  We present an algorithm for L1-norm kernel PCA and provide a convergence
analysis for it. While an optimal solution of L2-norm kernel PCA can be
obtained through matrix decomposition, finding that of L1-norm kernel PCA is
not trivial due to its non-convexity and non-smoothness. We provide a novel
reformulation through which an equivalent, geometrically interpretable problem
is obtained. Based on the geometric interpretation of the reformulated problem,
we present a fixed-point type algorithm that iteratively computes a binary
weight for each observation. As the algorithm requires only inner products of
data vectors, it is computationally efficient and the kernel trick is
applicable. In the convergence analysis, we show that the algorithm converges
to a local optimal solution in a finite number of steps. Moreover, we provide a
rate of convergence analysis, which has been never done for any L1-norm PCA
algorithm, proving that the sequence of objective values converges at a linear
rate. In numerical experiments, we show that the algorithm is robust in the
presence of entry-wise perturbations and computationally scalable, especially
in a large-scale setting. Lastly, we introduce an application to outlier
detection where the model based on the proposed algorithm outperforms the
benchmark algorithms.
