On the asymptotic properties of product-PCA under the high-dimensional setting

TL;DR

This paper investigates the high-dimensional asymptotic behavior of product-PCA, revealing its eigenvalue distribution and robustness advantages over traditional PCA, especially in the presence of outliers.

Contribution

It provides the first theoretical analysis of PPCA eigenvalues in high dimensions using random matrix theory, highlighting its robustness and spectral properties.

Findings

Derived critical values for distant spiked eigenvalues

Established limiting spectral distribution of PPCA eigenvalues

Confirmed theoretical results with numerical simulations

Abstract

Principal component analysis (PCA) is a widely used dimension reduction method, but its performance is known to be non-robust to outliers. Recently, product-PCA (PPCA) has been shown to possess the efficiency-loss free ordering-robustness property: (i) in the absence of outliers, PPCA and PCA share the same asymptotic distributions; (ii), in the presence of outliers, PPCA is more ordering-robust than PCA in estimating the leading eigenspace. PPCA is thus different from the conventional robust PCA methods, and may deserve further investigations. In this article, we study the high-dimensional statistical properties of the PPCA eigenvalues via the techniques of random matrix theory. In particular, we derive the critical value for being distant spiked eigenvalues, the limiting values of the sample spiked eigenvalues, and the limiting spectral distribution of PPCA. Similar to the case of PCA, the explicit forms of the asymptotic properties of PPCA become available under the special case of the simple spiked model. These results enable us to more clearly understand the superiorities of PPCA in comparison with PCA. Numerical studies are conducted to verify our results.