Generalizing Jensen and Bregman divergences with comparative convexity and the statistical Bhattacharyya distances with comparable means
This work provides a theoretical extension of divergence measures for statistical and machine learning applications, but it appears incremental as it builds on existing convexity concepts without addressing a specific practical bottleneck.
The paper tackles the generalization of Jensen and Bregman divergences using comparative convexity, resulting in explicit formulas for these divergences with quasi-arithmetic means and a new generalization of Bhattacharyya distances based on comparative means.
Comparative convexity is a generalization of convexity relying on abstract notions of means. We define the Jensen divergence and the Jensen diversity from the viewpoint of comparative convexity, and show how to obtain the generalized Bregman divergences as limit cases of skewed Jensen divergences. In particular, we report explicit formula of these generalized Bregman divergences when considering quasi-arithmetic means. Finally, we introduce a generalization of the Bhattacharyya statistical distances based on comparative means using relative convexity.