Robust Dictionary Learning on the Hilbert Sphere in Kernel Feature Space

Abstract

This paper presents a novel dictionary learning method in kernel feature space that is part of a reproducing kernel Hilbert space (RKHS). Our method focuses on several popular kernels, e.g., radial basis function kernels like the Gaussian, that implicitly map data to a Hilbert sphere, a Riemannian manifold, in RKHS. Our method exploits this manifold structure of the mapped data in RKHS, unlike typical methods for kernel dictionary learning that use linear methods in RKHS. We show that dictionary learning on a Hilbert sphere in RKHS is possible without the need of the explicit lifting map underlying the kernel, but using solely the Gram matrix. Unlike the typical $L^1$ norm sparsity prior, we incorporate the non-convex $L^p$ quasi-norm based penalty, with $p < 1$ , on coefficients to enforce a stronger sparsity prior and achieve more robust dictionary learning in the presence of corrupted training data. We evaluate our method for image classification on two large publicly available datasets and demonstrate the improved performance of our method over the state of the art dictionary learning methods.