Provable ICA with Unknown Gaussian Noise, with Implications for Gaussian Mixtures and Autoencoders

Abstract

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 = A x + η where A is an unknown n × n matrix and x is chosen uniformly at random from { + 1 , − 1 } n , 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 ϵ whose running time and sample complexity are polynomial in n and 1 / ϵ . 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 approximately 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.