- Research Article
- Open Access
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
We investigate the solution to the following function equation , which arises from the theory of divergence measures. Moreover, new results on divergence measures are given.
- Machine Learning
- Functional Equation
- Orthonormal Base
- Divergence Measure
- Statistical Inference
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.
then is the solution of a linear homogenous differential equation with constant coefficients. Moreover, new results on divergence measures are given.
Throughout this paper, we let be the set of real numbers and are a convex set.
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.
then we say the or is in the intersection of -divergence and general Bregman divergence.
for some .
The proof is then complete.
for some .
Thus, a modification of Theorem 2.1 implies the conclusion.
Moreover, it is not so hard to deduce the following theorem.
where is strictly monotone twice-continuously differentiable functions. Then the divergence is -divergence or vector -divergence times a positive constant .
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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