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

Abstract

We present a new algorithm for independent component analysis which has provable performance guarantees. In particular, suppose we are given samples of the form 𝑦=𝐴𝑥+𝜂 where 𝐴 is an unknown but non-singular 𝑛×𝑛 matrix, 𝑥 is a random variable whose coordinates are independent and have a fourth order 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 provably recovers 𝐴 and 𝛴 up to an additive 𝜖 and 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 applications where there is additive Gaussian noise whose covariance is unknown. 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.