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Sets
Stéphane Chrétien, Pascal Bondon
To cite this version:
Stéphane Chrétien, Pascal Bondon. Projection Methods for Uniformly Convex Expandable Sets.
Mathematics , MDPI, 2020, �10.3390/math8071108�. �hal-02896463�
Article
Projection Methods for Uniformly Convex Expandable Sets
Stéphane Chrétien
1,2,3,∗and Pascal Bondon
41
Laboratoire ERIC, Université Lyon 2, 69500 Bron, France
2
The Alan Turing Institute, London NW1 2DB, UK
3
Data Science Division, The National Physical Laboratory, Teddington TW11 0LW, UK
4
Laboratoire des Signaux et Systèmes, CentraleSupélec, CNRS, Université Paris-Saclay, 91190 Gif-sur-Yvette, France; pascal.bondon@l2s.centralesupelec.fr
* Correspondence: stephane.chretien@univ-lyon2.fr
Received: 21 February 2020; Accepted: 22 June 2020; Published: 6 July 2020
Abstract: Many problems in medical image reconstruction and machine learning can be formulated as nonconvex set theoretic feasibility problems. Among efficient methods that can be put to work in practice, successive projection algorithms have received a lot of attention in the case of convex constraint sets. In the present work, we provide a theoretical study of a general projection method in the case where the constraint sets are nonconvex and satisfy some other structural properties.
We apply our algorithm to image recovery in magnetic resonance imaging (MRI) and to a signal denoising in the spirit of Cadzow’s method.
Keywords: cyclic projections; nonconvex sets; uniformly convex sets; strong convergence
1. Introduction
1.1. Background and Goal of the Paper
Many problems in applied mathematics, engineering, statistics and machine learning can be reduced to finding a point in the intersection of some subsets of a real separable Hilbert space H . In mathematical terms, let ( S
i)
i∈Ibe a finite family of proximinal subsets (i.e., sets S such that any x ∈ H admits a closest point in S) [1] of H with a non-empty intersection, we address the problem of finding a point in the intersection of the sets ( S
i)
i∈Iusing a successive projection method.
When the sets are closed and convex, the problem is known as the convex feasibility problem, and is the subject of extensive literature; see [2–5] and references therein. In this paper, we take one step further into the scarcely investigated topic of finding a common point in the intersection of nonconvex sets. Convex feasibility problems have been applied to an extremely wide range of topics in systems analysis and control, signal and image processing. Important examples are: model order reduction [6], controller design [7], tensor analysis [8], image recovery [9], electrical capacitance tomography [10], MRI [11,12], and stabilisation of quantum systems and application to quantum computation [13,14].
Extension of this problem to the nonconvex setting also has many applications, related to sparse estimation and more specifically low rank constraints, such as in control theory [15], signal denoising [16], phase retrieval [17], structured matrix estimation [18,19] and has great potential impact on the design of scalable algorithm in many machine learning problems such as Deep Neural Networks [20]. Studies of projection methods in the nonconvex settings have been quite scarce in the literature [21–23] and a lot of work still remains to be done in order to understand the behavior of such methods for these difficult problems.
Mathematics2020,8, 1108; doi:10.3390/math8071108 www.mdpi.com/journal/mathematics
In the present paper, our goal is to investigate how the results in [22] can be improved in such a way that strong convergence can be obtained. The results proved in [22] make the assumption that the sets involved in the feasibility problem can be written as a possibly not countable union of closed convex sets, and one of the sets is boundedly compact. Our study will be based on the assumption that the convex sets in the family are uniformly convex, which allows to remove the boundedly compactness assumption. As will be seen in Section 4, uniform convexity can be obtained by simply modifying the algorithm even in the case where the sets S
iare not expandable into a family of uniformly convex sets.
1.2. Preliminary on Projections and Expansion into Convex Sets
The notation Card will denote the cardinality of a set. H will denote a real separable Hilbert space with scalar product h· | ·i , norm | · | , and distance d. Let S be a subset of H . S denotes the closure of S.
S is proximinal if each point in H has at least one projection onto S. If S is proximinal, then S is closed.
When S is proximinal, P
Sis the projection point-to-set mapping onto S defined by (∀ x ∈ H) P
S( x ) = { y ∈ S | | x − y | = inf
z∈S| x − z |} .
p
S( x ) denotes an arbitrary element of P
S( x ) and d ( x, S ) = | x − p
S( x )| . B ( x, ρ ) denotes the closed ball of center x and radius ρ. In the case of a projection onto a convex set C, Kolmogorov’s criterion is:
x
0− P
C( x ) , x − P
C( x ) ≤ 0 (1) for all x
0∈ C. Our work is a follow-on project to the work in [22] which introduced the notion of convex expandable sets in order to tackle certain feasibility problems involving rank-constrained matrices as described in [23]. We recall what it means for a nonconvex set to be expandable into a family of convex sets.
Definition 1. A subset S of H is said to be expandable into a family of convex sets when it is the union of non-trivial, i.e., not reduced to a single point, convex subsets C
jof H , i.e.,
S = S
j∈JC
j(2)
where J is a possibly uncountable index set. Any family ( C
j)
j∈Jsatisfying (2) is called a convex expansion of S.
Remark 1. Uncountable unions often appear in practice. An important example is the set of matrices of rank r in R
n×mfor 0 < r < min { n, m } . This set is the union of the rays passing through the null matrix and any matrix of rank r different from zero. This constraint often appears in signal processing problems such as signal denoising [16] and more generally, structured matrix estimation problems [19].
Uniform convexity is an important concept, which allows to prove many strong convergence results in the framework of successive projection methods ([4], Section 5.3).
Definition 2. A convex subset C of H is uniformly convex with respect to a positive-valued nondecreasing function b ( t ) if
∀( x, y ) ∈ C
2, B (( x + y ) /2, b (| x − y |)) ⊂ C.
Uniform convexity will be instrumental in our analysis. We define the following condition which is stronger than the condition of being expandable into a family of convex sets.
Definition 3. A subset S of H is said to be expandable in uniformly convex sets when the sets C
jin (2) are
moreover uniformly convex with the same function b (·) . Any family ( C
j)
j∈Jof uniformly convex sets satisfying
(2) is called a uniformly convex expansion of S.
1.3. The Projection Algorithm
For any point x in H , for the sake of simplicity, p
Si( x ) will be noted by p
i( x ) . We will assume throughout that the family { S
i}
i∈Iis finite.
Definition 4. (Method of Projection onto Uniformly Convex Expandable Sets)
Given a point x
0in H , two real numbers α and λ in ( 0, 1 ] , a nonnegative integer M and a sequence ( I
n) of subsets of I satisfying the following conditions
C
1. (∀ n ∈ N ) I
n⊂ I, I = S
n+Mj=nI
jand Card ( I
n) > 1 C
2. (∀ n ∈ N ) (∀ i ∈ I
n) α ≤ α
i,nand ∑
j∈Inα
j,n= 1 the projection-like method considered in this paper is iteratively defined by
(∀ n ∈ N )
1.
– If ∃ j ∈ I
nsuch that p
j( x
n) ∈ T
i∈InS
iThen
– If I
n6= I set I
n= I and go to 1
– If I
n= I set λ
n= 1, α
j,n= 1 and go to 2 – Else go to 2
2. x
n+1= x
n+ λ
n∑
i∈Inα
i,n( p
i( x
n) − x
n)
(3)
with λ
nsatisfying
C
3. (∀ n ∈ N ) λ ≤ λ
n≤ µ
nwhere µ
n=
∑i∈Inαi,n|xn−pi(xn)|2
|xn−∑i∈Inαi,npi(xn)|2
if x
n6∈ T
i∈InS
i1 if x
n∈ T
i∈InS
i.
Remark 2. Using the assumption Card ( I
n) > 1, the coefficients α
i,n, i ∈ I
n, can be chosen in such a way that the denominator in the definition of µ
nis not equal to zero. Note that one can relax the constraint Card ( I
n) > 1 in the case where µ
nis well defined at every iteration, thus allowing to recover the cyclic projection method.
The numbers α and λ ensure that the steps taken at each iteration are sufficiently large as compared to the distance of the iterations to their projections on each S
i, i ∈ I. The use of the integer M ensures that each S
iis involved at least every M iterations. The “If” condition in Step 1 of the algorithm ensures that x
nwill move at each iteration by enforcing that it does not yet belong to all the selected sets S
i, i ∈ I
n. Our method is an extension of Ottavy’s method [4] to the case of nonconvex sets. The idea of using a potentially variable index set I
nat each iteration allows to recover several variants, such as cyclic projections [2,21,22,24] or parallel projections [2,23,24]. Notice that, due to finiteness of the cardinality of I, finding an index j such that p
j( x
n) ∈ T
i∈InS
iis easy if such an index exists.
1.4. Our Contributions
The main contributions of our work are a proof that strong convergence holds in the proposed
setting, and a new constructive modification of the projection method in order to accommodate the case
of non-necessarily uniformly convex expandable sets. Based on our findings, we will then weaken our
assumptions and provide a new algorithm which does not need uniform convexity, while preserving
strong convergence to a feasible solution. Applications to an infinite dimensional MRI problem and to
low rank matrix estimation for signal denoising are presented in Section 5.
2. Ottavy’s Framework
2.1. Successive Projection Point-to-Set Mapping
Let α and λ be two real numbers chosen in ( 0, 1 ) . Define the point-to-set mapping D : x ∈ H → D ( x ) = { x ˜ ∈ H ; ˜ x = x + λ ( x )( x ¯ − x )}
where ¯ x is a weighted average,
x ¯ = ∑
i∈I(x)
α
i( x ) p
i( x ) I ( x ) is a subset of selected constraints such that
I ( x ) ⊂ I and I ( x ) 6= ∅ α
i( x ) , i ∈ I is a set of normalized nonvanishing weightings such that
α
i( x ) ∈ [ 0, 1 ] , ∑
i∈I(x)
α
i( x ) = 1, and p
i( x ) 6= x ⇒ α
i( x ) ≥ α
λ ( x ) is the relaxation parameter such that
λ ( x ) ∈ [ λ, + ∞ ) and x 6= x ¯ ⇒ λ ( x ) ∈ [ λ, µ ( x ¯ )]
with
µ ( x ¯ ) =
∑
α∈I(x)
α
i( x ) | x − p
i( x )|
2
/ | x − x ¯ |
2. 2.2. Ottavy’s Lemma
In [4], Ottavy proved the very useful lemma.
Lemma 1. For any x ∈ H , ˜ x ∈ D ( x ) , and z ∈ ∩
i∈IS
i, the following results hold:
( i ) | x ˜ − x |
2≤ β
| x − z |
2− | x ˜ − z |
2( ii ) 2 h x − z | x − x ˜ i ≤
1 + 2
λ
| x − z |
2− | x ˜ − z |
2( iii ) max
i∈I(x)
| p
i( x ) − x |
2≤ ( 2 + λ ) 2αλ
2| x − z |
2− | x ˜ − z |
2.
3. A Strong Convergence Result
Let ( x
n)
n∈Nbe the sequence of iterates of the projection method defined by (3). In the present section, we show strong convergence of this sequence to a point in ∩
i∈IS
i.
For every n in N and for each i ∈ I, define
T
i,n= S { C ⊂ S
i| C is convex and p
i( x
n) ∈ C } T
n= T
i∈InT
i,n.
When the (finite) family ( S
i)
i∈Iis expandable into a family of convex sets, T
i,n6= ∅ for every n in
N and for each i ∈ I.
Lemma 2. If the following conditions are satisfied C
4. (∀ n ∈ N ) T
n6= ∅
C
5. The finite family ( S
i)
i∈Iis expandable in convex sets then, for every n in N and for every z
nin T
n, we have
( i ) | x
n+1− x
n|
2≤ 2 λ
| x
n− z
n|
2− | x
n+1− z
n|
2( ii ) 2 h x
n− z
n| x
n− x
n+1i ≤ ( 1 + 2
λ ) | x
n− z
n|
2− | x
n+1− z
n|
2( iii ) sup
i∈In
| p
i( x
n) − x
n|
2≤ ( 2 + λ ) 2αλ
2| x
n− z
n|
2− | x
n+1− z
n|
2.
Proof. Fix n in N and z
nin T
n. According to C
5, each point p
i( x
n) belongs to a set C
i,jn⊂ S
i, a convex set in the uniformly convex expansion of S
iindexed by j
n. Hence p
i( x
n) is also the projection of x
nonto C
i,jn, and z
n∈ C
i,jn. Replacing respectively x, ˜ x and z by x
n, x
n+1and z
nin Ottavy’s Lemma (Lemma 1, or [4, Lemma 7.1]), we obtain the desired results.
Lemma 3. If the conditions C
4, C
5, and
C
6. (∃( z
n)
n∈N) such that z
n∈ T
nfor all n in N and ∑
n∈N| z
n+1− z
n| < + ∞ are satisfied, then
( i ) The sequences (| x
n− z
n|)
n∈Nand (| x
n+1− z
n|)
n∈Nare bounded.
( ii ) The series ∑
n∈N(| x
n− z
n|
2− | x
n+1− z
n|
2) is convergent.
( iii ) For each i in I, the sequence (| p
i( x
n) − x
n|)
n∈Nconverges to zero.
( iv ) The sequence (| x
n− z
n|)
n∈Nis convergent.
( v ) The sequence ( z
n)
n∈Nconverges strongly to a point in T
i∈IS
i.
Proof. (i) Let ( z
n)
n∈Nbe a sequence satisfying C
6, and let k in N . We deduce from Lemma 2(i) that
| x
k+1− z
k| ≤ | x
k− z
k| . Then we have
| x
k+1− z
k+1| ≤ | x
k+1− z
k| + | z
k+1− z
k| ≤ | x
k− z
k| + | z
k+1− z
k| . Therefore
| x
n+1− z
n+1| ≤ | x
0− z
0| + ∑
nk=0| z
k+1− z
k|
and then C
6ensures that the sequences (| x
n− z
n|)
n∈Nand (| x
n+1− z
n|)
n∈Nare bounded.
(ii) Now, the cosine law, followed by the Cauchy-Schwarz inequality give
| x
k+1− z
k+1|
2= | x
k+1− z
k|
2+ 2 h x
k+1− z
k| z
k− z
k+1i + | z
k− z
k+1|
2≤ | x
k+1− z
k|
2+ 2 | x
k+1− z
k| | z
k+1− z
k| + | z
k+1− z
k|
2. (4) Using Lemma 2(i), we then obtain that
0 ≤ | x
k− z
k|
2− | x
k+1− z
k|
2≤ | x
k− z
k|
2− | x
k+1− z
k+1|
2+ 2 | x
k+1− z
k| | z
k+1− z
k|
+ | z
k+1− z
k|
2.
Thus
0 ≤ ∑
nk=0| x
k− z
k|
2− | x
k+1− z
k|
2≤ | x
0− z
0|
2+ 2 ∑
nk=0| x
k+1− z
k| | z
k+1− z
k|
+ ∑
nk=0| z
k+1− z
k|
2. Since the sequence (| x
n+1− z
n|)
n∈Nis bounded and the series ∑
n∈N| z
n+1− z
n| is convergent, the result (i) follows at once.
(iii) Lemma 3(i) and Lemma 2(iii), imply that lim
n→∞sup
i∈In| p
i( x
n) − x
n|
2= 0. Thus for any k ∈ [ 0, M ] ,
n→
lim
∞sup
i∈In+k
| p
i( x
n) − x
n| = 0.
Then we deduce from C
1that lim
n→∞sup
i∈I| p
i( x
n) − x
n| = 0, which completes the proof of (iii).
(iv) Squaring the triangle inequality gives
| x
k+1− z
k|
2− | x
k+1− z
k+1|
2≤ 2 | x
k+1− z
k+1| | z
k+1− z
k| + | z
k+1− z
k|
2≤ 2 | x
k+1− z
k| | z
k+1− z
k| + 3 | z
k+1− z
k|
2. Using (4), we deduce that
| x
k+1− z
k+1|
2− | x
k+1− z
k|
2≤ 2 | x
k+1− z
k| | z
k+1− z
k| + 3 | z
k+1− z
k|
2.
Since the sequence (| x
n+1− z
n|)
n∈Nis bounded, and ∑
n∈N| z
n+1− z
n| is convergent, we deduce from the last inequality that ∑
n∈N(| x
n+1− z
n+1|
2− | x
n+1− z
n|
2) is convergent. Then using Lemma 3(i), we obtain that the series ∑
n∈N(| x
n+1− z
n+1|
2− | x
n− z
n|
2) is convergent, which is equivalent to the convergence of the sequence (| x
n− z
n|)
n∈N.
(v) Since, by C
6, ( z
n)
n∈Nis a Cauchy sequence, it is strongly convergent to a point z in the Hilbert space H . For every n in N , z
n∈ T
n= T
i∈InT
i,n⊂ T
i∈InS
i. Fix i in I. Due to condition C
1, there exists a subsequence ( z
σ(n))
n∈Nof ( z
n)
n∈Nsatisfying z
σ(n)∈ S
ifor all n ∈ N . Therefore, z ∈ S
i. Since this assertion is true for all i in I, and each S
iis closed, we obtain that z ∈ T
i∈IS
i.
Remark 3. In the convex case, Lemma 3(iii) is known in the simpler case where ( z
n)
n∈Nis chosen to be a constant sequence in T
i∈IS
iand is a consequence of the Fejér monotonicity [5].
We now state the main result of this section, which is strong convergence under the assumption that the sets S
i, i ∈ I, are expandable in uniformly convex sets.
Theorem 1. If the conditions C
4, C
6, and C
7.
The finite family ( S
i)
i∈Iis expandable in uniformly convex sets with respect to a function b are satisfied, then the sequence ( x
n)
n∈Nconverges strongly to a point in T
i∈IS
i.
Proof. Notice that C
7implies C
5. According to C
5, each point p
i( x
n) belongs to a convex set C
i,jn⊂ S
iin the uniformly convex expansion of S
i, indexed by j
n. Hence p
i( x
n) is also the projection of x
nonto C
i,jn, and z
n∈ C
i,jn. Let ( z
n)
n∈Nbe a sequence satisfying C
6. Fix n in N , and i in I. Define
H ( x
n, p
i( x
n)) = { x ∈ H | h x − p
i( x
n) | p
i( x
n) − x
ni ≥ 0 } . According to C
7, p
i( x
n) and z
nbelong to a uniformly convex set C
i,jn⊂ S
i. Thus
B (( z
n+ p
i( x
n)) /2, ρ
i,n) ⊂ C
i,jn,
where ρ
i,n= b (| z
n− p
i( x
n)|) . Since p
i( x
n) is also the projection of x
nonto C
i,jn, it results from the Kolmogorov criterion (1) that
C
i,jn⊂ H ( x
n, p
i( x
n)) . Then
B (( z
n+ p
i( x
n)) /2, ρ
i,n) ⊂ C
i,jn⊂ H ( x
n, p
i( x
n)) . (5) Now, consider the translation T : x 7→ y = x − ( p
i( x
n) − z
n) /2. Clearly,
x ∈ B (( z
n+ p
i( x
n)) /2, ρ
i,n) if and only if y ∈ B ( z
n, ρ
i,n) . (6) On the other hand, using z
n∈ C
i,jn, we get that x ∈ H ( x
n, p
i( x
n)) implies y ∈ H ( x
n, p
i( x
n)) . Indeed,
h y − p
i( x
n) | p
i( x
n) − x
ni= h x − ( p
i( x
n) − z
n) /2 − p
i( x
n) | p
i( x
n) − x
ni
= h x − p
i( x
n) | p
i( x
n) − x
ni − 1
2 h p
i( x
n) − z
n| p
i( x
n) − x
ni and since z
n∈ C
i,jn, Kolmogorov’s criterion gives
1
2 h p
i( x
n) − z
n| p
i( x
n) − x
ni ≤ 0, which implies
h y − p
i( x
n) | p
i( x
n) − x
ni= h x − ( p
i( x
n) − z
n) /2 − p
i( x
n) | p
i( x
n) − x
ni
≥ h x − p
i( x
n) | p
i( x
n) − x
ni and, after recalling that x ∈ H ( x
n, p
i( x
n)) ,
h y − p
i( x
n) | p
i( x
n) − x
ni ≥ 0
which implies that y ∈ H ( x
n, p
i( x
n)) as desired. Hence (5), together with (6) imply B ( z
n, ρ
i,n) ⊂ H ( x
n, p
i( x
n)) .
Now, assume x
n6∈ S
i, and define
y
n= z
n+ ( p
i( x
n) − x
n) / | p
i( x
n) − x
n| u
n= z
n+ ρ
i,n( z
n− y
n) .
One easily checks that | u
n− z
n| = ρ
i,n, i.e., u
n∈ B ( z
n, ρ
i,n) . Thus, using (2.2), we get u
n∈ H ( x
n, p
i( x
n)) , i.e., h u
n− p
i( x
n) | p
i( x
n) − x
ni ≥ 0. Therefore
h z
n− p
i( x
n) | p
i( x
n) − x
ni ≥ h z
n− u
n| p
i( x
n) − x
ni = ρ
i,n| p
i( x
n) − x
n| . (7) Since this inequality is obviously satisfied when x
n∈ S
i, it holds whether x
n6∈ S
ior not. Now, let
ρ
n= min
i∈In
ρ
i,n, and
ρ = inf
n∈N
ρ
n. The case: ρ = 0.
In the case where ρ = 0, we have
• either there exist n
0in N and i
0in I
n0, such that ρ
i0,n0= 0,
• or lim inf
n→∞ρ
n= 0.
In the first case, since b (·) vanishes only at zero, we have z
n0= p
i0( x
n0) and p
i0( x
n0) ∈ T
n0⊂ T
i∈In0
S
i. According to (3), p
i0( x
n0) ∈ T
i∈IS
i, and for all n > n
0, x
n= p
i0( x
n0) . Therefore, the sequence ( x
n)
n∈Nhas converged to a solution in a finite number of steps. In the second case, there exists a subsequence ( ρ
σ(n))
n∈Nwhich converges to zero. Fix i in I. Since b (·) is nondecreasing,
ρ
σ(n)≥ min
j∈Iσ(n)
b
| z
σ(n)− p
i( x
σ(n))| − | p
i( x
σ(n)) − p
j( x
σ(n))|
. (8)
According to Lemma 3(ii) and (iii), (| p
i( x
n) − p
j( x
n)|)
n∈Nconverges to zero for all j in I, and (| z
n− p
i( x
n)|)
n∈Nconverges to a limit c independent of i. Hence, for all e > 0 there exists N in N such that for all n ≥ N, | z
σ(n)− p
i( x
σ(n))| ≥ c − e/2 and | p
i( x
n) − p
j( x
n)| ≤ e/2 for all j in I. Thus, for all n ≥ N, and for all j in I, | z
σ(n)− p
i( x
σ(n))| − | p
i( x
σ(n)) − p
j( x
σ(n))| ≥ c − e.
Assume c > 0, and take e = c/2. Then, since b (·) is nondecreasing and vanishes only at zero, we deduce from (8) that for all n ≥ N, ρ
σ(n)≥ b ( c/2 ) > 0, which contradicts lim
n→∞ρ
σ(n)= 0.
Hence, c = 0, and lim
n→∞| z
n− p
i( x
n)| = 0 for all i in I. Using Lemma 3(ii), we deduce that lim
n→∞| x
n− z
n| = 0, and it results from Lemma 3(iv) that ( x
n)
n∈Nconverges strongly to a point in T
i∈I
S
i.
The case ρ 6= 0.
In the case ρ 6= 0, we deduce from (7) that
h z
n− p
i( x
n) | p
i( x
n) − x
ni ≥ ρ | p
i( x
n) − x
n| and since h p
i( x
n) − x
n| p
i( x
n) − x
ni = | p
i( x
n) − x
n|
2≥ 0,
h z
n− x
n| p
i( x
n) − x
ni ≥ ρ | p
i( x
n) − x
n| . (9) Therefore, combining the definition of x
n+1and (9), we have
λ
n∑
i∈In
α
i,n| p
i( x
n) − x
n| ≤ λ
nρ ∑
i∈In
α
i,nh z
n− x
n, p
i( x
n) − x
ni
= 1 ρ
*
z
n− x
n, λ
n∑
i∈In
α
i,n( p
i( x
n) − x
n) +
= 1
ρ h z
n− x
n, x
n+1− x
ni .
Using Lemma 2(ii) and Lemma 3(i), we deduce that the series ∑
n∈N| x
n+1− x
n| is convergent.
Then ( x
n)
n∈Nis a Cauchy sequence, and is therefore strongly convergent to a point x
∗in H . Using Lemma 3(ii), we deduce that ( p
i( x
n))
n∈Nis also strongly convergent to x
∗for any i in I.
Since each set S
iis closed, we immediately conclude that x
∗∈ T
i∈IS
i, which completes the proof.
4. Projections onto Stepwise Generated Uniformly Convex Sets
In this section, we show how the results of Section 3 may be used to define a strongly convergent
cyclic projection algorithm in the case where the sets S
iare only expandable in convex sets, but not into
uniformly convex sets. To our knowledge, this method is new, even in the convex case. The underlying
idea of the algorithm is as follows. First, note that the condition C
4is realistic for the type of nonconvex
sets considered in this paper, and may be interpreted as a strong consistency condition. On the other
hand, in many cases, a trivial sequence ( a
n)
n∈Nwhere a
n∈ T
nfor all n in N is already known, as in
the applications presented in Section 5. Using this sequence, we define a projection method which
converges strongly to a point in T
i∈IS
i. For every n in N , i in I, and a
nin T
n, B
i,nwill denote the closed ellipsoid with main axis [ a
n, p
i( x
n)] , with center 1/2 ( a
n+ p
i( x
n)) , which is rotationally invariant around this axis and with maximal radius equal to | a
n− p
i( x
n)| /2 and with minimal radius equal to γ | a
n− p
i( x
n)| /2 for some γ ∈ ( 0, 1 ) . We denote by p
0i( x
n) the projection of x
nonto B
i,n; see Figure 1.
Definition 5. (Method of Double Projection onto Convex Expandable Sets)
Assume C
4is satisfied. Given a point x
0in H , two real number α and λ in ( 0, 1 ] , a nonnegative integer M, a sequence ( I
n) of subsets of I and a sequence ( a
n)
n∈Nwhere a
n∈ T
nfor all n in N , the projection method onto stepwise generated uniformly convex sets is iteratively defined by
(∀ n ∈ N )
1. (∀ i ∈ I
n) p
0i( x
n) = p
Bi,n( x
n)
2.
– If ∃ j ∈ I
nsuch that p
0j( x
n) ∈ T
i∈InS
iThen
– If I
n6= I set I
n= I and go to 2
– If I
n= I set λ
n= 1, α
j,n= 1 and go to 3 – Else go to 3
3. x
n+1= x
n+ λ
n∑
i∈Inα
i,n( p
0i( x
n) − x
n) under the conditions C
1, C
2, and
C
03. (∀ n ∈ N ) λ ≤ λ
n≤ µ
nwhere µ
n=
∑i∈Inαi,n|xn−p0i(xn)|2
|xn−∑i∈Inαi,np0i(xn)|2
if x
n6∈ T
i∈InB
i,n1 if x
n∈ T
i∈InB
i,n.
In the remainder of this section, the sequence ( x
n)
n∈Nis defined by the algorithm of Definition 5.
The main result is the following.
Theorem 2. If the condition C
5is satisfied, and the sequence ( a
n)
n∈Nintroduced in Definition 5 satisfies C
6, then the sequence ( x
n)
n∈Nconverges strongly to a point in T
i∈IS
i.
Proof. The method introduced in Definition 5 is a general projection algorithm onto specific uniformly convex sets constructed at each iteration. For every n in N and each i in I, a
nand p
0i( x
n) belong to B
i,n. Then Lemma 2 and Lemma 3 where p
i( x
n) and ( z
n)
n∈Nare respectively replaced with p
0i( x
n) and ( a
n)
n∈Nhold true. In the same way, taking ( z
n)
n∈N= ( a
n)
n∈Nin the proof of Theorem 1, we deduce that the sequence ( x
n)
n∈Nconverges strongly to some point x
∗in H . Nevertheless, since p
i( x
n) is replaced with p
0i( x
n) , it is still not clear why x
∗∈ T
i∈IS
i. Let us now address this specific point. Fix n in N , i in I, and a
nin T
n. According to C
5, a
nand p
i( x
n) belong to a convex set C
i,jn⊂ S
i, and p
i( x
n) is also the projection of x
nonto C
i,jn. Hence the Kolmogorov criterion (1) gives h p
i( x
n) − a
n| x
n− p
i( x
n)i ≥ 0. Therefore
h p
i( x
n) − a
n| x
n− p
i0( x
n)i + h p
i( x
n) − a
n| p
0i( x
n) − p
i( x
n)i ≥ 0 which implies
h p
i( x
n) − a
n| x
n− p
i0( x
n)i ≥ h p
i( x
n) − a
n| p
i( x
n) − p
0i( x
n)i . (10) Moreover, since the segment [ a
n, p
i( x
n)] is the main axis of the ellipsoid B
i,nand p
0i( x
n) ∈ B
i,n, we have h a
n− p
i( x
n) | p
0i( x
n) − p
i( x
n)i ≥ 0. Combining this with (10), we get
0 ≤ h p
i( x
n) − a
n| p
i( x
n) − p
0i( x
n)i ≤ h p
i( x
n) − a
n| x
n− p
0i( x
n)i ≤ | p
i( x
n) − a
n| | x
n− p
0i( x
n)| . We deduce from Lemma 3(iii) where p
i( x
n) is replaced with p
0i( x
n) that lim
n→∞| p
0i( x
n) − x
n| = 0.
On the other hand, as p
i( x
n) is the projection of x
nonto C
i,jn, and a
n∈ C
i,jn, we have | p
i( x
n) − a
n| ≤
| x
n− a
n| . Moreover, following the same steps as in the proof of Lemma 3(i) gives that (| x
n− a
n|)
n∈Nis bounded, from which we deduce that (| p
i( x
n) − a
n|)
n∈Nis bounded as well. Therefore we deduce from the last inequality that
n→
lim
∞h a
n− p
i( x
n) | p
0i( x
n) − p
i( x
n)i = 0. (11) Let x = a
n− p
i( x
n) and y = p
0i( x
n) − p
i( x
n) . Then, we have
x − y = a
n− p
0i( x
n) and
x − ωy = ω 1
ω ( a
n+ ( ω − 1 ) p
i( x
n)) − p
0i( x
n)
(12) for all ω ∈ R .
Assume that p
i0( x
n) 6= p
i( x
n) . Using Claim 1, we get that
a
n− p
0i( x
n) | 1
ω
i,n( a
n+ ( ω
i,n− 1 ) p
i( x
n)) − p
0i( x
n)
≤ 0 as long as we enforce
ω
i,n≥ h p
0i( x
n) − a
n| p
i( x
n) − a
ni
h p
0i( x
n) − a
n| p
i( x
n) − p
0i( x
n)i . (13) In particular, we can take
ω
i,n= max
h p
0i( x
n) − a
n| p
i( x
n) − a
ni h p
0i( x
n) − a
n| p
i( x
n) − p
0i( x
n)i , e
(14) for any appropriately chosen e > 0. Using Niculescu’s Lemma 4 in Appendix B, we obtain that
s 1 ω
i,n+
r ω
i,n1
! h a
n− p
i( x
n) , p
0i( x
n) − p
i( x
n)i
2 | a
n− p
i( x
n)| ≥ | p
0i( x
n) − p
i( x
n)| . (15) Note that this inequality also holds without resorting to Claim 1 if p
0i( x
n) = p
i( x
n) . According to Theorem 1, ( x
n)
n∈Nand ( p
0i( x
n))
n∈Nare strongly convergent to a point x
∗in H . We now split the remainder of the argument into the following complementary cases involving an increasing function σ: N 7→ N which parametrises possible subsequences.
• If the sequence (| a
n− p
0i( x
n)|)
n∈Nconverges to zero, then using Claim 2, (| p
0i( x
n) − p
i( x
n)|)
n∈Nalso converges to zero. Therefore, ( p
i( x
n))
n∈Nis also strongly convergent to x
∗for each i in I.
Since each set S
iis closed, we again conclude that x
∗∈ T
i∈IS
i.
• If the sequence (| a
n− p
0i( x
n)|)
n∈Ndoes not converge to zero, and the sequence (h p
0i( x
n) − a
n| p
i( x
n) − p
0i( x
n)i)
n∈Nconverges to zero, then we must have | a
σ(n)− p
i0( x
σ(n))| > e for some e > 0 and some subsequence indexed by an appropriately chosen increasing function σ. This implies in particular that
n→+
lim
∞| p
i( x
σ(n)) − p
0i( x
σ(n))| | cos
p
0i( x
σ(n)) − a
σ(n), p
i( x
σ(n)) − p
0i( x
σ(n)) | = 0.
As a result, either
n→+
lim
∞cos
p
0i( x
σ(n)) − a
σ(n), p
i( x
σ(n)) − p
0i( x
σ(n)) = 0 (16) or
n→+∞
lim | p
i( x
σ(n)) − p
0i( x
σ(n))| = 0.
Notice further that (16) holds only if (| p
i( x
σ(n)) − p
0i( x
σ(n))|)
n∈Nconverges to zero. Thus, both cases simplify into the conclusion that (| p
i( x
σ(n)) − p
0i( x
σ(n))|)
n∈Nconverges to zero. Therefore ( p
i( x
σ(n)))
n∈Nis also strongly convergent to x
∗for each i in I. Since each set S
iis closed, we again conclude that x
∗∈ T
i∈IS
i.
• If both sequences (| a
n− p
0i( x
n)|)
n∈Nand (h p
i0( x
n) − a
n| p
i( x
n) − p
0i( x
n)i)
n∈Ndo not converge to zero, then | a
σ(n)− p
0i( x
σ(n))| > e and h p
0i( x
σ(n)) − a
σ(n)| p
i( x
σ(n)) − p
0i( x
σ(n)i > e for some e > 0 and for some subsequence indexed by an appropriately chosen increasing function σ.
Since | a
n− p
i( x
n)| is the length of the main axis of B
i,n, and, as such, bounds from above the distance between any two points in B
i,n, we deduce that | a
n− p
0i( x
n)| ≤ | a
n− p
i( x
n)| . Furthermore, since the sequences (| a
n− p
i( x
n)|)
n∈Nand (| a
n− x
n|)
n∈Nare bounded, we deduce that
| a
n− p
i( x
n)| ≤ | a
n− x
n| ≤ B
for some B > 0. Then using the Cauchy-Schwarz inequality in the numerator in (14), we get
ω
i,σ(n)∈ [ e, B
2e ]
for some e ∈ ( 0, B ) . Since | a
σ(n)− p
i( x
σ(n))| > e, we deduce from (11) and (15) that (| p
0i( x
σ(n)) − p
i( x
σ(n))|)
n∈Nconverges to zero. Therefore, ( p
i( x
σ(n)))
n∈Nis also strongly convergent to x
∗for each i in I. Since each set S
iis closed, we again conclude that x
∗∈ T
i∈IS
i.
These previous cases cover all the possible cases and all lead to the conclusion that x
∗∈ T
i∈IS
i. The proof is thus complete.
The two following claims were instrumental in the proof of Theorem 2. We now provide their proofs.
Claim 1. Assume that p
i0( x
n) 6= p
i( x
n) . Then
h p
0i( x
n) − a
n| p
i( x
n) − a
ni
h p
0i( x
n) − a
n| p
i( x
n) − p
0i( x
n)i (17) is a well defined nonnegative real number. Moreover, we have
a
n− p
0i( x
n) | 1
ω
i,n( a
n+ ( ω
i,n− 1 ) p
i( x
n)) − p
0i( x
n)
≤ 0 (18)
for all
ω
i,n≥ h p
0i( x
n) − a
n| p
i( x
n) − a
ni h p
0i( x
n) − a
n| p
i( x
n) − p
0i( x
n)i .
Proof. We have p
0i( x
n) 6∈ [ a
n, p
i( x
n)] . Indeed, assume for contradiction that p
0i( x
n) ∈ [ a
n, p
i( x
n)] . This would imply that p
0i( x
n) lies in the interior of B
i,nand therefore x
n= p
0i( x
n) . However, since by convexity of C
i,jn, [ a
n, p
i( x
n)] ⊂ C
i,jn, this would imply that x
n∈ C
i,jn, and therefore p
i0( x
n) = x
n= p
i( x
n) , the sought-after contradiction.
Consider now the 2D plane containing a
n, p
i( x
n) and p
0i( x
n) . Set g
nto be at the intersection of the line perpendicular at p
0i( x
n) to [ a
n, p
0i( x
n)] with the segment [ a
n, p
i( x
n)] (see Figure 1). First, we have
h p
0i( x
n) − a
n| p
i( x
n) − a
ni = cos ( α
i,n) | p
0i( x
n) − a
n|| p
i( x
n) − a
n| and
| p
0i( x
n) − a
n|
2= h g
n− a
n| p
0i( x
n) − a
ni = cos ( α
i,n) | g
n− a
n|| p
0i( x
n) − a
n|
from which we obtain
| g
n− a
n| = | p
0i( x
n) − a
n|
2| p
i( x
n) − a
n|
h p
0i( x
n) − a
n| p
i( x
n) − a
ni (19) Since B
i,nis an ellipse with maximal axis [ a
n, p
i( x
n)] , our next step is to express g
nas the convex combination
g
n= θ
i,na
n+ ( 1 − θ
i,n) p
i( x
n) (20) of a
nand p
i( x
n) for some θ
i,n∈ [ 0, 1 ] . Notice that (20) implies
| g
n− a
n| = ( 1 − θ
i,n)| p
i( x
n) − a
n| , which itself gives
θ
i,n= 1 − | g
n− a
n|
| p
i( x
n) − a
n| . Using (19), we get
θ
i,n= 1 − | p
0i( x
n) − a
n|
2h p
0i( x
n) − a
n| p
i( x
n) − a
ni = h p
0i( x
n) − a
n| p
i( x
n) − p
0i( x
n)i h p
0i( x
n) − a
n| p
i( x
n) − a
ni .
Notice further that, by construction, since [ a
n, p
i( x
n)] is the main axis of B
i,nand since, by assumption, p
0i( x
n) 6= p
i( x
n) , we have (see Figure 1)
h p
0i( x
n) − a
n| p
i( x
n) − p
0i( x
n)i > 0. (21) Then, setting
d
n= 1
ω
i,n( a
n+ ( ω
i,n− 1 ) p
i( x
n)) , we see that we need to take ω
i,n≥ 1/θ
i,n, i.e.,
ω
i,n≥ h p
0i( x
n) − a
n| p
i( x
n) − a
ni h p
0i( x
n) − a
n| p
i( x
n) − p
0i( x
n)i , (which, owing to (21), is well defined), in order to ensure that
a
n− p
0i( x
n) | d
n− p
0i( x
n) ≤ 0
i.e., (18) holds. Finally, (19) and (21) together ensure that (17) is nonnegative.
Claim 2. We have
| a
n− p
0i( x
n)| ≥ | p
0i( x
n) − p
i( x
n)| . Proof. By definition of p
i( x
n) , the fact that a
n∈ S
iimplies that
| p
i( x
n) − x
n| ≤ | a
n− x
n| . Thus x
nbelongs to the half space
H =
x ∈ H ; h x − 1
2 ( a
n+ p
i( x
n)) | p
i( x
n) − 1
2 ( a
n+ p
i( x
n))i ≥ 0
which is the set of points in H which are closer to p
i( x
n) than a
n. Clearly, p
0i( x
n) ∈ H as well,
which completes the proof.
Figure 1. Section of the ellipsoid B
i,n. 5. Applications
5.1. MRI Image Reconstruction: The Infinite-Dimensional Hilbert Space Setting
The field of inverse problems was extensively investigated in the last five decades, fueled in particular by the many challenged in medical imaging. Projection methods have played an important role in the development of efficient reconstruction techniques [25], a well studied example being the method of projection onto convex sets (POCS) [2]. In recent years, techniques from penalised estimation have gained increasing popularity since the discovery of the compressed sensing paradigm, allowing for fewer measurements to be collected whilst achieving remarkable reconstruction performance.
One penalisation approach which has been a center of focus since its introduction is the Rudin Osher Fatemi functional [26], which can be described as follows: the reconstructed image u is the solution of the minimization problem,
arg min
u∈BV(Ω)
k u k
TV(Ω)+ λ 2 Z
Ω
( y
obs( x ) − A[ u ]( x ))
2dx
where λ is a positive relaxation parameter and the term k u k
TV(Ω)is defined in Appendix A. The term R
Ω
( y
obs( x ) − A[ u ]( x ))
2dx is called the fidelity term and the term k u k
TV(Ω)is called the regularisation term. This infinite-dimensional minimization problem enforces the solution, which is a priori in L
2( Ω ) , to satisfy a finite bounded variation (BV) norm constraint.
One important remark is that the optimisation problem can be turned into a pure feasibility problem by imposing the TV-norm and the fidelity terms to be less than or equal to a certain tolerance.
Using this approach, one can define an alternating projection procedure as in Algorithm 1 below, where we will use the forward operator A = F
partial, the partial Fourier Transform which plays a central role in MRI. The usual Fourier transform will be denoted by F as usual.
In this example, we will use the projections on the sets S
1=
u ∈ L
2( Ω ) | Z
Ω
( λ y
obs− A[ u ]( x ))
2dx ≤ λη
f id, λ ∈ R
+,
S
2= n u ∈ L
2( Ω ) | k u k
p,TV(Ω)≤ η
TVo
,
S
3= n u ∈ L
2( Ω ) | F
partial[ u ] = λy
obs, λ ∈ R
+o
.
The set S
2is not convex for p ∈ ( 0, 1 ) , but, since the epigraph of | · |
p,TV(Ω)is a cone, the set S
2is expandable into convex sets as discussed in [22,23]. Notice that when the scaling factor λ is dropped in the definition of S
1and S
3, the problem is equivalent to the standard TV-penalised reconstruction approach. Introducing this scaling factor allows to recover a solution up to a scaling, which can easily be recovered by rescaling the solution by using the constraint F
partial[ u ] = y
obsas a post-processing step. The introduction of the scaling α in these definition allows the null function to belong to the intersection of the sets S
1, S
2and S
3. Thus, we will set a
n= 0, n ∈ N in the sequel. In what follows,
˜
y
obswill be the function will value equal to zero except for the observed frequencies which are taken as the observed values.
Our implementation is described in Algorithm 1. Experiments were made after incorporating the projection onto the TV ball into a freely available code provided in [27]. The fidelity term was set to η
f id= 10
−3and the regularisation term η
TVwas tuned using cross validation. The projection onto the TV-ball was computed using the method described in [28]. In order to avoid numerical instabilities, the iterates were rescaled every 10 iteration, which does not change the convergence analysis due to conic invariance of our formulation. We chose very thin ellipsoids B
i,nby setting α = 10
−6. A numerical illustration is given in Figure 2 below. Extensive numerical simulations will be presented in a forthcoming paper devoted to a thorough study of projection methods for MRI and X-ray computed tomography (XCT).
Algorithm 1: Alternating projection method for MRI for p = 1.
Result:
1
y = F
partial[ u
0] observed data;
2
η
TVbound on the TV norm;
3
η
f idbound on the fidelity term;
4
e convergence tolerance;
5
Compute y
(0)from y by filling in unobserved with zeros ;
6
Set k = 1, u
1= F
−1( y ˜
obs) ;
7
while Difference between two successive iterates larger than e do
8
Compute p
01( u
n) = P
B1,n( u
n) ;
9
Compute p
01( u
n) = P
B2,n( u
n) ;
10
Compute p
01( u
n) = P
B3,n( u
n) ;
11
Set u
n+1= u
n+ λ
n∑
3i=1α
i,n( p
i0( u
n) − u
n) ;
12