- Research Article
- Open Access

# Solution to a Function Equation and Divergence Measures

- Chuan-Lei Dong
^{1}and - Jin Liang
^{1}Email author

**2011**:617564

https://doi.org/10.1155/2011/617564

© C.-L. Dong and J. Liang. 2011

**Received:**1 January 2011**Accepted:**11 February 2011**Published:**10 March 2011

## Abstract

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.

## Keywords

- Machine Learning
- Functional Equation
- Orthonormal Base
- Divergence Measure
- Statistical Inference

## 1. Introduction

As early as in 1952, Chernoff [1] 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 more information on some basic concepts of divergence measures, we refer the reader to, for example, [2–5] and references therein.

## 2. Main Results

Theorem 2.1.

for some .

Proof.

So .

The proof is then complete.

Theorem 2.2.

for some .

Proof.

Thus, a modification of Theorem 2.1 implies the conclusion.

Moreover, it is not so hard to deduce the following theorem.

Theorem 2.3.

where is strictly monotone twice-continuously differentiable functions. Then the divergence is -divergence or vector -divergence times a positive constant .

## Declarations

### Acknowledgments

This work was supported partially by the NSF of China and the Specialized Research Fund for the Doctoral Program of Higher Education of China.

## Authors’ Affiliations

## References

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.