Composite distance based approach to von Mises mixture reduction

This paper presents a systematic approach for component number reduction in mixtures of exponential families, putting a special emphasis on the von Mises mixtures. We propose to formulate the problem as an optimization problem utilizing a new class of computationally tractable composite distance measures as cost functions, namely the composite Rényi αα-divergences, which include the composite Kullback–Leibler distance as a special case. Furthermore, we prove that the composite divergence bounds from above the corresponding intractable Rényi αα-divergence between a pair of mixtures. As a solution to the optimization problem we synthesize that two existing suboptimal solution strategies, the generalized k-means and a pairwise merging approach, are actually minimization methods for the composite distance measures. Moreover, in the present paper the existing joining algorithm is also extended for comparison purposes. The algorithms are implemented and their reduction results are compared and discussed on two examples of von Mises mixtures: a synthetic mixture and a real-world mixture used in people trajectory shape analysis.