Noisy Laplace deconvolution with error in the operator

We adress the problem of Laplace deconvolution with random noise in a regression framework. The time set is not considered to be fixed, but grows with the number of observation points. Moreover, the convolution kernel is unknown, and accessible only through experimental noise. We make use of a recent procedure of estimation which couples a Galerkin projection of the operator on Laguerre functions, with a threshold performed both on the operator and the observed signal. We establish the minimax optimality of our procedure under the squared loss error, when the smoothness of the signal is measured in a Laguerre-Sobolev sense and the kernel satisfies fair blurring assumptions. It is important to stress that the resulting process is adaptive with regard both to the target function's smoothness and to the kernel's blurring properties. We end this paper with a numerical study emphazising the good practical performances of the procedure on concrete examples.


Introduction
Laplace deconvolution is motivated by a wide set of practical applications, ranging from population dynamics or physics to computational tomography or fluorescence spectroscopy (Linz [22,Chap. 2], Ameloot et al. [4], Comte et al. [9]). In the corresponding setting we observe q, the result of the action of a kernel g on the function of interest f , according to the following equation Equation (1) is also refered to as Volterra integral equation. One of its main features is its causal property, since qptq is affected only by the values of f and g at times anterior to t. Of course, only finite samples of qptq are accessible in practice. Moreover, the presence of additional noise justifies the empirical modelization of (1) by the classical regression model, inspired by Abramovich et al. [3] ypt i q " ż t i 0 gpt i´τ qf pτ qdτ`ση i , i " 1, ..., n where 0 ď t 1 ď ... ď t n ď T n are the points of observation, pη i q i"1,...,n are independent standard gaussian variables, and σ is a fixed factor accounting for the precision of the observations. T n is supposed to grow with the number of observations n.
As pointed out in Abramovich et al. [3] and Comte et al. [9], in spite of its apparent similarity with the Fourier deconvolution problem, the theoritical features of equation (1), as well as the practical problems raised during its resolution are deeply different. More precisely, setting artificially gptq " f ptq " 0 for t ă 0 amounts to solving the classical Fourier deconvolution problem ypt i q " ż Tn 0 gpt i´τ qf pτ qdτ`ση i , i " 1, ..., n A first notable objection is that the framework of classical Fourier deconvolution assumes periodicity of the function f and the kernel g on r0, T s, a meaningless notion when applied to a varying time set r0, T n s. Even more problematic is the fact that this modelization totally ignores the causal feature of Laplace convolution, creating unwanted interferences between different time sets. To finish, the manipulation consisting in artificially expanding q and g for t ă 0 creates artifacts on the estimated function at times t ă 0 as well.
Another approach is to treat equation (2) as a general ill-posed problem and apply a Tikhonov regularization (Golubev [15]). However the direct implementation of this method also destroys the causal nature of equation (1), and tends to oversmooth the solution (Cinzori and Lamm [7]). Subsequent adaptations which remedy these shortcomings are present in Lamm [20] and Cinzori and Lamm [7]. However in these works the time set is considered to be fixed. A more suitable theoritical tool in solving (1) is the use of Laplace transform, which allows to derive a closed form of the solution. However, its direct implementation is compromised by numerical problems, since the generic expression of the inverse Laplace transform is not easily computable in general. This motivates the widespread use of inversion tables, unfortunately irrelevant when the image function is not known exactly but approximated via a numerical scheme. In this paper, following Comte et al. [9], we will exploit the properties of Laguerre functions, which can be used either to compute the inverse Laplace transform (Abate et al. [1], Lien et al. [21]), or to solve directly equation (1) (Keilson and Nunn [19]). More precisely, a Galerkin method applied to (2) shows that, even if their role is not entirely symmetric to the role played by harmonics in the framework of Fourier deconvolution, they allow a sparse analysis of equation (1). All the previous mentionned works only concerned the case of a deterministic noise at best. The presence of random noise requires an additionnal treatment, and calls for specific statistical tools. In the setting of random noise, Dey et al. [11] considered a kernel of the form e´a t and used a regularized inversion of the inverse Laplace transform. More recently, Abramovich et al. [3] conceived an optimal procedure in the minimax sense on Hölder spaces H s pR`q. This procedure used an exact expression of the solution involving the derivatives of q, which were then estimated via Lepskii's method. However a shortcoming of the procedure is its strong dependence on the kernel g, in the sense that a small error in g can translate into a wide difference in the result. In other words there seems to be a trade off between the closed form of the solution, and the unstability with regard to the kernel. Moreover, the fact that g is seldom observed directly in practice, but is usually subject to experimental noise should prompt us to privilege stability over exactitude.
In that spirit, Comte et al. [9] took advantage of the algebraic properties of Laguerre functions in the context of (2). With an adequate penalty term, they proposed an estimator which mimicks the oracle risk to within logarithmic terms. This modelization has the non negligible advantage of practical simplicity and efficiency, since solving equation (1) amounts to the inversion of a lower triangular Toeplitz matrix. Even if this latter procedure proves to be more stable with regard to g experimentally, no systematic study has been conducted on the subject yet. In this paper we attempt to fill in this gap: we suppose that the observation of g is contaminated by a gaussian white noise, and show how Laguerre functions allow to handle this issue. We place ourselves under the minimax point of view and suppose that f belongs to a Laguerre-Sobolev space and that g satisfies standard blurring assumptions. We apply recent techniques for the treatment of noisy operators in the context of inverse problems (Hoffmann and Reiß [17],Delattre et al. [10]), which consist in a preliminary processing of the operator K coupled with a classical thresholding procedure applied to y.

Laguerre functions
Suppose that the target function f and the kernel K both lie in L 2 pR`q. Define the Laguerre polynomials (see Gradshteyn and Ryzhik [16]) and, following Comte et al. [9], the ensuing Laguerre functions, depending on the parameter a ą 0, The parameter a is a tuning parameter used to fit experimental curves. The Laguerre functions constitute a Hilbert basis of L 2 pR`q. Any function f P L 2 pR`q satisfies The following proposition illustrates the conveniency of Laguerre functions in the framework of equation (1). @a ą 0, @t ě 0, From now on, except if explicitly mentionned, we will suppose a " 1 2 .

Galerkin method
Proposition 2.1 prompted Comte et al. [9] to apply a Galerkin scheme to equation (1). Galerkin schemes rely on the choice of a set of functions which discretize the inverse problem at stake in a convenient way. They were beneficially applied in the context of inverse problems (Cohen et al. [8]), and blind deconvolution (Efromovich and Koltchinskii [14], Hoffmann and Reiß [17] and Delattre et al. [10]). To this end we will remind briefly the underlying methodology of a Galerkin scheme and show how it conveniently applies to equation (1). Let f P L 2 pR`q and K an operator of L 2 pR`q, and suppose we want to recover f from the observation q " Kf . Note V ℓ the finite dimensional space spanned by the orthogonal set of Laguerre functions tϕ k u kďℓ . The Galerkin approximation f ℓ of f on V ℓ is the solution of the equation We shall note K ℓ the Galerkin matrix pK ℓ q i,j " xKϕ j , ϕ i y, i, j ď ℓ. Note hence K the operator of L 2 pR`q mapping f onto t Þ Ñ ş t 0 f pt´τ qgpτ qdτ . We can reformulate (1) as Moreover, Proposition 2.1 implies: Proposition 2.2 (Comte et al. [9], Lemma 1). The Galerkin matrix K ℓ is lower triangular, Toeplitz. More precisely, note 9 g the function with Laguerre coefficientš In the sequel, for any function f P L 2 pR`q, we will note T pf q the infinite Toeplitz matrix such that T pf q i,1 " f i`1 for all i ě 0, and T ℓ pf q the extracted matrix defined by T ℓ pf q i,1 " T pf q i,1 , i ď ℓ`1. In particular, The resolution of the linear system (8) now shows great practical conveniency, provided that K ℓ is invertible. This is equivalent toǧ 0 ‰ 0, an assumption we will make in the sequel.

Application to the regression model with irregular design
It remains to incorporate two supplementary features of equation (2) in the inversion of (9). First, the presence of the random noise η and secondly, the possible irregularity of the design points. This construction is due to Comte et al. [9]. Due to the fact that the observation points t i are imposed by the problem, the estimation of the Laguerre coefficientsq ℓ of the function q suffers from two potential drawbacks. First, the infinite support of the Laguerre polynomials as well as the function q which should not be too problematic, provided that T n is large enough and that the functions decrease sufficiently to infinity. More problematic is the fact that the observation points t i are sometimes subject to experimental constraints, which affect their repartition on R`. The consistency of the estimation ofq ℓ is hereby deteriorated. We will hence suppose that the following conditions are fulfilled: ‚ There exists an integer n 0 such that n Tn ą σ for all n ě n 0 . ‚ lim nÑ8 T n " 8, and lim nÑ8 T n n " 0 To take into account the irregularity of the design, we follow Comte et al. [9] and define P n : r0; T n s Ñ r0; T n s a regular non decreasing function such that P n p0q " 0, P n pT n q " T n , P n pt i q " i n T n for i ď n Note Φ ℓ the pℓ`1qˆn matrix with entries pΦ ℓ q k,i " ϕ k pt i q. For any function h P L 2 pRq, we have where P ℓ is the orthogonal projector onto V ℓ . We deduce that where η ℓ " N p0, I n q. Let us take a closer look to the matrix p t Φ ℓ Φ ℓ˘. Its general term is ż Tn 0 ϕ k pτ qϕ ℓ pτ qP 1 pτ qdτ for n, T n large enough. If the points t i are equispaced, taking P pτ q " τ in (10) entails that T n n´1p t Φ ℓ Φ ℓ˘i s close to the identity provided that T n is large enough. As in Comte et al. [9], we hence reformulate (11) as the sequential model , Ω ℓ q and Ω ℓ " nT´1 n p t Φ ℓ Φ ℓ˘´1 . In general, Ω ℓ somehow quantifies the distance to the uniform design case. To ensure that the design is not too ill conditionned, we will suppose that the following assumption is fulfilled.
Assumption 2.3. Let L P N. There exists C ě 0, such that for all ℓ ď L, for all λ P SppΩ ℓ q, λ ď C This assumption is dependent on the integer L, which plays the role of a maximal resolution level, and will be adapted to the case of interest later. The inversion of (11) now requires controls of the variable pK ℓ q´1ξ ℓ . Under suitable properties of f and g, we shall be able to apply a classical inverse/thresholding procedure, and derive rates of convergence over specific regularity spaces. These properties are the subject of Part 3.

Error in the operator
We already mentionned the fact that the resolution of (1) is usually unstable with respect to g (Abramovich et al. [3]). Furthermore, in practice, inference on the kernel g is possible only through experimental noise, and requires a preliminary step of estimation giving way to imprecision. This additionnal error might significantly contaminate the result of any procedure of estimation if not properly treated. Let us see how Laguerre functions ϕ ℓ allow to handle this issue: in section 2.2, we established that the discretization of (13) with Laguerre functions involved a Toeplitz matrix with entries constituted of the Laguerre coefficients of 9 g. We can thus consider 9 g as the finite impulse response of the operator K when applied to the system pϕ ℓ q ℓě0 . To take into account the imprecision in the observations of 9 g, we adopt the framework of blind deconvolution and suppose that 9 g is not known exactly, but that we have acces the noisy version 9 g δ " 9 g`δb (12) where b is a gaussian white noise on L 2 pR`q. The generic problem of blind deconvolution is motivated by numerous scientific fields, including for example electronic microscopy or astrophysics, where the corresponding kernel is seldom known nor directly observed. It was adequatly discussed in Efromovich and Koltchinskii [14] and Hoffmann and Reiß [17]. Taking into account the observations (12), the projection 9 g ℓ is changed to 9 g ℓ δ " 9 where B ℓ " T ℓ pbq is a random Toeplitz matrix. In the sequel, for the sake of clarity, we note ε " σ b Tn n . Remark 2.4. We could as well suppose that we observe g δ " g`δb, yet it is more convenient to work with 9 g (the entries of the noisy Toeplitz matrix B are directly i.i.d standard gaussian variables). In the former case, the rest of the paper however adapts with no change in the algorithms, since inequality (27) is satisfied as well. A modification of the proof of Theorem 4.6 should also provide the lower bound for the second procedure.

Sobolev spaces associated to Laguerre functions
We proceed to the description of regularity spaces associated with the resolution of (13). The following material is classical, we refer to Bongioanni and Torrea [5] or Rathnakumar [26] for example. Since f Þ Ñ ? 2af p2a.q is an isometry of L 2 pR`q, the structures defined for different values of a are equivalent. Hence we shall only concentrate on the mainstream case where a " 1{2. Define the operator L on L 2 pR`, dxq by The functions ϕ ℓ are the eigenfunctions of L associated with eigenvalues pℓ`1 2 q. We hence define the Sobolev space W s associated with Laguerre functions as For a function f P L 2 pR`, dxq, we have the straightforward equivalence xf, ϕ ℓ yˇˇ2 ă 8 and the associated norm For M ě 0, we shall note W s pMq the Sobolev ball of radius M. Finally, we remind that, as }ϕ ℓ } 8 ď 1 for all ℓ ě 0, we have s ą 1{2 ñ W s Ă C 0 pR`q. From now on, we will hence suppose that there exists s ą 1{2 such that f P W s .

Banded Toeplitz matrices
Before entering into details about the kernel features, we introduce basic material on Toeplitz matrices. Most of it is inspired by Böttcher and Grudsky [6] and Comte et al. [9]. Let a " pa ℓ q P ℓ 1 pZq be a sequence of real numbers. We remind from section 2.2 that we note T paq the infinite Toeplitz matrix defined by The Toeplitz matrices T paq and T ℓ paq are naturally linked to the two respective Laurent series apzq " We will indifferently refer to the vector a or the corresponding Laurent serie. The spectral norm of T paq is related to the behaviour of apzq, as illustrated in the following proposition.
Proposition 3.1. Let a P ℓ 1 pZq. Let C stand for the complex unit circle. We have a ℓ z ℓˇ. A simple corollary is the following inequality In particular, Proposition 3.1 applies to the case of truncated Toeplitz matrices T ℓ paq. Moreover, if a has no zero on the complex unit circle, we have lim sup ℓÑ8 }T ℓ paq} op ă 8 and lim ℓÑ8 }T ℓ paq} op " }T paq} op (15) Now suppose that a and a 1 both generate lower triangular Toeplitz matrices (i.e. a k " a 1 k " 0 if k ă 0). Then the following equalities hold for all ℓ ě 0: In other words, the matrix multiplication (resp. inversion) is equivalent to a power serie multiplication (resp. inversion).

Degree of ill posedness
We now need to precise the properties of K as a blurring operator of L 2 pR`q. Usually the operator K is not compact, and the problem (1) is ill-posed. This results in practical unstabilities when trying to invert equation (11) from discrete observations. The quantification of the ill-posedness of the problem is specified by the introduction of a constant, called degree of ill-posedness (DIP) of the problem (see Nussbaum and Pereverzev [25], Mathe and Pereverzev [23] for a generic review). We adapt this concept to our framework, and make the following assumption.
Assumption 3.2 (Degree of ill-posedness of g). There exists ν ě 0, Q ě 0 such that, for all ℓ ě 0, }pK ℓ q´1} op ď Qpℓ _ 1q ν ν is called degree of ill-posedness of g (or equivalently of K). We note K ν pQq the set of functions which satisfy this assumption.
We shall see examples of kernels satisfying this assumption further. For the moment, we concentrate on the treatment of observations (13) in the context we just described.

Algorithms and rates of convergence
The main challenge which remains to be treated now is to articulate the two critical steps of inversion and regularization, via adapted procedures. For example, let us give a brief overview of the methodology in Comte et al. [9]: Let ℓ P N, and let Λ be the following contrast function, defined on R ℓ by Λ : t Þ Ñ }t} 2´2 xt, pK ℓ q´1y ℓ y Note }.} op the spectral norm and }.} HS the Hilbert Schmidt norm. A model selection is performed on the maximal level L, by introducing the following penalizing factor (B ą 0 is an arbitrary constant): where Q ℓ " pK ℓ q´1Ω ℓ t pK ℓ q´1 and a Q ℓ is a lower triangular matrix satisfying a Q ℓ t a Q ℓ " Q ℓ . The maximal levelL is hence chosen as where ℓpnq is a large enough resolution level, possibly depending on n, and the ensuing estimator of f is pKLq´1yL We follow here a different path: we suppose that the target function belongs to a Sobolev-Laguerre space, and perform thresholding techniques in a minimax framework. Furthermore, our results are asymptotic with regard to ε, δ. Would g be known, the estimation of f from observations (13) amounts to solving a standard inverse problem with signal noise. To this end, a prolific litterature is at disposal (a selected list is Donoho [12], Abramovich and Silverman [2], Cohen et al. [8]). In order to take into account the presence of noise in the operator, we shall hence apply a preliminar regularizing thresholding procedure to the noisy operator K δ in order to ensure the stability of the further inversion step. To that end, define the maximal level as with λ a positive constant. Define also the two thresholding levels We call this procedure Algorithm I. The preliminary threshold performed on pK ℓ δ q´1 ensures its proximity with pK ℓ q´1 with high probability (see Lemma 6.2). We now study the squared loss performance of the procedure.
The rates in Theorem 3.3 reveal two components, accounting respectively for the imprecision in the observation of the operator and the signal. The latter is fairly classic in non parametric statistics (Nussbaum and Pereverzev [25], Johnstone et al. [18]) where it is also optimal, while the former is standard (and optimal too) in blind deconvolution on Hilbert spaces (Efromovich and Koltchinskii [14], Hoffmann and Reiß [17]). Thus, we do not study the optimality of these rates in this paper, but rather concentrate on a more specific framework related to the problem of interest.

Adaptation to the standard framework of Laplace deconvolution
We now discuss the adapation of our algorithm in the mainstream framework of Laplace deconvolution, as exposed in Abramovich et al. [3] or Comte et al. [9]. As we shall see, this more restrictive framework allows to treat observations (13) more efficiently. To this end, we first define a more restrictive version of the degree of ill-posedness. gq, ϕ k y, so that p1{ 9 gqpzq " ÿ kě0 γ k z k . There exists ν ą 0, there exists Q 2 , Q 1 ą 0, such that for all ℓ ě 0, For Q " pQ 1 , Q 2 q, we note G ν pQq the set of functions g P L 2 pR`q such that Assumption 4.1 holds. Note that the validity of this assumption automatically entails Q 1 ď`1`2 2ν 2ν˘Q 2 . Note also that the left term in (21) is the Hilbert-Schmidt norm of pK ℓ q´1. Thus, Assumption 4.1 is more restrictive that Assumption 3.2. However, it is satisfied by a natural class of functions g: Proposition 4.2 (Comte et al. [9], Lemma 3/ Lemma 5). Suppose that there exists C, ν ą 1{2, µ P C and wpzq " ś N i"1 pz´µ i q, |µ i | ą 1 a polynomial function with no pole inside of the complex unit disc, such that Then Assumption 4.1 is satisfied. Furthermore, if w " 1 and ν ě 0, then |γ ℓ | " ℓ ν´1 Γpνq . For completeness, we give a proof of Proposition 4.2 in section 6. We now turn to the standard framework of Laplace deconvolution, as exposed in Abramovich et al. [3] and Comte et al. [9]. To this end, we define the following assumptions concerning the kernel g. (A1) There exists an integer r ě 1 such that (A2) g P L 1`r 0,`8q˘is r times differentiable and g prq P L 1`r 0,`8q˘.
(A3) The Laplace transform of g has no zeros with non negative real parts except for the zeros of the form 8`ib.
The consequences of these assumptions are well formulated in the terms of the preceding framework: Hence, Assumption 4.1 is verified with ν " r and Algorithm I applies. However, Assumption 4.1 provides additional information on the behaviour of p1{ 9 gq. We adapt Algorithm I to this new framework, by operating the following changes: ‚ Set the maximal level to where }A} HS " a Trp t AAq is the Hilbert-Schmidt norm. We call the modified procedure Algorithm II and note r f II the corresponding estimated function. A notable gain of this new algorithm is its independence with regard to the parameter ν. Indeed, Assumption 4.1 allows us to use }pK ℓ δ q´1} HS in (19) as a substitute of ℓ ν , and to overesimate the 'true' maximal level L I . Its performances are exposed in the next theorem: Thus, in addition to the adaptivity over the parameter ν, the strengthening of Assumption 3.2 via (20) and (21) allows to improve on the rates of Theorem 3.3 with regard both to the operator and signal noise. Our next result shows that the rate achieved in Theorem 4.5 is indeed optimal, up to logarithmic terms. The lower bound will not decrease for increasing noise levels δ and ε, whence it suffices to provide separately the cases δ " 0 and ε " 0. Theorem 4.6. Let s ą 1{2, let M ě 0 ν ą 1{2 and Q 2 ě c ν Q 1 ą 0. Here c ν is a constant depending only on ν which will will not seek to precise. We have where the infimum is taken among all estimatorsf of f based on observations (13).
Combining Theorem 3.3 together with Theorem 4.6, we conclude that our algorithm is minimax over W s pMq to within logarithmic terms in ε and δ, uniformly with regard to the blurring kernel g P G ν pQq.

Practical performances
In this section we study the practical performances of the two procedures developped above. Note that three potential sources of errors may contaminate the quality of the observations in (13) : the signal precision σ b Tn n , the operator precision δ and the design quality }Ω ℓ } op . We shall hence emphasize their influence in the estimation of f , as well as their respective interactions. Our first aim is to study the interaction between the effect of signal and kernel noise in the two procedures of reconstruction. To this end, we will isolate them from the effect of the design, and suppose that the latter is ideally conditionned by setting Ω ℓ " I ℓ . Let us start by a few precisions concerning the tuning parameters of Algorithm I and II. The setting up of these procedures requires the preliminary definition of λ, κ, τ sig and τ op . Tuning parameters: for the definition of the maximal level of resolution, we set λ " 1 for both algorithms. The concrete choice of adequate thresholding constants κ and τ is a complex issue. Our practical choices will be based on the following remark, inspired by Donoho and Johnstone [13]: in the case of direct estimation on real line, the universal threshold which is both efficient and simple to implement, takes the form 2 a | log ε|. A consistent interpretation is to consider that this threshold should kill any pure noise signal. We will adapt this reasoning to the case of interest. Choice of κ : we use as a benchmark the case where g " 0. Given δ large enough, we define κ as the smallest value κ δ such that , for all ℓ ď 10, 1 t}pK ℓ δ q´1}opăO´1 δ,l u " 0. The results are reported in Table 1 and give κ " 0.3. Choice of τ sig and τ op : It is clear that the role of τ sig and τ op is to control the influence of the signal (resp. the operator) error. To choose τ sig (resp. τ op ), we therefore set ε sig ą δ sig ą 0 (resp. δ op ą ε op ą 0) large enough. We resort to the case f " 0 as a benchmark: we have xf , ϕ ℓ y " 0 for ℓ ě 1, consequently the observations xg ε sig , ϕ ℓ y, ℓ ě 0 are pure noise. We hence simulate K δ sig and, integrating the precedently computed value of κ, apply the procedure for increasing values of τ sig (resp. τ op q until all the computed coefficients xf i , ϕ ℓ y (i "I,II) are killed for ℓ ď 10. The results are reported in Table 2.
κ 0.1 0.2 0.3 N 3 1 0 Table 1. Choosing of κ. N is the average number, computed on a basis of 10 realizations, of levels ℓ ď 10 such that }pK ℓ δ q´1} op ă O´1 δ,l pκq. We have δ " 10´2.  Table 2. Choosing of τ . For pδ sig , ε sig q " pε op , δ op q " p10´2, 10´1q and each value of τ , we computed 10 times the described procedure and reported N i the average number of remaining Laguerre coefficients for Algorithm i. We now apply the two procedures to the case where f 1 ptq " pt 2´t q expp´tq and g " ϕ 0 (a graphical representation of these two functions is presented in Figure 1).
We have`1 { 9 g˘pzq " p1´zq´1 " ÿ ℓě0 z ℓ hence Assumptions 3.2 and 4.1 are both satisfied taking ν " 1. For several values of ε and δ, we report the corresponding squared loss, computed on a basis of 500 realisations with the use of Parseval's identity, in Table 3. The corresponding results are presented in Figure 2 for one particular realization of ξ, b. The results indicate that the transition on the two types of errors occur when δ is higher than ε, translating a prevailing effect of the signal noise ε over the operator error δ in practice. As Theorems 3.3 and 4.5 suggest, the second Algorithm overperforms the first in (almost) every case. Discussion on the design irregularity: to control the squared risk of the two  Table 3. Normalized mean squared error of the two procedures applied to the functions f 1 and g. The computations were performed using a monte carlo method on 500 realizations. procedures, one needs condition 2.3 to be fulfilled. If not, the eigenvalues of the matrix Ω L become potentially too large, and observations (11) are not conveniently treatable. In this case, it is preferable to lower the maximal level down to a point where }Ω L } op remains under control. To this end, we change the maximal level of the two respective procedures to where α is an arbitrary thresholding constant, set to 1.5 in the sequel. We now fix σ " δ " 10´2 and chose the design points t i as t i " 100i{n for n " 200, 250, 750 and 1000. Taking the same kernel g " ϕ 0 , and setting f 2 ptq " pt 1{2´t q expp´tq, we compare the performances of the new choice N i to the previous one L i , by computing the respective mean squared losses on a basis of 500 observations and report the result in  Table 4. Normalized mean squared error of the two procedures when the design is constituted of 200 equispaced points on the interval r0; 100s. We compare the performances of the two maximal resolution levels L i and N i for the parameters σ " δ " 10´2, g " ϕ 0 and f 2 ptq " pt 1{2´t q expp´tq. since L i is usually already smaller than N i . However, the gain is notable for Algorithm II when n ď 250. To illustrate this point, we plot in Figure 3 the corresponding results when n " 200.

Back to the regression model
We now turn back to the original model (2) to apply the two procedures. It is well known that this model is asymptotically equivalent to (11), in the sense that a fine  enough design will provide an estimation of the Laguerre coefficients with a negligible error when n Ñ 8. We work with f 3 pzq " p1´zq 1{2 , g " ϕ 0 , δ " 10´2, and suppose that the design is constituted of the points t i " i ÿ j"1 pstep`|X j |q where pX j q jďn is an i.i.d sequence of N p0, 10´2q variables. We observe the noisy values ypt i q " qpt i q`ση i where qpzq " p1´zq 3{2 , and compute the Laguerre coefficientsq ℓ via the approximatioň We apply the two procedures and present the results on Figure 4.

Proofs
In the sequel, for the sake of clarity, we suppose that τ sig " τ op ∆ " τ .

Proof of Proposition 4.2
Proof. We can restrict ourselves to the case where µ " 1. Proposition 16 applied to equality (22) entails As a consequence, and }pK ℓ q´1} HS ě C´1}T ℓ`w˘}´1 op }T ℓ`p 1´zq´ν˘} HS Since w is assumed to have no zeros on C, we deduce from Proposition 3.1 that both }w´1} circ and }w} circ are finite, and from (15) that It remains to treat the binomial serie p1´zq´ν. This serie can be expanded as Γpℓ`1qΓp´ν´ℓ`1q is the generalized binomial coefficient. Furthermore, we haveˆ´ν ℓ˙" ℓÑ8 p´1q ℓ Γpνqℓ´ν`1 (26) which is a direct consequence of Euler's definition of the Gamma function Γpzq " lim kÑ8 k!k z Π k i"0 pz`iq . Since ν ą 1{2, the serie ÿ kˆ´ν k˙2 is hence divergent, and there exists r Q 2 , r Q 1 ą 0 such that, for all ℓ ě 0, The proof is complete thanks to (24) and (25).

Preliminary lemmas
We begin with the following lemmas. Lemma 6.1 is a concentration inequality on the variable }B ℓ } op , which results from a concentration inequality on subgaussian processes. Lemma 6.2 states that }pK ℓ δ q´1} op behaves as }pK ℓ q´1} op on a set with large probability. Finally, Lemma 6.3 establishes deviations bounds on the variables ζ ℓ´f ℓ which will be useful throughout the proofs of Theorem 3.3 and Theorem 4.5.
Lemma 6.1. There exists β 0 , c 0 independent from ℓ ě 0, such that, for all ℓ ě 0, for all t ě β 0 , This readily entails the following moments control, available for all ℓ ě 0, p ě 1 Proof. The proof is a slight modification of Meckes [24,Theorem 1], to which we refer for a complete study. Lemma 6.1 is trivially satisfied if ℓ " 0, 1, hence we will suppose that ℓ ě 2. From Proposition 3.1, we derive that We claim the two following facts: ‚ Let a 0 , ..., a ℓ P R. There exists c ě 0 such that ofor all t ą 0, The first point is readily verified since pb k q kďℓ is a standard Gaussian vector, while the second point directly results from the bounďˇe 2iπkx´e2iπkyˇď 2^2πk|x´y| for all x, y P r0, 1s, k ě 0 (see Meckes [24] for the rest of the proof). The deviation bound is now a consequence of Talagrand [27,Lemma 5.3]. Indeed, for all x P r0, 1s, which ends the proof.

Proof of theorem 3.3
Lemma 6.3. Under Assumption 3.2, we have, for all ℓ ě 0, Proof. In order to prove Inequalities (33) and (34), it suffices to study the tails of the random variablesˇˇxpK ℓ δ q´11 A ℓ 1 B ℓ`´δ B ℓ f ℓ`ε ξ L I˘, ϕ ℓ yˇˇ. For convenience we will only treat the case where ℓ ě 2, otherwise the result follows by identical arguments. To this end, we study each term apart. On A ℓ X B ℓ , Lemma 6.2 and Assumption 3.2 entail Thus, combining Assumption 2.3 with the latter inequality, a brief conditionning argument readily yields et us study the second term. On A ℓ X B ℓ , we have Let's now bound separately r 1 and r 2 . We first apply equality (16) to get where pb ℓ q 1 k " pb ℓ q ℓ´k . The result is a centred gaussian variable with variance which hence satisfies et us study the term r 2 . Since the maximal level L verifies L ď λpδ| log δ|q´1 ν`1 , we have, for all ℓ ď L, δℓ ν`1 log ℓ À 1. We deduce that À expp´t`δℓ ν˘´1 q1 ttąβ 0 δℓ ν u`1ttďβ 0 δℓ ν u inequality (34) directly follows, and inequality (33) is now a direct application of the well known formula 2tPp|X| ą tqdt Proof of theorem 3.3. We apply Parseval's formula to derive The second term is easily handled. Remark first that, since s ą 1{2, we have 2s ν`1 ą 2s s`ν`1{2 and we can write ÿ In order to lighten the notations, we will only consider the indexes ℓ ě 2 in the first term. This is of course not problematic, since an identical reasoning allows to bound the two remaining summands by the desired rates of convergence. We hence write the following decomposition ÿ Hence we can decompose further I as Let us first treat the term V I. From Lemmas 6.2 and 6.3 and Cauchy-Schwarz inequality, we derive hich is less than the desired bound for τ large enough. As for term V , we split it in two and write ote ℓ δ "`δ| log δ|˘´2 2ps`νq`1 and write The ε-term is treated similarly by taking ℓ ε "`ε a | log ε|˘´2 2s`2ν`1 and leads to the desired convergence rate. As for the term II, a similar reasonning leads to The term V III is handled as the term V I. Indeed, which is less than the desired rate for τ large enough. Finally, we have Moreover, Lemma 6.1 entails P`B c ℓ˘ď δ κ 2 ρ 2 for all ℓ ě 1. It is hence clear that for κ large enough, the term III is less than the announced rate.
It remains to put together the bounds of the four terms above to get the desired rates of convergence in theorem 3.3.
Proof. The proof is very similar to Lemma 6.3, whence we will just mention the notable changes compared to it. Once more, we shall only treat the case ℓ ě 2. First, we have so that a brief contitionning argument, combined with (30) and Assumption 2.3 entails In order to treat the term Pp|xpK ℓ δ q´11 A ℓ 1 B ℓ δB ℓ f ℓ , ϕ ℓ y| ą tq, we first establish a useful result for the sequel: if g satisfies (20), then }pK ℓ q´1f ℓ } " }T ℓ pf qp1{ 9 gq ℓ } ď }T ℓ pf q} op }p1{ 9 gq ℓ } Furthermore, thanks to Proposition 3.1 we have since f P W s pMq and s ą 1{2. We derive that Let us now bound the term of interest. Once more, we decompose it as r 1`r2 where r 1 and r 2 are defined in (36). We now apply Proposition 16 and (32) and derive The latter is a gaussian random variable with variance δ 2 }pK ℓ q´1f ℓ } 2 À δ 2 ℓ 2ν´1 where we used (40). Turning to the term r 2 , we apply Proposition 16 to derive e now apply (30) and (40) to get }pK ℓ δ q´1f ℓ }} t pK ℓ q´1ϕ ℓ } À ℓ 2ν´1 Hence, et us take a look back to Lemma 6.2. On A ℓ X B ℓ we have prooved that . We deduce À exp`´t δℓ ν´1{2˘1 ttąβ 2 0 δℓ ν´1{2 u`1ttďβ 2 0 δℓ ν´1{2 u The end of the proof is identical to Lemma 6.3.
Proof of Theorem 4.5. The proof is very similar to Theorem 3.3, whence we will just emphasize the notable changes compared to it. First, we apply Parseval's formula to derive The second term is easily handled, since ÿ Thanks to Lemma 6.2 and the definition of S II ℓ,ε , we have on A ℓ X B ℓ . Thus, the Terms I and II can be treated identically to the preceding proof and yield the desired rates of convergence. The terms III and IV are treated exactly as in the preceding proof.

Proof of theorem 4.6
Proof. The lower bound will not decrease for increasing noise levels δ and ε, whence it suffices to provide the case δ " 0 and the case ε " 0 separately. In the sequel, c i will denote a positive constant to be adjusted later and L will play the role of a maximal level. Also, we will note K ν (resp. 9 g ν ) the operator (resp. the function) associated with the Laurent serie p1´zq ν L . The function 9 g´1 {2 will play an essential role in the sequel. Unfortunately it is not square integrable. We thus begin with a preliminary lemma, which states that a minor modification corrects this defect. Lemma 6.5. Let h be the function associated to the Laurent serie Then h is square integrable. Furthermore, for all ν ě 0, for all ℓ ď L, pℓ _ 1q ν´1{2 logpℓ _ 2q À |xK´νh, ϕ ℓ y| À pℓ _ 1q ν´1{2 Proof of Lemma 6.5. h is trivially squared integrable thanks to (26)  This entails the result.
‚ Case δ " 0. For more clarity, we will suppose that ξ is a white noise (the proof readily adapts otherwise). Let hence K 0 " c 1 K ν . Then K 0 P G ν pQq for an appropriate constant c 1 , thanks to Proposition 4.2. Following the arguments of Willer [28], it suffices to find f 0 , f 1 such that i) f 0 , f 1 P W s pMq ii) }f 0´f 1 } 2 Á ε 2s s`ν | log ε|´2 iii) KpP 1 , P 2 q À 1 where P i is the law of y under the hypothesis f i , and K is the Kullback-Leibler divergence.
‚ Case ε " 0. Let L " c 1 δ´1 s`ν . Following the lines of Hoffmann and Reiß [17], we set f 0 " c 2 ϕ 0 , K 0 " c 3 K ν and we only consider couples pK, f q such that Kf " q 0 for a fixed q 0 " K 0 f 0 . It is clear that, for well chosen c 2 and c 3 , we have f 0 P W s pMq and K 0 P G ν pQq. We thus define H the operator associated to the kernel h and introduce K δ " K ν`c4 δH a perturbation of K ν . We shall refer to 9 g δ for the corresponding kernel. Remark that we have f 1´f 0 " c 4 δpK δ q´1Hf 0 " c 4 δpK δ q´1h (41)