Stochastic Approximation for Online Tensorial Independent Component Analysis

Abstract

Independent component analysis (ICA) has been a popular dimension reduction tool in statistical machine learning and signal processing. In this paper, we present a convergence analysis for an online tensorial ICA algorithm, by viewing the problem as a nonconvex stochastic approximation problem. For estimating one component, we provide a dynamicsbased analysis to prove that our online tensorial ICA algorithm with a specific choice of stepsize achieves a sharp finite-sample error bound. In particular, under a mild assumption on the data-generating distribution and a scaling condition such that d 4 /T is sufficiently small up to a polylogarithmic factor of data dimension d and sample size T , a sharp finitesample error bound of Õ( d/T ) can be obtained. As a by-product, we also design an online tensorial ICA algorithm that estimates multiple independent components in parallel, achieving desirable finite-sample error bound for each independent component estimator.