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

Part of Advances in Neural Information Processing Systems 25 (NIPS 2012)

Authors

Sanjeev Arora, Rong Ge, Ankur Moitra, Sushant Sachdeva

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 $y = Ax + \eta$ where $A$ is an unknown $n \times n$ matrix and $x$ is chosen uniformly at random from $\{+1, -1\}^n$, $\eta$ is an $n$-dimensional Gaussian random variable with unknown covariance $\Sigma$: We give an algorithm that provable recovers $A$ and $\Sigma$ up to an additive $\epsilon$ whose running time and sample complexity are polynomial in $n$ and $1 / \epsilon$. 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.