Department of Mathematics
Exact Minimizers in Real Interpolation
Japhet Niyobuhungiro
Exact Minimizers in Real Interpolation
Some additional results
Japhet Niyobuhungiroa,b
aDivision of Mathematics and Applied Mathematics, Department of Mathematics,
Linköping University, SE–581 83 Linköping, Sweden
bDepartment of Mathematics, School of Pure and Applied Science, College of Science and Technology,
University of Rwanda, P.O. Box 3900 Kigali,Rwanda
Abstract
We present some extensions of results presented in our recent papers. First we extend the characterization of optimal decompositions for a Banach couple to optimal decompositions for a Banach triple. Next we show that our approach can apply when complex spaces are considered instead of real spaces. Finally we compare the performance of the algorithm that we have proposed for the ROF model with the Split Bregman algorithm. The Split Bregman algorithm can in principle be regarded as a benchmark algorithm for the ROF model. We find out that in most cases both algorithms behave in a similar way and that in some cases our algorithm decreases the error faster with the number of iterations.
Keywords: Regular Banach triple, Optimal decompositions, Complex Banach couple, Real Interpolation, Convex Duality
2010 MSC: 46B70, 46E35, 68U10
1. Optimal decomposition for a Banach triple 1.1. Introduction
In the paper [7], duality in convex analysis was used to characterize op-timal decompositions for functionals arising in real interpolation and only real Banach couples were considered. However in connection with applied problems in image processing (see, for example, the paper [1]) optimal de-composition for three spaces is sometimes needed. For example, models re-lying on the use of three different semi–norms such as the total variation for the geometrical component, the negative Sobolev norm for the texture and the neg-ative Besov norm for the noise, are often useful. For this reason we will inves-tigate decomposition for three spaces. However, description in this situation
Email address: jniyobuhungiro@ur.ac.rw, japhet.niyobuhungiro@liu.se (Japhet Niyobuhungiro)
starts to be more complicated, for example we no longer have the equality
(X0+X1+X2)∗ = X∗0∩X1∗∩X2∗even for the regular triple. Let(X0, X1, X2) be a compatible Banach triple. i.e., X0, X1 and X2 are all Banach spaces and there exists a Hausdorff topological vector space in which they are linearly and continuously embedded. Consider the the corresponding K–functional for s, t>0 and x∈X0+X1+X2defined by
K(s, t; x; X0, X1, X2) = inf x=x0+x1+x2 kx0kX0+skx1kX1+tkx2kX2 , . (1)
Its calculation is a difficult extremal problem, (see J. Peetre [10] for the Banach couples). More generally we define the corresponding Lp0,p1,p2–functional Lp0,p1,p2(s, t; x; X0, X1, X2) =x=xinf 0+x1+x2 1 p0 kx0kXp00 + s p1 kx1kXp11+ t p2 kx2kpX22 , (2) where s, t>0 and 1≤ p0, p1 p2<∞. The following question arises.
Problem 1. Suppose that for a given element x∈X0+X1+X2, 1≤ p0, p1, p2<∞ and s, t>0 there exists an optimal decomposition for the Lp0,p1,p2–functional, i.e. a decomposition x=x0,opt+x1,opt+x2,opt such that
Lp0,p1,p2(s, t; x; X0, X1, X2) =x=xinf 0+x1+x2 1 p0 kx0k p0 X0+ s p1 kx1k p1 X1+ t p2kx2k p2 X2 = p1 0 x0,opt p0 X0+ s p1 x1,opt p1 X1+ t p2 x2,opt p2 X2. How can this optimal decomposition be characterized (constructed)?
1.2. Preliminaries
Below we present some well–known definitions and results from convex analysis that are needed for the proofs of our main results. Throughout, E will denote a Banach space with the normk·kEand E∗will denote its topological
dual space. We will consider convex functions F : E → R∪ {+∞}. The
effective domain or simply domain of the function F is a convex set denoted by dom F, and defined by
dom F={x∈E : F(x) < +∞}.
The function F is said to be proper if dom F6= ∅. If the epigraph of a function F, i.e. the set
epi F={(x, α) ∈E×R | F(x) ≤α},
is closed then the function F is called lower semicontinuous (l.s.c.).
The definition of the operation of infimal convolution is the following. Definition 1. The infimal convolution of n functions Fi, i=0, 1, . . . , n−1 from E intoR∪ {+∞} is the function denoted by F0⊕F1⊕. . .⊕Fn−1 that maps E into R∪ {−∞,+∞}and is defined by
(F0⊕F1⊕. . .⊕Fn−1) (x) = inf x=x0+x1+...+xn−1
{F0(x0) +F1(x1) +. . .+Fn−1(xn−1)}, (3)
and it is exact at a point x∈E if the infimum is achieved, i.e.,
(F0⊕F1⊕. . .⊕Fn−1) (x) =x=x min
0+x1+...+xn−1
{F0(x0) +F1(x1) +. . .+Fn−1(xn−1)}. Suppose that(F0⊕F1⊕. . .⊕Fn−1) (x)is finite and exact. Then the decomposition x = x0+x1+. . .+xn−1, on which the infimum is attained will be called optimal and denoted as x=x0,opt+x1,opt+. . .+x(n−1),opt.
The following notion of conjugate function will be important for us. Definition 2. The conjugate function of F is the function F∗ : E∗ → R∪ {+∞}
defined by
F∗(y) =sup x∈E
{hy, xi −F(x)}. (4) Moreover, we will say that y is dual to x (y is also called a subgradient of the convex function F at the point x) with respect to F if F∗(y) =hy, xi −F(x)or, in symmetric form,
F(x) +F∗(y) =hy, xi.
The set of all dual elements to x is denoted by ∂F(x) and the function F is called subdifferentiable at the point x∈ E if the set ∂F(x)is not empty.
Note that (see [13], p. 24) for n = 2, if the set ∂F0(x0) ∩∂F1(x−x0) is nonempty then(F0⊕F1) (x)is exact at the point x.
We will also need the next two propositions. The proofs are straightfor-ward consequences of the definition and can be found in any standard book on convex analysis (see [11] or [4] for example).
Proposition 1. a) Let F(x) = 1pkxkEp where 1 < p < ∞, then F∗(y) =
1 p0kyk p0 E∗where 1p+ p10 =1. b) Let F(x) =kxkE, then F∗(y) = 0 if kykE∗≤1 +∞ if kykE∗>1. Proposition 2. If λ∈R\ {0}then (λF)∗(y) =sup x∈E {hy, xi −λF(x)} =λF∗(y λ).
The following nontrivial result (its proof is based on the Baire category theorem) is a very useful tool to check subdifferentiability (see [5]).
Theorem 1. Let F: E −→ (−∞, ∞]be a convex and lower semicontinuous
func-tion. Then F is subdifferentiable over the interior of its domain.
The next theorem demonstrates that the conjugate of infimal convolution is the sum of the conjugates.
Theorem 2. Let F0, F1, . . . , Fn−1be convex functions from E intoR∪ {+∞}. Then
(F0⊕F1⊕. . .⊕Fn−1)∗=F0∗+F1∗+. . .+Fn−1∗ .
The proof is a traightforward application of the definition and we omit it. Remark 1. The dual result
(F0+F1+. . .+Fn−1)∗=F0∗⊕F1∗⊕. . .⊕Fn−1∗ ,
is valid only under some additional restriction on F0, F1, . . . , Fn−1 (see [11] in the finite dimensional setting).
1.3. Optimal decomposition for the L–functional
Let (X0, X1, X2) be a regular Banach triple, i.e. X0∩X1∩X2 is dense in each one of X0, X1and X2. Let ϕj : Xj −→ R∪ {+∞}, j= 0, 1, 2, be convex and proper functions. If by
ϕj(u) =
ϕj(u) if u∈ Xj
+∞ if u∈ (X0+X1+X2) \Xj j=0, 1, 2 (5)
we define functions on X0+X1+X2, then we can express the infimal convo-lution ϕ0⊕ϕ1⊕ϕ2on the space X0+X1+X2as follows:
(ϕ0⊕ϕ1⊕ϕ2)(x) = inf
x=x0+x1+x2
(ϕ0(x0) +ϕ1(x1) +ϕ2(x2)), xj ∈Xj, j=0, 1, 2. (6) We will use the following result of [3] which states that, for a compatible
family of normed spaces X= Xj
j∈J such that ∀j ∈ J, ∩j∈JXj is dense in Xj then∑j∈JXj
∗
is a closed subspace of∩j∈JX∗j.
The next result provides a characterization of the optimal decomposition for infimal convolution ϕ0⊕ϕ1⊕ϕ2.
Theorem 3. Let ϕj : Xj −→ R∪ {+∞}, j = 0, 1, 2, be convex proper functions. Suppose also that ϕ0⊕ϕ1⊕ϕ2is subdifferentiable at a given element x∈dom(ϕ0⊕ ϕ1⊕ϕ2). Then the decomposition x = x0,opt+x1,opt+x2,opt is optimal for ϕ0⊕
ϕ1⊕ϕ2if and only if there exists y∗ ∈ (X0+X1+X2)∗ ⊆ X0∗∩X∗1∩X2∗that is dual to xj,optwith respect to ϕj, j=0, 1, 2, respectively, i.e.
ϕ0 x0,opt = hy∗, x0,opti −ϕ∗0(y∗) ϕ1 x1,opt= hy∗, x1,opti −ϕ∗1(y∗) ϕ2 x2,opt= hy∗, x2,opti −ϕ∗2(y∗). (7)
Proof. Let us assume that the decomposition x = x0,opt+x1,opt+x2,opt is optimal for(ϕ0⊕ϕ1⊕ϕ2) (x). It means that
(ϕ0⊕ϕ1⊕ϕ2) (x) = inf x=x0+x1+x2 ϕ0(x0) +ϕ1(x1) +ϕ2(x2) (8) =ϕ0 x0,opt +ϕ1 x1,opt +ϕ2 x2,opt .
Since ϕ0⊕ϕ1⊕ϕ2is subdifferentiable at x then there exists y∗∈ (X0+X1+ X2)∗⊆X∗0∩X1∗∩X2∗such that
y∗∈∂(ϕ0⊕ϕ1⊕ϕ2) (x),
i.e.
(ϕ0⊕ϕ1⊕ϕ2) (x) =hy∗, xi − (ϕ0⊕ϕ1⊕ϕ2)∗(y∗). (9) From (8) and the formula for the conjugate of infimal convolution (see Theo-rem 2), we have that (9) is equivalent to
ϕ0 x0,opt+ϕ1 x1,opt+ϕ2 x2,opt=hy∗, xi − (ϕ0)∗(y∗) − (ϕ1)∗(y∗) − (ϕ2)∗(y∗). Or equivalently,
ϕ0 x0,opt+ (ϕ0)∗(y∗) −y∗, x0,opt +ϕ1 x1,opt+ (ϕ1)∗(y∗) −y∗, x1,opt + (10)
+ϕ2 x2,opt+ (ϕ2)∗(y∗) −y∗, x2,opt =0 From the definition of the conjugate functions (see (4)) we have
ϕj xj,opt+ (ϕj)∗(y∗) −y∗, xj,opt ≥0, ∀j=0, 1, 2. Taking in account (??) we see that, in fact, we have the equalities
ϕj xj,opt+ (ϕj)∗(y∗) −y∗, xj,opt =0, ∀j=0, 1, 2. (11)
Since for y∗∈X0∗∩X1∗∩X∗2we have
(ϕj)∗(y∗) = sup x∈X0+X1+X2 hy∗, xi −ϕj(x) = sup x∈X0 hy∗, xi −ϕj(x)= ϕ∗j(y∗) for j=0, 1, 2. Since xj,opt∈Xj (j=0, 1, 2), therefore from (11) we obtain (7).
Conversely, let us assume there exist y∗ ∈ X0∗∩X∗1∩X2∗ and a decompo-sition x= ˜x0+ ˜x1+ ˜x2such that (7) is satisfied. Then from the definitions of
ϕ0, ϕ1, ϕ2and their conjugates we have
ϕ0(˜x0) = hy∗, ˜x0i − (ϕ0)∗(y∗) ϕ1(˜x1) = hy∗, ˜x1i − (ϕ1)∗(y∗) ϕ2(˜x2) = hy∗, ˜x2i − (ϕ2)∗(y∗). Then ϕ0(˜x0) +ϕ1(˜x1) +ϕ2(˜x2) = hy∗, ˜x0+ ˜x1+˜x2i − ((ϕ0)∗+ (ϕ1)∗+ (ϕ2)∗) (y∗). Since by Theorem 2, the conjugate of infimal convolution is the sum of conju-gates, then
By definition of infimal convolution (3), it follows that, in particular,
(ϕ0⊕ϕ1⊕ϕ2) (x) ≤ϕ0(˜x0) +ϕ1(˜x1) +ϕ2(˜x2). Then
(ϕ0⊕ϕ1⊕ϕ2) (x) ≤ hy∗, xi − (ϕ0⊕ϕ1⊕ϕ2)∗(y∗). Moreover, from the definition of a conjugate function we have
(ϕ0⊕ϕ1⊕ϕ2) (x) ≥ hy∗, xi − (ϕ0⊕ϕ1⊕ϕ2)∗(y∗). Combining this with the previous inequality we obtain
(ϕ0⊕ϕ1⊕ϕ2) (x) = hy∗, xi − (ϕ0⊕ϕ1ϕ2)∗(y∗). From that and (12) we conclude that
(ϕ0⊕ϕ1⊕ϕ2) (x) =ϕ0(˜x0) +ϕ1(˜x1) +ϕ2(˜x2).
Therefore, the decomposition x = ˜x0+ ˜x1+ ˜x2 is optimal for (ϕ0⊕ϕ1⊕
ϕ2)(x).
Below we assume that the triple(X0, X1, X2)is regular, i.e. X0∩X1∩X2is dense in each of Xj, j =0, 1, 2. Let x∈ X0+X1+X2and let s, t>0 be fixed parameters. We consider the following L–functional:
Lp0,p1,p2(s, t; x; X0, X1, X2) =x=xinf 0+x1+x2 1 p0 kx0kpX00+ s p1 kx1kpX11+ t p2 kx2kXp22 , (13) where 1≤ p0<∞, 1≤ p1<∞ and 1≤p2<∞.
If the functions ϕj: Xj −→R∪ {+∞}, j=0, 1, 2 are defined by
ϕ0(u) = 1 p0 kukp0 X0 , ϕ1(u) = s p1 kukp1 X1 and ϕ2(u) = t p2 kukp2 X2, (14)
then (see (5) and (6)) the L–functional (13) can be written as the infimal con-volution
Lp0,p1,p2(s, t; x; X0, X1, X2) = (ϕ0⊕ϕ1⊕ϕ2) (x),
on the space X0+X1+X2. So, as the function Lp0,p1,p2 is convex and con-tinuous on X0+X1+X2, we have by Theorem 1 that it is subdifferentiable over the interior of its domain, which is equal to X0+X1+X2. Next we use Propositions 1 and 2 to calculate the conjugate functions ϕ∗j of ϕj, j = 0, 1, 2 and apply Theorem 3 to obtain the following characterization of the optimal decomposition for the L–functional
Corollary 1. Let1 < p0, p1, p2 < ∞ and s, t > 0. Then the decomposition x =
an element y∗ ∈ (X0+X1+X2)∗⊆X∗0∩X∗1∩X2∗such that 1 p0 x0,opt p0 X0 = hy∗, x0,opti − 1 p00ky∗k p00 X∗ 0 s p1 x1,opt p1 X1 = hy∗, x1,opti − s p01 ys∗ p0 1 X1∗ t p2 x2,opt p2 X2 = hy∗, x2,opti − t p0 2 y∗ t p02 X2∗. (15) where p1 j + 1 p0j =1, j=0, 1, 2.
Remark 2. From the inequality app + bp
0
p0 ≥ ab, a, b ≥ 0, in which the equality
is attained only when b = ap−1, it follows that the first condition in (15) is equiv-alent to ky∗kX∗ 0 = x0,opt p0−1 X0 and hy∗, x0,opti = ky∗kX0∗ · x0,opt X 0. Simi-larly, the second condition in (15) is equivalent to ky∗kX∗
1 = s x1,opt p1−1 X1 and hy∗, x1,opti = ky∗kX∗ 1 · x1,opt X
1, and the third condition in (15) is equivalent to ky∗kX∗ 2 =t x2,opt p2−1 X1 andhy∗, x1,opti = ky∗kX∗2 · x2,opt X1.
Let us now consider the special case when p1 = p2 = 1. As in previous Corollary, in this case we obtain the following result
Corollary 2. Let1<p0< +∞ and s, t>0. Then the decomposition x =x0,opt+
x1,opt+x2,opt is optimal for the Lp0,1,1–functional if and only if there exists y∗ ∈
(X0+X1+X2)∗⊆X0∗∩X1∗∩X2∗such thatky∗kX∗1 ≤s;ky∗kX∗2 ≤t and 1 p0 x0,opt p0 X0 = hy∗, x0,opti − 1 p0 0 ky∗k p00 X∗0 s x1,opt X1 = hy∗, x1,opti t x2,opt X 2 = hy∗, x2,opti (16) where p1 0 + 1 p00 =1.
Remark 3. In a similar way as in Corollary2, one can show that the decomposition
x = x0,opt+x1,opt+x2,opt is optimal for the K–functional (1) if and only if there
exists y∗ ∈ (X0+X1+X2)∗ ⊆X∗0∩X1∗∩X2∗such thatky∗kX0∗ ≤1;ky∗kX∗1 ≤s andky∗kX∗ 2 ≤t and x0,opt X 0 = hy∗, x0,opti s x1,opt X1 = hy∗, x1,opti t x2,opt X 2 = hy∗, x2,opti.
1.4. A geometry of optimal decomposition for the triple(`p, X1, X2)onRn
Let us consider the spaceRnwith the normkxk`p and some Banach spaces X1 and X2on Rn with the norms k·kX1 and k·kX2 respectively. By X
∗
X2∗we denote the Banach spaces onRnwith the dual norms kykX∗ 1 = sup kxkX1≤1 hy, xi and kykX∗ 2 = sup kxkX2≤1 hy, xi, where hy, xi = n
∑
i=1 yixi.We consider the Lp,1,1–functional for the triple(`p, X1, X2), i.e. Lp,1,1(s, t; x;`p, X1, X2) =x=xinf 0+x1+x2 1 pkx0k p `p+skx1kX1+tkx2kX2 , where s, t>0 and 1<p< +∞. Let F0, F1and F2be functions defined onRn by F0(u) = 1 p kuk p `p , F1(u) =skukX1 and F2(u) =tkukX2. (17) To formulate the result let us define the setsΩs,X1,Ωt,X2 by
Ωs,X1 = n u∈Rn : ∇F0(u) ∈sBX∗ 1 o , Ωt,X2 = n u∈Rn : ∇F0(u) ∈tBX∗ 2 o , (18) where sBX∗
1 (resp. tBX∗2) is the ball of the dual space X
∗
1 (resp. X2∗) of radius s (resp. t) with its center at the origin.
Remark 4. (i) Notice that if element x0,opt of the optimal decomposition x =
x0,opt+x1,opt+x2,opt is equal to zero, then (see Corollary 2) so must be y∗,
x1,optand x2,optand therefore x=0. We assume in the sequel, that x0,opt6=0.
(ii) Note that the condition 1p x0,opt p X0 = hy∗, x0,opti − 1 p0ky∗kp 0 X∗0 in Corollary 2 can in this case be re-written as
1 p x0,opt p `p = hy∗, x0,opti − 1 p0 ky∗k p0 `p0, i.e. 1 p n
∑
i=1 x0,opt;i p = n∑
i=1 y∗,i· x0,opt;i− 1 p0 n∑
i=1 |y∗,i|p 0 . (19) But 1p|a|p+p10|b|p 0≥ab and the equality holds only when b=|a|p−1sgn(a), so the equality (19) is equivalent to
y∗= x0,opt
p−1
sgn x0,opt . Therefore, y∗ = ∇F0 x0,opt, where F0(x) = 1pkxk
p
`p and from Corollary 2 it follows that x0,opt∈Ωs,X1∩Ωt,X2.
Theorem 4. Let x∈Rnwith optimal decomposition x=x
0,opt+x1,opt+x2,optfor Lp,1,1(s, t; x;`p, X1, X2)–functional. Then
(1) If x0,opt ∈int Ωs,X1∩Ωt,X2 then the optimal decomposition for
Lp,1,1(s, t; x;`p, X1, X2)–functional is given by x0,opt=x and x1,opt=x2,opt= 0.
(2) If x0,opt ∈int Ωs,X1
∩bd Ωt,X2, then the optimal decomposition for Lp,1,1(s, t; x;`p, X1, X2)–functional is given by x = x0,opt+0+x2,opt and is such that
x2,opt,∇F0 x0,opt = x2,opt X 2 ∇F0 x0,opt X∗ 2 =t x2,opt X 2. (20) (3) If x0,opt ∈int Ωt,X2
∩bd Ωs,X1 then the optimal decomposition for Lp,1,1(s, t; x;`p, X1, X2)–functional is given by x = x0,opt+x1,opt+0 and is such that x1,opt,∇F0 x0,opt = x1,opt X1 ∇F0 x0,opt X∗1 =s x1,opt X1. (21) (4) If x0,opt ∈bd Ωs,X1
∩bd Ωt,X2 then the optimal decomposition for
Lp,1,1(s, t; x;`p, X1, X2)–functional is given by x=x0,opt+x1,opt+x2,optsuch that ( x 1,opt,∇F0 x0,opt = x1,opt X 1 ∇F0 x0,opt X∗ 1 =s x1,opt X 1 x2,opt,∇F0 x0,opt = x2,opt X 2 ∇F0 x0,opt X∗ 2 =t x2,opt X 2.
Proof. (1) Suppose that the element x0,opt is such that ∇F0(x0,opt) X∗
1 <s and ∇F0(x0,opt)
X∗2 <t. Then, from Corollary 2, if we take the element y∗= x0,opt p−1 sgn x0,opt= ∇F0(x0,opt), we have that s x1,opt X 1 =x1,opt,∇F0 x0,opt ≤ x1,opt X 1 ∇F0 x0,opt X∗ 1 <s x1,opt X 1, and t x2,opt X2 =x2,opt,∇F0 x0,opt ≤ x2,opt X2 ∇F0 x0,opt X∗2 <t x2,opt X2, which implies that x1,opt = x2,opt = 0 since s, t > 0. Thus the optimal
decomposition is x=x+0+0.
(2) Suppose that the element x0,optis such that ∇F0(x0,opt) X∗
1
<s;k∇F0(x0)kX∗
2 = t. Then, from Corollary 2, if we take the element
y∗= x0,opt p−1 sgn x0,opt = ∇F0(x0,opt),
we have that s x1,opt X1 =x1,opt,∇F0 x0,opt ≤ x1,opt X1 ∇F0 x0,opt X1∗<s x1,opt X1, which implies that x1,opt=0 since s>0. Note that in this case x2,opt6=0.
Indeed if x2,opt =0 then x0,opt =x. Therefore from the characterization of optimal decomposition, there exist y∗ ∈ ∂F0(x) = ∇F0(x)such that F1∗(y∗) = F2∗(y∗) = 0 which implies, by definition of functions F1 and F2that y∗ ∈sBX1∗∩tBX2∗, i.e., x0,opt =x ∈int Ωs,X1∩Ωt,X2. Therefore
x2,optmust be different from zero in this case and
x2,opt,∇F0 x0,opt = x2,opt X 2 ∇F0 x0,opt X∗ 2 =t x2,opt X 2 Therefore x=x0,opt+0+x2,optis optimal decomposition for Lp,1,1(s, t; x;`p, X1, X2)–functional.
(3) Suppose that the element x0,opt is such that ∇F0(x2,opt) X2∗<t; ∇F0(x2,opt)
X1∗=s. Then, from Corollary 2, if we take the element
y∗= x0,opt p−1 sgn x0,opt= ∇F0(x0,opt), we have that t x2,opt X2 =x2,opt,∇F0 x0,opt ≤ x2,opt X2 ∇F0 x0,opt X∗2 <t x2,opt X 2,
which implies that x2,opt=0 since t>0. As above, in this case x1,opt6=0 and x1,opt,∇F0 x0,opt = x1,opt X1 ∇F0 x0,opt X∗1 =s x1,opt X1 Therefore x=x0,opt+x1,opt+0 is optimal decomposition for Lp,1,1(s, t; x;`p, X1, X2)–functional.
(4) Suppose that the element x0,opt is such that ∇F0(x0,opt) X∗ 1 = s; ∇F0(x0,opt) X∗ 2
= t. As above, in this case both
x1,optand x2,optmust be different from zero. Then if we take the element
y∗=|x0|p−1sgn(x0), we have that ( x 1,opt,∇F0 x0,opt = x1,opt X1 ∇F0 x0,opt X1∗=s x1,opt X1 x2,opt,∇F0 x0,opt = x2,opt X 2 ∇F0 x0,opt X∗ 2 =t x2,opt X 2, and conclude that x=x0,opt+x1,opt+x2,optis optimal for
Lp,1,1(s, t; x;`p, X1, X2)–functional.
2. Optimal decomposition for complex Banach couples
In the papaer [7] it was shown that for the couple of real Banach spaces, convex analysis can be used to characterize optimal decomposition for K–, L– and E– functionals of real interpolation. However real interpoaltion is used for complex spaces also. Therefore the natural question that arises is how to characterize optimal decomposition for complex spaces. In this situation it appears some difficulties because convex analysis is developped mainy for real spaces. Nevertheless the natural correspondance existing between the space of functionals on real Banach space and the space of functionals on a complex Banach space makes it possible to use the same approach. Below we will outline this correspondance.
Let EC be a complex Banach space and let ER be the same space with the
same norm but considered real Banach space in the sense that we restrict multiplication by scalars to real numbers only, instead of complex numbers. Let (EC)∗ (resp. (ER)∗) be the dual space to EC (resp. ER) consisting of complex (resp. real) valued linear and bounded functionals f : EC→C (resp.
g : ER →R). We intent to illustrate in this paragraph, that the spaces(EC)∗
and (ER)∗ are isometric in some sense. Since f ∈ (EC)∗ is complex valued, then f(x) = f1(x) +i f2(x), x ∈ EC, where f1and f2 are real valued. If for the moment we regard EC as ER, then since f is linear and bounded on EC, then f1and f2are real valued linear and bounded functionals defined on ER. Furhermore, since f is linear then
i f(x) =i[f1(x) +i f2(x)] = f(ix) = f1(ix) +i f2(ix).
By comparing real and imaginary parts, we conclude that f2(x) = −f1(ix)so that
f(x) = f1(x) −i f1(ix). (22)
Clearly the mapping f 7−→ f1 is linear with norm not greater than 1 from
(EC)∗ considered as a real structured space to the space(ER)∗. Hence for all g∈ (ER)∗, set:
f(x) =g(x) −ig(ix), for all x∈EC, (23) and define the opeartor T : (ER)∗−→ (EC)∗defined by (23)
Lemma 1. The operator T: (ER)∗−→ (EC)∗defined by (23) is a bijective isometry from(ER)∗to(EC)∗.
Proof. First let us show that ∀g ∈ (ER)∗, the function f = Tg is linear has the same norm as g. To prove linearity, let λ=α+iβ∈C. Then by definition
of f ,
Since g is linear on ER, then
f(λx) =αg(x) +βg(ix) −i[αg(ix) −βg(x)] = (α+iβ)g(x) −i(α+iβ)g(ix)
= (α+iβ) [g(x) −ig(ix)] = (α+iβ)f (x) =λ f(x)
So f(λx) = λ f(x),∀λ ∈ EC and therefore the function f = Tg is linear on
EC.
To show that T preserves the norm, we need to prove thatkgk(E
R)∗ =kfk(EC)∗. We have on one hand that,
kgk(E R)∗ = sup kxkEC≤1 |g(x)| = sup kxkEC≤1 |<e f(x)| ≤ sup kxkEC≤1 |f(x)| = kfk(E C)∗. On the other hand, using polar form (where θ(x)means that θ depends on x):
f(x) = |f (x)|eiθ(x) so that|f(x)| = f(x)e−iθ(x). So, from linearity of f , we have that
|f (x)| = fe−iθ(x)x
Since|f(x)|is real then fe−iθ(x)xis also real, which means, from definition of f , that |f(x)| = fe−iθ(x)x=ge−iθ(x)x. Therefore kfk(E C)∗= sup kxkEC≤1 |f(x)| = sup kxkEC≤1 ge−iθ(x)x≤ sup kxkEC≤1 g e−iθ(x)x ≤ sup kykEC≤1 |g(y)| = kgk(E R)∗. We conclude thatkgk(E
R)∗ = kfk(EC)∗. It is clear that if g1, g2 ∈ (ER)∗ with g16= g2then Tg1 = f1 6= f2=Tg2. Moreover the operator T is onto. Indeed, from (22), for each f ∈ (EC)∗, there exist g∈ (ER)∗ such that f =Tg. Hence
is a bijective isometry from(ER)∗to(EC)∗.
2.1. Some preliminaries and formalizations
Below we remind some well–known results from convex analysis that are needed for the proofs of main results for a complex Banach space E with the norm k·kE and dual space E∗. We will consider convex functions F : E →
R∪ {+∞}. The domain of the function F is a convex set dom F, defined by dom F={x∈E : F(x) < +∞}.
The function F is said to be proper if dom F6= ∅. If the epigraph of a function F, i.e. the set
epi F={(x, λ) ∈E×R | F(x) ≤λ},
is closed then the function F is called lower semicontinuous (l.s.c.). We will define the conjugate function as follows:
Definition 3. The conjugate function of F is the function F∗ : E∗ → R∪ {+∞}
defined by
F∗(y∗) =sup x∈E
{<ehy∗, xi −F(x)}. (24) Hence the subdifferentiability of a function will be defined as follows Definition 4. The function F : E → R∪ {+∞}is said to be subdifferentiable at x∈dom F if the following set named subdifferential of F at x,
∂F(x) ={y∗∈E∗: F(w) ≥F(x) + <ehy∗, w−xi, ∀w∈E}, (25) is nonempty.
Proposition 3.
y∗ ∈∂F(x)if and only if F(x) = <ehy∗, xi −F∗(y∗), ∀x ∈E, y∗∈E∗. (26)
Proof. Assume that y∗∈ ∂F(x)for some x∈dom F. Then for any w∈ E, we
have that F(w) ≥F(x) + <ehy∗, w−xi =F(x) + <ehy∗, wi − <ehy∗, xi (27) Or equivalently, <ehy∗, xi −F(x) ≥ <ehy∗, wi −F(w), ∀w∈E. It follows that <ehy∗, xi −F(x) ≥sup w∈E (<ehy∗, wi −F(w)) =F∗(y∗). (28) But, by definition we also have that
F∗(y∗) =sup w∈E
{<ehy∗, wi −F(w)} ≥ <ehy∗, xi −F(x). (29) Combining (28) and (29), we conclude that
F(x) = <ehy∗, xi −F∗(y∗). (30) Conversely, let us assume that F(x) = <ehy∗, xi −F∗(y∗). Then it follows that
<ehy∗, xi −F(x) =F∗(y∗) =sup w∈E
{<ehy∗, wi −F(w)} ≥ <ehy∗, wi −F(w), for all w∈E. Therefore
<ehy∗, xi −F(x) ≥ <ehy∗, wi −F(w), ∀w∈E Or equivalently
F(w) ≥F(x) + <ehy∗, w−xi, ∀w∈E, (31)
We will say that y∗is dual to x with respect to F if y∗∈∂F(x).
Proposition 4. Let ϕ0and ϕ1be functions from E with values inR∪ {+∞}. Then
(ϕ0⊕ϕ1)∗(y∗) =ϕ0∗(y∗) +ϕ∗1(y∗). (32) Proof. By definition (ϕ0⊕ϕ1)∗(y∗) =sup x∈E <ehy∗, xi − inf x0∈E [ϕ0(x0) +τx0ϕ1(x)] , where τx0ϕ1(x) =ϕ1(x−x0). (33) We have that (ϕ0⊕ϕ1)∗(y∗) =sup x∈E <ehy∗, xi − inf x0∈E [ϕ0(x0) +τx0ϕ1(x)] =sup x∈E sup x0∈E (<ehy∗, xi − [ϕ0(x0) +τx0ϕ1(x)]) =sup x0∈E −ϕ0(x0) +sup x∈E [<ehy∗, xi −τx0ϕ1(x)] ! =sup x0∈E −ϕ0(x0) + (τx0ϕ1) ∗ (y∗) But (τx0ϕ1) ∗ (y∗) =sup x∈E (<ehy∗, xi −ϕ1(x−x0)) =sup w∈E (<ehy∗, w+x0i −ϕ1(w)) = ϕ∗1(y∗) + <ehy∗, x0i. So, (ϕ0⊕ϕ1)∗(y∗) = sup x0∈E (−ϕ0(x0) +ϕ∗1(y∗) + <ehy∗, x0i) =ϕ∗1(y∗) +sup x0∈E (<ehy∗, x0i −ϕ0(x0)) = ϕ∗1(y∗) +ϕ∗0(y∗) We conclude that (ϕ0⊕ϕ1)∗(y∗) =ϕ0∗(y∗) +ϕ∗1(y∗). (34)
The following theorem still holds. It guarantees subdifferentiability over the interior of the domain of the function.
Theorem 5. Let E be a complex Banach space and let F : E −→ −∞,+∞ be
a convex and lower semicontinuous function. Then F is subdifferentiable over the interior of its domain.
The result which gives characterization of optimal decomposition can then be written below:
Lemma 2. Let E be a complex Banach space and let ϕ0 : E −→ R∪ {+∞} and
ϕ1 : E −→ R∪ {+∞} be convex proper functions. Suppose also that ϕ0⊕ϕ1 is subdifferentiable for a given element x ∈ dom(ϕ0⊕ϕ1). Then the decomposition
x = x0,opt+x1,opt is optimal for ϕ0⊕ϕ1if and only if there exists y∗ ∈ E∗ that is
dual to both x0,opt and x1,optwith respect to ϕ0and ϕ1respectively, i.e.
ϕ0 x0,opt= <ehy∗, x0,opti −ϕ∗0(y∗)
ϕ1 x1,opt
= <ehy∗, x1,opti −ϕ∗1(y∗). (35) Proof. Let us assume that the decomposition x=x0,opt+x1,opt is optimal for
ϕ0⊕ϕ1(x). It means that
(ϕ0⊕ϕ1) (x) = inf x=x0+x1
{ϕ0(x0) +ϕ1(x1)} = ϕ0 x0,opt+ϕ1 x1,opt . (36) Since ϕ0⊕ϕ1is subdifferentiable at x then there exists y∗∈E∗ such that
y∗∈∂(ϕ0⊕ϕ1) (x), This is equivalent to
(ϕ0⊕ϕ1) (x) = <ehy∗, xi − (ϕ0⊕ϕ1)∗(y∗). (37) From (36) and the formula for the conjugate of infimal convolution (see Propo-sition 4), we have that (37) is equivalent to
ϕ0 x0,opt+ϕ1 x1,opt= <ehy∗, xi − (ϕ0)∗(y∗) − (ϕ1)∗(y∗). Or equivalently,
ϕ0 x0,opt+ (ϕ0)∗(y∗) − <ey∗, x0,opt +ϕ1 x1,opt+ (ϕ1)∗(y∗) (38)
− <ey∗, x1,opt
=0 From the definition of the conjugate functions (see (24)) we have
ϕ0 x0,opt+ (ϕ0)∗(y∗) − <ey∗, x0,opt ≥0, and
ϕ1 x1,opt+ (ϕ1)∗(y∗) − <ey∗, x1,opt ≥0. Taking into account (38) we see that, in fact, we have the equalities
ϕ0 x0,opt= <ehy∗, x0,opti − (ϕ0)∗(y∗)
ϕ1 x1,opt= <ehy∗, x1,opti − (ϕ1)∗(y∗). (39)
Conversely, let us assume there exist y∗∈E∗and a decomposition x= ˜x0+˜x1 such that (7) is satisfied. Then
Since by Proposition 4, the conjugate of infimal convolution is the sum of conjugates, then
ϕ0(˜x0) +ϕ1(˜x1) = <ehy∗, xi − (ϕ0⊕ϕ1)∗(y∗). (40) By definition of infimal convolution (3), it follows that, in particular,
(ϕ0⊕ϕ1) (x) ≤ϕ0(˜x0) +ϕ1(˜x1). Then
(ϕ0⊕ϕ1) (x) ≤ <ehy∗, xi − (ϕ0⊕ϕ1)∗(y∗). Moreover, from the definition of a conjugate function we have
(ϕ0⊕ϕ1) (x) ≥ <ehy∗, xi − (ϕ0⊕ϕ1)∗(y∗). Combining this with the previous inequality we obtain
(ϕ0⊕ϕ1) (x) = <ehy∗, xi − (ϕ0⊕ϕ1)∗(y∗). From that and (40) we conclude that
(ϕ0⊕ϕ1) (x) =ϕ0(˜x0) +ϕ1(˜x1).
Therefore, the decomposition x= ˜x0+ ˜x1is optimal for(ϕ0⊕ϕ1)(x). 2.2. Characterization of optimal decomposition
2.2.1. Optimal decomposition for the L–functional
Below we assume that the complex Banach couple(X0, X1)is regular, i.e. X0∩X1is dense in both X0 and X1. Let x∈ X0+X1and let t>0 be a fixed parameter. We consider the following L–functional:
Lp0,p1(t, x; X0, X1) =x=xinf 0+x1 1 p0 kx0kXp00 + t p1 kx1kXp11 , (41)
where 1≤ p0<∞ and 1≤ p1<∞. We are interested in a characterization of the optimal decomposition for this Lp0,p1-functional, i.e. a characterization of
x0,opt∈ X0and x1,opt ∈X1such that x=x0,opt+x1,optand
Lp0,p1(t, x; X0, X1) = 1 p0 x0,opt p0 X0+ t p1 x1,opt p1 X1 . (42)
Theorem 6. Let 1 < p0 < ∞ and 1 < p1 < ∞. Then the decomposition x =
x0,opt+x1,optis optimal for the L-functional (13) if and only if there exists an element
y∗∈X∗0∩X1∗such that 1 p0 x0,opt p0 X0 = <ehy∗, x0,opti − 1 p00 ky∗k p00 X∗ 0 t p1 x1,opt p1 X1 = <ehy∗, x1,opti − t p01 yt∗ p01 X∗ 1. (43)
Proof. The L-functional (13) can be written as the infimal convolution Lp0,p1(t, x; X0, X1) = (ϕ0⊕ϕ1) (x),
where the functions ϕ0and ϕ1are both defined on the sum X0+X1as follows:
ϕ0(u) = ( 1 p0kuk p0 X0 if u∈X0 +∞ if u∈ (X0+X1) \X0. (44) and ϕ1(u) = ( t p1kuk p1 X1 if u∈X1 +∞ if u∈ (X0+X1) \X1. (45) The conjugate functions ϕ∗0of ϕ0and ϕ∗1of ϕ1are defined on X0∗∩X1∗and are given by ϕ∗0(z) = 1 p00kzk p00 X0∗, z∈X ∗ 0∩X∗1, (46) ϕ∗1(z) = t p01 z t p01 X1∗, z∈X ∗ 0∩X∗1. (47)
As the function Lp0,p1 is convex and lower semicontinuous on X0+X1 there-fore from Theorem 5 it follows that it is subdifferentiable on the interior of its domain, which is equal to X0+X1. So from Lemma 3 we obtain that the de-composition x=x0,opt+x1,opt is optimal for the Lp0,p1-functional if and only if there exists y∗ ∈ (X0+X1)∗=X∗0∩X1∗such that
ϕ0 x0,opt= <ehy∗, x0,opti −ϕ∗0(y∗)
ϕ1 x1,opt
= <ehy∗, x1,opti −ϕ∗1(y∗). (48) Taking into account the formulas (44) for ϕ0, ϕ1, the fact that x∈dom(ϕ0⊕ϕ1) and the formulas (46), (47) for their conjugates, we see that conditions (48) can be written as 1 p0 x0,opt p0 X0 = <ehy∗, x0,opti − 1 p00 ky∗k p00 X∗ 0 t p1 x1,opt p1 X1 = <ehy∗, x1,opti − t p01 yt∗ p01 X∗ 1.
Remark 5. The conditions (43) are equivalent toky∗kX∗
0 = x0,opt p0−1 X0 ,ky∗kX∗1 = t x1,opt p1−1 X1 and <ehy∗, x0,opti = ky∗kX∗ 0 · x0,opt X0, <ehy∗, x1,opti = ky∗kX1∗ · x1,opt X1. Let us now consider a special but very important case.
2.2.2. The case when 1< p0< +∞ and p1=1. Consider the L-functional
Lp0,1(t, x; X0, X1) =x=xinf 0+x1 1 p0 kx0kpX00+tkx1kX1
where 1< p0< +∞. Let ϕ0and ϕ1be functions defined on the sum X0+X1 as follows ϕ0(u) = ( 1 p0kuk p0 X0 if u∈X0 +∞ if u∈ (X0+X1) \X0. (49) and ϕ1(u) = tkukX 1 if u∈X1 +∞ if u∈ (X0+X1) \X1. (50) Then the L-functional can be written as the following infimal convolution
Lp0,1(t, x; X0, X1) = (ϕ0⊕ϕ1) (x).
Theorem 7. Let 1 < p0 < +∞. Then the decomposition x = x0,opt+x1,opt is optimal for the Lp0,1-functional if and only if there exists y∗ ∈ X
∗ 0∩X1∗ such that ky∗kX∗ 1 ≤t and 1 p0 x0,opt p0 X0 = <ehy∗, x0,opti − 1 p00ky∗k p00 X∗0 t x1,opt X 1 = <ehy∗, x1,opti. (51)
Proof. As the function Lp0,1is convex and lower semicontinuous on X0+X1 therefore from Theorem 5 we obtain that it is subdifferentiable on the interior of its domain, which is equal to X0+X1. Then by Lemma 3, the decomposition
x =x0,opt+x1,opt is optimal for the Lp0,1-functional if and only if there exists
y∗∈ (X0+X1)∗=X0∗∩X1∗such that ϕ0 x0,opt= <ehy∗, x0,opti −ϕ∗0(y∗) ϕ1 x1,,opt = <ehy∗, x1,opti −ϕ∗1(y∗). (52) Taking into account the formulas (49) and (50) for ϕ0, ϕ1, the fact that x ∈ dom(ϕ0⊕ϕ1) and the formulas for their conjugates, we see that the con-ditions (52) are the same as the concon-ditions (51). Moreover, the condition
ky∗kX∗
1 ≤t follows from ϕ
∗
1(y∗) <∞.
Remark 6. Similarly to Remark2, we can demonstrate that the condition p10 x0,opt p0 X0 = <ehy∗, x0,opti −p10 0 ky∗kp 0 0 X0∗of Theorem 7 is equivalent toky∗kX∗ 0 = x0,opt p0−1 X0 and <ehy∗, x0,opti = ky∗kX∗ 0 · x0,opt X0.
Remark 7. A proof similar to the proof of Theorem7 shows that the decomposition
x = x0,opt+x1,opt is optimal for the K-functional if and only if there exists y∗ ∈
X0∗∩X1∗such thatky∗kX∗ 0 ≤1,ky∗kX∗1 ≤t and ( x0,opt X0 = <ehy∗, x0,opti t x1,opt X1 = <ehy∗, x1,opti. We would like to note that for the E-functional
E(t, x; X0, X1) = inf kx1kX1≤t
kx−x1kX0 the result is similar, but the proof is a bit different.
Proposition 5. The decomposition x = x0,opt+x1,opt, where x1,opt
X1 ≤ t, is optimal for the E-functional if and only if there exists y∗ ∈ X∗0∩X∗1 such that
ky∗kX∗ 0 ≤1 and ( x0,opt X 0 = <ehy∗, x0,opti <ehy∗, x1,opti =tky∗kX∗ 1.
3. ROF model on the graph and Split Bregman algorithms
A typical example of variational models for image and signal denoising based on the minimization of energy functionals, is known in image process-ing as the total variation (TV) regularization technique due to
Rudin-Osher-Fatemi (ROF) (see [12]) on a 2-dimensional domain Ω in R2 (for example
Ω = [0, 1]2). It suggests to take as an approximation to the original image f∗ the function fopt,t ∈ BV(Ω), which is the exact minimizer for the L2,1– functional for the couple L2, BV, namely
L2,1 t, fob; L2, BV = inf g∈BV 1 2kfob−gk 2 L2+tkgkBV , (53)
for some t>0, i.e., fopt,t∈BV is such that L2,1 t, fob; L2, BV = 1 2 fob−fopt,t 2 L2+t fopt,t BV. (54)
In [8, 9] we consider the discrete analogue of the model (53) on the graph and propose an algorithm to construct the element fopt,t. In the literature, there are many numerical methods for the ROF model (see for example [2]) and in our own point of view, Split Bregman iteration algorithm introduced in [6] is probably one of the best of such methods. The "Split Bregman" method can solve a very broad class of L1–regularized problems of the form
min
g H(g) +kφ(g)kL1, (55)
Remark 8. This category includes many important problems in engineering, com-puter science and imaging science.
It is clear that the ROF denoising problem can be posed as L1–regularized optimization problem: min g µ 2kf−gk 2 L2 +kgkBV. (56)
where the BV seminorm for the differentiable function g is equal tokgkBV = R |∇g|.
We will compare our algorithm with the Split Bregman on some test ex-amples. In our algorithm, the parameter used is denoted by s and is equal to s=N×t for an image of size N by N (see [8] or [9] for more details). We find out that both algorithms are comparable in quality of reconstructed images and that on average our algorithm takes fewer number of iterations to reach the same tolerance.
Original image Observed noisy image
Split Bregman ROF model on the graph
Figure 1: Lenna, the original is a 512×512 image with intensity values ranging from 0 to 255. Top left: Original image f∗; Top right: Noisy image fob=f∗+ε×randn(512,512), i.e image with
Gaussian additive noise of standard deviation ε=25. Bottom left: Split Bregman reconstruction
Residual from Split Bregman Residual from ROF graph
Split Bregman ROF model on the graph
Figure 2: Top: Residual image, i.e., the difference between the observed image and the recon-structed image fob−fopt,t. Top left: Residual from Split Bregman reconstruction. Top right:
Residual from ROF model on the graph reconstruction. Bottom: The white shows, matrix compo-nentwise, parts of the recontructed image fopt,twhere
f∗i,j−fopt,ti,j
≥15, i, j=1, 2, . . . , N. Left: in Split Bregman reconstruction. Right: in ROF model on the graph reconstruction.
0 20 40 60 80 100 120 140 −14 −12 −10 −8 −6 −4 −2 0 Iterations log(normalized error)
Error vs. iteration number
ROF on Graph Split Bregman
Figure 3: Error vs. iteration number for the split Bregman ROF minimization algorithm
and for our ROF model on the graph algorithm. The error at iteration k is defined as
log uk−u∗ `2/ku∗k`2
where uk is the approximation at iteration k, and u∗ is the
ex-act solution. Convergence results are for the test image "Lena" in Figure 1 with Gaussian
noise (e = 22.05). We see that for example at iteration 20 our algorithms achieves an
nor-malized error log
u20−u∗
`2/ku∗k`2
= −10 while the same error in Split Bregman is
log u20−u∗ `2/ku∗k`2 = −8. i.e., u20−u∗ `2/ku∗k`2approximately equal to 4.42×10−5
Original image Observed noisy image
Split Bregman ROF model on the graph
Figure 4: Geometric features with a cusp, the original is a 512×512 image with regions of
intensity values 150 and 100. Top left: Original image f∗; Top right: Noisy image fob= f∗+ε×
randn(512,512), i.e image with Gaussian additive noise of standard deviation ε=15.3. Bottom
left: Split Bregman reconstruction using µ= 18.068. Bottom right: ROF model on the graph
Residual from Split Bregman Residual from ROF graph
Split Bregman ROF model on the graph
Figure 5: Geometric features with a cusp. Top: Residual image, i.e., the difference between the observed image and the reconstructed image fob−fopt,t. Top left: Residual from Split Bregman
reconstruction. Top right: Residual from ROF model on the graph reconstruction. Bottom: The white shows, matrix componentwise, parts of the recontructed image fopt,twhere
f∗i,j−fopt,ti,j
≥ 15, i, j=1, 2, . . . , N. Left: in Split Bregman reconstruction. Right: in ROF model on the graph reconstruction.
0 20 40 60 80 100 120 140 −14 −12 −10 −8 −6 −4 −2 0 Iterations log(normalized error)
Error vs. iteration number
ROF on Graph Split Bregman
Figure 6: Error vs. iteration number for the split Bregman ROF minimization algorithm
and for our ROF model on the graph algorithm. The error at iteration k is defined as
log
uk−u∗
`2/ku∗k`2 where ukis the approximation at iteration k, and u∗is the exact
so-lution. Convergence results are for the test image "an artificial geometric image" in Figure 4 with Gaussian noise (ε=15.30). We see that for example at iteration 20 our algorithms achieves
an normalized error log
u20−u∗
`2/ku∗k`2
= −9.6 while the same error in Split Bregman is
log u20−u∗ `2/ku∗k`2 = −8.3. i.e., u20−u∗ `2/ku∗k`2approximately equal to 6.25×10−5
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