Tensor decompositions for learning latent variable models

TL;DR

This paper introduces an efficient tensor-based method for estimating parameters in various latent variable models, leveraging tensor decompositions of observable moments, with theoretical guarantees for robustness and tractability.

Contribution

It develops a robust tensor power method for parameter estimation in latent variable models, extending matrix SVD techniques to tensor decompositions with provable guarantees.

Findings

Efficient tensor decomposition algorithms for latent variable models.

Theoretical analysis showing robustness of the tensor power method.

Application to models like Gaussian mixtures, HMMs, and LDA.

Abstract

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.