[16]  J. Kruskal Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics , 1977 .

[17]  Ronitt Rubinfeld,et al.  On the learnability of discrete distributions , 1994, STOC '94.

[18]  Lieven De Lathauwer,et al.  Tensor-based techniques for the blind separation of DS-CDMA signals , 2007, Signal Process..

[19]  Bart De Moor,et al.  Subspace Identification for Linear Systems: Theory ― Implementation ― Applications , 2011 .