Skip to main content

Theory and Modern Applications

  • Research Article
  • Open access
  • Published:

Solution to a Function Equation and Divergence Measures


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.

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, [25] and references therein.

Motivated by these studies, we investigate in this paper the solution to the following function equation


which arises from the discussion of the theory of divergence measures, and show that for , if , , and satisfy


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.

If for every ,


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, [25] and references therein.

2. Main Results

Theorem 2.1.

Assume that there are differentiable functions


and such that


Then and


for some .


Since is differentiable functions, it is clear that




Then is a finite dimension space. So we can find differentiable functions


as the orthonormal bases of , where . Observing that




we have




Next we prove that


It is easy to see that we only need to prove the following fact:


Actually, if this is not true, that is,


then there exists such that






we get


that is,


Since is linearly independent, we see that




This is a contradiction. Hence (2.12) holds, and so does (2.11). Thus, there are such that




So we have






Let , and




Since is a symmetric matrix, we have


for an orthogonal matrix , and a diagonal matrix






So, for all ,


Without loss the generalization, we can assume that


Thus, for all ,


By the similar arguments as above, we can prove


So there is a matrix satisfying




By mathematical induction we obtain


So .



be the annihilation polynomial of . Then


Since , we can find such that


The proof is then complete.

Theorem 2.2.

Let the -divergence be in the section of -divergence and general Bregman divergence. Then satisfies


for some .


If are in the intersection of -divergence and general Bregmen divergence, then we have














Thus, a modification of Theorem 2.1 implies the conclusion.

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

Theorem 2.3.

Let a vector -divergence is are the intersection of vector -divergence and general Bregman divergence and satisfy


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


  1. Chernoff H: A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Mathematical Statistics 1952, 23: 493-507. 10.1214/aoms/1177729330

    Article  MathSciNet  MATH  Google Scholar 

  2. Amari S: Information geometry and its applications: convex function and dually flat manifold. In Emerging Trends in Visual Computing, Lecture Notes in Computer Science. Volume 5416. Edited by: Nielsen F. Springer, Berlin, Germany; 2009:75-102. 10.1007/978-3-642-00826-9_4

    Chapter  Google Scholar 

  3. Brègman LM: The relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming. Computational Mathematics and Mathematical Physics 1967, 7: 200-217.

    Article  MathSciNet  MATH  Google Scholar 

  4. Cichocki A, Zdunek R, Phan AH, Amari S: Non-Negative Matrix and Tensor Factorizations: Applications to Explanatory Multi-Way Data Analysis and Blind Source Separation. Wiley, New York, NY, USA; 2009.

    Book  Google Scholar 

  5. Csiszár I: Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. The Annals of Statistics 1991,19(4):2032-2066. 10.1214/aos/1176348385

    Article  MathSciNet  MATH  Google Scholar 

Download references


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

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jin Liang.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Dong, CL., Liang, J. Solution to a Function Equation and Divergence Measures. Adv Differ Equ 2011, 617564 (2011).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: