Solution to a Function Equation and Divergence Measures
© C.-L. Dong and J. Liang. 2011
Received: 1 January 2011
Accepted: 11 February 2011
Published: 10 March 2011
As early as in 1952, Chernoff  used the -divergence to evaluate classification errors. Since then, the study of various divergence measures has been attracting many researchers. So far, we have known that the Csiszár -divergence is a unique class of divergences having information monotonicity, from which the dual geometrical structure with the Fisher metric is derived, and the Bregman divergence is another class of divergences that gives a dually flat geometrical structure different from the -structure in general. Actually, a divergence measure between two probability distributions or positive measures have been proved a useful tool for solving optimization problems in optimization, signal processing, machine learning, and statistical inference. For more information on the theory of divergence measures, please see, for example, [2–5] and references therein.
Basic notations: ; is strictly convex and twice differentiable; is differentiable injective map; is the general vector Bregman divergence; is strictly convex twice-continuously differentiable function satisfying ; is the vector -divergence.
2. Main Results
The proof is then complete.
Thus, a modification of Theorem 2.1 implies the conclusion.
Moreover, it is not so hard to deduce the following theorem.
This work was supported partially by the NSF of China and the Specialized Research Fund for the Doctoral Program of Higher Education of China.
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