We consider the problem of learning with instances defined over a space of sets of vectors. We derive a new positive definite kernel f (A, B) defined over pairs of matrices A, B based on the concept of principal angles between two linear subspaces. We show that the principal angles can be recovered using only inner-products between pairs of column vectors of the input matrices thereby allowing the original column vectors of A, B to be mapped onto arbitrarily high-dimensional feature spaces. We

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... emonstrate the usage of the matrix-based kernel function f (A, B) with experiments on two visual tasks. The first task is the discrimination of "irregular" motion trajectory of an individual or a group of individuals in a video sequence. We use the SVM approach using f (A, B) where an input matrix represents the motion trajectory of a group of individuals over a certain (fixed) time frame. We show that the classification (irregular versus regular) greatly outperforms the conventional representation where all the trajectories form a single vector. The second application is the visual recognition of faces from input video sequences representing head motion and facial expressions where f (A, B) is used to compare two image sequences.