S.: Provable ICA with unknown Gaussian noise, with implications for Gaussian mixtures and autoencoders

We present a new algorithm for Independent Component Analysis (ICA) which has provable performance guarantees. In particular, suppose we are given samples of the form y = Ax+ η where A is an unknown n × n matrix and x is a random variable whose components are independent and have a fourth moment strictly less than that of a standard Gaussian random variable and η is an n-dimensional Gaussian random variable with unknown covariance Σ: We give an algorithm that provable recovers A and Σ up to an additive and whose running time and sam-ple complexity are polynomial in n and 1/. To accomplish this, we introduce a novel “quasi-whitening ” step that may be useful in other contexts in which the covariance of Gaussian noise is not known in advance. We also give a general framework for finding all local optima of a function (given an oracle for approx-imately finding just one) and this is a crucial step in our algorithm, one that has been overlooked in previous attempts, and allows us to control the accumulation of error when we find the columns of A one by one via local search.