Tensor Decompositions for Learning Latent Variable Models (A Survey for ALT)

Abstract

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.