The {\alpha}-divergences associated with a pair of strictly comparable quasi-arithmetic means

Frank Nielsen

doi:10.3390/a15110435

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arXiv · 2001.09660

The {\alpha}-divergences associated with a pair of strictly comparable quasi-arithmetic means

Frank Nielsen

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Abstract

We generalize the family of $\alpha$-divergences using a pair of strictly comparable weighted means. In particular, we obtain the $1$-divergence in the limit case $\alpha\rightarrow 1$ (a generalization of the Kullback-Leibler divergence) and the $0$-divergence in the limit case $\alpha\rightarrow 0$ (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 $\alpha$-divergences and its subfamily of bipower homogeneous $\alpha$-divergences which belong to the Csis\'ar'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.

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Frank Nielsen. 2020-01-27. The {\alpha}-divergences associated with a pair of strictly comparable quasi-arithmetic means. https://doi.org/10.3390/a15110435

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