Robust PCA for High Dimensional Data based on Characteristic Transformation

TL;DR

This paper introduces a robust PCA method for high-dimensional data that effectively handles heterogeneities like heavy tails and outliers by using a characteristic function-based transformation, improving robustness over classical PCA.

Contribution

The paper presents a novel robust PCA approach based on characteristic transformation, capable of managing heavy-tailed data and outliers, with theoretical analysis and practical validation.

Findings

Outperforms classical PCA in handling heavy-tailed and outlier data.

Provides theoretical bounds on excess error and eigenvalue behavior.

Demonstrates improved classification accuracy in biological data.

Abstract

In this paper, we propose a novel robust Principal Component Analysis (PCA) for high-dimensional data in the presence of various heterogeneities, especially the heavy-tailedness and outliers. A transformation motivated by the characteristic function is constructed to improve the robustness of the classical PCA. Besides the typical outliers, the proposed method has the unique advantage of dealing with heavy-tail-distributed data, whose covariances could be nonexistent (positively infinite, for instance). The proposed approach is also a case of kernel principal component analysis (KPCA) method and adopts the robust and non-linear properties via a bounded and non-linear kernel function. The merits of the new method are illustrated by some statistical properties including the upper bound of the excess error and the behaviors of the large eigenvalues under a spiked covariance model. In addition, we show the advantages of our method over the classical PCA by a variety of simulations. At last, we apply the new robust PCA to classify mice with different genotypes in a biological study based on their protein expression data and find that our method is more accurately on identifying abnormal mice comparing to the classical PCA.