R El Beainou
email: raya.elbeainou@femto-st.fr

J-Y Rauch

S Dembélé

O Lehmann

L Hirsinger

M Devel

Qualitative evidence of the flexoelectric effect in a single multiwall carbon nanotube by nanorobotic manipulation

Keywords: Flexoelectricity, nanomanipulation, carbon nanotubes, energy harvesting, polarization

The flexoelectric effect corresponds to the linear variation of the electric polarization of a material subjected to a strain gradient (i.e. during its mechanical bending). Unlike piezoelectricity, it also exists in non-centrosymmetric materials. Furthermore, due to the gradient term, its magnitude can increase as the size of the system decreases. Thanks to this effect, nanoscale systems could be used to harvest thermal vibration energy to power a microdevice. These could be multi-wall carbon nanotubes since they are known to bend easily in an elastic manner. However, it is very challenging to experimentally measure the flexoelectric behavior of a single multi-wall carbon nanotube due to its small size (less than 50 nm in diameter), to the low level of induced charges and to the need to vary the imposed stress.

To progress in this direction, a six-degree-of-freedom robot with a fiber tip is used inside a dual-beam microscope to pick up a few single carbon nanotubes from a tangle and connect them to the fiber tip. After ion-soldering the two tips, each carbon nanotube is dynamically bent several times while monitoring the brightness of the bending area and its effective radius of curvature. This allowed us to demonstrate qualitatively the flexoelectric effect at the level of a single MWCNT.

Flexoelectricity, as reviewed in [START_REF] Mashkevich | Electrical, optical and elastic properties of diamond crystals[END_REF], is the linear coupling between electrical polarization and strain gradient, predicted by Mashkevich and Tolpygo [START_REF] Mashkevich | Electrical, optical and elastic properties of diamond crystals[END_REF], and later phenomenologically described by Kogan [START_REF] Kogan | Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals[END_REF]. It can now be associated to the origin of peculiar features identified as being due to non-homogeneous strain by Scott in previously measured infra-red spectra [START_REF] Scott | Lattice Perturbations in CaWO4 and CaMoO4[END_REF], while Bursian and Zaikovskii measured the converse effect at the same time [START_REF] Bursian | Changes in the curvature of a ferroelectric film due to polarization[END_REF]. However, this was rarely considered for electromechanical transduction in solids due to its small relative magnitude in macroscopic solids. Hence, this field of research was pretty inactive until Ma and Cross [START_REF] Ma | Large flexoelectric polarization in ceramic lead magnesium niobate[END_REF][START_REF] Ma | Observation of the flexoelectric effect in relaxor Pb(Mg1/3Nb2/3)O3 ceramics[END_REF] got results significantly higher than expected according to the theory, after Fousek et al [START_REF] Fousek | Possible piezoelectric composites based on the flexoelectric effect[END_REF] proposed, that flexoelectric effect could be used to get an effective piezoelectric effect by vertically compressing an asymmetrically shaped material. Then, Gruverman et al. [START_REF] Gruverman | Mechanical stress effect on imprint behavior of integrated ferroelectric capacitors[END_REF] showed that it was possible to change the direction of polarization on PZT capacitors by bending the substrate of silicon. Han et al [START_REF] Han | Nanogenerators consisting of direct-grown piezoelectrics on multi-walled carbon nanotubes using flexoelectric effects[END_REF] showed that they obtained a nanogenerator by growing PZT directly on multi-wall carbon nanotube. Other authors (e.g. [START_REF] Chu | Flexure mode flexoelectric piezoelectric composites[END_REF]) developed a nanocomposite using non-homogeneous strains to obtain a behavior similar to a piezoelectric material. All these recent studies extend the potential of the flexoelectric effect.

The microscopic origin of flexoelectricity in ionic crystals can be attributed to the fact that, even in a centrosymmetric crystal (hence, one that cannot be piezoelectric), an inhomogeneous strain separates the centers of positive and negative charges and consequently induces a polarization [START_REF] Ma | Large flexoelectric polarization in ceramic lead magnesium niobate[END_REF]. More generally, the 4 th order tensor characterizing this effect, quantifies the ability of a dielectric material to polarize electrically under a strain gradient. The fact that high strain gradients can be obtained more easily at the nanoscale than at the macroscopic scale opens up possibilities for taking advantage of flexoelectricity in nanoscale systems. However, there is still no experimental report about the flexoelectricity of a single object at the nanoscale. A previous experimental study of the flexoelectric effect using thin films of Multi-Wall Carbon Nanotubes (or MWCNTs) bearing PZT microparticles [START_REF] Han | Nanogenerators consisting of direct-grown piezoelectrics on multi-walled carbon nanotubes using flexoelectric effects[END_REF] showed that it was possible to produce a peak output voltage of 8.6 V and an output current of 47 nA when a force of 20 N was applied. This prompted us to study this effect at the level of a single MWCNT. However, for single MWCNT manipulation and characterization, specific setups are required.

In the past years, Atomic Force Microscopy (or AFM) has appeared as a powerful and versatile tool for the characterization of nanomaterials in nanoscience and nanotechnology [START_REF] Butt | Force measurements with the atomic force microscope: Technique, interpretation and applications[END_REF][START_REF] Garcia | Dynamic atomic force microscopy methods[END_REF][START_REF] Giessibl | Advances in atomic force microscopy[END_REF]. Recently, Xiao Hu et al. [START_REF] Hu | Amplitude response of conical multiwalled carbon nanotube probes for atomic force microscopy[END_REF] used an AFM for MWCNT characterization in order to fabricate conical structure on a commercial conductive AFM probe with a single nanotube This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0065214

protruding from its end. However, a robotic platform inside a Scanning Electron Microscope (or SEM) seems to be a more versatile tool than AFM for nano-manipulation due to the magnification from x400 to over x100.000. Indeed, this kind of platform -with dual beam SEM/ Focused Ion Beam (or FIB) -enables its users to achieve, in an original way, nanomanipulation tasks offering disruptive characterization and nanofabrication potentials: characterization of nanowires [START_REF] Rauch | Smallest microhouse in the world, assembled on the facet of an optical fiber by origami and welded in the μRobotex nanofactory[END_REF][START_REF] Lugstein | Anomalous piezoresistance effect in ultrastrained silicon nanowires[END_REF][START_REF] Zhu | Size effects on elasticity, yielding, and fracture of silver nanowires: In situ experiments[END_REF] and graphene membranes [START_REF] Zimmermann | Automated robotic manipulation of individual colloidal particles using vision-based control[END_REF], assembly of photonic crystal device [START_REF] Aoki | Microassembly of semiconductor threedimensional photonic crystals[END_REF] and nano-wire based transistors [START_REF] Zhang | Automated nanomanipulation for nanodevice construction[END_REF], detailed study of DNA [START_REF] Chen | Nanodissection and sequencing of DNA at single sub-nuclear structures[END_REF], assembly of the smallest microhouse in the world at the tip of an optical fiber [START_REF] Rauch | Smallest microhouse in the world, assembled on the facet of an optical fiber by origami and welded in the μRobotex nanofactory[END_REF], the smallest camera operator [START_REF] Lehmann | Miniaturized robotics, the smallest camera operator bot pays tribute to David Bowie[END_REF] and many others. Thus, this approach has shown great interest compared to other techniques, offering a platform with great versatility and with ease of reconfiguration and adaptation to different environments [START_REF] Ru | Automated four-point probe measurement of nanowires inside a scanning electron microscope[END_REF][START_REF] Zhang | Automated pick-place of silicon nanowires[END_REF][START_REF] Zhang | Piezoresistivity Characterization of synthetic silicon nanowires using a MEMS device[END_REF][START_REF] Xie | High-efficiency automated nanomanipulation with parallel imaging/manipulation force microscopy[END_REF].

In this article, the flexoelectric response of some isolated MWCNTs is studied using a robotic approach, that is to say, the MWCNTs are bent in vacuum by micromanipulators, and observed using a Focused Ion Beam and a Scanning Electronic Microscope. MWCNTs are chosen because they can be reversibly bent, with a small radius of curvature, without breaking.

In addition, they can have good thermal and electrical conductivity [START_REF] Du | Effect of nanotube alignment on percolation conductivity in carbon nanotube/polymer composites[END_REF][START_REF] Astorga | Electrical conductivity of multiwall carbon nanotubes thin films[END_REF][START_REF] Haggenmueller | Single wall carbon nanotube/polyethylene nanocomposites: Thermal and electrical conductivity[END_REF], low threshold voltage for field emission [START_REF] Umnov | Field emission from flexible arrays of carbon nanotubes[END_REF][START_REF] Wang | Flexible field emitter made of carbon nanotubes microwave welded onto polymer substrates[END_REF][START_REF] Hojati-Talemi | Highly efficient low voltage electron emission from directly spinnable carbon nanotube webs[END_REF], and high mechanical strength [START_REF] Ramanathan | Functionalized SWNT/polymer nanocomposites for dramatic property improvement[END_REF][START_REF] Mccullen | Morphological, electrical, and mechanical characterization of electrospun nanofiber mats containing multiwalled carbon nanotubes[END_REF]. In a second paragraph, our experimental setup is described. The way in which the raw data was processed is then presented and finally the results obtained are discussed.

Different types of MWCNTs are deposited on a (100) Si substrate by chemical vapor deposition (CVD) based on heptane and acetylene as a catalyst. A SMF 128 optical fiber is used in order to manipulate the MWCNTs. The optical fiber is stretched in order to have a tip diameter less than 100 nm. The purpose is to be able to use an optical fiber as a support in order to weld the MWCNT to it. Silica optical fibers allow to have a high mobility and flexibility of handling at this scale without having the disadvantages of a 5 nm to 30 nm tungsten tip, which deforms plastically on contact.

The manipulations are carried out in µRobotex station [START_REF] Rauch | Smallest microhouse in the world, assembled on the facet of an optical fiber by origami and welded in the μRobotex nanofactory[END_REF]. It is based on an Auriga 60 microscope from Zeiss and comprises a Scanning Electron Microscopy (SEM) column, a Focused Ion Beam (FIB) column, a Gas Injection System (GIS) and two robotic arms (a 5degree-of-freedom stage, and a 6-degree-of-freedom robot). The FIB column is positioned at an angle of 54° from the z-axis of the SEM column. The robot consists of six stick/slip actuators This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0065214 from SmartAct. A homemade real-time controller synchronizes the movement of the actuators in velocity and acceleration every 500 µs, leading to high precision positioning of the tool center of the robotic arm with an accuracy of the order of 10 nm. The control is carried out from a human-machine interface (HMI) and can be performed either in the robot frame, the SEM frame or the FIB frame.

The SEM is used to visualize the surface during electron beam welding of the MWCNT.

The FIB can be used for etching, cutting, folding and imaging. The GIS air system can use three gases -platinum, carbon, xenon fluoride-and is used to deposit a thin layer of naphthalene onto the tip of the previously metallized optical fiber, which helped fix the MWCNT afterwards [START_REF] Calafiore | Campanile Near-Field Probes Fabricated by Nanoimprint Lithography on the Facet of an Optical Fiber[END_REF].

The optical fiber has an inclination of 54° with respect to the vertical axis -z axis of FIB-, in order to be in the intersection of the focal planes of SEM and FIB (see Figure 1).

Figure 1: a) CAD representation of vacuum chamber interior showing SEM, FIB, first and second robot (with the tool carrying the optical fiber) and a screen copy of the human machine interface (HMI) b) Photo of the same chamber

The manipulation procedure starts with the search for an isolated MWCNT in the MWCNT tangle, the free end of which is then welded to the thin naphthalene layer covering the tip of the metallized optical fiber. A low current of 1 pA is used in order to avoid damaging both the tip and the MWCNT. After extraction of the MWCNT from the tangle, its new free end is welded on a grounded Si substrate (Figures 2a and2b). This MWCNT is subjected to repetitive local bending by repeated horizontal translations of the optical fiber. Then, a variation of the brightness of the bent part is observed as the local curvature radius of the MWCNT This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0065214 changes over time (Figures 2c, 2d and2e). This effect is mainly attributed to a local variation in bound surface electric charges that appear due to the flexoelectric effect, as detailed in the next section. In order to study this effect, the apparent radius of curvature in the plane of observation is measured by first defining a square region of interest (ROI) containing the bent part. Then, before the nanotube is bent, the maximum level of brightness, in the ROI, is determined. The pixels with a positive variation of brightness when the nanotube is bent, are then used to fit a circle, which radius determines the apparent radius of curvature in the observation plane. This radius could also be estimated using the ImageJ software, by drawing an osculating circle tangent to the bent part and measuring its radius. This procedure is repeated ten times in order to average the errors (e.g. due to the operator). The standard deviations for these measures vary between 0.007 and 0.048 µm. Generally, there is an accumulation of free electric charges at sharp ends of conductors (tip effect). These electric charges create a local electric field that interacts with an incident beam of charged particles (i.e. electrons in the case of an SEM imaging and Gallium ions in the This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0065214 case of a FIB imaging), which can result in an increase of secondary electrons. Seen through an electron detector, these areas appear brighter than the other areas. As MWCNTs can have both semiconducting and metallic walls, this could be the explanation of the increase of brightness that we see in the bent parts. However, we have checked that, in our case, the outer parts of the MWCNTs we used, are electrically insulating enough so that they do not show a tip effect. This can be seen in the SEM image presented in Figure 3, where the nanotube tip on the left (area A) presents no relative increase of brightness but an almost uniform grey level (with a slight shading effect revealing that the incident beam of particles arrives from above the image on the left). However, in this same figure (i.e. with the same settings) one can see that the brightest areas correspond to the most strongly curved and best oriented areas with respect to the incident beam (see areas B and C). We tentatively attribute this local increase of brightness to the presence of bound (and not free) electric charges that are induced by the flexoelectric effect as explained below.

Figure 3) SEM observation of MWCNTs: tip of a single nanotube (area A on the left) and more or less strongly curved nanotubes (areas B and C on the right).

In a continuous medium model, the flexoelectric effect can be expressed as an additional term to piezoelectricity in the constitutive equation of the polarization of a material subjected to an external electric field and mechanical strain:

𝑃 𝑖 = 𝜀 0 𝜒 𝑖𝑗 𝐸 𝑗 + 𝑒 𝑖𝑗𝑘 𝑆 𝑗𝑘 + 𝜇 𝑖𝑗𝑘𝑙 𝜕𝑆 𝑗𝑘 𝜕𝑋 𝑙 (1)
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0065214

where 𝑖, 𝑗, 𝑘 are indices denoting directions (in {𝑥, 𝑦, 𝑧}), 𝑃 is the polarization vector, 𝜒 the dielectric susceptibility tensor, 𝑒 the piezoelectricity tensor, 𝑆 the symmetrized strain tensor, 𝜇 the 4 th order flexoelectricity tensor, and 𝜕𝑆 𝑗𝑘 𝜕𝑋 𝑙 ⁄ the strain gradient tensor. For a thin membrane, this gradient corresponds to the mean curvature 2 𝐶 = (1 𝑅 𝑚𝑖𝑛 ⁄ + 1 𝑅 𝑚𝑎𝑥 ⁄ ) where 𝑅 𝑚𝑖𝑛 and 𝑅 𝑚𝑎𝑥 are the principal radii of curvature. Therefore, in a non-piezoelectric material such as MWCNTs, the polarization 𝑃 ⃗ 𝑐 due to its curvature, in the radial direction is such that:

‖𝑃 ⃗ 𝑐 ‖ = 2 𝜇 𝑒𝑓𝑓 𝐶 (2)
where 𝜇 𝑒𝑓𝑓 denotes the effective flexoelectric coefficient, that can be evaluated using an [START_REF] Kalinin | Electronic flexoelectricity in low-dimensional systems[END_REF] to compute the volume per atom (𝑉 𝑎𝑡 ). We note that this value is in good agreement with the macroscopic order of magnitude (10 -10 C/m) estimated by Kogan [39] for crystalline materials.

Since a nanotube corresponds to a graphene sheet folded into a cylinder of radius 𝑟 0 , we have a mean curvature 2 𝐶 0 = 1 𝑟 0 ⁄ . When this nanotube is bent with a mean radius 𝑅, we have a mean curvature, for points inside the bend, 2 𝐶 𝑖𝑛 = (1 𝑟 0 ⁄ -1 (𝑅 -𝑟 0 ) ⁄ ), and, for points outside the bend, 2 𝐶 𝑜𝑢𝑡 = (1 𝑟 0 ⁄ + 1 (𝑅 + 𝑟 0 ) ⁄ ) [START_REF] Nguyen | Nanoscale Flexoelectricity[END_REF]. Therefore, the macroscopic variation of polarization due to curvature ∆𝑃 ⃗ 𝑐,𝑜 between points outside the bend and points in the straight part of the nanotube is such that

‖∆𝑃 ⃗ 𝑐,𝑜 ‖ = 2 𝜇 𝑒𝑓𝑓 (𝐶 𝑜𝑢𝑡 -𝐶 0 ) = 𝜇 𝑒𝑓𝑓 𝑅+𝑟 0 (3)
Similarly, ∆𝑃 ⃗ 𝑐,𝑖 = 𝜇 𝑒𝑓𝑓 (𝑅 -𝑟 0 ) ⁄ between points inside the bend and points in the straight part of the nanotube These macroscopic variations of polarization ∆𝑃 ⃗ 𝑐,𝑜 and ∆𝑃 ⃗ 𝑐,𝑖 (coming from the outward atomic dipoles created by the flexoelectric effect) can also be modelled by variations in the surface densities of bound electric charge ∆𝜎 such that (with 𝑛 ⃗ the unit vector normal to the surface from the center of curvature):

∆𝑃 ⃗ 𝑐 • 𝑛 ⃗ = ∆𝜎 (4)
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0065214 which is not exactly the same for the parts of the outer surface of the bent MWCNT that are inside and outside of the bend, since their radii of curvature are not exactly the same. For a typical curvature of 30 µm -1 , we can use our estimate of 𝜇 𝑒𝑓𝑓 to compute a corresponding variation of surface charge density of 1.32 • 10 -10 • 3 • 10 7 = 4 • 10 -3 C.m -2 . This variation of bound electric charge creates a local electric field which, if sufficiently intense, can induce a detectable increase of brightness in this bent area in the SEM and FIB images. For the sake of simplicity, a linear relationship between the change in brightness ∆B and the change in bound electric charge is tentatively assumed:

∆𝐵 ∝ ∆𝜎 ≈ 𝜇 𝑒𝑓𝑓 𝑅 (4)
This implies that the local change of brightness, caused by the bending, would be inversely proportional to its mean bending radius 𝑅, which can be validated (or not) by recording images of the MWCNT as it is bent.

As described in the previous section, the flexoelectric response of isolated MWCNTs is explored by manipulating them within our SEM-FIB-based microrobotic system [START_REF] Lugstein | Anomalous piezoresistance effect in ultrastrained silicon nanowires[END_REF]. Figure 4 presents FIB observations of a tangle of MWCNTs during its bending. Image 4a shows some brightness points on different MWCNTs deposited on the Si substrate. These MWCNTs are very flexible and tend to orient following the field of the FIB. One side of a MWCNT is attached on the tip of the optical fiber and extracted from the tangle of MWCNTs. Image 4b shows the same MWCNT bent as a result of translating the optical fiber. The MWCNT has a diameter between 15 nm and 20 nm. The FIB observations in images 4b 4c and 4d show that the bending location appears with a brighter white color than the surrounding parts of the MWCNT. One can clearly see that the more it is bent, the brighter it appears, thus demonstrating qualitatively the flexoelectric effect. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0065214 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0065214 On the whole, Figure 5 shows three peaks of brightness variations corresponding to three different translations of the tip fiber, held by the robot, and three radii of curvature. In addition, various phenomena related to electric charges, such as image drift or aberrations in the formation of the image were observed in FIB images. Furthermore, it is highly probable that the bending of the MWCNT is not in the plane of observation of the FIB therefore introducing a projection error. The fact that the MWCNT does not bend in a single plane (as can be easily seen by manipulating, for example, a shoelace) complicates the interpretation. Nevertheless, Figure 6 shows a good correlation between the brightness variations for each peak and the apparent curvature 1 𝑅 ⁄ , thus confirming our qualitative interpretation by a flexoelectric effect.

A more quantitative determination of the effective flexoelectricity coefficient requires a method to compute the changes of local electric charge from the brightness variations. We have been trying to measure the current that can be produced by the repeated bending of a MWCNT, with a nano-ammeter electrode located close to the tip of that MWCNT, but up to now the values obtained are still too noisy to be used. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0065214 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0065214

Figure 2 a

 2 Figure 2 a) Schematic representation of the setup: a MWCNT is welded on the tip of an optical fiber and on a Si cantilever. b) FIB observation of that setup. c) and d) schemes of the bent MWCNT after translation of the Si cantilever. e) FIB observation of the bent MWCNT. The green square defines the region of interest (ROI) used to measure the apparent radius of curvature.

Figure 4 FIB

 4 Figure 4 FIB observations of MWCNTs deposited with CVD. a) A single 20 nm diameter MWCNT extended by the tip of an optical fiber. b), c), d) the same MWCNT bent three times. In order to be more quantitative, computer vision tools are used to analyze the bending region in the successive images and to extract both the brightness variations in this region and the apparent radius of curvature. The variation of the brightness ∆𝐵 in the ROI, as a function of the index of the image in the recorded movie, is represented in Figure 5.

Figure 5 :

 5 Figure 5: Variation of brightness (solid curve) and apparent radius of curvature (dashed curve) in the region of bending with respect to the index of the image in the sequence of images. Peaks of brightness variation correspond to maximum of bending.

Figure 6 :

 6 Figure 6: Brightness variation ∆𝐵 as a function of the apparent curvature (µm -1 ) for the three peaks.

  atomic-scale effective flexoelectric coefficient 𝐹 = 0.831 D.Å = 0.173 e.Å 2 /atom computed by Kvashnin et al.[START_REF] Kvashnin | Flexoelectricity in carbon nanostructures: Nanotubes, Fullerenes, and Nanocones[END_REF] using Density Functional Theory (in good agreement with results between 0.157 e.Å 2 /atom et 0.187 e.Å 2 /atom, depending on the structure, obtained by Kalinin

	and Meunier [38], also using DFT), which gives 𝜇 𝑒𝑓𝑓 = 𝐹 𝑉 𝑎𝑡 ⁄	= 0.173 • 1.6 • 10 -19 •
	10 -20 ((3√3 • (1.42 • 10 -10 ) 2 4 ⁄ ) • 0.8 • 10 -10 ) ⁄	= 1.32 • 10

-10

C/m, using the thickness estimate of Kalinin and Meunier

This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0065214

Acknowledgements

This work was supported by the EIPHI Graduate School (contract "ANR-17-EURE-0002"), the Region of Bourgogne Franche-Comté, the French RENATECH network and the Equipex ROBOTEX project (contract ANR-10-EQPX-44-01). We thank Mohammad Arab Pour Yazdi and Thierry Barriere for the MWCNT samples deposited on silicon substrate.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.


Clément Marteau

Risk hull method for spectral regularization in linear statistical inverse problems

We consider in this paper the statistical linear inverse problem Y = Af + ξ where A denotes a compact operator, a noise level and ξ a stochastic noise. The unknown function f has to be recovered from the indirect measurement Y . We are interested in the following approach: given a family of estimators, we want to select the best possible one. In this context, the unbiased risk estimation (URE) method is rather popular. Nevertheless, it is also very unstable. Recently, Cavalier and Golubev (2006) introduced the risk hull minimization (RHM) method. It significantly improves the performances of the standard URE procedure. However, it only concerns projection rules. Using recent developments on ordered processes, we prove in this paper that it can be extended to a large class of linear estimators.

Introduction

This paper is devoted to statistical linear inverse problems. We want to recover an unknown function f from noisy and indirect measurements. Formally, consider H and K two Hilbert spaces and A : H → K a compact operator. We observe:

Y = Af + ξ, (1.1)
where f belongs to H and ξ denotes some noise. This representation arises in many mathematical and physical domains. In the numerical literature, the noise ξ is deterministic. For such a model, many recovering methods have been proposed. For a survey, we mention for instance [START_REF] Engl | On the choice of the regularization parameter for iterated Tikhonov regularization of ill-posed problems[END_REF], [START_REF] Hanke | Accelerated Lanweber iterations for the solution of ill-posed equations[END_REF], [START_REF] Engl | Regularization of Inverse Problems[END_REF] or [START_REF] Ghosh Roy | Inverse problems and inverse scattering of plane waves[END_REF]. In the statistical literature, one deals instead with stochastic perturbations. The Gaussian white noise model is the most used. The representation (1.1) is equivalent to:

Y, g = Af, g + ξ, g , ∀g ∈ K, (1.2)
where ξ, g ∼ N (0, g 2 ) and > 0. Given g 1 , g 2 ∈ K, E ξ, g 1 ξ, g 2 = g 1 , g 2 . We refer to [START_REF] Hida | Brownian Motion[END_REF] for more details concerning the Gaussian white noise. We assume throughout this paper that the noise level is known.

The singular value decomposition (SVD) setting provides a better understanding of the model (1.1). The operator A A is compact and selfadjoint. We denote by (b 2 k ) k∈N the associated sequence of eigenvalues. The set of eigenfunctions (φ k ) k∈N is assumed to be orthonormal. Then define

ψ k = b -1
k Aφ k , ∀k ∈ N. For all integer k, the following equality holds:

Aφ k = b k ψ k , A ψ k = b k φ k . (1.
3)

The triple (b k , φ k , ψ k ) k∈N is called the singular system for the operator A A. The associated representation matrix is diagonal. For all k ∈ N, set g = ψ k in (1.2) and use (1.3) in order to obtain the sequence space model:

y k = b k θ k + ξ k , ∀k ∈ N, (1.4)
where θ k = f, φ k . The ξ k are i.i.d. standard Gaussian random variables since the set (ψ k ) k∈N is also orthonormal. In the L 2 sense, the function f and θ = (θ k ) k∈N represent the same mathematical object. The sequence θ has to be recovered from the noisy observations (y k ) k∈N . Since A A is compact, b k → 0 as k → +∞. When k is large, the estimation of θ k is quite difficult: there is mainly noise in the observation y k . The SVD can explicitly computed for a large class of inverse problems, e.g. tomography (see [START_REF] Johnstone | Speed of estimation in positron emission tomography and related inverse problems[END_REF]) or Biophotonic imaging (see [START_REF] Bissantz | Testing for lack of fit in inverse regression -with applications to biophotonic imaging[END_REF]).

In some sense, the best approximate solution f minimizes the distance between the observation and the image of the operator A. It corresponds to the least square solution and verifies f = (A A) -1 A Y . Since A A is compact, it is not continuously invertible. The solution f will not be convenient: it does not necessarily converge to f as → 0. In order to solve (1.1), one may use instead:

ft = Φ t (A A)A Y,
where Φ t approximates the function x → x -1 on the spectrum of A A. This is a regularization method. The parameter t is a regularization parameter. The performances of ft are related to the choice of t which is crucial for both numerical and statistical approaches. Some examples are presented in Sections 2.1 and 3.2.

In the SVD setting, a regularization method Φ t corresponds to a filter λ(t) = (λ k (t)) k∈N . It is a real sequence with values in [0, 1]. The associated estimator is:

ft = +∞ k=1 λ k (t)b -1 k y k φ k .
The quality of ft is measured via its quadratic risk:

R(θ, λ(t)) = E θ ft -f 2 = +∞ k=1 (1 -λ k (t)) 2 θ 2 k + 2 +∞ k=1 λ 2 k (t)b -2 k . (1.5)
The behavior of R(θ, λ(t)) depends on both the function f and the regularization approach, i.e. the filter λ and the parameter t. For more details and some examples, we mention [START_REF] Ermakov | Minimax estimation of the solution of an ill-posed convolution type problem[END_REF], [START_REF] Johnstone | Speed of estimation in positron emission tomography and related inverse problems[END_REF], [START_REF] Donoho | Nonlinear solutions of linear inverse problems by wavelet-vaguelette decomposition[END_REF] or [START_REF] Efromovich | Robust and efficient recovery of a signal passed trough a filter and then contaminated by non-gaussian noise[END_REF].

In this context, the unbiased risk estimation (URE) method is rather popular for choosing t since it does not require a priori informations on the solution f . It has been studied for instance in [START_REF] Cavalier | Oracle inequalities for inverse problems[END_REF]. However, this method is rather unstable due to the ill-posedness of the problem. The risk hull minimization (RHM) method initiated in [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF] proposes to take into account the variability of the problem through the constrution of a risk hull. The definition is detailed below.

Definition 1 Let Λ a family of filters. A deterministic term V (θ, λ) such that:

E θ sup λ∈Λ ft -f 2 -V (θ, λ) ≤ 0,
is called a risk hull for Λ.

The principle of the RHM scheme is to minimize the hull, through an appropriate estimator, instead of the quadratic risk. Specific phenomena are related to the Gaussian white noise model. They require particular approaches that may not be useful in numerical areas. The RHM method is a good example of such a procedure. It provides answers to specific problems of the statistical model (1.2) and improves the standard URE approach. The risk hull method has been developed for projection estimators. As proved in this paper, it can be extended to a large class of linear estimators. This class contains for instance Tikhonov estimators or Landweber iterative methods. This paper is organized as follows. In Section 2, we present the RHM method and recall the main problems related to the choice of the regularization parameter t through some well-known properties of Tikhonov estimators. A risk hull for a wide family of spectral regularization schemes is constructed in Section 3. Section 4 contains the main results and the proofs are gathered in Section 5. Finally, Section 6 is devoted to ordered processes. A reader not familiar with this topic is advised to read this part before being interested in the proofs.

Consider the following problem: given a function f and a set of estimators Λ, we want to select the best possible one, i.e. that minimizes the quadratic risk. In most cases, the family Λ may be identified with a regularization operator Φ t indexed by t and a collection of parameters T . This collection may be finite or not. We want to select the parameter t in an adaptive way, i.e. without using some unknown information on the function f .

The Tikhonov estimator

In order to shed some light on this framework, we recall briefly some properties of the Tikhonov estimators. Obviously, these estimators are well-known. However, some steps may be useful in the sequel. For a survey on this procedure, both in the numerical and statistical domains, we mention for instance [START_REF] Engl | Regularization of Inverse Problems[END_REF], [START_REF] Mathé | Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods[END_REF], [START_REF] Bissantz | Convergence rates of general regularization methods for statistical inverse problems and applications[END_REF] or [START_REF] Bissantz | Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise[END_REF] for non-linear inverse problems.

The Tikhonov estimation is rather intuitive. One wants to control both the regularity of our estimator f and the fitting of A f with the data Y . Therefore, resolve the following optimization problem: ft = arg min

f ∈H { Af -Y 2 + t -1 f 2 a }, (2.1)
where t is a regularization parameter. For all f ∈ H and a ∈ R, f a = (AA ) -a f . In some sense, (AA ) -a is a differential operator. The parameter a is chosen a priori. It is related to the expected regularity of the function f . In the literature, one often uses α instead of t -1 . However, the notation (2.1) is consistent with Sections 3-6.

The solution of (2.1) is:

ft = Φ t (A A)A Y, with Φ t (A A) = (A A + t -1 (L a ) L a ) -1 , and L a = (AA ) -a . (2.2)
In the SVD representation, the Tikhonov estimator becomes:

ft = +∞ k=1 λ k (t)b -1 k y k φ k , with λ k (t) = 1 1 + t -1 (b -2 k ) 1+2a , ∀k ∈ N.
(2.

3)

The choice of t is crucial and has a real impact on the performances of ft . It is a trade-off between the two sums in the right hand side of (1.5). Assume for instance that the function f belongs to a ball H µ Q of radius Q:

f ∈ H µ Q = g ∈ H : (A A) -µ g 2 ≤ Q ⇔ θ ∈ Θ(s, Q) = ϑ : +∞ k=1 b -4µ k ϑ 2 k ≤ Q , (2.4)
for some µ > 0. In what follows, we note b 1 b 2 when there exists C > 0 such that b 1 ≤ Cb 2 . Using simple algebra:

+∞ k=1 (1 -λ k (t)) 2 θ 2 k t -2µ (1+2a) , (2.5)
provided µ < 1 + 2a. For µ larger than 1 + 2a, g α cannot be optimally bounded. The Tikhonov estimator does not attain the minimax rate of convergence on H µ Q . It is said to be underqualified. For the special case a = 0, we obtain the well-known condition µ < 1. The term 1 + 2a is called qualification of the Tikhonov regularization (see [START_REF] Engl | Regularization of Inverse Problems[END_REF] or [START_REF] Bissantz | Convergence rates of general regularization methods for statistical inverse problems and applications[END_REF] for more details). Now consider the second sum. Assume that the sequence of eigenvalues possesses a polynomial behavior: (b k ) k∈N ∼ (k -β ) k∈N for some β > 0. The problem is said to be mildly ill-posed. Let n = n t ∈ N which will be chosen later:

2 +∞ k=1 λ k (t) 2 b -2 k = 2 nt k=1 λ k (t) 2 b -2 k + 2 k>nt λ k (t) 2 b -2 k 2 n 2β+1 t + 2 t -2 n -8βa-2β+1 t .
Using simple algebra, we obtain the following bound:

2 +∞ k=1 λ k (t) 2 b -2 k 2 t 2β+1 2β(1+2a) , setting n t = t 1 2β(1+2a) . (2.6)
Consider the parameter t 0 that make the trade-off between (2.5) and (2.6). The associated estimator attains the minimax rate of convergence on H µ Q for µ < 1 + 2a. The parameter t 0 can easily be constructed with the a priori knowledge of µ and Q. If it is not the case, it is necessary to use a data-driven parameter choice rule.

The risk hull method

In a minimax sense, the optimal regularization parameter depends essentially on the regularity and on the norm of f . Since these informations are in most cases unknown, the choice of t is somewhat difficult.

Several data driven regularization methods have been proposed. The unbiased risk estimation procedure is rather popular. The principle is intuitive: given f and a set of estimators Λ (or equivalently a collection T of regularization parameters), we want to select the best possible one, i.e. that minimizes the quadratic risk R(θ, λ(t)) (see (1.5)). Since this quantity depends on the unknown function f , this estimator is not available. It is called the oracle. However, we can construct an estimator U (y, λ(t)) of R(θ, λ(t)) and then, minimize U (y, λ(t)) instead of R(θ, λ(t)). A natural estimator for R(θ, λ(t)) is:

U (y, λ(t)) = +∞ k=1 {λ 2 k (t) -2λ k (t)}(b -2 k y 2 k -2 b -2 k ) + 2 +∞ k=1 λ 2 k (t)b -2 k .
This method has been applied by [START_REF] Cavalier | Sharp adaptation for inverse problems with random noise[END_REF] on the set of blockwise monotone estimator. In a model selection context, [START_REF] Cavalier | Oracle inequalities for inverse problems[END_REF] dealt with finite families of estimators. Sharp oracle inequalities have been obtained. However, numerical simulations are somewhat disappointing. This has been illustrated in [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF] for the projection regularization. The same phenomenon occurs for Tikhonov estimators. For ill-posed inverse problems, the URE algorithm selects large regularization parameters when the oracle is typically small.

The instability of the URE procedure has already been discussed in the literature. In the last decades, some authors were interested in the criterion U (y, λ(t)) + pen(t) where pen(t) → +∞ as t → +∞. A penalty is introduced in the estimator of R(θ, λ(t)). This penalty is chosen in order to control the variability of the problem. We expect that smaller parameters will be selected. Several penalizations have been proposed for both direct and inverse problems. We mention for instance [START_REF] Barron | Risk bounds for model selection via penalization[END_REF], [START_REF] Birgé | Gaussian model selection[END_REF], [START_REF] Yu | The principle of penalized empirical risk in severely ill-posed problems[END_REF] or [START_REF] Loubes | Penalized estimators for non-linear inverse problems[END_REF].

Other approaches leading to oracle inequalities in a more general context have been proposed. For instance, concerning the Lepski balancing principle we may mention [START_REF] Bauer | A Lepskij-type stopping rule for regularized Newton methods[END_REF], [START_REF] Mathé | The Lepskij principle revisited[END_REF] or [START_REF] Bissantz | Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise[END_REF].

The risk hull minimization method (also denoted RHM) initiated by [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF] provides an interesting alternative. Consider the simple example of projection filter (also called spectral cut-off): λ k (t) = 1 {k≤t} , for all k ∈ N. Denote by θt the related estimator. The loss is:

l(θ, λ(t)) = θt -θ 2 = k>t θ 2 k + 2 t k=1 b -2 k ξ 2 k .
Due to the ill-posedness of the problem, the variance of l(θ, t) is very large and explodes with t. However, this variability is neglected in the URE procedure. Indeed, we deal with R(θ, λ(t)) = E θ l(θ, λ(t)). In order to take account of this variability, one may look after a risk hull, i.e. a deterministic quantity V (θ, λ(t)) verifying:

E θ sup t {l(θ, λ(t)) -V (θ, λ(t))} ≤ 0.
Then, we estimate and minimize V (θ, λ(t)) instead of R(θ, λ(t)). The challenge is to find an explicit hull. It should not be too large in order to provide a good quality of estimation although a small hull will not be sufficiently stable. The hull V (θ, λ(t)) of [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF] can be explicitly constructed. It is easily computable using for instance Monte-Carlo approximations. The related performances are significantly better than the standard URE procedure, both from a theoretical and numerical point of view. Note that the risk hull method leads in fact to a penalized URE criterion where the penalty is explicitly computable.

However, the projection estimation is rather rough. There exist several regularization methods with better performances. We may mention for instance the Tikhonov estimators presented in this section. The generalization of the RHM algorithm to a wide family of linear estimators may produce interesting results but is not obvious. It requires in particular the control of the process:

η(t) = 2 +∞ k=1 λ 2 k (t)b -2 k (ξ 2 k -1), t ≥ 0.
When dealing with projection procedures, η(t), t ≥ 0 can be controled as a Wiener process. These objects have been intensively studied and many results are available. In this paper, the assumptions concerning the sequence (λ k (t)) k∈N are rather weak. In such a situation, we will see that the process η(t), t ≥ 0 is said to be ordered. Some interesting properties have been established in [START_REF] Kneip | Ordered linear smoother[END_REF] and [START_REF] Cao | On oracle inequalities related to smoothing splines[END_REF].

It presents a different behavior compared to the well-known Wiener process. The generalization of the RHM algorithm requires advanced probabilistic tools, gathered in Section 6.

3 Risk hull

Assumptions and construction

Here and in the sequel, assume that Λ is a family of monotones filters indexed by a positive parameter t, i.e. Λ = (λ(t)) t∈T , where T ⊆ R + . Each filter λ(t) may be noted λ or identified with the regularization parameter t. The meaning will be clear following the context. The associated linear estimator is denoted by θt . For all k ∈ N, the function t → λ k (t) is assumed to be monotone non-decreasing. We require some additional assumptions on the family Λ.

Polynomial hypotheses. There exist C 1 , C 2 , C 3 , C 4 positive constants and d > 0 independent of t and such that:

max k 2 λ 2 k (t)b -2 k σ 2 (t) ≤ C 1 t -d/2 , where σ 2 (t) = 4 +∞ k=1 λ 4 k (t)b -4 k , (3.1)
exp -1 4

+∞ p=1 log 1 + 2s 2 4 λ 4 p (t)b -4 p σ 2 (t) ≤ 1 + C 2 s 2 t d -t d , ∀s ∈ R, (3.2)
and for all l ∈ N,

(C 3 t d ) -l/2+1 ≤ σ 2 (t) -l/2 2l +∞ p=1 λ 2l p (t)b -2l p ≤ (C 4 t d ) -l/2+1 . (3.3)
In some sense, the inequalities (3.1)-(3.3) generalize the polynomial hypotheses of [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF]. Some examples of regularization method satisfying (3.1)-(3.3) are presented in Section 3.2.

For all λ ∈ Λ, i.e. for all t ∈ T , introduce:

U 0 (t) = inf{u > 0 : Eη(t)1 {η(t)>u} ≤ 2 }, with η(t) = 2 +∞ k=1 λ 2 k (t)b -2 k (ξ 2 k -1). (3.4)
The term σ 2 (t) denotes the variance of the process η(t), t ≥ 0 up to some constant. This process is ordered: see Section 6 for more details. In particular, the function σ 2 is monotone non-decreasing.

The following lemma provides a lower bound for U 0 (t). This quantity will be useful in the following.

Lemma 1 Assume that (3.1)-(3.3) hold. There exists T 0 > 0 independent of such that for all t > T 0 ,

U 0 (λ(t)) = U 0 (t) ≥ 2σ 2 (t) log σ 2 (t) 2π 4 ,
where σ 2 (t) is defined in (3.1).

A proof of this lemma is available in Section 5. We are now able to propose a risk hull. First introduce:

λ(t 0 ) = λ 0 = arg inf λ∈Λ R(θ, λ), (3.5)
where R(θ, λ) is the quadratic risk defined in (1.5). Hence, λ 0 corresponds to the oracle filter for the family Λ. For all x ∈ R + , define also:

ω(x) = sup t∈T sup k λ 2 k (t)b -2 k 1 { P +∞ l=1 λ 2 l (t)b -2 l ≤x sup l λ 2 l (t)b -2 l } , (3.6)
and,

L T = log 2 S, with S = max t∈T sup k b -2 k λ 2 k (t) min t∈T sup k b -2 k λ 2 k (t)
.

(3.7)

These terms have been introduced in [START_REF] Cavalier | Oracle inequalities for inverse problems[END_REF]. The function ω is explicitly computed in Section 3.2 for some particular families of estimators. The quantity S provides an information on the homogeneity of T . For S < +∞, the collection T may be identified with an interval [t 1 , t max ] and S is linked to the ratio t max /t 1 .

Theorem 1 Assume that (3.1)-(3.3) hold. Then, there exist C , C > 0 such that, for all θ and α, γ > 0,

V α (θ, λ(t)) = (1 + γ) +∞ k=1 (1 -λ k (t)) 2 θ 2 k + 2 +∞ k=1 λ 2 k (t)b -2 k + (1 + α)U 0 (λ(t)) + C 2 α + C 2 γ -1 L T ω(γ -2 L T ) + γR(θ, λ(t 0 )), (3.8)
is a risk hull, i.e.:

E θ sup t∈T θt -θ 2 -V α (θ, λ(t)) ≤ 0.
The hull of [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF] is constructed for projection estimators. Here, the main difference is contained in the residual term C 2 γ -1 L T ω(γ -2 L T ). The hull is somewhat less precise. This can be explained by the structure of the stochastic processes involved in the loss. Indeed, when considering projection estimators, one essentially deals with Wiener processes. These processes are well-known and may be easily controlled. In particular:

P max t≥0 [W (t) -γt] ≥ x ≤ exp(-2γx), ∀γ > 0 and x ∈ R + ,
where W (t), t ≥ 0 denotes a Wiener process. The proof of Theorem 1 is based on the theory of ordered processes summarized in Section 6 (see also [START_REF] Kneip | Ordered linear smoother[END_REF] or [START_REF] Cao | On oracle inequalities related to smoothing splines[END_REF]). These objects are well understood but more difficult to control. At best, it is (at the moment) possible to prove that for all p ∈ N:

P max t≥0 ρ(t) -γΣ 2 (t) ≥ x ≤ Cp p (γx) p , ∀γ > 0 and x ∈ R + , (3.9)
for a given ordered process ρ(t), t ≥ 0 of variance Σ 2 (t). Here, C denotes a positive constant independent of x. In light of the proofs of Theorem 3.2, it seems difficult to improve (3.9) in order to obtain an exponential inequality as for the Wiener process.

Examples

In this section, we illustrate the polynomial hypotheses (3.1)- (3.3). Assume that the sequence (b k ) k∈N possesses a polynomial behavior, i.e. (b k ) k∈N ∼ (k -β ) k∈N for some positive parameter β.

The degree of ill-posedness β is assumed to be known. We consider three different procedures: projection, Tikhonov and Landweber iterative methods. 3) are satisfied for projection filters since the aim of this paper is to generalize [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF]. Consider the set:

Λ Proj = λ = (λ k (t)) k∈N : ∀k ∈ N, λ k = 1 {k≤t} , t ∈ N .
The projection estimator is also called spectral cut-off in the literature.

Assumption (3.2) is verified in the proof of Lemma 1 of [9] with d = 1. Since (b k ) k∈N ∼ (k -β ) k∈N : σ 2 (t) = t k=1 b -4 k t 4β+1 , and +∞ p=1 λ 2l p (t)b -2l p = t p=1 b -2l p t 2βl+1 , ∀l ∈ N,
uniformly in t ∈ N. Therefore, (3.3) holds with d = 1. Then, remark that:

max k λ 2 k (t)b -2 k σ 2 (t) t 2β t 2β+1/2 = t -1/2 .
[8] proved that ω(x) x 2β for all x ∈ R + and for some C > 0. This concludes Example 1.

EXAMPLE 2. (Tikhonov estimators)

We use the same notations of Section 2.1. Consider the family:

Λ Tikh = λ = (λ k (t)) k∈N : ∀k ∈ N, λ k = 1 1 + t -1 (b -2 k ) (1+2a) , t ∈ R + . It is possible to show: σ 2 (t) = 4 +∞ k=1 λ 4 k (t)b -4 k 4 (t d ) 4β+1 and +∞ p=1 λ l p (t)b -l p (t d ) 2βl+1 , ∀l ∈ N.
with d = 1/(2β(1 + 2a)) (see (2.6)). Hence, (3.3) holds. For all t, let n t defined in (2.6). For all k ∈ {n t /2, . . . , n t }:

λ 2 k b -2 k = b -2 k [1 + t -1 (b -2 k ) (1+a) ] 2 ≥ Ck 2β ≥ C n 2β t 2 2β .
Therefore, for all t ∈ R + and s ∈ R,

exp - 1 4 +∞ l=1 log 1 + 2s 2 4 λ 4 l (t)b -4 l σ 2 (t) ≤ exp   - 1 4 nt l=nt/2 log 1 + 2s 2 4 λ 4 l (t)b -4 l σ 2 (t)   , ≤ exp   - 1 4 nt l=nt/2 log 1 + Cs 2 n 4β t n 4β+1 t   , ≤ 1 + C 1 s 2 n t -nt/8 = 1 + C 1 s 2 t d -t d /8
. (3.10)

This prove (3.2). Finally, remark that:

max k λ 2 k (t)b -2 k σ 2 (t) λ 2 nt (t)b -2 nt σ 2 (t)
n 2β t n 2β+1/2 t = t -d/2 . (3.11)
Moreover, we can prove, using for instance (3.11), that ω(x) ≤ Cx 2β for all x ∈ R + and for some C > 0.

EXAMPLE 3. (Iterated Tikhonov regularization)

This method presents an interesting alternative to the previous one. The qualification of the standard Tikhonov estimator is 1 if we set a = 0 in (2.1). This can be enhanced using iterations.

In the following, we refer to fj as the iterated Tikhonov estimator of order j. Set f0 = 0. Given j ∈ N and fj-1 , fj is defined as the unique solution of:

(A A + t -1 I) fj = A T + t -1 fj-1 ,
where I denotes the identity operator. In the SVD setting, the following representation arises:

fj = +∞ k=1 λ k (t)b -1 k y k φ k , with λ k (t) = (t -1 + b 2 k ) j -t -j (t -1 + b 2 k ) j
, ∀k ∈ N.

The qualification corresponds to the number of iterations. It is possible to prove that max

k λ 2 k (t)b -2 k n 2β t and σ 2 (t) ≥ n 4β+1 t
, with n t = t -1/2β . Hence, (3.1) holds. Then, for all k ∈ {n t /2, . . . , n t },

λ 2 k (t)b -2 k ≥ (1 + t -1 b -2 k ) j -t -j b -2j k b -2 k ≥ Cn 2β t ,
for some C > 0. Using the same algebra as in (3.10), one obtain (3.2). The proof of (3.3) and the bound of ω essentially follows the same lines.

EXAMPLE 4. (Landweber iterative method)

This procedure is rather interesting from a numerical point of view since it does not require the inversion of an operator. Consider the equation g = Af for some g ∈ K. It can be rewritten as follows:

f = f + A (g -Af ). (3.12)
The Landweber iterative method is constructed in the following way. Define f0 = 0. Then, for all t ∈ N, t ≥ 1, set:

ft = ft-1 + A (g -A ft-1 ). (3.13)
The number t of iterations plays the role of the regularization parameter. With the model (1.1), replace g by Y . In the SVD setting, the Landweber iterative estimator is:

ft = +∞ k=1 λ k (t)b -1 k y k φ k , with λ k (t) = (1 -(1 -b 2 k ) t ), ∀k ∈ N.
The estimator ft is defined only if A ≤ 1 (i.e. max k b k ≤ 1). Otherwise, the method can easily be modified via the introduction of a relaxation parameter in (3.12). The qualification of the Landweber method is ∞, i.e. there is no restriction on the regularity µ.

Using simple algebra, it is possible to prove that σ 2 (t) 4 (n t ) 4β+1 with n t = t 1/2β and:

max k λ 2 k (t)b -2 k = max x∈(0,1) (1 -(1 -x) t ) 2 x , ≤ max x∈(0,1) s (x), with s(x) = 1 -(1 -x) t , = t
Hence, (3.1) holds with d = 1/2β. The proof of (3.2) and (3.3) follows essentially the same lines.

For the sake of convenience, only four regularization schemes satisfying the polynomial hypotheses are presented in this paper. The families of estimators covered by these assumptions is certainly larger.

In order to conclude these examples, a few words on the ν-methods and the Pinsker estimators. These filters are slightly different from the previous one since both the regularization parameter and the structure of the estimator may depend on the regularity of the function f . For instance, a Pinsker filter is of the form λ P k = (1 -c a k ) + for all k ∈ N, where both c and (a k ) k depend on the regularity of f . It is easy to prove that assumptions (3.1)-(3.3) hold for a Pinsker type sequence of the form λ k (t) = (1 -t -1 k a ) + , for some fixed a > 0. It is not so clear for (λ P k ) k∈N . The same observation holds for the ν-methods. Nevertheless, it seems that the behavior of these estimators is close to the examples presented in this section. Hence, the polynomial hypotheses may certainly be modified in order to take into account these specificities.

Oracle efficiency

Due to the ill-posedness of the problem, the standard URE method is rather unstable. Therefore, estimate and minimize the hull V α (θ, λ(t)) (introduced in (3.8)), instead of R(θ, λ(t)). Remark that:

arg min λ∈Λ V α (θ, λ) = arg min t∈T +∞ k=1 λ 2 k (t) -2λ k (t) 2 θ 2 k + 2 +∞ k=1 λ 2 k (t)b -2 k + (1 + α)U 0 (λ(t)) . (4.1)
For all α > 0, the right-hand side of (4.1) can be estimated by

V α (y, λ(t)) = +∞ k=1 λ 2 k (t) -2λ k (t) (b -2 k y 2 k -2 b -2 k ) + 2 +∞ k=1 λ 2 k (t)b -2 k + (1 + α)U 0 (λ(t)). (4.2)
Hence consider:

λ = arg min λ∈Λ V α (y, λ), (4.3)
and denote by θ the associated estimator. This approach corresponds in fact to a penalized URE method. The penalty (1 + α)U 0 (λ(t)) is explicitly computable via Monte-Carlo approximations.

The performances of θ are summarized in the following theorem.

Theorem 2 Let θ the estimator defined in (4.3) with α > 1. Assume that there exists a positive constant C T such that, uniformly in t ∈ T :

+∞ k=1 λ 2 k (t)b -4 k ≤ C T +∞ k=1 λ 4 k (t)b -4 k . (4.4)
Then, there exist B 1 , D 1 and γ 1 > 0 independent of , such that, for all λ ∈ Λ, (i.e. t ∈ T ) and

0 < γ < γ 1 , E θ θ -θ 2 ≤ (1 + B 1 γ)R α (θ, λ(t)) + D 1 2 γ -1 L T ω(γ -2 L T ) + D 1 2 (α -1) ,
where L T is defined in (3.7) and

R α (θ, λ(t)) = +∞ k=1 (1 -λ k (t)) 2 θ 2 k + 2 +∞ k=1 λ 2 k (t)b -2 k + (1 + α)U 0 (λ(t)). (4.5)
The proof is presented in Section 5. It is based on Theorem 1 and on the theory of ordered processes (summarized in Section 6). Clearly, inequality (4.4) is verified for projection estimators. This is also the case for Tikhonov estimators, provided a > 0, Tikhonov iterated procedure (j > 1) and for the Landweber iterative method. The constant C T has a real impact on the quality of estimation. Indeed, the penalty controls the stochastic terms in the loss and in V α (y, λ). For all λ ∈ Λ, the variance of these terms is of order 4 k λ 4 k b -4 k and 4 k λ 2 k b -4 k , respectively. If C T is too large, the penalty is not sufficient for V α (y, λ). The efficiency of θ will not necessarily be improved compared to the standard URE method.

In Theorem 2, we require α to be greater than 1. It seems that this condition is too restrictive. Indeed, acceptable bounds for the risk are available for α < 1. This needs more precision in the proofs. The obtained oracle inequality is not sharp: one cannot consider γ → 0 as → 0 in this case. On the other hand, large choices are related to a poor efficiency since R α (θ, λ) will be significantly greater than the quadratic risk R(θ, λ). From the proofs, it is possible to see that α has a small influence on the quality of estimation. In such a situation, a choice of α close to 2 seems to be reasonnable. We refer to [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF] for a complete discussion concerning this choice of α.

Several bounds in the proof of Theorem 2 are derived from the theory of ordered processes. These results can easily be applied to the unbiased risk estimator, called θ in the following.

Theorem 3 Assume that (3.1)-(3.3) and inequality (4.4) hold. Then, there exists B 2 , D 2 and γ 2 > 0 independent of , such that, for all λ ∈ Λ, (i.e. t ∈ T ) and 0 < γ < γ 2 ,

E θ θ -θ 2 ≤ (1 + B 2 γ)R(θ, λ(t)) + D 2 2 γ -1 L T ω(γ -2 L T ),
where R(θ, λ(t)) and L T are defined in (1.5) and (3.7) respectively.

In the oracle inequality of [START_REF] Cavalier | Oracle inequalities for inverse problems[END_REF], the term L T is of order log N where N denotes the size of the family Λ, i.e. the number of considered paramters. Thanks to the theory of ordered processes, the oracle inequalities of Theorems 4.1 and 4.2 are free of assumptions on the size of Λ. In particular, there is no theoretical price to pay for considering large collection of estimators. Moreover, the obtained results concerns continuous intervals for the regularization parameter of the form T = [t min ; t max ].

We have proved that the RHM method initiated by [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF] can be enhanced by using more precise estimators like Tikhonov or Landweber iterative methods. The principle of risk hull minimization is not restricted to projection procedures. This generalization is mainly due to recent developments on ordered processes. Section 6 presents the main lines of this theory. It contains important results from [START_REF] Kneip | Ordered linear smoother[END_REF] and [START_REF] Cao | On oracle inequalities related to smoothing splines[END_REF] and some additional lemmas useful for the proofs of Theorems 1-3.

Comparing different oracle inequalities in a transparent way is always a very difficult task. The constants and residual terms are derived from successive upper bounds. Nevertheless, the URE method does not take into account the variability of the problem. Hence, the RHM procedure may lead to better performances. The differences between the URE and RHM procedures will be certainly significant for the cases where the variability of the sequence space model is the most important, e.g. when the signal to noise ratio is small or the degree of ill-posedness increases.

In order to conclude this paper, some words on the numerical implementation of the RHM method. The term U 0 (λ(t)) may be approximated by using Monte-Carlo replications. In this case, the penalty will not be exactly the same than in Theorems 1 and 2. This approximation is equivalent to a perturbation of α. From proofs of Section 5, it is clear that a small variation in α does not affect the efficiency of θ . Therefore, the numerical approximation of (1 + α)U 0 (λ(t)) is pertinent from a theoretical point of view. Remark also that the lower bound obtained in Lemma 3.1 and used in the most part of the proofs may be used for a numerical implementation.

From a practical point of view, given a regularization scheme and an interval T for the parameter, finding λ may be a difficult task since (4.3) is a non-convex minimization problem. Remark that this inconvenience occurs also for the URE method. A possible outcome is then to consider a finite grid T and to construct λ on T . From Theorems 4.1 and 4.2, no restriction occurs on the size of the grid. Hence, the limitations are only due to practical considerations (e.g. computer performances). In order to deal with continuous intervals, an interesting alternative would be to develop and use a simulated annealing type algorithm for this setting.

Proofs

Proof of Lemma 1

The proof follows the same lines of [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF]. We shall sometimes omit some technical steps since they have already been developed.

Let η(t), t ≥ 0, the stochastic process defined in (3.4). For all t ∈ T , introduce:

K(t) = η(t) 2σ 2 (t) and u 1 (t) = log σ 2 (t) 2π 4 . (5.1)
For all t ≥ 0, the function x → E θ K(t) 1 {K(t)≥x} is monotone decreasing. In order to prove Lemma 1, it suffices to show that:

E θ K(t)1 {K(t)≥u1(t)} ≥ 2 2σ 2 (t) , (5.2)
at least for sufficiently large t. Using integration by part,

E θ K(t)1 {K(t)≥u1(t)} ≥ u 1 (t)P (K(t) ≥ u 1 (t)) + u1(t)+1 u1(t) P (K(t) ≥ x)dx. (5.3)
First, study the characteristic function Υ of the process K(t), t ≥ 0. For all s ∈ R, using (3.2):

|Υ t (s)| = |Ee isK(t) | ≤ exp - 1 4 +∞ l=1 log 1 + 2s 2 4 λ 4 l (t)b -4 l σ 2 (t) ≤ 1 + C 1 s 2 t d -t d . Setting x = t d /C 1 ,
Υ t (s) = exp - s 2 2 + M -1 l=3 (-i) l 2 l/2 R l (t)s l l + O CR M (t)2 M/2 s M M ,
where M ∈ N and for all l ∈ N:

R l (t) = (σ 2 (t)) -l/2 2l +∞ p=1 λ 2l p (t)b -2l p .
The behavior of R l (t), l ∈ N is controlled by (3.3). Expending Υ t (s) exp(s 2 /2) into a Taylor series, construct the following approximation of Υ t (s):

Υ M t (s) = exp - t 2 2 1 + t d M -1 l=3 Q M (l, t) is √ t d p ,
where Q M (l, t), l = 3 . . . M are function uniformly bounded in l and t. Therefore, the probability P (K(t) > x) can be approximated by:

P M t (x) = φ(x) - k √ 2π M -1 s=3 (-1) s Q M (s, k)k -s/2 d s-1 dx s-1 exp - x 2 2 , (5.5)
where φ(x) denotes the repartition function of a standard Gaussian random variable at the point x. Using Parseval identity, (5.4) and (5.5), we prove that:

|P (K(t) > x) -P M t (x)| ≤ C (t d ) M/2 , (5.6)
for some positive constant C: see [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF] for more details. Then, using (5.1)-(5.6), integration by part and choosing M large enough:

EK(t)1 {K(t)≥u1(t)} ≥ u 1 (t)φ(u 1 (t)) + +∞ u1(t) φ(x)dx - +∞ u1(t)+1 φ(x)dx -(1 + u 1 (t)) max u1(t)≤x≤u1(t)+1 |P M t (x) -φ(x)| - Cu 1 (t) (t d ) M/2 , ≥ 1 √ 2π e -u1(t) 2 /2 -Ce -(u1(t)+1) 2 /2 -(1 + u 1 (t)) max u1(t)≤x≤u1(t)+1 |P M t (x) -φ(x)|- Cu 1 (t) (t d ) M/2 , ≥ 2 σ 2 (t) + o 2 2σ 2 (t) , as t → +∞.
Indeed, (3.1) and (5.1) provide u 1 (t) C log(t) for some positive constant C > 0. Hence, (5.1) is satisfied. This concludes the proof of Lemma 1. Remark that:

E θ sup λ∈Λ θλ -θ 2 -V α (θ, λ) = E θ sup λ∈Λ +∞ k=1 (λ k b -1 k y k -θ k ) 2 -V α (θ, λ) , = E θ +∞ k=1 (1 -λk ) 2 θ 2 k + 2 +∞ k=1 λ2 k b -2 k ξ 2 k + 2 +∞ k=1 ( λk -1) λk θ k b -1 k ξ k -V α (θ, λ) , with, λ = λ( t) = arg sup λ∈Λ θλ -θ 2 -V α (θ, λ) .
Use the following decomposition:

2E θ +∞ k=1 ( λk -1) λk θ k b -1 k ξ k = 2E θ +∞ k=1 λ2 k -λk θ k b -1 k ξ k , = E θ +∞ k=1 λ2 k θ k b -1 k ξ k + E θ +∞ k=1 { λ2 k -2 λk }θ k b -1 k ξ k , = E θ +∞ k=1 λ2 k θ k b -1 k ξ k + E θ +∞ k=1 (1 -λk ) 2 θ k b -1 k ξ k .
Corollary 1 in Section 6 provides, for all γ > 0:

2E θ +∞ k=1 ( λk -1) λk θ k b -1 k ξ k ≤ γE θ R(θ, λ) + γR(θ, λ 0 ) + C 2 γ -1 L T ω(γ -2 L T ),
for some positive constant C > 0. Therefore,

E θ sup λ∈Λ { θλ -θ 2 -V α (θ, λ)} ≤ E θ sup λ∈Λ (1 + γ) +∞ k=1 (1 -λ 2 k )θ 2 k + 2 +∞ k=1 λ 2 k b -2 k ξ 2 k + γ 2 +∞ k=1 λ 2 k b -2 k +Cγ -1 2 L T ω(γ -2 L T ) + γR(θ, λ 0 ) -V α (θ, λ) , (5.7
)

≤ E sup λ∈Λ 2 +∞ k=1 λ 2 k b -2 k (ξ 2 k -1) -(1 + γ)(1 + α)U 0 (λ) - C 2 α , ≤ E sup λ∈Λ 2 +∞ k=1 λ 2 k b -2 k (ξ 2 k -1) -(1 + α)U 0 (λ) - C 2 α .
(5.8)

Let σ 2 (t) and η(t), t > 0 the quantities defined in (3.1) and (3.4), respectively. Recall that the function t → σ 2 (t) is monotone non-decreasing since it denotes the variance of the ordered process η(t), t ≥ 0 up to some constant. Without loss of generality, we consider that T =]0; +∞[ and σ 2 (t) → +∞ as t → +∞. Let (t s ) s∈N be a grid in R which will be chosen later. Using Lemma 1, it is easy to see that:

E sup λ∈Λ 2 +∞ k=1 λ 2 k b -2 k (ξ 2 k -1) -(1 + α)U 0 (λ) ≤ +∞ s=0 E sup t∈[ts,ts+1] η(t) -(1 + α) 2σ 2 (t) log(C -4 σ 2 (t)) + ,
for some positive constant C. For all x ∈ R, the notation x + denotes the positive part of x, i.e.

x + = x 1 {x>0} . Let s ∈ N be fixed:

E sup t∈[ts,ts+1] η(t) -(1 + α) 2σ 2 (t) log(C -4 σ 2 (t)) + = E sup t∈[ts,ts+1] η(t s ) -η(t s ) + η(t) -(1 + α) 2σ 2 (t) log(C -4 σ 2 (t)) + , ≤ E η(t s ) -(1 + α) 2σ 2 (t s ) log(C -4 σ 2 (t s )) + E(t s ) + , (5.9)
where, for all s ∈ N,

E(t s ) = sup t∈[ts,ts+1]
{η(t) -η(t s )} .

(5.10)

Since the variables η(t s ) and E(t s ) are not independent, the same method as [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF] cannot be applied. Instead, remark that for all 0 < µ < 1, using (5.9),

E sup t∈[ts,ts+1] η(t) -(1 + α) 2σ 2 (t) log(C -4 σ 2 (t)) + , ≤ +∞ Bs P (η(t s ) + E(t s ) ≥ x)dx, ≤ +∞ Bs P (η(t s ) ≥ µx)dx + +∞ Bs P (E(t s ) ≥ (1 -µ)x)dx = A 1 + A 2 , (5.11) with, B s = (1 + α) 2σ 2 (t s ) log(C -4 σ 2 (t s )), ∀s ∈ N.
(5.12)

We are first interested in the study of A 2 . For all p > 1, using a Markov inequality,

A 2 = +∞ Bs P (E(t s ) ≥ (1 -µ)x)dx = +∞ Bs P sup t∈[ts,ts+1] {η(t) -η(t s )} ≥ (1 -µ)x dx, ≤ 1 (1 -µ) p +∞ Bs E sup t∈[ts,ts+1] |η(t) -η(t s )| p x p dx, = 1 (1 -µ) p E sup t∈[ts,ts+1] |η(t) -η(t s )| p 1 (p -1)B p-1 s .
(5.13)

The process η(t), t > 0 is ordered (see Section 6). By Lemma 2, there exists a positive constant C > 0 independent of and t, such that, for all p > 0:

E sup t∈[ts,ts+1] |η(t) -η(t s )| p ≤ C p p p (σ 2 (t s+1 ) -σ 2 (t s )) p/2 . (5.14)
Therefore, using (5.12)-(5.14), we obtain:

A 2 ≤ C (1 -µ)(1 + α) p p p p -1 (σ 2 (t s+1 ) -σ 2 (t s )) p/2 [σ 2 (t s ) log(C -4 σ 2 (t s ))] (p-1)/2 .
(5.15)

Now, choose the grid (t s ) s∈N as follows:

σ 2 (t s ) = 4 1 + 1 s ν s , ∀s ∈ N, (5.16)
where ν ∈]3/4, 1[. This choice is a trade off between an exponential grid similar to [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF] and polynomial grids of [START_REF] Cao | On oracle inequalities related to smoothing splines[END_REF]. Remark that:

σ 2 (t s+1 ) -σ 2 (t s ) = 4 1 + 1 (s + 1) ν s+1 -4 1 + 1 s ν s ≤ 4 1 + 1 s ν s 1 s ν . (5.17)
Moreover, for all s ∈ N, using Taylor formula,

log(C -4 σ 2 (t s )) = s log(C 1/s ) + log 1 + 1 s ν ≥ s log 1 + 1 s ν ≥ Cs 1-ν . (5.18)
Hence, (5.15)-(5.18) yield:

A 2 ≤ 2 C (1 -µ)(1 + α) p p p p -1 (s -ν ) p/2 (1 + s -ν ) sp/2 (1 + s -ν ) s(p-1)/2 (s 1-ν ) (p-1)/2 , = 2 C (1 -µ)(1 + α) p p p p -1 (1 + s -ν ) s/2 (s ν ) p/2 (s 1-ν ) (p-1)/2 , = 2 C (1 -µ)(1 + α) p p p p -1 s (1-ν)/2 × (1 + s -ν ) s/2 s p/2 .
For all s ∈ N, with simple algebra,

(1 + s -ν ) s ν = exp{s ν log(1 + s -ν )} exp(s ν × s -ν ) = O(1)
, as s → +∞.

Therefore:

(1 + s -ν ) s/2 = exp s 1-ν 2 log (1 + s -ν ) s ν ≤ C s 1-ν ,
for some positive constant C > 0. We eventually obtain:

A 2 ≤ 2 g(µ, α) p s (1-ν)/2 p -1 × p p √ s p C s 1-ν , with g(µ, α) = C (1 -µ)(1 + α) . (5.19)
Now, choose the parameter p in a properly way. For example, set p = s 1/4 . Then:

A 2 ≤ 2 s 1-ν 2 -1 4 g(µ, α) s 1/4 s 1/4 s 1/2 s 1/4 C s 1-ν , = 2 s 1 4 -ν 2 g(µ, α) s 1/4 (s 1/4 ) -s 1/4 C s 1-ν , = 2 s 1 4 -ν 2 exp s 1/4 log(g(µ, α)) + s 1-ν log C -s 1/4 log(s 1/4 ) , = 2 s 1 4 -ν 2 exp s 1/4 log(g(µ, α)) + s 1-ν-1/4 log C - 1 4 log(s) .
It is then easy to see that, provided

ν ∈]3/4, 1[, log(g(µ, α)) + s 1-ν-1/4 log C - 1 4 log(s) < 0,
at least for s large enough. Therefore, for all µ ∈ (0, 1):

A 2 ≤ 2 Ce -s 1/4 and +∞ s=1 +∞ Bs P (E(t s ) ≥ (1 -µ)x)dx ≤ C 1 2 , (5.20)
where C 1 is a positive constant independent of .

We are now interested in the study of A 1 in inequality (5.11). For all δ > 0, using a Markov inequality:

A 1 = +∞ Bs P (η(t s ) ≥ µx)dx, = +∞ Bs P (exp(δη(t s )) ≥ exp(δµx))dx, ≤ +∞ Bs E exp{δη(t s )} exp{δµx} dx = 1 δµ E exp{δη(t s )}e -δµBs . (5.21)
Inequality (4.14) of [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF] yields:

E exp{δη(t s )} ≤ exp δ 2 σ 2 (t s ) + 4δ 3 +∞ k=1 6 λ k (t s ) 6 b -6 k (1 -2δλ 2 k (t s )b -2 k ) + . (5.22)
Setting,

δ = log(C -4 σ 2 (t s )) 2σ 2 (t s ) , (5.23)
we obtain,

A 1 ≤ Cµ -1 2σ 2 (t s ) log(C -4 σ 2 (t s )) exp 1 2 log(C -4 σ 2 (t s )) × e -µ(1+α) log(C -4 σ 2 (ts)) , ≤ Cµ -1 2 1 log(C -4 σ 2 (t s )) exp -[µ(1 + α) -1] log(C -4 σ 2 (t s )) .
Indeed, using (3.1)-(3.3) and (5.23):

δ 3 +∞ k=1 λ 6 k (t s )b -6 k (1 -2δλ 2 k (t s )b -2 k ) + → 0, as s → +∞.
With (5.16) and (5.18), we eventually obtain:

+∞ s=1 +∞ Bs P (η(t s ) ≥ µx)dx ≤ +∞ s=1 C 2 s (1-ν)/2 e -s log(1+s -ν ) µ(1+α)-1 , ≤ +∞ s=1 C 2 s (1-ν)/2 exp{-C(µ(1 + α) -1)s 1-ν } ≤ C 2 2 α , (5.24)
setting for example µ = (1 + α/2)/(1 + α). Therefore,

E sup λ∈Λ 2 +∞ k=1 λ 2 k b -2 k (ξ 2 k -1) -(1 + α)U 0 (λ) ≤ C 2 α . (5.25)
Theorem 1 is obtained using (5.8) and (5.25).

2

Proof of Theorem 2

Let µ > 0. By Theorem 1, V µ (θ, λ) is a risk hull. Therefore,

E θ θ -θ 2 ≤ E θ V µ (θ, λ ). (5.26)
Moreover, (4.3) provides that, ∀λ ∈ Λ,

E θ V α (y, λ ) ≤ E θ V α (y, λ).
(5.27)

We will combine inequalities (5.26) and (5.27). First, rewrite V α (y, λ ) in terms of V µ (θ, λ ). By simple algebra:

V α (y, λ ) = +∞ k=1 (λ k ) 2 -2λ k (b -2 k y 2 k -2 b -2 k ) + 2 +∞ k=1 (λ k ) 2 b -2 k + (1 + α)U 0 (λ ), = +∞ k=1 (1 -λ k ) 2 θ 2 k + 2 (λ k ) 2 b -2 k + (1 + µ)U 0 (λ ) + (α -µ)U 0 (λ ), +2 +∞ k=1 (λ k ) 2 -2λ k θ k b -1 k ξ k + +∞ k=1 (λ k ) 2 -2λ k 2 b -2 k (ξ 2 k -1) -θ 2 , = R µ (θ, λ ) + (α -µ)U 0 (λ ) -θ 2 +2 +∞ k=1 (λ k ) 2 -2λ k θ k b -1 k ξ k + +∞ k=1 (λ k ) 2 -2λ k 2 b -2 k (ξ 2 k -1), (5.28)
where R µ (θ, λ ) is defined in (4.5). Using (5.27) and (5.28), for all λ ∈ Λ,

E θ R µ (θ, λ ) ≤ E θ V α (y, λ) -(α -µ)E θ U 0 (λ ) + θ 2 -2E θ +∞ k=1 (λ k ) 2 -2λ k θ k b -1 k ξ k -E θ +∞ k=1 (λ k ) 2 -2λ k 2 b -2 k (ξ 2 k -1). (5.29)
It is then easy to see that, for all λ ∈ Λ,

E θ V α (y, λ) = R α (θ, λ) -θ 2 .
This yields:

E θ R µ (θ, λ ) ≤ R α (θ, λ) -2E θ +∞ k=1 (λ k ) 2 -2λ k θ k b -1 k ξ k (5.30) +E θ +∞ k=1 2λ k -(λ k ) 2 2 b -2 k (ξ 2 k -1) -(α -µ)U 0 (λ ) .
In order to complete the proof, we bound the two last terms of (5.30). First use Corollary 1 in Section 6. For all γ > 0 and for all λ ∈ Λ:

E θ +∞ k=1 (λ k ) 2 -2λ k θ k b -1 k ξ k = E θ +∞ k=1 (1 -λ k ) 2 θ k b -1 k ξ k ≥ -γE θ R(θ, λ ) -γR(θ, λ) -Cγ -1 2 L T ω(γ -2 L T ) (5.31)
Therefore, using (5.29)-(5.31), we obtain:

E θ R µ (θ, λ ) ≤ (1 + γ)R α (θ, λ) + γE θ +∞ k=1 (1 -λ k ) 2 θ 2 k + 2 +∞ k=1 (λ k ) 2 b -2 k +Cγ -1 2 L T ω(γ -2 L T ) + E θ +∞ k=1 {2λ k -(λ k ) 2 } 2 b -2 k (ξ 2 k -1) -(α -µ)U 0 (λ ) , = (1 + γ)R α (θ, λ) + γE θ +∞ k=1 (1 -λ k ) 2 θ 2 k + 2 +∞ k=1 (λ k ) 2 b -2 k ξ 2 k + Cγ -1 2 L T ω(γ -2 L T ), +E θ +∞ k=1 {2λ k -(1 + γ)(λ k ) 2 } 2 b -2 k (ξ 2 k -1) -(α -µ)U 0 (λ ) .
(5.32)

Now, study the last term in the right-hand side of (5.32). It can be rearranged as follow:

E θ +∞ k=1 2λ k -(1 + γ)(λ k ) 2 2 b -2 k (ξ 2 k -1) -(α -µ)U 0 (λ ) , = E θ 2 +∞ k=1 (1 -γ)(λ k ) 2 b -2 k (ξ 2 k -1) -(α -µ)U 0 (λ ) +2E θ +∞ k=1 λ k -(λ k ) 2 2 b -2 k (ξ 2 k -1) = T 1 + T 2 .
We are first interested in T 1 . The bound of this term represents the gain on the traditional URE method. Indeed, if the penalty is zero, only less precise bounds are available (see proof of Theorem 3). Remark that if the constant C T is large, the term T 1 will be negligible compared to T 2 . The efficiency will not be enhanced by the RHM method. Here, using (5.25):

T 1 = (1 -γ)E θ 2 +∞ k=1 (λ k ) 2 b -2 k (ξ 2 k -1) - (α -µ) 1 -γ U 0 (λ ) ≤ (1 -γ) 2 C 2 (α -µ + γ -1) + .
(5.33)

Then study the stochastic term T 2 :

T 2 = 2 2 E θ +∞ k=1 λ k b -2 k (ξ 2 k -1) -2 2 E θ +∞ k=1 (λ k ) 2 b -2 k (ξ 2 k -1). (5.34)
For all λ ∈ Λ and B > 0, using Lemma 3 and the same technics of proof of Corollary 1 (see Section 6) and (6.9):

2 2 E θ +∞ k=1 λ k b -2 k (ξ 2 k -1) ≤ C log 2 S × E θ 4 +∞ k=1 λ 2 k (t )b -4 k + 1 √ S 4 E θ +∞ k=1 λ 2 k (t )b -4 k , ≤ C log 2 S × 2 E θ sup k∈N (λ k ) 2 b -2 k +∞ k=1 λ 2 k (t )b -2 k + C 2 E θ sup k∈N (λ k ) 2 b -2 k E θ +∞ k=1 λ 2 k (t )b -2 k , ≤ γ 2 2 E θ +∞ k=1 λ 2 k (t )b -2 k + C 2 γ -1 log 2 S × E θ sup k∈N (λ k ) 2 b -2 k , ≤ γ 2 E θ +∞ k=1 λ 2 k (t )b -2 k + C 2 γ -1 L T ω(γ -2 L T ).
(5.35) Indeed, the process

ρ(t) = 2 +∞ k=1 λ k (t)b -2 k (ξ 2 k -1), t ≥ 0,
is ordered. The same bound for the second term in the right hand side of (5.34) occurs since the process -η(t), t ≥ 0 is also ordered. Using (3.8), (5.7) and (5.32)-(5.35), we eventually find:

(1 -cγ)E θ V µ (θ, λ ) ≤ (1 + γ) 2 R α (θ, λ) + γE θ +∞ k=1 (1 -λ k ) 2 θ 2 k + 2 +∞ k=1 (λ k ) 2 b -2 k ξ 2 k +Cγ -1 2 L T ω(γ -2 L T ) + C(1 -γ 2 ) 2 (α -µ + γ -1) + + C 2 µ , ≤ (1 + γ) 2 R α (θ, λ) + CγE θ V µ (θ, λ ) +Cγ -1 2 L T ω(γ -2 L T ) + C(1 -γ 2 ) 2 (α -µ + γ -1) + + C 2 µ , (5.36)
for some c > 0. Combining inequalities (5.26), (5.36), choosing γ < γ 0 for some γ 0 , we eventually obtain:

E θ θ -θ 2 ≤ E θ V µ (θ, λ ), ≤ (1 + γ) 2 1 -cγ R α (θ, λ) + C 2 µ + (1 -γ 2 )C 2 (α -µ + γ -1) + + Cγ -1 2 L T ω(γ -2 L T ).
Set µ = γ in order to conclude the proof. The proof essentially follows the same lines of Sections 5.2 and 5.3. Let γ > 0 be fixed and W (θ, λ) defined by:

W (θ, λ(t)) = (1 + 2γ) +∞ k=1 (1 -λ k (t)) 2 θ 2 k + 2 +∞ k=1 λ 2 k (t)b -2 k + C 2 γ -1 L T ω(γ -2 L T ) + γR(θ, λ 0 ), (5.37)
where C denotes a positive constant independent of . First remark that using (5.7):

E θ sup λ∈Λ θλ -θ 2 -W (θ, λ) ≤ E θ sup λ∈Λ (1 + γ) +∞ k=1 (1 -λ k ) 2 θ 2 k + 2 +∞ k=1 λ 2 k b -2 k ξ 2 k + γ 2 +∞ k=1 λ 2 k b -2 k +Cγ -1 2 ω(γ -2 L T ) + γR(θ, λ 0 ) -W (θ, λ) , = E θ sup t∈T (1 + γ) +∞ k=1 (1 -λ k (t)) 2 θ 2 k + η(t) + (1 + γ) 2 +∞ k=1 λ 2 k (t)b -2 k +Cγ -1 2 ω(γ -2 L T ) + γR(θ, λ 0 ) -W (θ, λ) ,
The process η(t), t ≥ 0 is ordered. Using (5.35):

E θ η( t) ≤ γ 2 E θ +∞ k=1 λ 2 k ( t)b -2 k + C 2 γ -1 L T ω(γ -2 L T ),
for some positive constant C and γ > 0, with,

λ( t) = arg sup t∈T θλ -θ 2 -W (θ, λ) .
This inequality can easily be derived from Lemma 3 and proof of Corollary 1. Hence:

E θ sup λ∈Λ θλ -θ 2 -W (θ, λ) ≤ E θ sup λ∈Λ (1 + γ) +∞ k=1 (1 -λ 2 k )θ 2 k + (1 + 2γ) 2 +∞ k=1 λ 2 k b -2 k +Cγ -1 2 L T ω(γ -2 L T ) + γR(θ, λ 0 ) -W (θ, λ) ≤ 0. (5.38)
Therefore, W (θ, λ) can also be considered as a risk hull. Hence, the traditional URE procedure is in some sense a risk hull method. Nevertheless, the hull W (θ, λ) is rather rough compared to V α (θ, λ). The variability of the problem is neglected.

Using the same principle as in (5.26)-(5.32) and (5.38), we get:

E θ R(θ, λ ) ≤ (1 + γ)R(θ, λ) + γE θ R(θ, λ ) + Cγ -1 2 L T ω(γ -2 L T ) +E θ +∞ k=1 {2λ k -(λ k ) 2 } 2 b -2 k (ξ 2 k -1), ≤ (1 + γ)R(θ, λ) + 2γE θ R(θ, λ ) + Cγ -1 2 L T ω(γ -2 L T ).
(5.39)

Then use (5.39), (5.38) in order to conclude the proof.

6 Ordered processes

The control of processes like, for instance:

η(t) = +∞ k=1 2 λ 2 k (t)b -2 k (ξ 2 k -1), t ≥ 0 or ρ(t) = +∞ k=1 (1 -λ k (t))θ k b -1 k ξ k , t ≥ 0, ( 6.1)
is rather important for the construction of a data-driven parameter choice rule. These processes have a strong influence on the behavior of the related estimators. For all t > 0, the sequence (λ k (t)) k∈N is assumed to be monotone decreasing. We will see in this section that they are embedded in a more general class: the ordered processes. These stochastic objects have been introduced in [START_REF] Kneip | Ordered linear smoother[END_REF] and are intensively studied in [START_REF] Cao | On oracle inequalities related to smoothing splines[END_REF]. In this section we recall the definition and present some results useful for the proofs of Theorems 1-3.

(t). It is called ordered if for all t 2 ≥ t 1 ≥ 0, Σ 2 (t 2 ) ≥ Σ 2 (t 1 ), and E[ζ(t 2 ) -ζ(t 1 )] 2 ≤ Σ 2 (t 2 ) -Σ 2 (t 1 ). (6.2)
The class of ordered processes is thus rather vast. In particular, it contains the well-known Wiener processes.

Proposition 1 The process η(t), t ≥ 0, defined in (6.1) is ordered.

PROOF. For all t ≥ 0,

Eη(t) = 0 and Σ 2 (t) = 2 4 +∞ k=1 λ 4 k (t)b -4 k = 2σ 2 (t).
Assume that for all k ∈ N, λ k (t) → 1 as t → +∞ and let t 1 ≤ t 2 be fixed. Then Σ 2 (t 1 ) ≤ Σ 2 (t 2 ) since, for all k ∈ N, λ k (t 1 ) ≤ λ k (t 2 ). Moreover,

Eη(t 1 ) 2 = E +∞ k=1 2 λ 2 k (t 1 )b -2 k (ξ 2 k -1) 2 = 4 +∞ k=1 λ 4 k (t 1 )b -4 k E (ξ 2 k -1) 2 , ≤ +∞ k=1 2 λ 2 k (t 1 )b -2 k 2 λ 2 k (t 2 )b -2 k E (ξ 2 k -1) 2 , = E +∞ k=1 2 λ 2 k (t 1 )b -2 k (ξ 2 k -1) × +∞ k=1 2 λ 2 k (t 2 )b -2 k (ξ 2 k -1) = E[η(t 1 )η(t 2 )]. (6.
3)

The property (6.2) is a consequence of (6.3). The process η is ordered.

2

The main characteristic of ordered processes is contained in the following lemma.

Lemma 2 Let ζ(t), t ≥ 0 an ordered process and suppose that there exists κ > 0 such that

ϕ(κ) = sup t1,t2 E exp κ ζ(t 1 ) -ζ(t 2 ) E[ζ(t 1 ) -ζ(t 2 )] 2 < +∞. (6.4)
Then, there exists a constant C depending on κ such that, for all S, T > 0 and all p ≥ 1,

E sup s,t∈[S,T ] |ζ(t) -ζ(s)| p ≤ C p p p Σ 2 (T ) -Σ 2 (S) p . (6.5)
This result is an extension of Lemma 1 of [START_REF] Cao | On oracle inequalities related to smoothing splines[END_REF]. They consider the particular case S = 0. This lemma is rather important. It may be useful in many situations. In particular, many processes encountered in the proofs of Theorems 1 and 2 are ordered and satisfy (6.4). The proof follows essentially the same lines of [START_REF] Cao | On oracle inequalities related to smoothing splines[END_REF].

The following result is a consequence of the previous lemma.

Lemma 3 Let ζ(t), t ≥ 0 be an ordered process satisfying (6.4) such that ζ(0) = 0 and t measurable with respect to ζ. Then there exists a positive constant C depending on κ and τ > 0 such that for all K > 1:

E|ζ(t )| ≤ log 2 (K)EΣ(t ) + C EΣ 2 (t ) K .
PROOF. First remark that:

Eζ(t ) = Eζ(t )1 {ζ(t )≤(log K) 2 Σ(t )} + Eζ(t )1 {ζ(t )≥(log K) 2 Σ(t )} , ≤ (log K) 2 EΣ(t ) + Eζ(t )1 {ζ(t )≥(log K) 2 Σ(t )} . (6.6)
We can assume without loss of generality that Σ(t) → +∞ as t → +∞. Let (t k ) k∈N the real sequence verifying:

Σ(t k ) = k d EΣ 2 (t ) K , ∀k ∈ N, (6.7)
where d > 0 will be chosen later. Using Lemma 2, C(s + q) (s+q) 2 d(s+q) K s(p-1) k ds(p-1) EΣ 2 (t )

Eζ(t )1 {ζ(t )≥log 2 (K)Σ(t )} = Eζ(t )
s(p-1) × 1 (log K) 2q 1/s , = EΣ 2 (t ) p EΣ 2 (t ) p-1 × 1 K × +∞ k=1 1 k d(p-1)
C(s + q) (s+q) 2 d(s+q) K sp (log K) 2q 1/s .

Setting for instance d = 2/(p -1), Eζ(t )1 {ζ(t )≥log 2 (K)Σ(t )} 1 {t >t1} ≤ C EΣ 2 (t ) K × q q/s 2 dq K p (log K) 2q/s .

Recall that the parameter 1 < p < 2 is fixed. Set q = s log K. Therefore:

Eζ(t )1 {ζ(t )≥log 2 (K)Σ(t )} 1 {t >t1} ≤ C EΣ 2 (t ) K × K p (log K) log K 2 d log K/s (log K) 2 log K , = C EΣ 2 (t ) K × e p 2 d log K log K ≤ C EΣ 2 (t ) K ,
for some C > 0 independent of K. This concludes the proof of Lemma 3.

Corollary 1 Let t > 0 measurable w.r.t. the sequence (ξ k ) k∈N and t 0 the oracle defined in (3.5). For all γ > 0, the following inequalities hold:

(i) E θ +∞ k=1 λ 2 k ( t)θ k b -1 k ξ k ≤ γE θ R (θ, λ) + γR (θ, λ 0 ) + C 2 γ -1 L T ω(γ -2 L T ), (ii) E θ +∞ k=1 (1 -λ k ( t)) 2 θ k b -1 k ξ k ≤ γE θ R (θ, λ) + γR (θ, λ 0 ) + C 2 γ -1 L T ω(γ -2 L T ),
for some C > 0 independent of . PROOF. In a first time introduce:

∆ [λ] = 2 sup k∈N λ 2 k b -2 k , ∀λ ∈ Λ. (6.8)
First prove inequality (i).

E θ +∞ k=1 λ 2 k ( t)θ k b -1 k ξ k = E θ +∞ k=1 {λ 2 k ( t) -λ 2 k (t 0 )}θ k b -1 k ξ k .
The process:

ρ(t) = +∞ k=1 {λ 2 k (t) -λ 2 k (t 0 )}θ k b -1 k ξ k , t > 0,
is ordered on [t 0 ; +∞[. Moreover, ρ(t) = ρ(t -1 ), t ≥ 0 is also ordered on [t -1 0 ; +∞[. Both processes satisfy (6.4) for some κ > 0. Hence, using Lemma 3 with K = √ S:

E θ +∞ k=1 λ 2 k ( t)θ k b -1 k ξ k ≤ C log 2 (S).E θ +∞ k=1 λ 2 k (t 0 ) -λ 2 k ( t) 2 θ 2 k 2 b -2 k + C √ S E θ +∞ k=1 λ 2 k (t 0 ) -λ 2 k ( t) 2 θ 2 k 2 b -2 k ,
where λ 0 = λ(t 0 ) is the oracle defined in (3.5) and C a positive constant independent of . Then, remark that the following inequality holds:

{λ 2 k (t 0 ) -λ 2 k ( t)} 2 = [(1 -λ k ( t)) -(1 -λ k (t 0 ))] 2 (λ k ( t) + λ k (t 0 )) 2 , ≤ 4 1 -λ k ( t) 2 + (1 -λ k (t 0 ) 2 λ 2 k (t 0 ) + λ 2 k ( t) .
For all γ > 0, using the elementary inequality 2ab ≤ γa 2 + γ -1 b 2 , we obtain:

E θ +∞ k=1 λ 2 k ( t)θ k b -1 k ξ k ≤ γE θ +∞ k=1 (1 -λk ) 2 θ 2 k + Dγ -1 log 2 (S).E θ ∆ [ λ] + γ +∞ k=1 (1 -λ 0 k ) 2 θ 2 k + Dγ -1 log 2 (S).∆ [λ 0 ],
for some positive constant D. With simple algebra, for all x > 0 and λ ∈ Λ:

Dγ -1 log 2 (S).∆ [λ] = Dγ -1 log 2 (S). where the function ω is introduced in (3.6) and computed for some particular examples in Section 3.2. Then set x = Dγ -2 log 2 S in order to obtain (i).

The proof of (ii) exactly follows the same lines. It uses in particular Corollary 1 and inequality (34) of [START_REF] Cavalier | Oracle inequalities for inverse problems[END_REF].

EXAMPLE 1 .

 1 (Projection estimators) Obviously, Assumption (3.1)-(3.

2 5. 2

 22 Proof of Theorem 3.2

2 5. 4

 24 Proof of Theorem 4.2

Definition 2

 2 Let ζ(t), t ≥ 0 a separable random process with Eζ(t) = 0 and finite variance Σ 2

≤

  Dγ -1 log 2 (S).x -1 2 +∞ k=1 λ k b -2 k + Dγ -1 log 2 (S). 2 ω(x),(6.9)

  Let p ∈ N such that 1 < p < 2. Using Hölder inequality:Eζ(t )1 {ζ(t )≥log 2 (K)Σ(t )} 1 {t >t1} )1 {ζ(t )≥log 2 (K)Σ(t )} 1 {t∈[t k ,t k+1 ]} , )≥log 2 (K)Σ(t )} 1 {t∈[t k ,t k+1 ]} ,, where r and s are such that r -1 + s -1 = 1. In the following, set r = 2/p and s = 2/(2 -p). Such a choice leads to:Eζ(t )1 {ζ(t )≥log 2 (K)Σ(t )} 1 {t >t1} t∈[t k ,t k+1 ] |ζ(t)| s Σ ps (t k ) 1 n sup t∈[t k ,t k+1 ] |ζ(t)|≥log 2 (K)Σ(t k ) oEζ(t )1 {ζ(t )≥log 2 KΣ(t )} 1 {t >t1}

		+∞
		= Eζ(t = k=1 +∞ k=1 E Σ p (t ) 1 {ζ(t ≤ ζ(t ) Σ p (t ) +∞ k=1 [EΣ pr (t )] 1/r × E Σ ps (t ) |ζ(t )| s	1 {ζ(t )≥log 2 (K)Σ(t )} 1 {t∈[t k ,t k+1 ]}	1/s
							1/s
	≤	EΣ 2 (t )
							1/s
				×	1 (log K) 2q	,
		≤	EΣ 2 (t )	×	1 (log K) 2q	1/s	.
	Now, use (6.7) in order to obtain:
							1/s
	≤	EΣ 2 (t )				×	1 (log K) 2q	,
	≤	EΣ 2 (t )

1 {ζ(t )≥log 2 (K)Σ(t )} 1 {t <t1} + Eζ(t )1 {ζ(t )≥log 2 (K)Σ(t )} 1 {t >t1} , ≤ E sup t<t1 |ζ(t)| + Eζ(t )1 {ζ(t )≥log 2 (K)Σ(t )} 1 {t >t1} , ≤ Σ(t 1 ) + Eζ(t )1 {ζ(t )≥log 2 (K)Σ(t )} 1 {t >t1} , = EΣ 2 (t ) K + Eζ(t )1 {ζ(t )≥log 2 (K)Σ(t )} 1 {t >t1} . p × +∞ k=1 E sup .

Let q > 0 which will be chosen later. Using a Markov inequality and Lemma 2, we obtain:

Eζ(t )1 {ζ(t )≥(log K)Σ(t )} 1 {t >t1} ≤ EΣ 2 (t ) p × +∞ k=1 E sup t∈[t k ,t k+1 ] |ζ(t)| s+q Σ ps+q (t k ) p × +∞ k=1 C(s + q) (s+q) Σ s+q (t k+1 ) Σ ps+q (t k ) p × +∞ k=1 C(s + q) (s+q) (k + 1) d(s+q) EΣ 2 (t ) s+q K -(s+q) k d(ps+q) EΣ 2 (t ) ps+q K -(ps+q) p × +∞ k=1