On the Uniqueness of Inverse Problems with Fourier-domain Measurements and Generalized TV Regularization
aa r X i v : . [ m a t h . O C ] S e p On the Uniqueness of Inverse Problems with Fourier-domainMeasurements and Generalized TV Regularization
Julien Fageot, Thomas Debarre, and Quentin DenoyelleSeptember 25, 2020
Abstract
We study the super-resolution problem of recovering a periodic continuous-domain functionfrom its low-frequency information. This means that we only have access to possibly corruptedversions of its Fourier samples up to a maximum cut-off frequency K c . The reconstruction taskis specified as an optimization problem with generalized total-variation regularization involvinga pseudo-differential operator. Our special emphasis is on the uniqueness of solutions. Weshow that, for elliptic regularization operators ( e.g. , the derivatives of any order), uniqueness isalways guaranteed. To achieve this goal, we provide a new analysis of constrained optimizationproblems over Radon measures. We demonstrate that either the solutions are always made ofRadon measures of constant sign, or the solution is unique. Doing so, we identify a generalsufficient condition for the uniqueness of the solution of a constrained optimization problemwith TV-regularization, expressed in terms of the Fourier samples. In recent years, total-variation regularization techniques for continuous-domain inverse problemshave shown to be very fruitful, with rapidly-growing theoretical developments [1, 2, 3, 4], algorith-mic progress [5, 6, 7], and data science applications [8, 9, 10]. As is well known for its discrete-domaincounterpart ( i.e. , ℓ optimization), this leads to variational problems whose solutions are not neces-sarily unique. Our goal in this paper is to provide a systematic study of the uniqueness of TV-basedoptimization problems for the special case of Fourier sampling measurements.The torus is the interval T = R / π Z = [0 , π ] where the two points and π are identified. Westudy the reconstruction of an unknown periodic real function f : T → R from the knowledge of itspossibly corrupted low-frequency Fourier series coefficients. Let K c ≥ be the maximum frequency,we therefore have access to y = ( y , y , . . . , y K c ) ∈ R × C K c (1)such that y k is approximately the k th Fourier series coefficient b f [ k ] of f . Note that, since f isa real function, y ∈ R is the (approximated) mean b f [0] = h f , i of f , while y k ∈ C for k = 0 .Moreover, the Fourier series of f is Hermitian symmetric, meaning that b f [ − k ] = b f [ k ] ∈ C forevery k ∈ Z . The observation y in (1) therefore has K c + 1 (real) degrees of freedom: one for thereal mean y and two for each other complex Fourier series coefficients in C . Finally, we will alwaysassume that y ≥ (otherwise, we can consider the reconstruction of − f instead of f ). The recovery of a periodic function from finitely many observations is of course ill-posed. Wetherefore formulate the reconstruction task as a regularized optimization problem. More precisely,1he reconstruction f ∗ of f is defined as a solution of f ∗ ∈ arg min f E ( y , ν ( f )) + λ k L f k M , (2)where y ∈ R + × C K c is the observation vector, ν ( f ) is the measurement vector ν ( f ) = ( b f [0] , b f [1] , . . . , b f [ K c ]) ∈ R × C K c , (3) E ( · , · ) : ( R × C K c ) → R + ∪ {∞} is a data-fidelity functional, strictly convex over its domain , lowersemi-continuous (lsc), and coercive with respect to its second argument, k·k M is the total-variationnorm on periodic Radon measures, and L is a regularization operator acting on periodic functions.The data-fidelity term encourages the measurement vector ν ( f ) to be close to the observation y . A typical example is the quadratic functional E ( y , ν ( f )) = 12 k y − ν ( f ) k = 12 K c X k =0 | y k − b f [ k ] | . (4)The data fidelity (4) corresponds to an additive noise model where the measurements y are generatedvia the model y = ν ( f ) + n with n a complex Gaussian vector (see [12, Section IV-B] for moredetails). We will also be interested in constrained optimization problems of the form arg min f, ν ( f )= y k L f k M , (5)which corresponds to E ( y , ν ( f )) = 0 if y = ν ( f ) and ∞ otherwise . Other classical data-fidelityfunctionals can be found in [9, Section 7.5].The choice of the total-variation norm promotes sparse and adaptive continuous-domain recon-struction, and has recently received a lot of attention (see Section 1.3). The operator L controlsthe transform domain in which sparsity is enforced together with the regularity properties of therecovery: Dirac recovery corresponds to L = Id [1], and higher-order operators induce smootherreconstructions [4].
The existence of solutions for problems such as (2) is well established, and simply follows from theconvexity of the cost functional. However, the solution is in general not unique (the simplest caseof non-uniqueness is with K c = 0 and L = Id , see Section 3.1). It is actually well know that evenfinite-dimensional ℓ -regularization, of which the total-variation norm for Radon measures is thecontinuous-domain generalization, can lead to non-unique solutions [13, 14]. In this paper, we focuson characterizing the cases of uniqueness for (2). Our contributions can be detailed as follows. (i) Optimization over Radon measures: positivity versus uniqueness. We first consider the con-strained problem V ( y ) = arg min ν ( w )= y k w k M (6)over periodic Radon measures w ∈ M ( T ) , with y ∈ R + × C K c and ν ( w ) = ( b w [0] , b w [1] , . . . , b w [ K c ]) .We show that two (not necessarily mutually exclusive) scenarios are possible: either (i) the solution The domain of a convex function g : X → R + ∩ {∞} is the set { x ∈ X, g ( x ) < ∞} [11] In this case, the value of the regularization parameter λ > plays no role.
2s unique, or (ii) the solution set consists of all positive Radon measures satisfying the constraints.The two scenarios are characterized by a simple positive-definiteness condition on the vector y (seeTheorem 2). We then link (2) with regularized optimization problems of the form (6), leading toour second main contribution. (ii) Uniqueness of the solution for elliptic operators. Our main theoretical result focuses on thecase where L is a periodic elliptic operator, meaning that its null space consists of constant functions(see Definition 1 thereafter). An L -spline is a function f such that L f = K X k =1 a k X ( · − x k ) (7)is a finite sum of Dirac combs, the distinct Dirac locations x k being the knots of the spline (seeSection 2.1). Theorem 1.
Let L be an elliptic periodic operator, K c ≥ , and y ∈ R + × C K c . Then, there existsa unique solution to (2) . Moreover, this solution is a L -spline whose number of knots is bounded by K c . To the best of our knowledge, Theorem 1 is the first systematic uniqueness result for the analysisof a variational problem of the form (2).
Optimization over Radon measures:
The historical motivation to consider total-variation as a regu-larization norm was to extend discrete ℓ regularization techniques, used in the theory of compressedsensing to recover sparse vectors [15, 16], for continuous-domain Dirac recovery. The goal is to re-cover point sources, modelled as a sum of Dirac masses, from finitely many measurements. This hasreceived a considerable attention in the 21st century, including FRI (finite rate of innovation) tech-niques [17, 18] and Prony’s methods [19]. Several data-science problems can indeed be formulatedas a Dirac recovery problem, including radio-astronomy [20], super-resolution microscopy [6], or 3Dimage deconvolution [10].After the development of the theory of compressed sensing [15, 21], inherently discrete in its sem-inal formulation, several infinite-dimensional extensions of the compressed sensing framework havebeen proposed [22, 23, 24, 25, 26]. In this context, in the early 2010’s, De Castro and Gamboa [27],Candès and Fernandez-Granda [1, 28], and Bredies and Pikkarainen [2] all considered optimizationtasks of the form (6) (or its penalized version), with both theoretical analyses and novel algorithmicapproaches to recover a sparse measure solution. From sparse measures to splines and beyond:
The study of optimization problems of the form (6) canbe traced back to the pioneering works of Beurling [29], where Fourier-sampling measurements werealso considered. The existence of sparse measure solutions, i.e. , solutions of the form P Kk =1 a k δ ( · − x k ) , seems to have been proven for the first time in [30] and was later improved by Fisher andJerome in [31]. Since then, a remarkable revival around TV optimization has occurred recently [32,33, 34, 35, 7].These works have revealed that the signal model induced by the total-variation norm is a streamof Dirac. Several authors extended this framework to smoother continuous-domain signals by consid-ering generalized total-variation regularization. In [4], Unser et al. revealed the connection between3onstrained problems (2) (in a non-periodic setting) and spline theory for general measurement func-tionals: the extreme-point solutions are necessarily L -splines. This result was revisited, extended,and refined by several authors [9, 36, 37, 38, 39, 40, 41, 42, 43, 44]. This manuscript will stronglyrely on the periodic theory of TV-based optimization problem recently developed in [44]. Uniqueness results for generalized TV optimization:
It is well-known that replacing the total-variation norm by the L norm in (2) (Tikhonov regularization, also known as ridge regression) leadsto optimization problems whose solution is unique (see [12] for a detailed study of the quadratic casein the periodic setting with general operators L and measurements ν ). This is no longer true forTV regularization, and it becomes important to understand the cases of uniqueness. Many unique-ness results for constrained or penalized TV-based optimization problems have been given in theliterature, but from different perspectives than the one studied in this paper. In [27], de Castroand Gamboa introduced the concept of extrema Jordan type measure (see [27, Definition 1]), whichgives sufficient conditions on a given signed sparse measure to be the unique solution of a TV-basedoptimization problem. Candès and Fernandez-Granda also studied the super-resolution problem ofrecovering a ground-truth sparse Radon measure w from its low-frequency measurements. Theyhave shown that if the minimal distance between the spikes of w is large enough, then the optimiza-tion problem has a unique solution, which is w itself [1, Theorem 1.2]. Duval and Peyré identifiedthe so-called non-degenerate source condition [3, Definition 5], under which the uniqueness of thereconstruction together with the support recovery of the underlying ground-truth sparse measureare shown. These results are based on the key notion of dual certificates, will play an important rolein our work. This notion has been introduced for discrete compressed sensing problems in [45] andconnected to TV-based optimization problems in [27].All these works are clearly related to this paper. However, the perspective we propose is different:we aim at characterizing the cases of uniqueness directly over the measurement vector y , and areoblivious to the ground-truth signal that generated it. The closest work in this direction is our recentpublication [46], where we provide a full description of the solution set of TV optimization problemswith a regularization operator L = D where D is the derivative operator (which leads to piecewise-linear reconstructions), and spatial sampling measurements, which includes the characterization ofthe cases of uniqueness [46, Proposition 6 and Theorem 2]. The paper is organized as follows. Section 2 introduces the mathematical material used in this paper.Section 3.1 is dedicated to an illustrative toy optimization problem with K c = 1 , highlightinginteresting phenomena which also occurs in the general case. In the rest of Section 3, we fullycharacterize the solution set of (6) when there exists a non-negative measure solution (Section 3.2),and show that the solution is unique when no non-negative measure is a solution (Section 3.3).Section 4 uses the results of the previous section to study (2) for invertible (Section 4.1) and ellipticoperators (Section 4.2). We conclude in Section 5. We first introduce some notations and recall some basic facts concerning periodic functions and theirFourier series. More details can be found in [44, Section 2]. The space of infinitely smooth periodicfunction is denoted by S ( T ) , endowed with its usual Fréchet topology. Its topological dual is thespace of periodic generalized functions S ′ ( T ) . 4or k ∈ Z , let e k : T → C be the complex exponential function e k ( x ) = exp(2i πkx ) , which isclearly in S ( T ) . The Fourier series coefficients of f ∈ S ′ ( T ) are given by b f [ k ] = h f, e k i ∈ C . For areal function f , these coefficients are Hermitian symmetric, i.e. , such that b f [ − k ] = b f [ k ] for all k ∈ Z ,which implies in particular that b f [0] ∈ R . We then have that f = P k ∈ Z b f [ k ] e k for any f ∈ S ′ ( T ) ,where the convergence is in S ′ ( T ) . The Dirac stream is defined as X = P n ∈ Z δ ( · − πn ) . Its Fouriercoefficients are c X [ k ] = 1 for each k ∈ Z . We consider periodic, linear, and shift-invariant operators
L : S ′ ( T ) → S ′ ( T ) . As is well known, L is fully characterized by its Fourier sequence ( b L [ k ]) k ∈ Z such that L e k = b L [ k ] e k , in the sense that L f = X k ∈ Z b L [ k ] b f [ k ] e k . (8)In this paper, we will only consider operators L that are invertible or elliptic, as defined thereafter.The space of periodic linear and shift-invariant operators is denoted by L SI ( S ′ ( T )) . Definition 1.
An operator L ∈ L SI ( S ′ ( T )) is invertible if there exists an operator L − ∈ L SI ( S ′ ( T )) such that LL − f = L − L f = f, ∀ f ∈ S ′ ( T ) . (9) A periodic operator is elliptic if b L [0] = 0 and if L + 1 is invertible.
The operator
K = L + 1 in Definition 1 has the same Fourier sequence as L except that b K [0] = b L [0] + 1 . It is such that K f = L f + b f [0] . Adding this constant allows the operator to be invertible,which is excluded for L by the condition b L [0] = 0 . Invertibility and ellipticity can be characterizedby the Fourier sequence of L as follows. The proof is given in Appendix A. Proposition 1.
A periodic operator is invertible if and only if there exist
A, B > and α, β ∈ R such that A (1 + | k | ) α ≤ | b L [ k ] | ≤ B (1 + | k | ) β , ∀ k ∈ Z . (10) In this case, the Fourier sequence of L − is (1 / b L [ k ]) k ∈ Z .A periodic operator is elliptic if and only if there exist A, B > and α, β ∈ R such that A | k | α ≤ | b L [ k ] | ≤ B | k | β , ∀ k ∈ Z . (11) Then, the null space of L is the one-dimensional space of constant functions. Definition 2.
We say that L † ∈ L SI ( S ′ ( T )) is a pseudoinverse of L ∈ L SI ( S ′ ( T )) if we have LL † L = L , and L † LL † = L † . (12)Any elliptic operator has a unique pseudoinverse whose Fourier sequence is given by b L † [0] = 0 and b L † [ k ] = 1 / b L [ k ] for k = 0 . This is a particular case of [44, Proposition 2.4] but it can easily beproved in this case.The pseudoinverse operator allows to define the Green’s function of L ∈ L SI ( S ′ ( T d )) . This notionis here adapted to the periodic setting, as discussed after [44, Definition 4]. We borrow the terminology for pseudo-differential operators in partial differential equations [47] whose Fouriersymbols are assumed to be non-vanishing except at the origin. Elliptic operators are generalizations of the Laplaceoperator. A pseudoinverse should also satisfy the self-adjoint relations (LL † ) ∗ = LL † and (L † L) ∗ = L † L , but they areautomatically satisfied in this case, as shown in [44, Section 2.2]. efinition 3. We fix L ∈ L SI ( S ′ ( T )) and we assume that it admits a pseudoinverse L † . Then, the Green’s function of L is ρ L = L † X . The Green’s function of an invertible periodic operator is simply ρ L = L − X . For an ellipticoperator, it is the generalized function such that b ρ L [0] = 0 and b ρ L [ k ] = 1 / b L [ k ] for k = 0 . Definition 4.
Let L be an elliptic or invertible periodic operator. We say that f is a periodic L -spline (or simply an L -spline) if L f = w = N X n =1 a n X ( · − x n ) (13) where N ≥ , a n ∈ R \{ } , and distinct knots x n ∈ T . We call w the innovation of the L -spline f . For invertible operators, (13) is equivalent to f = P Nn =1 a n ρ L ( · − x n ) where ρ L = L − X theGreen’s function of L . When L is elliptic, f satisfies (13) if and only if f = a + P Nn =1 a n ρ L ( · − x n ) where ρ L = L † X and a ∈ R . In this case, we necessarily have that P Nn =1 a n = 0 . This is aparticular case of [44, Proposition 2.8] and simply follows from taking the mean (or th Fouriercoefficient) in (13), giving b L [0] b f [0] = P Nn =1 a n . It is worth noting that the Green’s function ofan elliptic operator is not a L -spline. However, ρ L − ρ L ( · − / is a periodic L -spline. Examples.
Invertible operators include differential operators
D + α I with α ∈ R \{ } and Sobolevoperators L = ( α I − ∆) γ/ for γ > and α > (whose Fourier sequence is b L [ k ] = ( α + k ) γ/ ).The fractional derivatives D γ and fractional Laplacians ( − ∆) γ/ with γ > are examples of ellipticoperators. More examples can be found in [44, Section 5.1], together with the representation of thecorresponding Green’s functions and splines. Let M ( T ) be the space of periodic Radon measures. By the Riesz-Markov theorem [48], it is thecontinuous dual of the space C ( T ) of continuous periodic functions endowed with the supremumnorm. The total-variation norm on M ( T ) , for which it forms a Banach space, is given by k w k M = sup f ∈C ( T ) , k f k ∞ ≤ h w, f i . (14)We denote by M ( T ) the set of Radon measures with zero mean, i.e. M ( T ) = { w ∈ M ( T ) , b w [0] =0 } . It is the continuous dual of the space C ( T ) = { f ∈ C ( T ) , b f [0] = 0 } of continuous functions withzero mean. we also consider the set of positive Radon measures M + ( T ) which are Radon measures w such that h w, ϕ i ≥ for any positive continuous function ϕ . The set of probability measures, i.e. non-negative measures w with total-variation k w k M = 1 , is denoted by P ( T ) .The total-variation norm upper-bounds the Fourier coefficients of a Radon measure, as stated inProposition 2 which also provides elementary characterizations for non-negative Radon measures.The proof is provided in Appendix A. Proposition 2.
Let w ∈ M ( T ) . Then,1. For any k ∈ Z , we have | b w [ k ] | ≤ k w k M .2. w ∈ M + ( T ) if and only if k w k M = b w [0] .3. w ∈ P ( T ) if and only if k w k M = b w [0] = 1 . L be a periodic operator. We define the native space associated to L as M L ( T ) = { f ∈ S ′ ( T ) , L f ∈ M ( T ) } . (15)Periodic native spaces have been studied for general spline-admissible operators ( i.e. , periodic oper-ators with finite-dimensional null space and which admit a pseudoinverse) in [44, Section 3]. Proposition 3 (Theorem 3.2 in [44]) . Let L be a periodic operator. If L is invertible, then M L ( T ) =L − ( M ( T )) inherits the Banach structure of M ( T ) for the norm k f k M L = k L f k M . (16) If L is elliptic, then we have that the direct sum relation M L ( T ) = L † M ( T ) ⊕ Span { } (17) and any f ∈ M L ( T ) has a unique decomposition as f = L † w + a (18) where w ∈ M ( T ) and a ∈ R are given by w = L f and a = b f [0] . Then, M L ( T ) is a Banach spacefor the norm k f k M L = k w k M + | a | . (19)The case of invertible operators is particularly simple: the bijection induces an isometry between M ( T ) and the native space M L ( T ) . Native spaces of elliptic operators require to deal with constantsfrom the null space of L . Note that the measurement functional ν in (3) is well defined over M L ( T ) ,and more generally over S ′ ( T ) , since the complex exponentials are infinitely smooth.It is known that TV-based optimization problems with regularization operators lead to splinessolutions [4]. This is both an existence result and a representer theorem , which provides the formof the (extreme point) solutions of the optimization task. We now recall the representer theoremassociated to (2). This follows the same line as many recent works on representer theorems for TVoptimization [4, 36, 40, 39, 38], that has been recently adapted to the periodic setting in [44]. Proposition 4 (Theorem 4 in [44]) . Let L be an invertible or elliptic periodic operator, K c ≥ bethe cut-off frequency of the low-pass filter ν : M L ( T ) → R + × C K c defined in (3) , y ∈ R + × C K c , E ( · , · ) : ( R × C K c ) → R + be a functional which is strictly convex over its domain, lsc, and coercivewith respect to its second argument, and λ > . Then, the solution set V ( y ) = arg min f ∈M L ( T ) E ( y , ν ( f )) + λ k L f k M (20) is non-empty, convex, weak* compact, and its extreme points are periodic L -splines whose numberof knots is upper-bounded by K c + 1 .Proof. Proposition 4 is a direction application of [44, Theorem 4] to the case d = 1 and for Fouriersampling linear functionals ν m = e m . The latter, being in S ( T ) , fulfil the hypotheses of the theoremas justified in [44, Section 6.1]. The number of real measurements is given by M = 2 K c + 1 : y isone measurement, while each y k ∈ C for k = 1 , . . . , K c provides two real measurements via their realand imaginary parts.The main result of our paper can be seen as the uniqueness counterpart of Proposition 4. We shallidentify general sufficient conditions ensuring that the problem only admits one solution. Proposition4 then implies that this unique solution is necessarily a periodic L -spline.7 TV-based Constrained Problems over Radon Measures
As we have seen in Section 1.3, many works deal with the reconstruction of Dirac streams fromFourier-domain measurements. This is a super-resolution problem , because one aims to recovera Dirac stream from its low-frequency information. In this section, we focus on the constrainedoptimization problem V ( y ) = arg min w ∈M ( T ) , ν ( w )= y k w k M , (21)where K c ≥ , y = ( y , y , . . . , y K c ) ∈ R + × C K c , and ν ( w ) = ( b w [0] , b w [1] , . . . , b w [ K c ]) . Our goal is toderive new results on the solution set V ( y ) . We first provide a useful lower bound on the minimalvalue of (21). Lemma 1.
For any K c ≥ and y ∈ R + × C K c , min w ∈M ( T ) , ν ( w )= y k w k M ≥ max ≤ k ≤ K c | y k | . (22) Proof.
We know that V ( y ) is non-empty, so the minimum value is reached by at least one measure w . Then, we have k w k M ≥ | b w [ k ] | = | y k | for all ≤ k ≤ K c according to Proposition 2, whichgives (22). K c = 1 We start our investigation on constrained problems over Radon measures ( i.e. , no regularizationoperator L ), as in (6). First of all, there exist cases of non-uniqueness, as exemplified with themoderately interesting problem of reconstructing a Radon measure w uniquely from its mean b w [0] = y > . In our framework, this corresponds to solving V ( y ) = arg min w ∈M ( T ) , b w [0]= y k w k M . (23)The solution set is V ( y ) = { w ∈ M + ( T ) , b w [0] = y } = y · P ( T ) . Of course, this set is infinite, andits extreme points are of the form y · X ( · − x ) for x ∈ T , which is itself uncountably infinite. Wenow consider the case K c = 1 in the following result, whose proof is given in Appendix B. Proposition 5.
Let y ∈ R + and y = r e i α ∈ C such that ( y , y ) = (0 , . Consider the optimiza-tion problem V ( y ) = V ( y , y ) = arg min w ∈M ( T ) , b w [0]= y & b w [1]= y k w k M . (24) Then, the measure w ∗ = y + r X ( · + α ) + y − r X ( · + α + π ) (25) is always a solution to (24) . Moreover, we have the following scenarios. • If y ≤ r , then w ∗ is the unique solution. Note that w ∗ = y X ( · + α ) is a single Dirac massfor y = r and is a non-negative measure in this case. • If y > r , then the problem has uncountably infinitely many extreme point solutions, which arenecessarily non-negative measures. K C ≥ and y ≥ , either the solution set consists of non-negativemeasures, or the solution is unique. Note moreover that the two situations are not exclusive: with y = r , the unique solution is a non-negative measure. For y > r , our proof of Proposition 5provides more than what is stated, with a full characterization of the extreme point solutions with Dirac masses (see Appendix B).
We first study the case where the optimization problem (21) admits a solution that is a non-negativemeasure. Theorem 2 reveals that this has strong implications on the solution set.
Theorem 2.
Let K c ≥ and y = ( y , y , . . . , y K c ) ∈ R + × C K c . We also set, for ≤ k ≤ K c , y − k = y k . Then, the following conditions are equivalent:1. There exists w ∈ M + ( T ) such that ν ( w ) = y .2. We have the equality y = min w ∈M ( T ) , ν ( w )= y k w k M . (26)
3. The solution set (21) is V ( y ) = { w ∈ M + ( T ) , ν ( w ) = y } .4. For any complex numbers z , z , . . . z K c ∈ C , K c X k,ℓ =0 y k − ℓ z k z ℓ ≥ . (27)
5. The Toeplitz Hermitian matrix T y ∈ C ( K c +1) × ( K c +1) given by T y = y y y . . . y K c y − y y . . . y K c − y − y − y . . . y K c − ... . . . y − K c y − K c +1 y − K c +2 . . . y = [ y ℓ − k ] ≤ k,ℓ ≤ K c (28) is positive semi-definite.Proof. Set m = min w ∈M ( T ) , ν ( w )= y k w k M . We have the equivalence, for any w ∈ M ( T ) : w ∈ M + ( T ) ⇐⇒ k w k M = b w [0] . (29) . ⇒ . The existence of w ensures that m ≤ k w k M = b w [0] = y according to (29). FromLemma 1, we know moreover that m ≥ y , hence m = y . . ⇒ . Any w ∈ V ( y ) is such that b w [0] = y = k w k M , hence, by Proposition 2, w is anon-negative measure satisfying the constraints ν ( w ) = y . Conversely, non-negative measures w satisfying the constraints are such that k w k V = b w [0] = y = m , and are therefore solutions. . ⇒ . This is obvious since we know (for instance using Proposition 4) that V ( y ) is non-empty. . ⇒ . Let w be a non-negative measure solution. According to the Herglotz theorem (seeProposition 8 above), a measure in M ( T ) is positive if and only if its Fourier sequence is positive-definite in the sense of (56); that is, if and only if for any sequence ( z k ) k ∈ Z with finitely many9on-zero terms, P k,ℓ ∈ Z b w [ k − ℓ ] z k z ℓ ≥ . In particular, restricting to sequences such that z k = 0 for k < and k > K c , we have that K c X k,ℓ =0 b w [ k − ℓ ] z k z ℓ ≥ . (30)Applying this relation to w , we remark that | k − ℓ | ≤ K c for ≤ k, ℓ ≤ K c so that b w [ k − ℓ ] = y k − ℓ ,hence (30) is equivalent to (27). . ⇒ . Recall that P K ( T ) is the space of real trigonometric polynomials of degree at most K .Consider the mapping Φ : P K c ( T ) → R such that Φ( p ) = Φ X | k |≤ K c c k e k = X | k |≤ K c c k y k . (31)Then, Φ is linear and positive. Indeed, let p ∈ P K c ( T ) such that p ≥ . According to Proposition9, p can be written as p = | q | for some complex trigonometric q = P K c k =0 z k e k with q k ∈ C . Thisimplies that p = | q | = P K c k,ℓ =0 z k z ℓ e k − ℓ and therefore Φ( p ) = K c X k,ℓ =0 y k − ℓ z k z ℓ ≥ (32)due to (27). According to Proposition 10, Φ can be extended as a positive, linear, and continuousfunctional from C ( T ) to R . Hence, Φ ∈ ( C ( T )) ′ = M ( T ) specifies a positive Radon measure Φ = w due to the Riesz-Markov theorem [48]. Then, w ∈ M + ( T ) satisfies ν ( w ) = y and 1. is proved. . ⇔ . The relation (27) can be written in matrix form as P K c k,ℓ =0 y k − ℓ z k z ℓ = h z , T y z i , with z = ( z , · · · , z K c ) ∈ C K c +1 , with h· , ·i the canonical inner product over C K c +1 . Hence, (27) isequivalent to h z , T y z i ≥ for any z ∈ C K c +1 , which is equivalent to being positive semi-definite.Theorem 2 characterizes the situations where (21) admits a non-negative measure as solution.Conditions 1. to 3. are expressed in term of the solution set V ( y ) , while 4. and 5. give directcharacterizations on the observation vector y . The condition 3. implies that the solution set V ( y ) is completely understood when there exists a non-negative measure solution. The next sectionconsiders the case where the conditions of Theorem 2 do not occur. Remarkably, if the conditions of Theorem 2 do not occur, the optimization problem (21) has aunique solution. This is shown in the next theorem. This section also highlights some interestingconsequences of this fact.
Theorem 3.
Let K c ≥ and y ∈ R + × C K c such that y = 0 . Assume that the equivalent conditionsof Theorem 2 are not satisfied. Then, the optimization problem (21) has a unique solution, which isa sum of at most K c periodic Dirac impulses.Proof. By assumption, there is no non-negative measure solution. Let η be a dual certificate of theoptimization problem (21) (it always exists due to Proposition 11 in Appendix D). Then, we havethat k w ∗ k M = h w ∗ , η i for any w ∗ ∈ V ( y ) (see again Proposition 11).Thanks to Proposition 12, we know that either the dual certificate is constant, or the solutionof (21) is unique. We assume by contradiction that η is constant. We therefore have that h w ∗ , η i = k w ∗ k M = ǫ b w [0] = ǫy . (33)10his means in particular that ǫy > . Yet, y ≥ , which implies ǫ = 1 . This shows that b w [0] = k w ∗ k M , which together with Proposition 2 implies that w ∗ ∈ M + ( T ) . This contradicts ourinitial assumption.This shows by contradiction that η is necessarily non-constant, and therefore, due to Proposition12, that the solution is unique and is a sum of at most K c Dirac masses, as expected.
Remark.
When the solution is unique, Theorem 1 provides a slight refinement of existing results,summarized in Proposition 4. Indeed, it is known that extreme points of V ( y ) consists of at most (2 K c + 1) Dirac masses, which is the dimension of the observation space. In this case, the uniquesolution has at most K c Dirac masses.
Corollary 1.
Let K c ≥ and y ∈ R + × C K c be such that y < | y k | for some k = 0 . Then, (21) has a unique solution, which is a sum of at most K c periodic Dirac impulses.Proof. According to Lemma 1, we have min ν ( w )= y k w k M ≥ | y k | > y . This means in particularthat (26) does not hold, and therefore, by the equivalence between (26) and (27) in Proposition 2,we are in a case of uniqueness described by Proposition 3.Corollary 1 gives a deceptively simple sufficient condition ensuring that the solution of (21) isunique. It will play a crucial role in our proof of Theorem 1. We explore the consequences of the previous section for penalized problems of the form V λ ( y ) = arg min f ∈M L ( T ) E ( y , ν ( f )) + λ k L f k M (34)with an observation vector y ∈ R + × C K c , a measurement vector ν ( f ) ∈ R × C K c , a data-fidelityfunctional E ( · , · ) : ( R × C K c ) → R + ∪ {∞} which is strictly convex over its domain, lsc, and coercivewith respect to its second argument, λ > , and a regularization operator L . We consider invertibleoperators in Section 4.1 and elliptic operators (with the proof of Theorem 1) in Section 4.2. Proposition 6.
Let L be an invertible periodic operator, y ∈ R + × C K c , and λ > . Then, (34) isnon-empty and all the solutions f ∗ ∈ V λ ( y ) share an identical measurement vector y λ = ν ( f ∗ ) . Weset z λ ∈ R × C K c such that z λ,k = b L [ k ] y λ,k for all ≤ k ≤ K c . Then, the following statements hold,assuming first that z λ, ≥ .(1) The problem (34) admits a non-negative measure solution if and only if the Toeplitz Hermitianmatrix T z λ is positive semi-definite, where T z λ is given by (28) with y = z λ . In this case, V λ ( y ) = { f ∈ M L ( T ) , L f ∈ M + ( T ) and ν ( f ) = y } (35) (2) If V ( z λ ) ∩ M + ( T ) = ∅ , then (34) has a unique solution, which is a periodic L -spline with atmost K c knots.If z λ, ≤ , then the same conclusions remain except that non-negative measures are substitutedby non-positive measures and that the positive semi-definiteness of T z λ is replaced by the negativesemi-definiteness. roof. We treat the case z λ, ≥ . The case z λ, ≤ is deduced by remarking that V ( − y ) = −V ( y ) .The existence of a common y λ is classic and uses the strict convexity of the data fidelity E ( y , · ) ;see for instance [46, Proposition 7]. It implies in particular that the optimization problem (34) hasexactly the same solution set than arg min f ∈M L ( T ) , ν ( f )= y λ k L f k M . (36)Using the invertibility of L , we then remark that V λ ( y ) = L − ( V ( z λ )) , where V ( z λ ) = arg min w ∈M ( T ) , ν ( w )= z λ k w k M , (37)as in (21). Then, the case (1) follows by applying Theorem 2 and the case (2) is a direct consequenceof Theorem 3. Remark.
Once we have recognized the link between (21) and (34), Proposition 6 is a reformulationof the results of Section 3. However, one cannot easily deduce the modified vector y λ from y , andthus adjudicate on uniqueness using Theorem 2. In order to compute y λ , we need to solve (34), findone solution f ∗ , compute ν ( f ) = y λ , and then apply the criterion on y λ . This section is dedicated to the proof of Theorem 1. We therefore consider Problem (34) for ellipticoperators L , such as the (fractional) derivative or the (fractional) Laplacian. We start with thefollowing preparatory result. We recall that M ( T ) is the space of Radon measure with zero mean. Lemma 2.
Let L be an elliptic periodic operator and y ∈ R + × C K c . Let f ∗ = L † w ∗ + a ∗ ∈ M L ( T ) where ( w ∗ , a ∗ ) ∈ M ( T ) × R are uniquely determined as in (18) . Then, we have the equivalence: f ∗ ∈ arg min f ∈M L ( T ) , ν ( f )= y k L f k M ⇐⇒ a ∗ = y and w ∗ ∈ arg min w ∈M ( T ) , ν ( w )= z k w k M (38) with z = ( z , z , . . . , z K c ) ∈ R × C K c such that z = 0 and z k = b L [ k ] y k for k = 0 .Proof. The uniqueness of the decomposition f ∗ = L † w ∗ + a ∗ ∈ M L ( T ) with w ∗ ∈ M ( T ) and a ∗ ∈ R implies that f ∗ ∈ arg min f ∈M L ( T ) , ν ( f )= y k L f k M ⇐⇒ ( w ∗ , a ∗ ) ∈ arg min ( w,a ) ∈M ( T ) × R , ν (L † w + a )= y k w k M . (39)Moreover, ν (L † w ∗ + a ∗ ) = ( a ∗ , b w ∗ [1] / b L [1] , . . . , b w ∗ [ K c ] / b L [ K c ]) , hence ν ( f ∗ ) = y ⇐⇒ a ∗ = y and b w ∗ [ k ] = b L [ k ] y k , ∀ ≤ k ≤ K c ⇐⇒ a ∗ = y and ν ( w ∗ ) = z . (40)Coupling (39) and (40) implies the equivalence of Lemma 2. Proof of Theorem 1.
As was briefly discussed in the proof of Proposition 6, the solutions of (34)share a common value ν ( f ∗ ) = y λ . Hence, the optimization problem is equivalent to V λ ( y ) = arg min f ∈M L ( T ) , ν ( f )= y λ k L f k M . (41)12oreover, the optimization problem arg min w ∈M ( T ) , ν ( w )= z k w k M (42)has a unique solution. Indeed, this is obvious for z = (the unique solution being w ∗ = 0 ). For z = 0 , we are in the conditions of Corollary 1 since z = 0 < | z k | for some k , which impliesuniqueness. Due to Lemma 2, the optimization problem on the right in (38) therefore also admits aunique solution. Hence, (41) also admits a unique solution, as expected.Theorem 1 guarantees the uniqueness of the spline solution for (34). For many classic data-fidelity, one can actually be slightly more precise and show that the mean of the solution is known. Proposition 7.
We assume that we are under the conditions of Theorem 1 and moreover that thedata fidelity E is such that y = arg min z ∈ R E ( y , z ) (43) where y = ( y , y , . . . , y K c ) ∈ R + × C K c and z = ( z , z , . . . , z K c ) ∈ C K c . Then, the unique solution f ∗ to (34) is such that b f ∗ [0] = y .Proof. We use the unique decomposition f = a + L † w ∈ M L ( T ) with a ∈ R and w ∈ M ( T ) . Thisdecomposition implies that f ∗ = a ∗ + L † w ∗ where ( a ∗ , w ∗ ) = arg min a ∈ R , w ∈M ( T ) E ( y , ν ( a + L † w )) + k w k TV . (44)Then, we have that ν ( a + L † w ) = ( a, d L † w [1] , · · · , d L † w [ K c ]) . Moreover, w being fixed, (43) impliesthat E ( y , ν ( a + L † w )) ≥ E ( y , ν ( y + L † w )) , with equality if and only if a = y . The constant a does not impact the regularization hence we always have that a ∗ = b f ∗ [0] = y as expected. Remark.
The relation (43) is typically satisfied by the quadratic data-fidelity (4), and actuallyby most of the data-fidelity encountered in practice. Proposition 7 hence ensures that the mean ofthe solution f ∗ is given by the observation y . It then suffices to focus on the measure w ∗ ∈ M ( T ) in the unique decomposition f ∗ = L † w ∗ + a ∗ (see Proposition 3). This paper deals with continuous-domain inverse problems, where the goal is to recover a periodicfunction from its low-pass measurements. The reconstruction task is formalized as an optimizationproblem with a TV-based regularization involving a pseudo-differential operator. It was known thatspline solutions always exist (representer theorem), and our goal was to investigate the uniquenessissue. We have shown that two scenarios occur in general: either the solution is unique, or theinnovation of any solution is a measure of constant sign. Once applied to elliptic regularizationoperator, we were able to exclude the second scenario, implying that the solution is always uniquein this case. 13 cknowledgments
The authors are grateful to Christian Remling for his help regarding the extension of positive linearfunctionals in Proposition 10. Julien Fageot is supported by the Swiss National Science Foundation(SNSF) under Grant P2ELP2_181759. The work of Thomas Debarre is supported by the SNSFunder Grant 200020_184646 / 1. Quentin Denoyelle is supported by the European Research Council(ERC) under Grant 692726-GlobalBioIm.
A Deferred Proofs from Section 2
Proof of Proposition 1.
The Fourier sequence of L is also the one of L { X } ∈ S ′ ( T ) , which is boundedby a polynomial (as any Fourier sequence of a periodic generalized function; see [49, Chapter VII].This implies the existence of B and β in (10). For an invertible L , the Fourier sequence of L − is (1 / b L [ k ]) k ∈ Z and is also bounded by a polynomial, giving A and α in (10). The relation (11) is thenobvious for k = 0 (all the quantities are vanishing) and easily deduced from (10) applied to L + 1 for k = 0 . Finally, L f = 0 if and only if b L [ k ] b f [ k ] = 0 for any k ∈ Z . For L elliptic, due to (11), thisis equivalent to b f [ k ] = 0 for any k = 0 . Hence, the null space of L is Span { } . Proof of Proposition 2.
The first relation is obvious for k = 0 since | b w [0] | = |h w, i| ≤ sup k ϕ k ∞ ≤ h w, ϕ i = k w k M (45)by picking both ϕ ≡ and ϕ ≡ − . It is worth noting that this argument does not work for k = 0 because b w [ k ] = h w, e k i with e k a complex continuous function. Let us fix k = 0 and assume that b w [ k ] = r e i α = 0 with r > and α ∈ [0 , π ) (the case b w [ k ] = 0 is obvious). Let ψ = e − i α e k .Then, we have h w, ψ i = e − i α b w [ k ] = r and h w, ψ i = h w, ψ i = h w, ψ i = r , where we used the factthat w is real on the second equality. Hence, setting ϕ = ℜ ( ψ ) , we have that ϕ ∈ C ( T ) with k ϕ k ∞ ≤ sup x ∈ T | ψ ( x ) | = 1 and | b w [ k ] | = r = h w, ψ i + h w, ψ i h w, ϕ i ≤ sup k ϕ k ∞ ≤ h w, ϕ i = k w k M , (46)as expected.Any w ∈ M ( T ) can be uniquely decomposed as w = w + − w − where w + and w − ∈ M + ( T ) (Jordan decomposition). Moreover, k w k M = k w + k M + k w − k M = h w + , i + h w − , i = b w + [0]+ b w − [0] .Then, w ∈ M + ( T ) ⇔ w − = 0 ⇔ b w − [0] = 0 ⇔ k w k M = b w + [0] = b w [0] , (47)as expected. Finally, w ∈ M + ( T ) is a probability measure if and only if k w k M = 1 , which concludesthe proof. B Proof of Proposition 5
We first observe that V ( y , e i θ y ) = { w ∈ M ( T ) , w ( · − θ ) ∈ V ( y , y ) } . (48)One can therefore assume that y = r e i α is such that α = 0 , the solution set with a general y thenbeing obtained by a translation of ( − α ) . 14ccording to Lemma 1, we have that min b w [0]= y , b w [1]= r k w k M ≥ max( y , r ) . In particular, if w satisfies b w [0] = y , b w [1] = r , and k w k M = max( y , r ) , then w ∈ V ( y , r ) . We then verify that thisis the case for w ∗ given by (25). Indeed, we have b w [0] = y + r y − r y , (49) b w [1] = y + r y − r i π = y + r − y − r r, and (50) k w k M = (cid:12)(cid:12)(cid:12)(cid:12) y + r (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) y − r (cid:12)(cid:12)(cid:12)(cid:12) = max( y , r ) , (51)which shows that w ∗ is solution and that min b w [0]= y , b w [1]= r k w k M = max( y , r ) . Moreover, thisimplies that V ( y , r ) = { w ∈ M ( T ) , b w [0] = y , b w [1] = r, and k w k M = max( y , r ) } . (52) The case y < r . According to Corollary 1 , this implies that the solution to (24) is unique, andis therefore given by w ∗ in (25). The case y = r . According to Proposition 4, the extreme point solutions are sum of at most K c + 1 = 3 Dirac masses. Moreover, (24) has a unique solution if and only if it has a uniqueextreme point (since V ( y ) is the weak-* closure of the convex hull of its extreme points). Let w = a X ( · − α ) + a X ( · − α ) + a X ( · − α ) be an extreme point solution with distinct α p .Any solution is such that k w k M = y = b w [0] , which is only possible for non-negative measures (seeProposition 2. This implies that a , a , a are non-negative. Without loss of generality, we assumethat a ≥ a ≥ a ≥ , with a = 0 . Then, y = | a e i α + a e i α + a e i α | ≤ | a | + | a | + | a | = a + a + a = y . (53)Hence, | a e i α + a e i α + a e i α | = a + a + a , which is only possible if the four points a p e i α p , p = 1 , , and z = 1 of the 2d plan C are positively colinear. Because the α p are distinct, thisimplies that α = 0 and a = a = 0 , and then a = y . Hence, w ∗ = y X : the solution is uniqueand we recover w ∗ in (25). The case y > r . Our goal is to characterize the solutions of the form w = a X ( · − α ) + a X ( · − α ) . (54)One easily sees that there is no solution with only one Dirac mass ( w = a X ( · − α ) is such that | b w [0] | = | a | = | b w [1] | ), hence the solutions with Dirac masses are extreme points. As for y = r , thesolutions are non-negative measures, hence a , a > . Without loss of generality, one can assumethat a ≥ a .Then, if a < y − r , w cannot be solution. Indeed, we have that r = a e i α + a e i α with a = y − a . Hence, r − y e i α = a (e i α − e i α ) and therefore y − r ≤ | r − y e i α | = a | e i α − e i α | ≤ a (55)We therefore fix α ∈ (cid:2) y − r , y (cid:3) , meaning that α = y − α ∈ (cid:2) y , y + r (cid:3) . Then, denoting by C ( z, ρ ) the circle of center z ∈ C and radius ρ > , the point a e i α is at the intersection between C (0 , a ) and C ( r, a ) . Due to a ≥ y − r , this intersection is non-empty: it contains element if a = y − r and if a > y − r . This gives one or two possibilities for α , and once α is determined, α has onlyone possibility. This gives a complete geometric description of the solutions with Dirac masses.This construction shows in particular that there are uncountably infinitely many extreme points in V ( y , r ) , as expected. It is possible to give an elementary demonstration of this fact without using the machinery behind Corollary 1. Trigonometric Toolbox
In this section is dedicated to known theoretical results (or easily deducible from known ones) thatplay a crucial role for the main result of this paper.A sequence ( a k ) k ∈ Z of complex numbers is positive-definite if a ∈ R + , a − k = a k for any k ≥ ,and for any sequence ( z k ) k ∈ Z of complex numbers with finitely many non-zero terms, we have X k,ℓ ∈ Z a k − ℓ z k z ℓ ≥ . (56) Proposition 8 (Herglotz Theorem) . A sequence ( a k ) k ∈ Z is positive-definite if and only if thereexists a non-negative measure w ∈ M + ( T ) such that b w [ k ] = a k for all k ∈ Z . This theorem was obtained by Herglotz in [50]. For a modern exposition, we refer to [51, Theorem7.6].For K ≥ , we denote by P K ( T ) the set of real trigonometric polynomial of degree at most K ; i.e. , functions of the form p = P | k |≤ K c k e k with z k such that c ∈ R and c − k = c k ∈ C for any ≤ k ≤ K . Proposition 9 (Fejér–Riesz Theorem) . Let p = P | k |≤ K c k e k ∈ P K ( T ) be a positive trigonometricpolynomial of degree K ≥ . Then, there exists a complex trigonometric polynomial q = P K c k =0 z k e k such that p = | q | . The Fejér–Riesz theorem was conjectured by Fejér [52] and shown by Riesz [53]. See [54, p.26] for a recent exposition of this classic result. The next proposition deals with the extension ofpositive linear functionals from trigonometric polynomials to the space of continuous functions.
Proposition 10.
Let K ≥ . Let Φ : P K ( T ) → R be a linear and positive functional ( i.e. , Φ( p ) ≥ for any p ≥ ). Then, there exists an extension Φ : C ( T ) → R which is still linear and positive.Moreover, any such extension is continuous on ( C ( T ) , k·k ∞ ) .Proof. Let E be an ordered topological vector space, C its positive cone, and M ⊂ E . Then,according to [55, Corollary 2 p. 227], if C ∩ M contains an interior point of C , then any continuous,positive, and linear form over M can be extended as a continuous, positive, and linear form over E .We apply this result to E = C ( T ) , whose positive cone is the space of positive continuous functions C + ( T ) , and to M = P K ( T ) . Then, C ∩ M = C + ( T ) ∩ P K ( T ) contains the constant function p = 1 ,which is an interior point of C + ( T ) due to { f ∈ C ( T ) , k f − k ∞ ≤ } ⊂ C + ( T ) .In our case, Φ is continuous over ( P K ( T ) , k·k ∞ ) , since it is a linear functional over a finite-dimensional space. Hence, Φ is continuous, positive, and linear, and admits the desired extension. D Duality and Certificates for Convex Optimization on MeasureSpaces
The analysis of (2) and (6) benefits from the theory of duality for infinite dimensional convexoptimization, as exposed for instance by Ekeland and Temam in [56]. This line of research hasproven to be extremely fruitful for optimization on measure spaces [3, 33, 57, 46]. We focus on thefollowing problem for fixed K c ≥ and y ∈ R + × C K c : V ( y ) = arg min w ∈M ( T ) , ν ( w )= y k w k M (57)16e will mostly rely on the concepts and results exposed in [3, 46]. We recall that a Radon measure w can be uniquely decompose as w = w + − w − where w + and w − are non-negative measures (Jordandecomposition). Definition 5.
Let w ∈ M ( T ) . We define the signed support of w by supp ± ( w ) = supp( w + ) × { } ∪ supp( w − ) × {− } . (58) Let η ∈ C ( T ) be such that k η k ∞ ≤ . The positive and negative saturation sets of η are given by sat + ( η ) = η − ( { } ) and sat − ( η ) = η − ( {− } ) , (59) respectively. Finally, we define the signed saturation set of η by sat ± ( η ) = sat + ( η ) × { } ∪ sat − ( η ) × {− } . (60)We summarize the main results which help us to characterize the cases of uniqueness for (57). Proposition 11.
Let K c ≥ and y ∈ R + × C K c and w ∈ M ( T ) . The following statements areequivalent. • The measure w is a solution of (57) . • There exists a real trigonometric polynomial η ∈ C ( T ) of degree at most K c such that k η k ∞ ≤ and k w k M = h w, η i . • There exists a real trigonometric polynomial η ∈ C ( T ) of degree at most K c such that k η k ∞ ≤ and supp ± ( w ) ⊂ sat ± ( η ) .In this case, for any other solution w ∗ ∈ V ( y ) , we have that supp ± ( w ) ∗ ⊂ sat ± ( η ) . (61)The function η in Proposition 11 is called a dual certificate of (57). Note that a dual certificatealways exist for Fourier sampling [1]. Proposition 11 is an application of the main results of [3],where dual certificates of optimization problems of the form (57) are studied . A key role is playedby the adjoint operator ν ∗ : R + × C K c → C ( T ) (denoted by Φ ∗ in [3]). For the Fourier samplingscenario, we have that ν ∗ ( c , c , . . . , c K c ) = X | k |≤ K c c k e k (62)with the convention that c − k = c k for ≤ k ≤ K c . This explains the role of trigonometricpolynomials in Proposition 11. We do not provide a detailed proof of Proposition 11 since it hasalready been exposed elsewhere. It is for instance done in [46, Propositions 1 & 2] in a differentsetting, but the arguments can be readily adapted.An important consequence for the uniqueness is the following proposition, that can also bededuced from [3]; see also [27]. We provide a proof for the sake of completeness. Duval and Peyré consider more general measurement operators that can even be infinite dimensional and exemplifytheir results for low-frequency measurements. Strictly speaking, the fact that we use complex
Fourier series measurements implies that the dual certificats ofProblem (57) are complex trigonometric polynomials of the form P K c k =0 c k e k , which differs from (62). However, onecan modify the initial problem into an equivalent one by imposing that b w [ k ] = y k for − K c ≤ k ≤ K c with theconvention y − k = y k , leading to (62). It is indeed more convenient to work with real dual certificates when dealingwith real Radon measures. roposition 12. If there exists a non-constant dual certificate for the optimization problem (57) ,then it has a unique solution of the form w = P Kk =1 a K X ( · − x k ) with K ≤ K c .Proof. We start with three observations:1. With Proposition 11, we know that for any certificate η and any solution w ∗ to (57), we havethat supp ± ( w ∗ ) ⊂ sat ± ( η ) .2. Assume that a non-constant certificate η exists. Firstly, η is a trigonometric polynomial ofdegree at most K c and so is its derivative η ′ . Each saturation point x ∈ T is a local optimum of η and is therefore a root of η ′ . The non-degeneracy of η implies that η ′ has finitely many roots, thenumber of which is then bounded by K c [58, p. 150].3. Let τ = ( τ , . . . , τ P ) ∈ T P be the distinct roots of η ′ , with P ≤ K c . For a ∈ R P , weintroduce w a , τ = P Pp =1 a p X ( · − τ p ) . Consider the mapping Φ : R P → R × C K c such that Φ( a ) = ν ( w a , τ ) . (63)Then, Φ is injective. Indeed, let a such that Φ( a ) = . Then, for each − K c ≤ k ≤ K c , we have that b w a , τ [ k ] = 0 (this is due to (63) for k ≥ and to Hermitian symmetry, w being real by assumption,for k < ). This provides K c + 1 linear equations over a that can be written in matrix form as M a = with M = e − i K c τ . . . e − i K c τ P ... . . . e − i τ . . . e − i τ P . . . +i τ . . . e +i τ P ... . . . e +i K c τ . . . e +i K c τ P = [e i kτ p ] ≤ p ≤ P − K c ≤ k ≤ K c ∈ C (2 K c +1) × P . (64)One recognizes a Vandermonde-type matrix, which is therefore of full rank P since the e i τ p are dis-tinct and P < K c + 1 . This implies that a = and thus that Φ is injective.Let η be a non-constant certificate. Due to 2., it has at most K c saturation points in sat ± ( η ) ,that we denote by τ , . . . , τ P with P ≤ K c . According to 1., any solution is a sparse measure whosesupport is included in sat ± ( η ) . In particular, it is of the form w a , τ for some a ∈ R K . However, dueto 3., there is at most one measure w a , τ such that ν ( w a , τ ) = y . Hence, the solution, if it exists, isunique. Finally, we already know from Proposition 4 that solutions exist, which concludes the proof. References [1] E. Candès and C. Fernandez-Granda, “Super-resolution from noisy data,”
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