doi:10.1155/2011/617564
Research Article
Solution to a Function Equation and
Divergence Measures
Chuan-Lei Dong and Jin Liang
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
Correspondence should be addressed to Jin Liang,jinliang@sjtu.edu.cn
Received 1 January 2011; Accepted 11 February 2011
Academic Editor: Toka Diagana
Copyrightq2011 C.-L. Dong and J. Liang. 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.
We investigate the solution to the following function equationf1xg1y· · ·fnxgny Gx
y, which arises from the theory of divergence measures. Moreover, new results on divergence measures are given.
1. Introduction
As early as in 1952, Chernoff 1used theα-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 Csisz´arf-divergence is a unique class of divergences having information monotonicity, from which the dual α 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 α-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,2–5and references therein.
Motivated by these studies, we investigate in this paper the solution to the following function equation
f1xg1
y· · ·fnxgn
yGxy, 1.1
which arises from the discussion of the theory of divergence measures, and show that for
n >1, iffi:a, b → R,gi:a, b → R, i1,2, . . . , n, andG:2a,2b → Rsatisfy n
i1
fixgi
thenGis the solution of a linear homogenous differential equation with constant coefficients. Moreover, new results on divergence measures are given.
Throughout this paper, we letRbe the set of real numbers andF ⊂ Rn are a convex
set.
Basic notations:Rn : {x ∈Rn :xi >0, i 1,2, . . . , n};φ :F → Ris strictly convex
and twice differentiable; π : Rn
→ F is differentiable injective map; Dφπ is the general
vector Bregman divergence; f : 0,∞ → 0,∞ is strictly convex twice-continuously differentiable function satisfyingf1 0, f1 0;Df is the vectorf-divergence.
If for everyp, q∈Rn,
Dφπp:qDf
p:q, 1.3
then we say the Dπφ or Df is in the intersection of f-divergence and general Bregman
divergence.
For more information on some basic concepts of divergence measures, we refer the reader to, for example,2–5and references therein.
2. Main Results
Theorem 2.1. Assume that there are differentiable functions
fi:a, b−→R, gi:a, b−→R, i1,2, . . . , n, 2.1
andG:2a,2b → Rsuch that
n
i1
fixgi
yGxy, for everyx, y∈a, b. 2.2
ThenG∈C∞2a,2band
anGnan−1Gn−1· · ·a1Ga0G0, 2.3
for somean, an−1, . . . , a0∈R.
Proof. Sincefi, giis differentiable functions, it is clear that
fi, gi∈L2a, b, i1,2, . . . , n. 2.4
Let
Mspanf1, f2, . . . , fn
. 2.5
ThenMis a finite dimension space. So we can find differentiable functions
as the orthonormal bases ofM, wherem≤n. Observing that
n
i1
fixgi
y n
i1 ⎡ ⎣m
j1
aijsjxgi
y
⎤ ⎦
m
j1
sjx n
i1
aijgi
y
m
j1
sjxtj
y,
2.7
where
aij∈R, tj
y n
i1
aijgi
y, i1,2, . . . , n, j1,2, . . . , m, 2.8
we have
Gxy n
i1
fixgi
y m
j1
sjxtj
y, for everyx, y∈a, b. 2.9
Clearly,
tj ∈L2a, b, j1, . . . , m. 2.10
Next we prove that
tj∈M, j1, . . . , m. 2.11
It is easy to see that we only need to prove the following fact:
span{s1, s2, . . . , sm, t1, t2, . . . , tm}M. 2.12
Actually, if this is not true, that is,
span{s1, s2, . . . , sm, t1, t2, . . . , tm}/M, 2.13
then there existst /0 such that
Therefore,
SinceCis a symmetric matrix, we have
for an orthogonal matrixQ, and a diagonal matrix
Without loss the generalization, we can assume that
λ1, λ2, . . . , λm/0. 2.33
By the similar arguments as above, we can prove
spanr1, . . . , rm, r1, . . . , rm
span{r1, . . . , rm}. 2.35
So there is a matrixAsatisfying
Thus,
Gxy ∂G
xy
∂x RxAΛR
yT. 2.37
By mathematical induction we obtain
GixyRxAiΛRyT, ∀i0,1, . . . 2.38
SoG∈C∞2a,2b. Let
b0b1λ· · ·bmλm 2.39
be the annihilation polynomial ofA. Then
b0G
xyb1G
xy· · ·bmGm
xy
m
i0
biRxAiΛR
yT
Rx m
i0
biAiΛR
y
0.
2.40
Sincen≥m, we can findan, an−1, . . . , a0∈Rsuch that
anGnan−1Gn−1· · ·a1Ga0G0. 2.41
The proof is then complete.
Theorem 2.2. Let the f-divergence Df be in the section of f-divergence and general Bregman divergence. ThenGx fexsatisfies
n
i0
aiGi0, 2.42
for somean, . . . , a0∈R.
Proof. IfDf, Dφπare in the intersection off-divergence and general Bregmen divergence, then
we have
xfy x
nφπX−φπY− n
i1
∂φπY ∂xi
where
X x, x, . . . , x∈Rn, Y y, y, . . . , y∈Rn. 2.44
Let
∂φπY ∂xi si
y, πiX tix. 2.45
Then
∂2xfy/xn
∂x∂y
∂2φπX−φπY−n i1si
ytix−ti
y
∂x∂y . 2.46
Hence
y x2f
y x
n
i1
siytix. 2.47
Let
Gx fex, f ix
siex
ex , gix ti
e−xe−2x. 2.48
Then
Gxy n
i1
fixgi
y. 2.49
Thus, a modification ofTheorem 2.1implies the conclusion.
Moreover, it is not so hard to deduce the following theorem.
Theorem 2.3. Let a vectorf-divergence is are the intersection of vectorf-divergence and general Bregman divergence andπsatisfy
πx π1x1, . . . , πnxn, ∀x∈Rn, 2.50
where π1, . . . , πn is strictly monotone twice-continuously differentiable functions. Then the f divergence isα-divergence or vectorα-divergence times a positive constantc.
Acknowledgments
References
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