The α-divergences associated with a pair of strictly comparable quasi-arithmetic means

01/27/2020

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by   Frank Nielsen, et al.

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We generalize the family of α-divergences using a pair of strictly comparable weighted means. In particular, we obtain the 1-divergence in the limit case α→ 1 (a generalization of the Kullback-Leibler divergence) and the 0-divergence in the limit case α→ 0 (a generalization of the reverse Kullback-Leibler divergence). We state the condition for a pair of quasi-arithmetic means to be strictly comparable, and report the formula for the quasi-arithmetic α-divergences and its subfamily of bipower homogeneous α-divergences which belong to the Csisár's f-divergences. Finally, we show that these generalized quasi-arithmetic 1-divergences and 0-divergences can be decomposed as the sum of generalized cross-entropies minus entropies, and rewritten as conformal Bregman divergences using monotone embeddings.