Overcomplete order-3 tensor decomposition, blind deconvolution and Gaussian mixture models

TL;DR

This paper introduces a new efficient algorithm for overcomplete order-3 tensor decomposition, enabling improved parameter estimation in Gaussian mixture models and blind deconvolution, with robustness guarantees.

Contribution

It presents a simple, robust tensor decomposition algorithm and extends its application to a broad class of Gaussian mixture models and blind deconvolution problems.

Findings

Algorithm successfully decomposes symmetric overcomplete tensors.

Enhanced parameter estimation for Gaussian mixture models.

Applicable to a wide range of symmetric distributions with finite moments.

Abstract

We propose a new algorithm for tensor decomposition, based on Jennrich's algorithm, and apply our new algorithmic ideas to blind deconvolution and Gaussian mixture models. Our first contribution is a simple and 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 algorithm comes with a detailed robustness analysis. 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 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.