TCA: High Dimensional Principal Component Analysis for non-Gaussian Data

We propose a high dimensional semiparametric scale-invariant principle component analysis, named TCA, by utilize the natural connection between the elliptical distribution family and the principal component analysis. Elliptical distribution family includes many well-known multivariate distributions like multivariate t and logistic and it is extended to the meta-elliptical by Fang (2002) using the copula techniques. In this paper we extend the meta-elliptical distribution family to a even larger family, called transelliptical. We prove that TCA can obtain a near-optimal s(log d/n)^{1/2} estimation consistency rate in the transelliptical distribution family, even if the distributions are very heavy-tailed, have infinite second moments, do not have densities and possess arbitrarily continuous marginal distributions. A feature selection result with explicit rate is also provided. TCA is also implemented in both numerical simulations and large-scale stock data to illustrate its empirical performance. Both theories and experiments confirm that TCA can achieve model flexibility, estimation accuracy and robustness at almost no cost.