Trace and Determinant Kernels between Matrices

Kernels between ensembles (or a collection of entities) have recently attracted growing interests in the literature on machine learning . In this paper, we focus on the ‘ensemble’ that is defined as a collection of vectors. One natural way to interpret such an ensemble is through the notion of matrix. We present two basic reproducing kernels between matrices: namely trace and determinant kernels that can be interpreted using a ‘vector’ viewpoint as they are in an inner product form between two vectors, explaining the required positive definiteness for a reproducing kernel. Using the ‘vector’ viewpoint and generic kernel construction rules, we are able to construct more kernels between matrices based on the basic trace and determinant kernels. Further, we also consider column space matrices, possibly arising from matrices of different column sizes, and ‘kernerlized’ matrices whose columns are mapped to a reproducing kernel Hilbert space.