Principal Component Analysis with Contaminated Data: The High Dimensional Case

TL;DR

This paper introduces HR-PCA, a robust high-dimensional PCA method that effectively handles contaminated data, achieving high robustness, bounded deviation, and optimality as corruption diminishes.

Contribution

The paper presents a tractable, robust PCA algorithm for high-dimensional contaminated data with a 50% breakdown point and kernelizability, outperforming existing methods.

Findings

Achieves a breakdown point of 50% for contaminated data

Bounded deviation from the true subspace in high dimensions

Optimal performance as the proportion of corrupted points approaches zero

Abstract

We consider the dimensionality-reduction problem (finding a subspace approximation of observed data) for contaminated data in the high dimensional regime, where the number of observations is of the same magnitude as the number of variables of each observation, and the data set contains some (arbitrarily) corrupted observations. We propose a High-dimensional Robust Principal Component Analysis (HR-PCA) algorithm that is tractable, robust to contaminated points, and easily kernelizable. The resulting subspace has a bounded deviation from the desired one, achieves maximal robustness -- a breakdown point of 50% while all existing algorithms have a breakdown point of zero, and unlike ordinary PCA algorithms, achieves optimality in the limit case where the proportion of corrupted points goes to zero.