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 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 is an n-dimensional Gaussian random variable with unknown covariance : We give an algorithm that provable recovers A and up to an additive  and whose running time and sample complexity are polynomial in n and 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.