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

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. Vempala and Xiao (2011) proposed a local search algorithm, and showed that it was able to estimate $E$ accurately with polynomial time and sample complexity, if the dimension of $E$ is treated as a constant and with the assumption that all one-dimensional marginals of the non-Gaussian distribution over $E$ have non-Gaussian moments. In this paper, we propose a simple spectral algorithm called \textsc{Reweighted PCA}, and prove that it possesses the same guarantee. The principle that underlies this approach is a new characterization of multivariate Gaussian distributions.