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The generalized Bregman distance
- Publication Year :
- 2021
- Publisher :
- US : SIAM, 2021.
-
Abstract
- Refereed/Peer-reviewed Recently, a new kind of distance has been introduced for the graphs of two point-to-set operators, one of which is maximally monotone. When both operators are the subdifferential of a proper lower semicontinuous convex function, this kind of distance specializes under modest assumptions to the classical Bregman distance. We name this new kind of distance the generalized Bregman distance, and we shed light on it with examples that utilize the other two most natural representative functions: the Fitzpatrick function and its conjugate. We provide sufficient conditions for convexity, coercivity, and supercoercivity: properties which are essential for implementation in proximal point type algorithms. We establish these results for both the left and right variants of this new kind of distance. We construct examples closely related to the Kullback--Leibler divergence, which was previously considered in the context of Bregman distances and whose importance in information theory is well known. In so doing, we demonstrate how to compute a difficult Fitzpatrick conjugate function, and we discover natural occurrences of the Lambert ${\mathcal W}$ function, whose importance in optimization is of growing interest.
- Subjects :
- 0211 other engineering and technologies
010103 numerical & computational mathematics
02 engineering and technology
Subderivative
Bregman divergence
01 natural sciences
Regularization (mathematics)
Theoretical Computer Science
Representative function
Combinatorics
FOS: Mathematics
0101 mathematics
Fitzpatrick distance
Mathematics - Optimization and Control
Mathematics
Fitzpatrick function
convex function
021103 operations research
Bregman distance
Functional Analysis (math.FA)
Mathematics - Functional Analysis
regularization
Monotone polygon
representative function
Optimization and Control (math.OC)
generalized Bregman distance
Convex function
Software
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....f9729d6e7ea306fea713ea70226f5567