Overcomplete order-3 tensor decomposition, blind deconvolution and Gaussian mixture models
This work addresses tensor decomposition and its applications in machine learning, offering incremental improvements to existing methods for specific domains.
The authors tackled the problem of decomposing symmetric overcomplete order-3 tensors and applied it to expand efficient parameter estimation for Gaussian mixture models and blind deconvolution, achieving robustness and applicability to centrally symmetric distributions with finite 6th moment.
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.