Hierarchical Tensor Decomposition of Latent Tree Graphical Models

Le Song, Mariya Ishteva, Ankur Parikh, Eric Xing, Haesun Park
Proceedings of the 30th International Conference on Machine Learning, PMLR 28(3):334-342, 2013.

Abstract

We approach the problem of estimating the parameters of a latent tree graphical model from a hierarchical tensor decomposition point of view. In this new view, the marginal probability table of the observed variables in a latent tree is treated as a tensor, and we show that: (i) the latent variables induce low rank structures in various matricizations of the tensor; (ii) this collection of low rank matricizations induce a hierarchical low rank decomposition of the tensor. Exploiting these properties, we derive an optimization problem for estimating the parameters of a latent tree graphical model, i.e., hierarchical decomposion of a tensor which minimizes the Frobenius norm of the difference between the original tensor and its decomposition. When the latent tree graphical models are correctly specified, we show that a global optimum of the optimization problem can be obtained via a recursive decomposition algorithm. This algorithm recovers previous spectral algorithms for hidden Markov models (Hsu et al., 2009; Foster et al., 2012) and latent tree graphical models (Parikh et al., 2011; Song et al., 2011) as special cases, elucidating the global objective these algorithms are optimizing for. When the latent tree graphical models are misspecified, we derive a better decomposition based on our framework, and provide approximation guarantee for this new estimator. In both synthetic and real world data, this new estimator significantly improves over the-state-of-the-art.

Cite this Paper

BibTeX
@InProceedings{pmlr-v28-song13, title = {Hierarchical Tensor Decomposition of Latent Tree Graphical Models}, author = {Song, Le and Ishteva, Mariya and Parikh, Ankur and Xing, Eric and Park, Haesun}, booktitle = {Proceedings of the 30th International Conference on Machine Learning}, pages = {334--342}, year = {2013}, editor = {Dasgupta, Sanjoy and McAllester, David}, volume = {28}, number = {3}, series = {Proceedings of Machine Learning Research}, address = {Atlanta, Georgia, USA}, month = {17--19 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v28/song13.pdf}, url = {https://proceedings.mlr.press/v28/song13.html}, abstract = {We approach the problem of estimating the parameters of a latent tree graphical model from a hierarchical tensor decomposition point of view. In this new view, the marginal probability table of the observed variables in a latent tree is treated as a tensor, and we show that: (i) the latent variables induce low rank structures in various matricizations of the tensor; (ii) this collection of low rank matricizations induce a hierarchical low rank decomposition of the tensor. Exploiting these properties, we derive an optimization problem for estimating the parameters of a latent tree graphical model, i.e., hierarchical decomposion of a tensor which minimizes the Frobenius norm of the difference between the original tensor and its decomposition. When the latent tree graphical models are correctly specified, we show that a global optimum of the optimization problem can be obtained via a recursive decomposition algorithm. This algorithm recovers previous spectral algorithms for hidden Markov models (Hsu et al., 2009; Foster et al., 2012) and latent tree graphical models (Parikh et al., 2011; Song et al., 2011) as special cases, elucidating the global objective these algorithms are optimizing for. When the latent tree graphical models are misspecified, we derive a better decomposition based on our framework, and provide approximation guarantee for this new estimator. In both synthetic and real world data, this new estimator significantly improves over the-state-of-the-art.} }
Endnote
%0 Conference Paper %T Hierarchical Tensor Decomposition of Latent Tree Graphical Models %A Le Song %A Mariya Ishteva %A Ankur Parikh %A Eric Xing %A Haesun Park %B Proceedings of the 30th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2013 %E Sanjoy Dasgupta %E David McAllester %F pmlr-v28-song13 %I PMLR %P 334--342 %U https://proceedings.mlr.press/v28/song13.html %V 28 %N 3 %X We approach the problem of estimating the parameters of a latent tree graphical model from a hierarchical tensor decomposition point of view. In this new view, the marginal probability table of the observed variables in a latent tree is treated as a tensor, and we show that: (i) the latent variables induce low rank structures in various matricizations of the tensor; (ii) this collection of low rank matricizations induce a hierarchical low rank decomposition of the tensor. Exploiting these properties, we derive an optimization problem for estimating the parameters of a latent tree graphical model, i.e., hierarchical decomposion of a tensor which minimizes the Frobenius norm of the difference between the original tensor and its decomposition. When the latent tree graphical models are correctly specified, we show that a global optimum of the optimization problem can be obtained via a recursive decomposition algorithm. This algorithm recovers previous spectral algorithms for hidden Markov models (Hsu et al., 2009; Foster et al., 2012) and latent tree graphical models (Parikh et al., 2011; Song et al., 2011) as special cases, elucidating the global objective these algorithms are optimizing for. When the latent tree graphical models are misspecified, we derive a better decomposition based on our framework, and provide approximation guarantee for this new estimator. In both synthetic and real world data, this new estimator significantly improves over the-state-of-the-art.
RIS
TY - CPAPER TI - Hierarchical Tensor Decomposition of Latent Tree Graphical Models AU - Le Song AU - Mariya Ishteva AU - Ankur Parikh AU - Eric Xing AU - Haesun Park BT - Proceedings of the 30th International Conference on Machine Learning DA - 2013/05/26 ED - Sanjoy Dasgupta ED - David McAllester ID - pmlr-v28-song13 PB - PMLR DP - Proceedings of Machine Learning Research VL - 28 IS - 3 SP - 334 EP - 342 L1 - http://proceedings.mlr.press/v28/song13.pdf UR - https://proceedings.mlr.press/v28/song13.html AB - We approach the problem of estimating the parameters of a latent tree graphical model from a hierarchical tensor decomposition point of view. In this new view, the marginal probability table of the observed variables in a latent tree is treated as a tensor, and we show that: (i) the latent variables induce low rank structures in various matricizations of the tensor; (ii) this collection of low rank matricizations induce a hierarchical low rank decomposition of the tensor. Exploiting these properties, we derive an optimization problem for estimating the parameters of a latent tree graphical model, i.e., hierarchical decomposion of a tensor which minimizes the Frobenius norm of the difference between the original tensor and its decomposition. When the latent tree graphical models are correctly specified, we show that a global optimum of the optimization problem can be obtained via a recursive decomposition algorithm. This algorithm recovers previous spectral algorithms for hidden Markov models (Hsu et al., 2009; Foster et al., 2012) and latent tree graphical models (Parikh et al., 2011; Song et al., 2011) as special cases, elucidating the global objective these algorithms are optimizing for. When the latent tree graphical models are misspecified, we derive a better decomposition based on our framework, and provide approximation guarantee for this new estimator. In both synthetic and real world data, this new estimator significantly improves over the-state-of-the-art. ER -
APA
Song, L., Ishteva, M., Parikh, A., Xing, E. & Park, H.. (2013). Hierarchical Tensor Decomposition of Latent Tree Graphical Models. Proceedings of the 30th International Conference on Machine Learning, in Proceedings of Machine Learning Research 28(3):334-342 Available from https://proceedings.mlr.press/v28/song13.html.

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