Novel Algorithms for Lp-Quasi-Norm Principal-Component Analysis

Abstract

We consider outlier-resistant Lp-quasi-norm (p ≤ 1) Principal-Component Analysis (Lp-PCA) of a D-by-N matrix. It was recently shown that Lp-PCA (p ≤ 1) admits an exact solution by means of combinatorial optimization with computational cost exponential in N. To date, apart from the exact solution to Lp-PCA (p ≤ 1), there exists no converging algorithm of lower cost that approximates its exact solution. In this work, we (i) propose a novel and converging algorithm that approximates the exact solution to Lp-PCA with significantly lower computational cost than that of the exact solver, (ii) conduct formal complexity and convergence analyses, and (iii) propose a multi-component solver based on subspace-deflation. Numerical studies on matrix reconstruction and medical-data classification illustrate the outlier resistance of Lp-PCA.