Quantum Annealing for Robust Principal Component Analysis

TL;DR

QAPCA is proposed, a new method for finding principal components using quantum annealing hardware which will optimize over the robust L1-norm and is found that the reconstruction error when using QAPCA is comparable to that when using L1-BF.

Abstract

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 L2-norm, however, the L2-norm is known to exaggerate the contribution of errors and outliers. When optimizing over the L1-norm, the components generated are known to exhibit robustness or resistance to outliers in the data. The L1-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 paper we propose QAPCA, a new method for finding principal components using quantum annealing hardware which will optimize over the robust L1-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 techniques 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.

Figures (8)

• Figure 1: Example of the $\hat{\mathbf J}$ matrix used for QAPCA when (a) $N > N_\text{limit}$ (b) $K > 1$
• Figure 2: Average Rank vs. $\epsilon$, $N=20$, $K=4$ of algorithms on Gaussian example problem before \ref{['Procrustes']} is applied.
• Figure 3: Reconstruction Error vs. $\epsilon$, $N=20$, $K=4$ of algorithms on Gaussian example problem. Red indicates QAPCA, Green indicates QAPCA-R, blue indicates SVD, and black indicates L1-BF.
• Figure 4: Reconstruction Error vs. $N$, $K=4$, $\epsilon=100$ of algorithms on WBCD. Red indicates QAPCA, Green indicates QAPCA-R, blue indicates SVD, and black indicates L1-BF.
• Figure 5: Time vs. $N$, $K=4$, $\epsilon=100$ of algorithms on WBCD. Red indicates QAPCA, Green indicates QAPCA-R, blue indicates L2-PCA using SVD, and black indicates L1-PCA using L1-BF.