Stochastic Approximation for Online Tensorial Independent Component Analysis

• URL: http://arxiv.org/abs/2012.14415v1
• Date: Mon, 28 Dec 2020 18:52:37 GMT
• Title: Stochastic Approximation for Online Tensorial Independent Component Analysis
• Authors: Chris Junchi Li, Michael I. Jordan
• Abstract summary: Independent component analysis (ICA) has been a popular dimension reduction tool in statistical machine learning and signal processing. In this paper, we present a by-product online tensorial algorithm that estimates for each independent component.
• Score: 98.34292831923335
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 dynamics-based 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 finite-sample error bound of $\tilde O(\sqrt{d / T})$ 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.

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