On information projections between multivariate elliptical and location-scale families

Frank Nielsen

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arXiv · 2101.03839

On information projections between multivariate elliptical and location-scale families

Frank Nielsen

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Abstract

We study information projections with respect to statistical $f$-divergences between any two location-scale families. We consider a multivariate generalization of the location-scale families which includes the elliptical and the spherical subfamilies. By using the action of the multivariate location-scale group, we show how to reduce the calculation of $f$-divergences between any two location-scale densities to canonical settings involving standard densities, and derive thereof fast Monte Carlo estimators of $f$-divergences with good properties. Finally, we prove that the minimum $f$-divergence between a prescribed density of a location-scale family and another location-scale family is independent of the prescribed location-scale parameter. We interpret geometrically this property.

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Frank Nielsen. 2021-01-11. On information projections between multivariate elliptical and location-scale families. https://arxiv.org/abs/2101.03839

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