Tensor decompositions for learning latent variable models

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

Finding a low-rank basis in a matrix subspace

For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained by the tensor CP decomposition. For the higher rank case, the situation is not as straightforward. In this work we present an algorithm based on a greedy process applicable to higher rank problems. Our algorithm first estimates the minimum rank by applying soft singular value thresholding to a nuclear norm relaxation, and then computes a matrix with that rank using the method of alternating projections. We provide local convergence results, and compare our algorithm with several alternative approaches. Applications include data compression beyond the classical truncated SVD, computing accurate eigenvectors of a near-multiple eigenvalue, image separation and graph Laplacian eigenproblems