Combinatorial Search for the Lp-Norm Principal Component of a Matrix

We study Lp-norm Principal-Component Analysis (Lp-PCA) of a matrix. For p = 2 (standard PCA), the problem can be solved with standard Singular-Value Decomposition (SVD). For p = 1 (L1-PCA), the problem was recently solved exactly and approximately with efficient iterative algorithms. For general values of p, the exact solution to Lp-PCA remains to date unknown. In this work, for the first time in the literature, we prove that, for p ≤ 1, Lp-PCA can be solved exactly through combinatorial optimization and present the first exact solver. Our experimental studies on medical data demonstrate the significant robustness of Lp-PCA, p ≤ 1, against outliers.