## Tensor decompositions for learning latent variable models (A survey for ALT)

This note is a short version of that in [1]. It is intended as a survey for the 2015 Algorithmic Learning Theory (ALT) conference. This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models— including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation—which exploits a certain tensor structure in their low-order observable moments (typically, of second- and third-order). Specifically, parameter estimation is reduced to the problem of extracting a certain (orthogonal) decomposition of a symmetric tensor derived from the moments; this decomposition can be viewed as a natural generalization of the singular value decomposition for matrices. Although tensor decompositions are generally intractable to compute, the decomposition of these specially structured tensors can be efficiently obtained by a variety of approaches, including power iterations and maximization approaches (similar to the case of matrices). A detailed analysis of a robust tensor power method is provided, establishing an analogue of Wedin’s perturbation theorem for the singular vectors of matrices. This implies a robust and computationally tractable estimation approach for several popular latent variable models.

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- Artificial Intelligence & Image Processing
- 46 Information and computing sciences

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*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 9355, pp. 19–38). https://doi.org/10.1007/978-3-319-24486-0_2

*Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, 9355:19–38, 2015. https://doi.org/10.1007/978-3-319-24486-0_2.

*Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 9355, 2015, pp. 19–38.

*Scopus*, doi:10.1007/978-3-319-24486-0_2.

## Published In

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## ISSN

## Publication Date

## Volume

## Start / End Page

## Related Subject Headings

- Artificial Intelligence & Image Processing
- 46 Information and computing sciences