Matrix-valued Kernels for Shape Deformation Analysis

The main purpose of this paper is providing a systematic study and classification of non-scalar kernels for Reproducing Kernel Hilbert Spaces (RKHS), to be used in the analysis of deformation in shape spaces endowed with metrics induced by the action of groups of diffeomorphisms. After providing an introduction to matrix-valued kernels and their relevant differential properties, we explore extensively those, that we call TRI kernels, that induce a metric on the corresponding Hilbert spaces of vector fields that is both translation- and rotation-invariant. These are analyzed in an effective manner in the Fourier domain, where the characterization of RKHS of curl-free and divergence-free vector fields is particularly natural. A simple technique for constructing generic matrix-valued kernels from scalar kernels is also developed. We accompany the exposition of the theory with several examples, and provide numerical results that show the dynamics induced by different choices of TRI kernels on the manifold of labeled landmark points.