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

Conference Paper

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.

Full Text

Duke Authors

Cited Authors

  • Anandkumar, A; Ge, R; Hsu, D; Kakade, SM; Telgarsky, M

Published Date

  • January 1, 2015

Published In

Volume / Issue

  • 9355 /

Start / End Page

  • 19 - 38

Electronic International Standard Serial Number (EISSN)

  • 1611-3349

International Standard Serial Number (ISSN)

  • 0302-9743

International Standard Book Number 13 (ISBN-13)

  • 9783319244853

Digital Object Identifier (DOI)

  • 10.1007/978-3-319-24486-0_2

Citation Source

  • Scopus