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

Sanjeev Arora, Rong Ge, Ankur Moitra, Sushant Sachdeva

Research output: Contribution to journalArticle

7 Scopus citations

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 $$A$$A one by one via local search.

Original languageEnglish (US)
Pages (from-to)215-236
Number of pages22
JournalAlgorithmica
Volume72
Issue number1
DOIs
StatePublished - May 1 2015

All Science Journal Classification (ASJC) codes

  • Computer Science(all)
  • Computer Science Applications
  • Applied Mathematics

Keywords

  • Cumulants
  • Independent component analysis
  • Method of moments
  • Mixture models

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