Polynomial Time and Sample Complexity for Non-Gaussian Component Analysis: Spectral Methods

TL;DR

This paper introduces a polynomial-time spectral method called Reweighted PCA for Non-Gaussian Component Analysis, providing effective tests for non-gaussianity and recovering the relevant subspace with theoretical guarantees.

Contribution

It offers a new characterization of high-dimensional Gaussian distributions and proposes a simple, provably effective algorithm for NGCA with polynomial sample and time complexity.

Findings

Reweighted PCA recovers at least one non-gaussian direction.

The method has polynomial sample complexity.

The approach effectively tests for non-gaussianity in high dimensions.

Abstract

The problem of Non-Gaussian Component Analysis (NGCA) is about finding a maximal low-dimensional subspace $E$ in $\mathbb{R}^n$ so that data points projected onto $E$ follow a non-gaussian distribution. Although this is an appropriate model for some real world data analysis problems, there has been little progress on this problem over the last decade. In this paper, we attempt to address this state of affairs in two ways. First, we give a new characterization of standard gaussian distributions in high-dimensions, which lead to effective tests for non-gaussianness. Second, we propose a simple algorithm, \emph{Reweighted PCA}, as a method for solving the NGCA problem. We prove that for a general unknown non-gaussian distribution, this algorithm recovers at least one direction in $E$, with sample and time complexity depending polynomially on the dimension of the ambient space. We conjecture that the algorithm actually recovers the entire $E$.