Semiparametric Principal Component Analysis

We propose two new principal component analysis methods in this paper utilizing a semiparametric model. The according methods are named Copula Component Analysis (COCA) and Copula PCA. The semiparametric model assumes that, after unspecifiedmarginally monotone transformations, the distributions are multivariate Gaussian.The COCA and Copula PCA accordingly estimate the leading eigenvectors of the correlation and covariance matrices of the latent Gaussian distribution. Therobust nonparametric rank-based correlation coefficient estimator, Spearman's rho, is exploited in estimation. We prove that, under suitable conditions, althoughthe marginal distributions can be arbitrarily continuous, the COCA and Copula PCA estimators obtain fast estimation rates and are feature selection consistent in the setting where the dimension is nearly exponentially large relative to the sample size. Careful numerical experiments on the synthetic and real data are conducted to back up the theoretical results. We also discuss the relationship with the transelliptical component analysis proposed by Han and Liu (2012).