Smoothed analysis of tensor decompositions

Abstract

Low rank decomposition of tensors is a powerful tool for learning generative models. The uniqueness results that hold for tensors give them a significant advantage over matrices. However, tensors pose serious algorithmic challenges; in particular, much of the matrix algebra toolkit fails to generalize to tensors. Efficient decomposition in the overcomplete case (where rank exceeds dimension) is particularly challenging. We introduce a smoothed analysis model for studying these questions and develop an efficient algorithm for tensor decomposition in the highly overcomplete case (rank polynomial in the dimension). In this setting, we show that our algorithm is robust to inverse polynomial error -- a crucial property for applications in learning since we are only allowed a polynomial number of samples. While algorithms are known for exact tensor decomposition in some overcomplete settings, our main contribution is in analyzing their stability in the framework of smoothed analysis.