ECA: High-Dimensional Elliptical Component Analysis in Non-Gaussian Distributions

ABSTRACT

We present a robust alternative to principal component analysis (PCA)—called elliptical component analysis (ECA)—for analyzing high-dimensional, elliptically distributed data. ECA estimates the eigenspace of the covariance matrix of the elliptical data. To cope with heavy-tailed elliptical distributions, a multivariate rank statistic is exploited. At the model-level, we consider two settings: either that the leading eigenvectors of the covariance matrix are nonsparse or that they are sparse. Methodologically, we propose ECA procedures for both nonsparse and sparse settings. Theoretically, we provide both nonasymptotic and asymptotic analyses quantifying the theoretical performances of ECA. In the nonsparse setting, we show that ECA’s performance is highly related to the effective rank of the covariance matrix. In the sparse setting, the results are twofold: (i) we show that the sparse ECA estimator based on a combinatoric program attains the optimal rate of convergence; (ii) based on some recent developments in estimating sparse leading eigenvectors, we show that a computationally efficient sparse ECA estimator attains the optimal rate of convergence under a suboptimal scaling. Supplementary materials for this article are available online.

KEYWORDS:

• Multivariate Kendall’s tau
• Elliptical component analysis
• Sparse principal component analysis
• Optimality property
• Robust estimators
• Elliptical distribution

Supplementary Materials

The online supplementary materials contain the appendices for the article.

Funding

Fang Han’s research was supported by NIBIB-EB012547, NSF DMS-1712536, and a UW faculty start-up grant. Han Liu’s research was supported by the NSF CAREER Award DMS-1454377, NSF IIS-1546482, NSF IIS-1408910, NSF IIS-1332109, NIH R01-MH102339, NIH R01-GM083084, and NIH R01-HG06841.

Notes

1 The Tyler’s M estimator cannot be directly applied to study high-dimensional data because of both theoretical and empirical reasons. Theoretically, to the authors’ knowledge, the sharpest sufficient condition to guarantee its consistency still requires d = o(n^1/2) (Duembgen 1997). Empirically, our simulations show that Tyler’s M estimator always fails to converge when d > n.