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

TL;DR

ECA is a robust high-dimensional analysis method for elliptically distributed data, effectively handling heavy tails and sparsity, with proven theoretical performance guarantees.

Contribution

This paper introduces ECA, a novel robust PCA alternative for elliptically distributed data, with methods for both sparse and non-sparse cases and comprehensive theoretical analysis.

Findings

ECA's performance depends on the effective rank of the covariance matrix.

Sparse ECA achieves the optimal convergence rate.

Efficient sparse ECA attains near-optimal performance under certain conditions.

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 non-sparse or that they are sparse. Methodologically, we propose ECA procedures for both non-sparse and sparse settings. Theoretically, we provide both non-asymptotic and asymptotic analyses quantifying the theoretical performances of ECA. In the non-sparse 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.