Exactly Robust Kernel Principal Component Analysis
This addresses the limitation of robust PCA for high-rank data in applications like noise removal and subspace clustering, offering a novel unsupervised nonlinear method robust to sparse noises.
The paper tackles the problem of recovering high-rank matrices corrupted by sparse noises, which existing robust PCA methods cannot handle, by proposing robust kernel PCA (RKPCA) that decomposes such matrices into sparse and high-rank components with low latent dimensionality, achieving high recovery accuracy as shown in theoretical analysis and outperforming others in noise removal and subspace clustering tasks.
Robust principal component analysis (RPCA) can recover low-rank matrices when they are corrupted by sparse noises. In practice, many matrices are, however, of high-rank and hence cannot be recovered by RPCA. We propose a novel method called robust kernel principal component analysis (RKPCA) to decompose a partially corrupted matrix as a sparse matrix plus a high or full-rank matrix with low latent dimensionality. RKPCA can be applied to many problems such as noise removal and subspace clustering and is still the only unsupervised nonlinear method robust to sparse noises. Our theoretical analysis shows that, with high probability, RKPCA can provide high recovery accuracy. The optimization of RKPCA involves nonconvex and indifferentiable problems. We propose two nonconvex optimization algorithms for RKPCA. They are alternating direction method of multipliers with backtracking line search and proximal linearized minimization with adaptive step size. Comparative studies in noise removal and robust subspace clustering corroborate the effectiveness and superiority of RKPCA.