Mathematics and Statistics 4(1): 40-45, 2016
DOI: 10.13189/ms.2016.040105
http://www.hrpub.org
The Constructive Implicit Function Theorem and Proof
in Logistic Mixtures
Xiao Liu
Methods in Empirical Educational Research, TUM School of Education and Centre for International Student Assessment (ZIB),
TU München, Arcisstr. 21, 80333 Munich, Germany
c
Copyright ⃝2016
by authors, all rights reserved. Authors agree that this article remains permanently
open access under the terms of the Creative Commons Attribution License 4.0 International License
Abstract There is the work by Bridges et al (1999) on the key features of a constructive proof of the implicit function
theorem, including some applications to physics and mechanics. For mixtures of logistic distributions such information is
lacking, although a special instance of the implicit function theorem prevails therein. The theorem is needed to see that the
ridgeline function, which carries information about the topography and critical points of a general logistic mixture problem,
is well-defined [2]. In this paper, we express the implicit function theorem and related constructive techniques in their multivariate extension and propose analogs of Bridges and colleagues’ results for the multivariate logistic mixture setting. In
particular, the techniques such as the inverse of Lagrange’s mean value theorem [4] allow to prove that the key concept of a
logistic ridgeline function is well-defined in proper vicinities of its arguments.
Keywords Constructive Implicit Function Theorem, Logistic Mixture, Lagrange Mean Value Theorem, Ridgeline
1 Introduction
In this paper, we focuse on constructive techniques that are based on expressing the proof of implicit function theorem
in their multivariate extension and propose analogs of Bridges and colleagues’ results for the multivariate logistic mixture
setting.
According to [2], applying the implicit function theorem, we can prove that a unique explicit formula for the ridgeline
function is possible locally in Theorem 1; In this paper, we propose analogs of Bridges et al.’s results (see [1]) for the
multivariate logistic distribution.
Applying Lagrange’s mean value theorem (see [4]), we can get the first lemma. Moreover, we are interested in uniform
differentiability of 2 variables on Lemma 3 due to [5], which carries important information about the proof of our goal.
2 Applications to Topography
Theorem 1 later frequently allows us to show some ridgeline and contour plots, where the ridgeline function satisfies
that the left side of formula (1) is equal to null (see [3]). The following example is in the case of two dimensions and three
components.
Example 1. The mixture logistic density with D
( )
0
µ1 =
,
0
( )
1
µ2 =
,
1
( )
2
µ3 =
,
2
= 2 and K = 3, and the parameters
(
)
1
s1 =
,
0.07
(
)
0.07
s2 =
,
1
(
)
1
1
s3 =
,
π1 = π2 = π3 = .
0.07
3
Figure 1 shows the contours of the density given in Example 1.
Mathematics and Statistics 4(1): 40-45, 2016
41
Figure 1. Density contour plot for the three component mixture density of Example 1.
3
The Constructive Implicit Function Theorem
For the most part we confine our attention to the following special case of the Implicit Function Theorem.
Theorem 1. Let
ψ(α, x) :=
K
∑
αi
i=1
D
∑
k=1
(
)
D+1
s−1
1
−
ik
Eik
(1)
x −µ
xk −µik
xk −µik ∑
− js ij
D
ij
where Eik = 1 + e sik + e sik
e
,
j̸=k
be a differentiable mapping of a neighbourhood of (α0 , x0 ) ∈ RK ×RD into Rp , let ψ(α0 , x0 ) = 0, and let det(D2 ψ(α0 , x0 )) ̸=
0. Then there exists r > 0 and a differentiable function f : B̄(α0 , r) ⊂ RK → RD such that for each α ∈ B̄(α0 , r),
(α, f (α)) is the unique solution x of the equation ψ(α, x) = 0 in some neighbourhood of (α0 , x0 ).
Where we use standard modern notations for derivatives, such as D for the derivative itself, and Dl for the lth partial
derivative (l = 1, 2), of a mapping from a subset of RK × RD to Rp .
This result, given here for the logistic mixture case, has obvious generalizations to any general implicit function as
following corollary. To keep the presentation short we refrain from presenting them here, as their treatment follows the same
strategy analogously to [1].
Corollary 1. Let ψ be a differentiable mapping of a neighbourhood of (x0 , y0 ) ∈ RK × RD into Rp , let ψ(x0 , y0 ) = 0, and
let det(D2 ψ(x0 , y0 )) ̸= 0. Then there exists r > 0 and a differentiable function f : B̄(x0 , r) ⊂ RK → RD such that for
each x ∈ B̄(x0 , r), (x, f (x)) is the unique solution y of the equation ψ(x, y) = 0 in some neighbourhood of (x0 , y0 ).
In next section, we will give the proof of Theorem 1. At last, we will show some necessary lemmata for preparing the
proof of our main theorem in appendix.
4
A New Proof of The Constructive Implicit Function Theorem in The Case of
Logistic Mixtures
4.1
Proof of Theorem 1
Proof.
(i) Suppose that the assumption of Theorem 1 are satisfied, and choose r, s > 0 as in Lemma 2. Assume
J := B × C = {(α, x) : |α − α0 | ≤ r, |x − x0 | ≤ s},
which is a compact set.
Fix ξ with |ξ − α0 | ≤ r, and there ∃ ε > 0 such that
0<ε<m≡
inf
|h|≤r,|g|≤s
|D2 ψ(α0 + h, x0 + g)|.
(2)
42
The Constructive Implicit Function Theorem and Proof in Logistic Mixtures
Consider x, x∗ such that |x − x0 | ≤ s, |x∗ − x0 | ≤ s, and |ψ(ξ, x) − ψ(ξ, x∗ )| < ε2 .
If |x − x∗ | ≥ ε, then due to Lemma 1, ∃ η between x and x∗ such that
|D2 ψ(ξ, η)| ≤ |x − x∗ |−1 ε2 ≤ ε < m ≤ |D2 ψ(ξ, η)|,
(3)
a contradiction.
Thus |x − x∗ | < ε.
In particular, if |ψ(ξ, x)| < ε2 and |ψ(ξ, x∗ )| < ε2 , then |x − x∗ | < 2ε.
(ii) Next give a hypothesis of that
0 < γ :=
If |x − x0 | ≤ s, then
inf
|x−x0 |≤s
|ψ(ξ, x)|.
|D2 ψ 2 (ξ, x)| = 2|ψ(ξ, x)||D2 ψ(ξ, x)| ≥ 2γm > 0.
(4)
(5)
According to the function D2 ψ 2 (ξ, ·) is continuous on the segment
C := [x0 − m, x0 + m], without loss of generality, the Intermediate Value Theorem (see [4], Ch. 2) allows us to suppose
that D2 ψ 2 (ξ, x) ≥ 2γm for ∀ x ∈ C.
This implies that ψ 2 (ξ, ·) is strictly increasing on C, and hence ψ 2 (ξ, x0 ) > ψ 2 (ξ, x0 − m); this is contradictory due to (12)
in view of the choice of s in Lemma 2.
1
Therefore γ = 0, then we can choose xn for each n such that |xn − x0 | ≤ s and |ψ(ξ, xn )| < 2 . Now the work in part (i)
n
2
of this proof shows that ∃ N , such that when nj , nl > N , we can get |xnj − xnl | < ; then the sequence {xn } is a Cauchy
n
sequence, and hence converges to a limit x∞ on the segment C = [x0 − m, x0 + m]. The same argument shows that x∞ is
also the unique solution x on C of the equation ψ(ξ, x) = 0, thus we can define a function f : [α0 − n, α0 + n] → C due
to f (ξ) = x∞ .
On account of Lemma 3, we complete the proof.
A Definition of Uniform Differentiability in 2 Variables
Definition 1. Let f : Rm ×Rn → R be differentiable and such that ∇f is uniformly continuous. We define that f is uniformly
differentiable, i.e., for any ε > 0, there is a δ > 0 such that for all a, x ∈ Rm and b, y ∈ Rn , we have
|f (a, b) − f (x, y) − D1 f (x, y)(a − x) − D2 f (x, y)(b − y)|
<ε
∥(a, b) − (x, y)∥
√
whenever ∥(a, b) − (x, y)∥ := ∥a − x∥2 + ∥b − y∥2 < δ.
B Three Lemmata
Although, logically unnecessary for the inverse of Lagrange’s Mean Value Theorem, it is, for pedagogical reasons, interesting to see that we show the following lemma. Moreover, the result will be also useful for the proof of Theorem 1.
Lemma 1. For every function f : Rn → R with continuous partial derivatives fx1 , · · · , fxn and for all distinct pairs a and
b in Rn , there exists an intermediate point x on the line segment joining a and b which we denote as x ∈ [a, b] such that
inf |Df (x)| ≤ |b − a|−1 |f (b) − f (a)|.
(6)
x∈[a,b]
Proof. According to [4], we can get
f (b) − f (a) =< b − a, Df (x) >
then
⟨
b − a,
inf |Df (x)|
x∈[a,b]
Thus
⟩
≤ f (b) − f (a).
inf |Df (x)| ≤ |b − a|−1 |f (b) − f (a)|.
x∈[a,b]
(7)
(8)
(9)
Mathematics and Statistics 4(1): 40-45, 2016
43
The following example of Lemma 1 can be applied for calculating the average velocity of non-uniform motion in kinematics.
Example 2. Let
f (x) = ax2 + bx
(a ̸= 0)
then
Df (x) = 2ax + b.
So we have
a(a + b − a)2 + b(a + b − a) − aa2 − ba
}
{
1
= (b − a) 2a[a + (b − a)] + b .
2
The core to our mathematical expression of the existence of an implicit function in the logistic mixtures case is provided by
the following lemma.
Lemma 2. Under the hypotheses of Theorem 1, there exist m ∈ RD , |m| = s, n ∈ RK , |n| = r, and r, s > 0 such that
2
s|D2 ψ(α0 , x0 )|,
3
(10)
|D2 ψ(α0 + h, x0 + g)| > 0,
(11)
|ψ(α0 , x0 ± m)| ≥
for ∀ h ∈ RK , g ∈ RD
inf
|h|≤r,|g|≤s
and
inf |ψ(α0 + h, x0 ± m)| ≥
|h|≤r
1
s|D2 ψ(α0 , x0 )| > sup |ψ(α0 + h, x0 )|.
2
|h|≤r
(12)
Proof. Choose an open ball B, with centre (α0 , x0 ) and radius R. Following formula (1), obviously D2 ψ(α, x) ∈ C 0 (B).
Then we can get
1
|D2 ψ(α, x)| > |D2 ψ(α0 , x0 )|
(13)
2
for all (α, x) ∈ B.
According to that ψ is differentiable at (α0 , x0 ), there exists s ∈ (0, R) such that if |x − x0 | ≤ s, we obtain
ψ(α0 , x) − ψ(α0 , x0 ) 4
≤ |D2 ψ(α0 , x0 )|,
3
x − x0
then
|ψ(α0 , x) − D2 ψ(α0 , x0 )(x − x0 )| ≤
1
|D2 ψ(α0 , x0 )(x − x0 )|
3
(14)
and therefore
2
|D2 ψ(α0 , x0 )(x − x0 )|.
(15)
3
In particular, choose x = x0 ± m, we obtain inequality (10). Since ψ(α0 , x0 ) = 0 and ψ is continuous, we can now choose
r ∈ (0, R) such that inequality in (12) hold. According to r < R, our choice of R ensures that we can obtain inequality (11).
|ψ(α0 , x)| ≥
Next we separate out the proof of the differentiability of the implicit function. It will be convenient to establish the
existence of Theorem 1 before.
Lemma 3.
Let B be a compact ball in RK , C a compact domain in RD , and ψ : B × C → Rp be a uniformly differentiable function
such that
0 < m := inf |D2 ψ|.
(16)
B×C
Suppose that there exists a function f : B → C such that ψ(α, x) = 0 for ∀ α ∈ B, x := f (α) ∈ C. Then f is uniformly
differentiable on B, and
D1 ψ(ξ, f (ξ))
(17)
f ′ (ξ) = −
D2 ψ(ξ, f (ξ))
for any ξ ∈ B.
44
The Constructive Implicit Function Theorem and Proof in Logistic Mixtures
Proof. Let 0 < ε < 17 m, and let α(1) , α(2) be points of B, we define
√
(1)
(2) 2
(1)
(2)
∥α − α ∥ := ΣK
i=1 (αi − αi ) .
(18)
According to Definition 1, we have the definition of uniform differentiability in 2 variables (also see [5]), so we can get
(
ψ α(1) , f (α(1) )) − ψ (α(2) , f (α(2) )) − D ψ (α(2) , f (α(2) )) (α(1) − α(2) )
1
(19)
(
)(
)
−D2 ψ α(2) , f (α(2) ) f (α(1) ) − f (α(2) ) √
∥α(1) − α(2) ∥2 + ∥f (α(1) ) − f (α(2) )∥2 ≤ ε.
Then
≤
≤
(
)
(
)
ψ α(1) , f (α(1) ) − ψ α(2) , f (α(2) )
(
)(
)
−D1 ψ α(2) , f (α(2) ) α(1) − α(2)
(
)(
)
−D2 ψ α(2) , f (α(2) ) f (α(1) ) − f (α(2) ) √
ε ∥α(1) − α(2) ∥2 + ∥f (α(1) ) − f (α(2) )∥2
)
(
ε α(1) − α(2) + f (α(1) ) − f (α(2) ) .
(
)
(
)
(
)
Otherwise ψ α(1) , f (α(1) ) = ψ α(2) , f (α(2) ) = 0 and D2 ψ α(2) , f (α(2) ) ≥ m due to formula (16), so
D ψ (α(2) , f (α(2) )) (
)
1 (
(1)
(2)
(1)
(2)
) α −α
+
f
(α
)
−
f
(α
)
D2 ψ α(2) , f (α(2) )
)
(
≤ m−1 ε α(1) − α(2) + f (α(1) ) − f (α(2) )
)
1 (
(1)
≤
α − α(2) + f (α(1) ) − f (α(2) ) .
7
Therefore
(21)
D ψ (α(2) , f (α(2) )) (
) 1
(1)
(2) (
) α(1) − α(2) + −
α
α
.
f (α(1) ) − f (α(2) ) ≤ 7 D2 ψ α(2) , f (α(2) )
Choosing a bound M for |D1 ψ| on the compact set B × C, we see that
(
)
f (α(1) ) − f (α(2) ) ≤ 7M m−1 + 1 α(1) − α(2) .
It follows from formula (21) that
≤
D ψ (α(2) , f (α(2) )) (
)
1 (
) α(1) − α(2) + f (α(1) ) − f (α(2) )
D2 ψ α(2) , f (α(2) )
(
) m−1 7M m−1 + 2 ε α(1) − α(2) → 0
when ε → 0.
Thus f is uniformly differentiable on B, with
f ′ (ξ) = −
for ∀ ξ ∈ B.
(20)
D1 ψ(ξ, f (ξ))
D2 ψ(ξ, f (ξ))
(22)
(23)
Mathematics and Statistics 4(1): 40-45, 2016
45
REFERENCES
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Mechanics, Chaos Solitons and Fractals, Vol.10, 927-934, 1999.
[2] X. Liu, A. Ünlü. Multivariate Logistic Mixtures, European Conference on Data Analysis (ECDA) , The University of
Bremen, 105, 2014.
[3] X. Liu. Multivariate Logistic Mixtures, Universal Journal of Applied Mathematics, Vol.3, No.4, 77-87, 2015.
[4] P. K. Sahoo, T. Riedel. Mean Value Theorems and Functional Equations, World Scientific Press, New Jersey, 1998.
[5] E. M. Stein. Singular Integrals and Differentiability Properties of Functions, Princeton University Press, New Jersey,
1970.