  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.
