Re-examination of Bregman functions and new properties of their divergences. (English) Zbl 1407.52008
Summary: The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the ‘Bregman function’). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, operations research, information theory, nonlinear analysis, machine learning and more. This paper re-examines various aspects related to the theory of Bregman functions and divergences. In particular, it presents many sufficient conditions which allow the construction of Bregman functions in a general setting and introduces new Bregman functions (such as a negative iterated log entropy). Moreover, it sheds new light on several known Bregman functions such as quadratic entropies, the negative Havrda-Charvát-Tsallis entropy, and the negative Boltzmann-Gibbs-Shannon entropy, and it shows that the negative Burg entropy, which is not a Bregman function according to the classical theory but nevertheless is known to have ‘Bregmanian properties’, can, by our re-examination of the theory, be considered as a Bregman function. Our analysis yields several by-products of independent interest such as the introduction of the concept of relative uniform convexity (a certain generalization of uniform convexity), new properties of uniformly and strongly convex functions, and results in Banach space theory.
MSC:
| 52A41 | Convex functions and convex programs in convex geometry |
| 52B55 | Computational aspects related to convexity |
| 46N10 | Applications of functional analysis in optimization, convex analysis, mathematical programming, economics |
| 90C25 | Convex programming |
| 90C30 | Nonlinear programming |
| 46T99 | Nonlinear functional analysis |
| 47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |
| 49M37 | Numerical methods based on nonlinear programming |
| 26B25 | Convexity of real functions of several variables, generalizations |
| 58C05 | Real-valued functions on manifolds |
Keywords:
Bregman divergence; Bregman function; gauge; negative Boltzmann-Gibbs-Shannon entropy; negative Burg entropy; negative Havrda-Charvát-Tsallis entropy; negative iterated log entropy; relative uniform convexity; strongly convex; uniformly convexReferences:
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