Difference between revisions of "Siddiqi et al 2009 Reduced-Rank Hidden Markov Models"
Line 5: | Line 5: | ||
== Summary == | == Summary == | ||
− | This paper introduces Reduced-Rank Hidden Markov Models (RR-HMMs). RR-HMMs are similar to standard HMMs, except the rank of the transition matrix is less than the number of hidden states. | + | This paper introduces Reduced-Rank Hidden Markov Models (RR-HMMs). RR-HMMs are similar to standard HMMs, 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 == | == 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]] | [[File:Siddiqi et al 2009 Definition of P.png]] | ||
Line 16: | Line 16: | ||
[[File:Siddiqi et al 2009 Algorithm.png]] | [[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. m 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>. | + | <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>. |
Line 24: | Line 24: | ||
== Experimental Result == | == Experimental Result == | ||
− | + | [[File:Siddiqi et al 2009 Results.png]] | |
== Related Papers == | == Related Papers == | ||
In progress by [[User:Jmflanig]] | In progress by [[User:Jmflanig]] |
Revision as of 12:57, 10 October 2011
Citation
Online version
Summary
This paper introduces Reduced-Rank Hidden Markov Models (RR-HMMs). RR-HMMs are similar to standard HMMs, 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 be the observed output of the RR-HMM and let:
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.
is the initial state distribution, is the final state distribution, and is the transition matrix when x is observed. is the rank of the reduced state space. Note that denotes the Moore-Penrose pseudo-inverse of the matrix .
Inference can be performed using the model parameters:
Experimental Result
Related Papers
In progress by User:Jmflanig