Overcomplete Order-3 Tensor Decomposition, Blind Deconvolution, and Gaussian Mixture Models

Abstract

We expand the family of tensors that can be efficiently decomposed and apply our new algorithmic ideas to blind deconvolution and Gaussian mixture models. Our basic contribution is an efficient algorithm to decompose certain symmetric overcomplete order-3 tensors, that is, three dimensional arrays of the form $T = \sum_{i=1}^n a_i \otimes a_i \otimes a_i$ where the $a_i$s are not linearly independent. Our second contribution builds on top of our tensor decomposition algorithm to expand the family of Gaussian mixture models whose parameters can be estimated efficiently. These ideas are also presented in a much more general framework of blind deconvolution that makes them applicable to mixture models of identical but very general distributions, including all centrally symmetric distributions with finite 6th moment.