Abstract
We describe a framework to build distances by measuring the tightness of
inequalities, and introduce the notion of proper statistical divergences and
improper pseudo-divergences. We then consider the Hölder ordinary and reverse
inequalities, and present two novel classes of Hölder divergences and
pseudo-divergences that both encapsulate the special case of the Cauchy-Schwarz
divergence. We report closed-form formulas for those statistical
dissimilarities when considering distributions belonging to the same
exponential family provided that the natural parameter space is a cone (e.g.,
multivariate Gaussians), or affine (e.g., categorical distributions). Those new
classes of Hölder distances are invariant to rescaling, and thus do not
require distributions to be normalized. Finally, we show how to compute
statistical Hölder centroids with respect to those divergences, and carry out
center-based clustering toy experiments on a set of Gaussian distributions that
demonstrate empirically that symmetrized Hölder divergences outperform the
symmetric Cauchy-Schwarz divergence.
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