Quantum Annealing for Robust Principal Component Analysis

Principal component analysis is commonly used for dimensionality reduction, feature extraction, denoising, and visualization. The most commonly used principal component analysis method is based upon optimization of the <inline-formula><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula>-norm; however, the <inline-formula><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula>-norm is known to exaggerate the contribution of errors and outliers. When optimizing over the <inline-formula><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula>-norm, the components generated are known to exhibit robustness or resistance to outliers in the data. The <inline-formula><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula>-norm components can be solved for with a binary optimization problem. Previously, L1-BF has been used to solve the binary optimization for multiple components simultaneously. In this article, we propose quantum annealing principal component analysis (QAPCA), a new method for finding principal components using quantum annealing hardware that will optimize over the <inline-formula><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula>-norm. The conditions required for convergence of the annealing problem are discussed. The potential speedup when using quantum annealing is demonstrated through complexity analysis and experimental results. To showcase performance against classical principal component analysis technique experiments upon synthetic Gaussian data, a fault detection scenario and breast cancer diagnostic data are studied. We find that the reconstruction error when using QAPCA is comparable to that when using L1-BF.