Timothée Dumas
email: dumas@emse.fr

Olivier Bonnefoy

Philippe Grosseau

Gérard Thomas

Sébastien Nebut

Laurent Guy

NEW METHODS TO ANALYSE FRAGMENTATION MECHANISMS OF PRECIPITATED SILICAS

Keywords: Structure-property, Fluidized bed, Ultrasounds, Breakage, Attrition, Silica

Precipitated silica is traditionally used as reinforcing filler in rubber applications. In pneumatic manufacture, it offers several advantages compared to carbon black. Indeed, in tyres treads, precipitated silica can yield a lower rolling resistance and better wet grip at equal wear resistance than carbon black.

The development of a new method to study the fragmentation mechanism of precipitated silica is investigated. Usually, the dispersion of granules is done in high viscous media (elastomer or oils). In this paper, the study is carried out in a friendlier medium (air or water).

INTRODUCTION

Precipitated silica is constantly gaining importance as reinforcing filler for rubber compounds. It offers several advantages compared to carbon black. In tyres treads, precipitated silica can yield a lower rolling resistance and better wet grip at equal wear resistance than carbon black (Figure 1). The capacity of precipitated silica in rubber is due to its high aptitude to disperse (decrease of granules size) and distribute (spatial repartition) in the elastomer during the mixing steps. Previous studies of agglomerates break-up have revealed different fragmentation mechanisms. One mechanism, denoted rupture [START_REF] Bolen | Intensive mixing[END_REF][START_REF] Bohin | Kinetics of dispersion for sparse agglomerates in simple shear flows: application to silica agglomerates in silicone polymers[END_REF][START_REF] Scurati | Influence of powder surface treatment on the dispersion behavior of silica into polymer materials[END_REF], is characterized by an abrupt breakage of the agglomerate into a few large pieces. The second mechanism, named erosion [START_REF] Shiga | Processability of EPR in an internal mixer (II) -Morphological changes of carbon black agglomerates during mixing[END_REF][START_REF] Rwei | Observation of carbon black agglomerate dispersion in simple shear flows[END_REF][START_REF] Rwei | Characterization of agglomerate dispersion by erosion in simple shear flows[END_REF][START_REF] Collin | Etude rhéo-optique des mécanismes de dispersion du noir de carbone dans des élastomères[END_REF][START_REF] Astruc | Etude rhéo-optique des mécanismes de dispersion de mélanges sous cisaillement simple[END_REF], is characterized by the detachment of small fragments from the outer surface of the agglomerate. More recently, a third mechanism has been observed and called disintegration [START_REF] Collin | Etude rhéo-optique des mécanismes de dispersion du noir de carbone dans des élastomères[END_REF]. In this mechanism, a large number of very small fragments are created in a short period of time.

These fragmentation mechanisms are observed in viscous media (elastomers and oils) similar to the medium in which they will be dispersed (tyre rubber). In this paper, new and easier methods (in air or water) are suggested for the observation of fragmentation mechanisms and compared with previous results in viscous media. The proposed experimental protocols give the possibility to study in detail the dispersion kinetics.

MATERIALS AND METHOD

Products

The materials used in this study are precipitated silicas named Z1115MP and Z1165MP (supplied by Rhodia). Observations using scanning electron microscopy (SEM) give information on the morphology (Figure 2). Silica granules appear as spherical micro-balls. The mean diameter of the two populations is similar (about 280 µm). Physico-chemical properties of silica agglomerates are known to play a leading role during the mixing step with elastomer and other compounds. The main difference between the two precipitated silicas is their specific surface area which is more important for Z1165MP (160 m²/g) than for Z1115MP (110 m²/g). Z1115MP silica presents a bigger median pore diameter than the Z1165MP one. This pore plays an important role in agglomerate infiltration by the elastomer [START_REF] Roux | Caractérisation in-situ des mécanismes de dispersion de la silice dans une matrice élastomère soumise à un cisaillement[END_REF]. The physical properties are presented in Table 1.

Devices for studying fragmentation in air

In order to evaluate the behaviour of precipitated silica granules in air, two tests have been designed: the first one is a high energy breakage test and the second one a low energy breakage test.

The first test consists in a wall impact experiment. In this test, some granules are accelerated thanks to a Venturi system and undergo high energy collisions against a wall. The Venturi system is characterized by its pressure drop. When the pressure drop increases, the granule velocity increases. The granule breaks into small fragments, which sizes are measured by laser granulometry (Malvern Mastersizer Scirocco). We can modify the pressure drop to analyze the influence of the granule velocity on the fragmentation.

The second test is led in a fluidized bed. Usually in this test, a large number of granule-granule low energy collisions are observed. The fluidization device is composed of a glass tube fed at its bottom by an air jet (flow speed: 6 cm/s). A mass of 10 grams of silica is fluidized for a time between 1 and 8 minutes. Then, the air jet is stopped and the size distribution is measured (Malvern Mastersizer HydroS).

Device for studying fragmentation in water

In water, we have been using ultrasounds (US) to solicit the granules. The experiments are performed directly in the Malvern Mastersizer HydroS which is equipped with an internal ultrasound probe (max 26 W, 20 kHz, with a liquid volume of 160 mL, 100 mg of silica).

After each US excitation, measurements of the particles' size are performed by laser granulometry.

Fragmentation description

Two complementary parameters are chosen to describe the fragmentation.

The first parameter is the weight fraction P 50 of grains whose size remains under 50 µm. It is especially convenient to describe the generation of fine particles during the impacts.

The second parameter is the reduction index K, chosen to quantify the overall size reduction during the fragmentation.

Eq (1)

d 43 0 and d 43 t are respectively the mean particle size (volume) at initial time and after a solicitation time t. This parameter is mostly influenced by the biggest granules and is relatively unaware of the existence of small particles created during the solicitation.

RESULTS AND DISCUSSION

Fragmentation in air

High energy breakage test (impact on a wall)

The PSD curves obtained for different initial grains velocities give information on the fragmentation mechanisms (Figure 3). For a high impact velocity (∆P=2 bars), the main population appears at 7 µm for Z1115MP while Z1165MP has its main population at 140 µm. We observed, between ∆P=0.1 bar and ∆P=1 bar, that the main peak of the initial population is shifted towards smaller sizes for Z1165MP while, for Z1115MP, only the fraction of granule of this size is reduced. According to Sahoo's [START_REF] Sahoo | Review : an investigation of single particle breakage tests for coal handling system of the gladstone port[END_REF] definition, it matches to an erosion mechanism for Z1165MP. In addition, the Z1115MP granules break quickly and with a lower impact velocity (∆P=0.5 bar).

Low energy breakage test (fluidized bed)

In a fluidized bed, inter-granules collisions are predominant, albeit some impacts between granules and the column are present. This second type of impact is much less violent than in the impact breakage test. The PSD of the two precipitated silicas after different solicitation times are presented in Figure 5 and Figure 6. After solicitation in fluidized bed, we can observe the same kind of results as with the previous high energy test, that is to say an important population appears at 10 µm for Z1115MP. The initial population disappears more quickly for Z1115MP than for Z1165MP. Now, we shall use the two parameters, the reduction index K and P 50 previously introduced, to study the fragmentation.

In a fluidized bed (Figure 7), we do not observe a major variation of K and P50 for a solicitation time between 1 and 4 minutes. For a much longer time (8 min), Z1115MP generates many small fragments with an important reduction of size. An energy gap seems to appear between 4 and 8 min. Indeed, for the three first points (at 1, 2, and 4 min) there is no large variation of the two parameters. But between 4 and 8 min, increases of K (for the two silicas) and P 50 (for Z1115MP) are observed. In a second step, results of impact breakage (Figure 8) are analyzed with the two descriptors. For Z1115MP a larger reduction of size and a more important formation of fine fragments than for Z1165MP are observed. The reduction index is always 20% bigger for Z1115MP for a pressure drop greater than 1 bar. Whatever the test and the associated collision energy, we observe that the Z1115MP silica creates more fine particles than the Z1165MP one, and that a fraction of the main peak of initial population is always present for Z1165MP. That suggests an erosion or rupture in small fragment mechanism. For Z1165MP, the fine generation is less intense but a reduction of size is observed with a shift of the main peak of initial population. That is consistent with a rupture mechanism.

Put all together, these results suggest that:

-The Z1115MP is more brittle than Z1165MP for high energy collisions.

-Fracture is favoured vs. Erosion when the impact energy increases.

-The energy of solicitation plays a role in the fragmentation mechanisms.

-Collision energy in fluidized bed is approximately equal to collision energy in impact test with a pressure drop of 0.5 to 1 bar.

Fragmentation in water

An observation of the fragmentation in water is done with an ultrasound probe (0.26 W) to disperse the agglomerates. The silica Z1165MP seems to break slowly in comparison with Z1115MP. During the solicitation by ultrasounds, two modes of population appear for the Z1165MP (maximum at 10 and 60 µm) whereas the Z1115MP presents only one population mode (at 10 µm). This observation leads us to postulate the existence of two different fragmentation mechanisms linked to the solicitation route.

When analyzing with the descriptors introduced before (K and P 50 : Figure 11) more information about the mechanisms can be derived. A very quick size reduction is observed for Z1115MP since more than 50% of granules have a size under 50 µm after 110 seconds of solicitation. Again, this is compatible with erosion mechanism. The Z1165MP is slower to break, the fraction of granules with size under 50 µm increases linearly during the first minutes of solicitation. 850 seconds are needed to have 50 % of the granules under 50 µm. The initial population of Z1165MP disappears slowly while a population of fine particles (10 µm) is created. This analysis for Z1165MP confirms the rupture mechanism.

Size evolution equation

From the PSD, the evolution of particle size (d 43 ) as a function of the solicitation time t is studied (Figure 12). A fit of the curve by an exponential law (Eq. 2) with a characteristic time τ is performed and gives a fairly good agreement. For a given set of experimental conditions, this equation brings us quantification of the limit size (thermodynamics data) and characteristic time of granule fragmentation (kinetics data) (Table 2). Thanks to this characteristic time, we find again that Z1165MP breaks-up slowly in comparison with Z1115MP. The asymptotic fragment size is also different for these silicas. Z1165MP seems harder to fragment with a limit fragment size of 22 µm and only 12 µm for Z1115MP. These two results show that the Z1165MP silica presents a better strength resistance to breakage.

CONCLUSION

Silica agglomerates can be dispersed by rupture and erosion. Many studies have observed these mechanisms of dispersion in viscous media. In this paper, we have presented new methods (in air and in water) to analyse the dispersion of precipitated silica. We have observed the same dispersion mechanism as in elastomer. These new methods give also access to the kinetics of dispersion. Finally a size evolution equation introduced a new parameter, the characteristic time of dispersion.

Figure 1 .

 1 Figure 1. Advantages of precipitated silica against carbon black.

Figure 2 .

 2 Figure 2. MEB observations of precipitated silica Z1165MP (left) and Z1115MP (right).

Figure 3 .

 3 Figure 3. Impact breakage test for Z1165MP.

Figure 4 .

 4 Figure 4. Impact breakage test for Z1115MP.

Figure 5 .

 5 Figure 5. Fluidized bed test for Z1165MP.

Figure 6 .

 6 Figure 6. Fluidized bed test for Z1115MP.

Figure 7 .

 7 Figure 7. Fluidized bed results about reduction index K and P 50 .

Figure 8 .

 8 Figure 8. Impact breakage results about the reduction index K and P 50 .

Figure 9 .

 9 Figure 9. PSD evolution of Z1115MP during ultrasonic excitation (16 s of ultrasound between each curves).

Figure 10 .

 10 Figure 10. PSD evolution of Z1165MP during ultrasonic excitation (16 s of ultrasound between each curves).

Figure 11 .

 11 Figure 11. Time evolution of the average size reduction (K) and the fines generation (P 50 ).

Figure 12 .

 12 Figure 12. Evolution of particle diameter with solicitation time t.

Table 1 .

 1 Precipitated silica characteristics.

	Typical analysis	Z1165MP	Z1115MP
	BET specific surface area (m²/g)	160	110
	Median particles size (µm)	279±2	282±2
	Median pore diameter (nm)	29	48
	Elementary particle diameter (nm)	17	24

Table 2 .

 2 Characteristic time for Z1165MP and Z1115MP.

		Z1165MP	Z1115MP
	Characteristic time τ (s)	510	100
	Limit size d ∞ @ 0.26 W (µm)	22	12

ACKNOWLEDGMENTS

This work is done in the framework of the DURAMAT project which is one of the AXELERA's programmes for innovative, durable and environmental friendly materials. We thank our industrial partners: Rhodia and Michelin, our partners in other laboratories CEMEF/EMP in Nice, LTDS/ECL in Lyon, IMP/UJM in Saint Etienne and CNRS who are also involved in this work.


Akira Sato

Eric Serris

Philippe Grosseau

Gérard Thomas

Alain Chamayou

Laurence Galet
email: laurence.galet@mines-albi.fr

Michel Baron

Experiment and simulation of the dry particle coating

Keywords: Dry particle coating, Discrete Element Method, Cyclomix

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Dry particle coating is a method to modify the properties of powders by attaching the fine particles (invited particles) onto the surface of core particles (host particles) by mechanical forces using neither organic solvent nor aqua binder. Because of its environmental friendliness and simple process, it is getting more and more importance for industries. Even though a lot of experimental works of the dry particle coating are reported in the literature [START_REF] Alonso | Mechanism of the combined coating-mechanofusion processing of powders[END_REF][START_REF] Lefevre | Dry coating of talc particles with fumed silica: Influence of the silica concentration on the wettability and dispersibility of the composite particles[END_REF][START_REF] Honda | Preparation of monolayer particle coated powder by the dry impact blending process utilizing mechanochemical treatment[END_REF][START_REF] Watano | Surface modification of food fiber by dry particle coating[END_REF][START_REF] Ouabbas | Surface modification of silica particles by dry coating: Characterization and powder ageing[END_REF][START_REF] Lefevre | Dry Coating of Talc Particles: Effect of Material and Process Modifications on Their Wettability and Dispersibility[END_REF], the theoretical modelling for such process [START_REF] Iwasaki | Determination of optimum operating conditions based on energy requirement for particle coating in a dry process[END_REF], that is also very important to optimize the process or to design new geometry of the coating apparatus, is found to be less comprehensive. The objective of this study is also to understand the dry powder coating by experimental and numerical methods. Discrete Element Method (DEM), proposed by Cundall and Strack [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF], has been applied to simulate the motion of the particles inside the coating apparatus. DEM is one of the most popular techniques for simulating and analysing the solid particle behaviour and has been successfully applied in many fields [START_REF] Cleary | DEM modelling of industrial granular flows: 3D case studies and the effect of particle shape on hopper discharge[END_REF][START_REF] Kano | A method for simulating three dimensional motion of balls under presence of powder in a tumbling mill[END_REF][START_REF] Matchett | Vibrating powder beds: a comparison of experimental and Distinct Element method simulated data[END_REF]. Then this work describes our research to reveal the effect of operating condition such as rotation speed of the mixer on the particles coating and its numerical approach.

EXPERIMENT

Materials

For the powder couple, the Suglets and magnesium stearate (MgSt) were chosen as host/invited particles respectively. Suglets were provided by NP pharm company. This powder consists of spherical sugar core composed of mainly sucrose and maize starch. Also it has highly hydrophilic characteristic. This material is mainly used as excipient in capsule and tablet formulation, particularly in multiparticulate systems. They form cores upon which API are coated for controlled or sustained release drug delivery technologies. Figure 1 (a) shows a SEM picture of Suglets particles. They are spherical and their surface seems to have crystal-like form. The median particle size of Suglet, D 50 is about 250 µm. MgSt were used as invited particles. This material is a fine, white, greasy, cohesive and hydrophobic powder widely used in pharmaceutical formulations as a lubricant. The shape of the magnesium is relatively random, including needle-like and plate-like configuration (Figure 2(b)). The D 50 is about 4 µm. These properties of the samples are summarized in Table 1.

Coating process

A 1L Cyclomix has been chosen to perform a dry coating in this study. This apparatus is defined as a high shear mixer/granulator, manufactured by Hosokawa Micron B.V. As it can be seen from Figure 2, it consists of a conical vessel with an axial impeller in the centre equipped with four sets of special designed blades. It applies high mechanical impact and shearing forces on the particles in order to break the fine agglomerates and coat them on the host particles. This device has also been used successfully for dry particle coating experiment [START_REF] Ouabbas | Particle-particle coating in a cyclomix impact mixer[END_REF]. Assuming the invited and host particles have uniform sizes respectively and spherical shape, the mass ratio of invited to host particles to obtain a 100% monolayer surface coverage, w could be expressed using the size ratio of host and invited particle, K H [START_REF] Thomas | Modeling the mean interaction forces between powder particles. Application to silica gel-magnesium stearate mixtures[END_REF]:

Eq(1)
Eq [START_REF] Lefevre | Dry coating of talc particles with fumed silica: Influence of the silica concentration on the wettability and dispersibility of the composite particles[END_REF] Where, C 2D is the packing fraction of invited particles on the surface of host particle, ρ H and ρ I are the densities of host and invited particles, and R H and R I are the radii of host and invited particle respectively. Estimating the packing fraction shall be that of compact hexagonal structure: C 2D =0.906, the ideal mass ratio for the 100 % surface coverage is about 3.92 %. In this research, the mass ratio has been chosen to be 5 %. To focus on the effect of the operation time and rotational speeds, the mass of the host particles and the mass fraction was fixed. The rotational speeds were in the range from 500 to 1500 rpm and operation time was varied from 1 to 30 min. The operation conditions are summarized in Table 2.

Characterization

The various characterization methods were used after dry particle coating. The surface morphology of the particles has been observed by scanning electron microscopy (SEM). The SEM allows us to analyse the surface morphology of the particles and also identify the guest particles on the surface of the host particles. Malvern Mastersizer 2000 laser diffraction granulometer has been used in order to analyse the different size distributions of the particles and also to evaluate qualitatively the interactions between guest and host particles by changing dispersing air pressure. Modification of different surface properties of the particles was analysed using various techniques. Freeman technology powder rheometer (FT-4) has been used in order to characterise the flowability of products. This device analyses the flow rate index (FRI) that represents the flowability of a powder, high values are characteristic of cohesive powders. The sessile drop test has been done to evaluate the wettability of the products.

SIMULATION OF THE PARTICLE MOTION IN CYCLOMIX

The particle motion in the Cyclomix has been simulated the discrete element method (DEM). Figure 3 shows the geometry of Cyclomix modelled for the DEM simulation. The particles with the properties of the host particles given in Table 3 were placed in the vessel of the Cyclomix with hexagonal-close packed structure. Gravity has been introduced settle the assembly. The current computer power is not enough to simulate the actual number of particles. Then smaller number of particles (about 40000), but with larger size (2.0 mm) has been applied. To estimate the coating efficiency, several values that could be related to the coating has been measured, such as the collision frequency, average rotational distance and the distribution of the collision forces. Especially, the average rotational distance shown in Eq. 2 could be very important for the coating. When the host particle rotates (like in Figure 4), the invited particles near this host particle would be attached like as making a snowman.

Eq (2)

RESULT AND DISCUSION

Surface morphology

The SEM images of the particle surface after the dry coating operation are shown in Figure 5. At the coated particle at 500 rpm and 1min operation, it can be seen that the almost all the surface of the particle is covered by MgSt. It might be because the experiments were carried out with excess of ideal mass ratio, the surface of the host particles was also easily covered by the invited particles. However, after operating much more time, surface is getting smoother. That could be because the MgSt deforms film-like coating by shear force or impaction, that is to say, dry coating progressed to film coating. As for the coated product at 1000 rpm and 1min operation, the surface seems rougher than that of 500rpm operation. After 10 and 30 min operation, those surfaces seem to present the agglomeration on itself. During the operation, a little flake of host particle could separate from the main part, agglomerated each other, and then finally attached to the surface of the host particles again. Therefore, the surface would be a mixture of the host particle flake and MgSt. At 1min and 1500rpm operation, the surface is seen to be similar to that of 30 min and 500 rpm operation. After 10 and 30 min operation, they seem to have similar surface of the particle after 10 and 30 min and 1000 rpm.

Flowability

The value of Flow Rate Index (FRI) was measured after the dry coating treatment. Figure 6 shows FRI as a function of operating time at each rotational speed. The initial value of FRI is that of host particle. At 500 rpm, FRI decreases with an increase in the operating time. In the other word, the flowability of the composite is getting better. In addition, the surface is getting smoother along with the operation time at 500 rpm as seen in Figure 5. That could help the particles to have better flowability. As for 1000 rpm, FRI decreases with an increase in the operation time more rapidly than that of 500 rpm up to 5min, and then it seems to be stable. At 1500 rpm, FRI decreases with an increase in the operation time up to just 1min, and then it increases rapidly to the certain value. That could be because of the fragmentation or breakage of the host particles happen due to the high shear stress and impaction between particle-particle and particle to the wall/impeller.

Wettability

The water drop test has been carried out to analyse the wettability of the products. Figure 7 shows that the images of the water drop test for raw Suglets and for the product submitted to a 1 min at 1000 rpm treatment. As can be seen, the water drop has been absorbed as soon as it dropped on the powder bed of the raw Suglets due to its high hydrophilic property. On the other hand, the water drop on the product remains even after 30 s. It could be said that as for the wettability, the 1 min operation at 1000 rpm is enough to change the surface property from hydrophilic to hydrophobic. Figure 8 shows the contact angle of the products 30 s after the water drop on the surface. The initial point of this figure is calculated and induced identically by the following equations.

θ(t 0 ) = S H0 θ H + (1-S H0 ) θ I Eq(3)
Eq(4)

Where, θ H and θ I are contact angles of host and invited particle respectively, and S H is surface fraction of host particles. As can be seen in Figure 8, at 500 rpm the contact angle increase with an increase of operation time until 5 min and then, it seems to become stable and the value is almost as same as that of MgSt, which indicates that the surface of the host particle is almost covered by the MgSt at 5 min operation. At 1000 rpm, the contact angle increases more rapidly than that of 500 rpm until 1 min operation, however, the contact angle decreases slightly along with the operation time after that. That might be because of the fragmentation or the breakage of the host particles. At 1500 rpm, it also increases until 1 min operation, then, after that, it starts to decrease along with the operation time with more remarkable rapidity. That also might be due to the breakage of the host particles.

Particle size distribution analysis

Figure 9 shows the number distribution obtained by a Malvern Mastersizer dry feed system with 0.5bar air pressure. All of the peaks are around 250 μm that is near to the median diameter of the Suglets. Figure 10 shows the volume distribution obtained with 0.5 bar. Almost all the peaks are around 250 μm, however, there is one difference observed. The peaks of the samples corresponding to 1 min and 3 min at 500 rpm operations are smaller than those observed at the other samples. It can be estimated that the very small particles are measured in these samples, that is to say, there are 30s after already separated MgSt or fragment of the Suglets in the sample. Figure 11 shows the volume distribution obtained with 2.5 bar air pressure. At 500 rpm there are small peaks around 7-8 μm.

From their sizes, it could be estimated that these peaks are that of magnesium stearate separated due to the air pressure. At 1000 rpm, on the other hand, there is no remarkable change observed. At 1500 rpm, there also can be seen smaller peaks around 60 μm. These peaks might be the broken or abraded Suglets. The number distribution would show these tendencies clearer. Figure 12 shows the number distributions obtained with 2.5 bar air pressure. At 500 rpm, majority of the peaks is around 250 μm that is almost as same as that of Suglets itself. However, for the products obtained after 1min and 3min operations, there is smaller peak, which is around 7-8 μm. That could be estimated that it is MgSt separated during the analysis under pressure of 2.5 bar. At 1000 rpm, the majority of the peak remains around the 50 % particle size of Suglets. Only the particles after 1min operation has smaller peak, which is similar to that of 500 rpm. That tendency indicates that 1000 rpm operation gives stronger adhesion between host and invited particles than that of 500 rpm. At 1500 rpm, a different tendency was observed. There are the peaks around 60 μm, which couldn't be seen at the others rotation speed. Hence, they should be the consequence of breakage or abrasion.

Numerical result

An example snapshot of DEM simulation is shown in Figure 13. The colour shows the velocity distribution of the particles, blue particles have lower velocity and red one has higher velocity. The particles around the impeller have high velocity. There are few particles at the bottom part of the vessel, which indicates that impellers force the particles to the wall and then, due to the effect of the conical shape wall, the particles go to the upper side of the vessel. Finally, the particles around the vertical rotor fall down to the bottom. The parameters that would be related to the dry coating phenomena such as number of collision and average rotational distance are shown in Figure 14. Number of collision increases with an increase in the rotational speed. It increases rapidly up to 1000 rpm, however over 1000 rpm, it increases moderately. The rotational distance increases linearly with an increase in the rotational speed up to 2500 rpm. Over 3000 rpm, it decreases a little. That is because at high rotational speed, too many particles go up to the upper side of the vessel, and then the particles interrupt other particles motion each other. Those results indicate that the dry coating effect becomes lager along with the rotational speeds. On the other hand, the efficiency could be down. Figure 15 shows that the distribution of the colliding forces of the particles. The orders of forces range from 10 -2 to 10 2 . The peak of the distribution increases with an increase in the rotational speed. In the same time, the forces at the peak increase too. This result indicates that the colliding forces increase with an increase in the rotational speed. However the rate is not as same rapidity as the initial rotational speed. In addition, if the yield stress of the particles is given, it is able to predict if the breakage of the particles happen or not by this approach.

CONCLUSION

In this study, dry coating experiments were carried out by using a Cyclomix high shear mixer for the purpose of coating Suglets by MgSt using different operating conditions. SEM images of particle surface show that the invited particles initially cover the host particles and then they deform to appear the film-like shape coating. In addition, giving an excessive treatment to the particles should induce attrition, which makes the dry coating performance worse. The flowability becomes better along with the operation. However it gets worse when breakage or abrasion of host particles occurs. Particle size distribution indicates the presence of the separation of the invited particles and breakage of the host particles. The DEM simulation of these experiments has been performed. The results show that the dry coating performance becomes better along the rotational speed until an asymptotic level.

LIST OF SYMBOLS

Figure 1 .

 1 Figure 1. SEM images of Suglets(a) and MgSt(b).

Figure 3 .

 3 Figure 3. The geometry of the modelled Cyclomix.

Figure 4 .

 4 Figure 4. Definition of rotational distance of the i-th particle.

Figure 5 .

 5 Figure 5. SEM images of particle surface after each operation.

Figure 6 .

 6 Figure 6. Flow rate index as a function of operating time.

Figure 7 .

 7 Figure 7. The images of the water drop test : (a) Raw suglets, (b) Sulget + 5% MgSt 1min at 1000rpm.

Figure 8 .

 8 Figure 8. Contact angles as a function of operation time.

Figure 9 .

 9 Figure 9. Number distribution of the particles after operation using dispersion pressure of 0.5 bar: (a) 500 rpm, (b) 1000 rpm, (c) 1500 rpm.

Figure 10 .

 10 Figure 10. Volume distribution of the particles after operation using dispersion pressure of 0.5 bar: (a) 500 rpm, (b) 1000 rpm, (c) 1500 rpm.

Figure 11 .Figure 12 .

 1112 Figure 11. Volume distribution of the particles after operation using dispersion pressure of 2.5 bar: (a) 500 rpm, (b) 1000 rpm, (c) 1500 rpm.

Figure 13 .Figure 14 .

 1314 Figure 13. An example of snapshot of the DEM simulation at 1500rpm.

Figure 15 .

 15 Figure 15. The distribution of the colliding forces of the particles.

Table 1 .

 1 Properties of suglets and MgSt.

	particles	D 50 (µm)	solid density (kg/m3)	water affinity
	Suglets	250	158.0	hydrophilic
	MgSt	4	114.0	hydrophobic

Table 2 .

 2 Operating conditions.

	Coating device	Rotation speed (rpm)	Time (min)	Mass % of MgSt
	Cyclomix	500,1000,1500	1-30	5%

Table 3 .

 3 Parameters for DEM simulation.

		i	1
		N	i
	avg	N
			i-th particle
			Young's modulus	E [GPa]	4.5
			Poisson's ratio	[-]	0.30
			Density	[kg/m3]	160
			Coefficient of restitution	[-]	0.16
			Time step	t [sec]	2.0 x 10-6
			Total simulation time	T [sec]	1.5


Thanh Mai

Pham Ngoc

Vincent Rivoirard
email: vincent.rivoirard@dauphine.fr

The dictionary approach for spherical deconvolution

Keywords: Density deconvolution, Dictionary, Lasso estimate, Oracle inequalities, Calibration, Sparsity, Second generation wavelets AMS subject classification : 62G07, 62G05, 62G20

We consider the problem of estimating a density of probability from indirect data in the spherical convolution model. We aim at building an estimate of the unknown density as a linear combination of functions of an overcomplete dictionary. The procedure is devised through a well-calibrated 1-penalized criterion. The spherical deconvolution setting has been barely studied so far, and the two main approches to this problem, namely the SVD and the hard thresholding ones considered only one basis at a time. The dictionary approach allows to combine various bases and thus enhances estimates sparsity. We provide an oracle inequality under global coherence assumptions. Moreover, the calibrated procedure that we put forward gives very satisfying results in the numerical study when compared with other procedures.

Introduction

We consider the spherical deconvolution problem. We observe:

Z i = ε i X i , i = 1, . . . , N (1)
where the ε i are i.i.d. random variables of SO(3) the rotation group in R 3 and the X i 's are i.i.d. random variables of S 2 , the unit sphere of R 3 . We suppose that X i and ε i are independent. We also assume that the distributions of Z i and X i are absolutely continuous with respect to the uniform measure on S 2 and we set f Z and f the densities of Z i and X i respectively. The distribution of ε i is absolutely continuous with respect to the Haar measure on SO(3) and we will denote it f ε . In this paper, we consider that f ε is known. Then we have

f Z = f ε * f,
where * denotes the convolution product which is defined below in [START_REF] Candès | The Dantzig selector: statistical estimation when p is much larger than n[END_REF].

The aim of the present paper is to recover the unknown density f from the noisy observations Z i 1 thanks to a well-calibrated 1 -penalised least squares criterion. Roughly speaking, each genuine observation X i is contaminated by a small random rotation. Although the problem of deconvolution has been extensively addressed in the case of the real line, it has been barely the case on the sphere. The spherical geometry has its own characteristics and includes more complex analytical tools. The model of spherical convolution, as expressed in [START_REF] Bertin | Adaptive Dantzig density estimation[END_REF], has applications in medical imaging and in astrophysics. In medical imaging, people are interested in estimating a fiber orientation density from high angular resolution diffusion MRI data (MRI stands for Magnetic Resonance Imaging) see the work of Tournier, Calamante, Gadian and Connelly [START_REF] Tournier | Direct estimation of the fiber orientation density function form diffusion-weihted MRI data using spherical deconvolution[END_REF]. In their paper, the spherical convolution (1) models the situation where the density of the MRI data is viewed as a convolution of a response function and the density of interest. In astrophysics, the socalled UHECR (Ultra High Energy Cosmic Rays) are at the core of astrophysics concerns. In order to understand the mechanisms of the UHECR, a crucial challenge is the estimation of the density probability of the incidence directions with which the UHECR arrive on the earth. The convolution model takes into account a natural noise which corrupts the genuine observations X i .

The first authors who actually solved this problem were Healy, Hendriks and Kim in their pioneering work, see [START_REF] Healy | Spherical deconvolution[END_REF]. They introduced an orthogonal series method based on the Fourier basis of L 2 (S 2 ) namely the spherical harmonics and assessed its theoretical performances by presenting convergence rates for Sobolev type regularities. Moreover, the spherical harmonics constitute the SVD (Singular Value Decomposition) basis in the spherical deconvolution setting and hence allow to invert the convolution operator f ε in a stable way. Subsequently, Kim and Koo [START_REF] Kim | Optimal spherical deconvolution[END_REF] proved that those rates of convergence were optimal and refined those results by enhancing sharp minimaxity under a super-smooth condition on the error distribution, see [START_REF] Kim | Sharp minimaxity and spherical deconvolution for super-smooth error distributions[END_REF]. The SVD procedure is of course appealing for its simplicity and its ability to invert quickly the operator f ε but it has poor local performances. Indeed, the spherical harmonics which are spread all over the sphere might be a drawback if one is interested in highlighting some local features of the density of interest. It is the case whenever one concentrates on the infinity norm or on adapting to inhomogeneous smoothness. To circumvent these problems, Kerkyacharian, Pham Ngoc and Picard [START_REF] Kerkyacharian | Localized spherical deconvolution[END_REF] considered a thresholding procedure on needlets. The needlets due to Narcowich, Petrushev and Ward [START_REF] Narcowich | Localized tight frames on spheres[END_REF] is a tight frame constructed on the spherical harmonics. They enjoy very good localization properties. This procedure turned out to be profitable both in theory with consideration of L p loss with 1 ≤ p ≤ ∞ and in practice.

Nonetheless as one may have noticed, each approach mentioned above leans on only one basis, the spherical harmonics or the needlet one. Consequently, instead of sticking to only one basis, it may be relevant to consider an overcomplete dictionary. Moreover, K can be larger than the number of observations N contrary to thresholding techniques where K ≤ N .

With this aim in view, we would like to build an estimate of f as a linear combination of functions of a dictionary (ϕ 1 , . . . , ϕ K ) with ϕ k ∈ L 2 (S 2 ). Denote by f λ the linear combination

f λ (x) = K k=1 λ k ϕ k (x), x ∈ S 2 , λ = (λ 1 , . . . , λ K ) ∈ C K . ( 2
)
By considering an overcomplete dictionary which cardinality K can be larger than the sample size N , we tacitly believe that the estimates of f is sparse, namely that very few coordinates of λ is non zero. To the best of our knowledge, the dictionary approach has not been used to face the spherical convolution model [START_REF] Bertin | Adaptive Dantzig density estimation[END_REF] or its analogous on the real line expressed as Y = X + ε . A question immediately comes into sight. Because we precisely treat a convolution problem which provides a relevant setup where observations may come from one source observed through some noise, is there some hope to keep a sparse structure of the estimates of λ? In other words, is the dictionary approach capable to retrieve sparsity despite the action of the convolution operator? The answer seems to be affirmative at least in the present numerical study that we conducted with a 1 -penalized criterion.

Indeed, we suggest a data-driven choice of λ that will be obtained by a well-calibrated 1penalized criterion. The so-called popular Lasso first introduced by Tibshirani [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF] has been widely used since then in the statistical literature. In a fairly general Gaussian framework we may cite the recent work of Massart and Meynet [START_REF] Massart | An 1 oracle inequality for the Lasso[END_REF], in the linear model regression, see [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF][START_REF] Donoho | Stable recovery of sparse overcomplete representations in the presence of noise[END_REF][START_REF] Meinshausen | High dimensional graphs and variable selection with the Lasso[END_REF][START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF] and for nonparametric regression with general fixed or random design, see [START_REF] Bickel | Simultaneous analysis of Lasso and Dantzig selector[END_REF][START_REF] Bunea | Sparsity Oracle Inequalities for the LASSO[END_REF][START_REF] Bunea | Aggregation for Gaussian regression[END_REF][START_REF] Bunea | Aggregation and sparsity via 1 penalized least squares[END_REF]. The Lasso performances were also studied in the density estimation framework by Bunea, Tsybakov and Wegkamp [START_REF] Bunea | Sparse density estimation with l 1 penalties[END_REF][START_REF] Bunea | Spades and Mixture Models[END_REF], van de Geer [START_REF] Van De Geer | High dimensional generalized linear models and the Lasso[END_REF] and Bertin, Le Pennec and Rivoirard [START_REF] Bertin | Adaptive Dantzig density estimation[END_REF]. In addition, many efforts have also been provided to prove model selection consistency of the Lasso, see [START_REF] Bickel | Simultaneous analysis of Lasso and Dantzig selector[END_REF][START_REF] Lounici | Sup-norm convergence rate and sign concentration property of Lasso and Dantzig estimators[END_REF][START_REF] Meinshausen | High dimensional graphs and variable selection with the Lasso[END_REF][START_REF] Yu | On model selection consistency of Lasso estimators[END_REF][START_REF] Zhang | The sparsity and bias of the Lasso selection in highdimensional linear regression[END_REF][START_REF] Zou | The adaptive Lasso and its oracle properties[END_REF].

1 penalty methods have also been investigated to solve inverse problems. We may cite among others the work of Loubes (see [START_REF] Loubes | 1 penalty for ill-posed inverse problems[END_REF]) who tackled the classical inverse regression model with independent errors by minimizing an empirical contrast built upon the SVD basis of the operator with an 1 penalty term. But we stress that it was by no means a dictionary approach. In addition, we point out that the model of interest in [START_REF] Loubes | 1 penalty for ill-posed inverse problems[END_REF] and papers devoted to inverse problems in general are much more linked to regression problem whereas our convolution model expressed in [START_REF] Bertin | Adaptive Dantzig density estimation[END_REF] is to be more connected to a density estimation problem.

In this paper, we aim at showing that the dictionary approach conducted thanks to the Lasso minimization algorithm can be used successfully to face the spherical deconvolution problem in theory but especially in practice. Indeed, in the simulation study, we compare it with the hard thresholding procedure on needlets of Kerkyacharian, Pham Ngoc and Picard [START_REF] Kerkyacharian | Localized spherical deconvolution[END_REF], showing that the dictionary approach does pretty well both in graphics reconstructions and in terms of quadratic and sup-norm losses. The Lasso estimates actually enhance the sparsity of the representation of the signal. Moreover, the choice of tuning parameters turns out to be easy to calibrate and match what the theory states contrary to thresholding techniques where theorems are too conservative about the allowed values of tuning parameters. This constitutes an undeniable advantage of the present Lasso procedure especially when simulations are time consuming which is the case on the sphere.

Here is the outline of the present paper. In section 2 we give some basic tools of Fourier analysis on L 2 (SO(3)) and L 2 (S 2 ) and the construction of the Lasso estimates. In section 3 we obtain oracle inequalities under mild assumptions on the dictionaries. In section 4 we present our simulation results. Section 5 is devoted to the proofs of our results.

2 Lasso-type estimates of the density f 2.1 Preliminaries about harmonic analysis on S 2 and SO [START_REF] Birgé | Minimal penalties for Gaussian model selection[END_REF] Let us begin with some notations and some elements of harmonic analysis on S 2 and SO(3) which will be useful throughout the paper.

For two functions g and h we denote < g, h > the L 2 -hermitian product between g and h: < g, h >= For any vector λ ∈ C K and any set of indices J, we denote λ J ∈ C K the vector which has the same coordinates as λ on J and 0 elsewhere. We set for any 1 ≤ q < ∞,

||λ|| q = K k=1 |λ k | q 1 q
.

We shall now recall some elements of harmonic analysis on SO(3) and S 2 . We shall refer the reader to Healy, Hendriks and Kim [START_REF] Healy | Spherical deconvolution[END_REF] and Kim and Koo [START_REF] Kim | Optimal spherical deconvolution[END_REF] for more precisions. Consider the functions, known as the rotational harmonics,

D l mn (φ, θ, ψ) = e -i(mφ+nψ) P l mn (cos θ), (m, n) ∈ I 2 l l = 0, 1, . . . (3)
where

θ ∈ [0, π), φ ∈ [0, 2π), ψ ∈ [0, 2π) and I l = [-l, -l + 1, . . . , l -1, l].
The generalized Legendre associated functions P l mn are fully described in Vilenkin [START_REF] Vilenkin | Fonctions spéciales et théorie de la représentation des groupes[END_REF]. The √ 2l + 1D l mn form a complete orthonormal basis of L 2 (SO(3)) with respect to the Haar probability measure. For f ∈ L 2 (SO(3)), we define the rotational Fourier transform on SO(3) by

(f * l ) mn = SO(3) f (g)D l mn (g)dg. (4)
Then (f * l ) is a matrix of size (2l + 1) × (2l + 1) which entrance is given by the element (f * l ) mn with m ∈ I l and n ∈ I l . The rotational inversion can be obtained by

f (g) = l -l≤m, n≤l (f * l ) mn D l mn (g) = l -l≤m, n≤l (f * l ) mn D l mn (g -1 ), (5)
Equation ( 5) is to be understood in L 2 -sense although with additional smoothness conditions, it can hold pointwise. A parallel spherical Fourier analysis is available on S 2 . Any point on S 2 can be represented by ω = (cos φ sin θ, sin φ sin θ, cos θ) t , with , φ ∈ [0, 2π), θ ∈ [0, π). We also define the functions:

Y l m (ω) = Y l m (θ, φ) = (-1) m (2l + 1) 4π (l -m)! (l + m)! P l m (cos θ)e imϕ , m ∈ I l , l = 0, 1, . . . , (6)
with φ ∈ [0, 2π), θ ∈ [0, π) and P l m (cos θ) are the associated Legendre functions. The functions Y l m obey Y l -m (θ, φ) = (-1) m Y l m (θ, φ). (7)
The set {Y l m , m ∈ I l , l = 0, 1, . . .} is forming an orthonormal basis of L 2 (S 2 ), generally referred to as the spherical harmonic basis.

Again, as above for f ∈ L 2 (S 2 ), we define the spherical Fourier transform on S 2 by

(f * l ) m = S 2 f (x)Y l m (x)dx, ( 8
)
where dx is the uniform probability measure on the sphere S 2 . Then (f * l ) is a vector of size 2l + 1 which entrance is given by the element (f * l ) m with m ∈ I l .

The spherical inversion can be obtained by

f (x) = l m∈I l (f * l ) m Y l m (x). (9)
The bases detailed above are important because they realize a singular value decomposition of the convolution operator created by our model. In effect, we define for

f ε ∈ L 2 (SO(3)), f ∈ L 2 (S 2 )
the convolution by the following formula:

f ε * f (x) = SO(3) f ε (u)f (u -1 x)du, (10)
and we have for all m ∈ I l , l = 0, 1, . . .,

(f ε * f ) * l m = n∈I l (f * l ε ) mn (f * l ) n . (11)
2.2 The lasso estimator of the density f .

In the sequel, the estimate of f will be a linear combination of functions of the dictionary Υ = (ϕ k ) k=1,...,K . For any λ ∈ C K we set:

f λ = K k=1 λ k ϕ k , λ = (λ k ) k=1...K .
We assume that for any k, ||ϕ k || 2 = 1. We set for any k,

β k = S 2 ϕ k (x)f (x)dx.
If we denote for any function ϕ k , (ϕ * l k ) m the (l, m)-Fourier coefficient of ϕ k :

(ϕ * l k ) m =< ϕ k , Y l m >= S 2 ϕ k (x)Y l m (x)dx,
the Parseval equality yields

β k = ∞ l=0 m∈I l (ϕ * k ) l m (f * ) l m .
The SVD method yields the following unbiased estimate of (f * ) l m (see Healy, Hendriks and Kim [START_REF] Healy | Spherical deconvolution[END_REF], Kim and Koo [START_REF] Kim | Optimal spherical deconvolution[END_REF] and Kerkyacharian, Pham Ngoc and Picard [START_REF] Kerkyacharian | Localized spherical deconvolution[END_REF]).

( f * l ) m = 1 N N i=1 n∈I l (f * l ε ) -1 mn Y l n (Z i ),
where (f * l ε ) -1 mn denotes the inverse of the rotational Fourier transform of f ε . Precisely, one considers the matrix f * l ε of size (2l + 1) × (2l + 1) which entrance at lign m and column n is given by the Fourier transform (f * l ε ) mn , then one inverts this matrix and take the entrance at lign m and column n given by (f * l ε ) -1 mn . Consequently,

βk = 1 N N i=1 ∞ l=0 m∈I l n∈I l (ϕ * l k ) m (f * l ε ) -1 mn Y l n (Z i ),
is an unbiased estimate of β k . In particular, if we set for any x ∈ S 2 ,

φ k (x) = ∞ l=0 m∈I l n∈I l (ϕ * l k ) m (f * l ε ) -1 mn Y l n (x), then βk = 1 N N i=1 φ k (Z i ).
We now introduce the Lasso estimator f L of the density f . Definition 1. The lasso estimate is f L = max ( (f λL ), 0 where λL is the solution of the following minimization problem

λL = argmin λ∈C K C(λ) + 2 K k=1 η 1,k | (λ k )| + η 2,k | (λ k )| , (12)
where

C(λ) = ||f λ || 2 2 -2 K k=1 λ k βk .
and (η 1,k ) k∈{1,...,K} and (η 2,k ) k∈{1,...,K} are two sequences of positive real numbers chosen in ( 21) and ( 22) subsequently.

The next proposition shows that λL is obtained by minimizing a 1 -penalized empirical contrast.

Proposition 1. For any λ ∈ C K , E[C(λ)] = ||f λ -f || 2 2 -f 2 2 , which yields argmin λ∈C K E[C(λ)] = argmin λ∈C K ||f λ -f || 2 2 .
Let G the Gram matrix associated to the dictionary Υ given for any 1 ≤ k, k ≤ K by

G kk = S 2 ϕ k (x)ϕ k (x)dx. (13)
We have for any k and k , G kk = G k k which entails that the matrix G is hermitian. Now, the key point for establishing oracle properties of f L is the following result.

Proposition 2. A necessary condition for λ to be a solution of ( 12) is

| ((Gλ) k -βk )| ≤ η 1,k and | ((Gλ) k -βk )| ≤ η 2,k ∀k ∈ {1, . . . , K}. (14)
In particular, for any k,

(Gλ) k -βk ) ≤ |η k |.
where

η k = η 1,k + iη 2,k . (15)
Now, we are ready to propose values for the parameters (η 1,k ) k and (η 2,k ) k . We rely on following heuristic arguments. Let us denote Π Υ (f ) the projection of f on the linear space spanned by the functions of the dictionary Υ. There exists λ(f

) ∈ C k such that Π Υ (f ) = K k=1 λ(f ) k ϕ k . ( 16
)
Our goal is to choose η 1,k and η 2,k as small as possible such that λ(f ) satisfies [START_REF] Donoho | Ideal spatial adaptation via wavelet shrinkage[END_REF]. If, as expected, our wealthy dictionary Υ provides a sparse linear combination of the functions of Υ that approximates f accurately, we can hope that the well calibrated Lasso procedure does a good job for estimating f . Let us justify these points. On the one hand, for fixed k, we have:

(Gλ(f )) k = ϕ k (x) K k =1 λ(f ) k ϕ k (x)dx = ϕ k (x)Π Υ (f )(x)dx = ϕ k (x)f (x)dx = β k .
On the other hand, when N goes to infinity,

βk -β k = O P (1).
More precisely, we have the following result providing the values of the Lasso parameters

(η 1,k ) k and (η 2,k ) k . Theorem 1. We set σ2 1,k = 1 2N (N -1) i =j ( (φ k (Z i )) -(φ k (Z j ))) 2 (17)
and σ2 2,k = 1 2N (N -1) i =j ( (φ k (Z i )) -(φ k (Z j ))) 2 (18)
the unbiased estimates of

σ 2 1,k = Var( (φ k (Z 1 ))) and σ 2 2,k = Var( (φ k (Z 1 ))).
Then we introduce

σ2 1,k = σ2 1,k + 2 (φ k ) ∞ 2σ 2 1,k γ log K N + 8 (φ k ) 2 ∞ γ log K N , (19)
σ2 2,k = σ2 2,k + 2 (φ k ) ∞ 2σ 2 2,k γ log K N + 8 (φ k ) 2 ∞ γ log K N ( 20
)
η 1,k = 2σ 2 1,k γ log K N + 2 (φ k ) ∞ γ log K 3N ( 21
)
and

η 2,k = 2σ 2 2,k γ log K N + 2 (φ k ) ∞ γ log K 3N . ( 22
)
Let us assume that K satisfies

N ≤ K ≤ exp(N δ ) for δ < 1. Let γ > 1.
Then, for any ε > 0, there exists a constant C 1 (ε, δ, γ) depending on ε, δ and γ such that if Ω is the random set such that for any k ∈ {1, . . . , K}

| (β k -βk )| ≤ η 1,k and | (β k -βk )| ≤ η 2,k , then P (Ω c ) ≤ C 1 (ε, δ, γ)K 1-γ 1+ε .
The probability bounds of Theorem 1 were established under the condition γ > 1. It is a well-known fact that the conditions on tuning parameters provided by the theory proved to be very often too conservative in practice. For instance, in thresholding techniques, one is often lead to consider smaller tuning parameter values than what the theory allows to. Here, we set in the sequel γ = 1.01 the smallest value authorized by the theory which conducts to a full calibrated Lasso procedure.

3 Oracle and minimax properties satisfied by lasso-type estimates

Oracle inequalities under coherence assumptions for general dictionaries

In the sequel, we establish oracle inequalities under classical assumptions on the dictionary. We first introduce the minimal "restricted" eigenvalue of the Gram matrix G: for 1 ≤ l ≤ K, we denote

ξ min (l) = min |J|≤l min λ∈C K λ J =0 ||f λ J || 2 2 ||λ J || 2 2 .
Since the functions of the dictionary satisfy ||ϕ k || 2 = 1 for any k, we have ξ min (l) ∈ [0, 1] for any l. When the dictionary constitutes an orthonormal system, we have ξ min (l) = 1 for any 1 ≤ l ≤ K. By contrast, if two functions of the dictionary are proportional, then ξ min (l) = 0 for any 2 ≤ l ≤ K. So, assuming that ξ min (l) is close to 1 means that every set of columns of G with cardinality less than l behaves like an orthonormal system. We also consider the restricted correlations: for 1 ≤ l, l ≤ K, we denote

θ l,l = max |J|≤l |J |≤l J∩J =∅ max λ,λ ∈C K λ J =0,λ J =0 f λ J , f λ J ||λ J || 2 ||λ J || 2 .
Small values of θ l,l mean that two disjoint sets of columns of G with cardinality less than l and l span nearly orthogonal spaces. We shall use the following assumption:

Assumption 1. For s an integer such that 1 ≤ s ≤ K/2 and c 0 a positive real number, we have

ξ min (2s) > c 0 θ s,2s .
Oracle inequalities for the Dantzig selector were established under Assumption 1 with c 0 = 1 in the parametric linear model by Candès and Tao in [START_REF] Candès | The Dantzig selector: statistical estimation when p is much larger than n[END_REF]. It was also considered by Bickel, Ritov and Tsybakov [START_REF] Bickel | Simultaneous analysis of Lasso and Dantzig selector[END_REF] for non-parametric regression and for the Lasso estimate for larger values of c 0 .

For any J ∈ {1, . . . , K}, let us set J C = {1, . . . , K}\J and define λ J the vector which has the same coordinates as λ on J and zero coordinates on J C .

We now have:

Theorem 2. Let us assume that Assumption 1 is true for some positive integer s and with c 0 = 1. On Ω, the random set introduced in Theorem 1, we have for any α > 0, any λ that satisfies the Lasso constraint ( 14)

|| f L -f || 2 2 ≤ inf λ∈C K λ 1 ≤ λ 1 inf J⊂{1,...,K} |J|=s ||f λ -f || 2 2 + α 1 + 2µs κs 2 λ J C 2 1 s + 16s 1 α + 1 κ 2 s ||η|| 2 ∞ ( 23
)
with, using [START_REF] Donoho | Adaptating to unknown smoothness via wavelet shrinkage[END_REF],

||η|| ∞ = max k∈{1,...,K} |η k |,
and κ s and µ s are defined as follows:

µ s = θ s,2s ξ min (2s) , κ s = ξ min (2s) - θ s,2s
ξ min (2s) .

Let us give an interpretation of the right hand side of Inequality [START_REF] Lounici | Sup-norm convergence rate and sign concentration property of Lasso and Dantzig estimators[END_REF]. The value of the infimum depends on three terms. The first two terms are approximation terms that naturally appear since our procedure is based on minimization of an 1 -penalized L 2 -criterion and the third one can be viewed as a variance term.

Concerning the behaviour of this variance term ||η|| 2 ∞ our result can be closely connected to the one recently obtained by Dalalyan and Salmon [START_REF] Dalalyan | Sharp Oracle Inequalities for Aggregation of Affine Estimators[END_REF] (see in their technical report the Theorem 1, the Remark 4 and the section 3 devoted to ill-posed inverse problems and group weigthing). Indeed, their paper deals with non-parametric regression model with heteroscedastic Gaussian noise which is known to well describe ill-posed inverse problems and they obtain the same behaviour for their remaining term.

Numerical results

In this section, we present some numerical experiments which make a comparison in practice between the Lasso procedure described in this paper and the thresholding algorithm on needlets of Kerkyacharian, Pham Ngoc and Picard [START_REF] Kerkyacharian | Localized spherical deconvolution[END_REF]. We aim at reconstructing a density defined on S 2 from noisy data. This density presents one principal mode and minor fluctuations at the basis which means that the observations are mainly concentrated in one direction and otherwise a little bit spread all over the sphere. In directional statistics, the unimodal density is of general interest as a common density to model spherical data is given by the well-known von Mises-Fisher distribution caractherized by a mean direction and a concentration parameter.

For each method, we consider a data set of 800 observations generated from our target density. Then we contaminate each data by some noise which consists in a rotation about the 0z axis by a random angle. For each observation, the random angle is of course different and follows a uniform law on a certain interval U [0, α], α > 0. Of course the larger the interval of the uniform law is, the larger the amount of noise is.

The dictionary combines spherical harmonics and needlets which are so far the two approximations basis which have been put forward in the spherical deconvolution issue. The dictionary is actually composed of 81 spherical harmonics and 1020 needlets, hence the cardinality of the dictionary is K = 1101. The number of spherical harmonics used corresponds to a maximal degree L = 8 and the needlets to a maximal resolution J = 3. Our dictionary thus mixes an orthonormal family the spherical harmonics and a semi-orthogonal family the needlets, namely, every two needlets which are from levels at least two levels apart are orthogonal. Hence the needlets are close to an orthonormal basis. The ensueing overcomplete dictionary ensures a property of mutual incoherence following the terminology of [START_REF] Donoho | Stable recovery of sparse overcomplete representations in the presence of noise[END_REF].

The construction of the needlets has been conducted using the spherical pixelization HEALPix software package. HEALPix provides an approximate quadrature of the sphere with a number of data points of order 12.2 2J and a number of quadrature weights of order 1 12.2 2J . This approximation is considered as reliable enough and commonly used in astrophysics.

More precisely, let us describe the scheme for the Lasso procedure.

1. Compute the βk for all k = 1 . . . K

Compute σ2

1,k , σ2 2,k , σ2 1,k and σ2 2,k given by ( 17)-( 20).

3. Compute the η 1,k and η 2,k defined in ( 21) and ( 22) with γ = 1.01.

4.

Compute the coefficients λL by the Lasso minimization described in [START_REF] Dalalyan | Sharp Oracle Inequalities for Aggregation of Affine Estimators[END_REF].

5. Select the support ĴL of the estimate λL . ĴL defines a subset of the dictionary on which the density is regressed:

λL ĴL = G -1 ĴL ( βk ) ĴL ,
where G ĴL is the submatrix of the Gram matrix G corresponding to the subset ĴL . The values of λL outside ĴL are set to 0.

Compute the final estimate f

L = (f λL ) = ( K k=1 λL k ϕ k ).
We shall give some comments about this scheme. The Gram matrix have been pre-computed, the scalar products between the dictionary functions being computed thanks to the spherical quadrature formula see [START_REF] Narcowich | Localized tight frames on spheres[END_REF].

Step 5 is a least squares step as advocated in Candes and Tao [START_REF] Candès | The Dantzig selector: statistical estimation when p is much larger than n[END_REF] which is intended to decrease the bias introduced by the Lasso. As already pointed out in the comments of Theorem 1, the tuning paramater γ in the expression of η 1,k and η 2,k is set to 1.01. Let us describe now briefly the needlet thresholding algorithm, all details for this procedure can be found in Kerkyacharian, Pham Ngoc and Picard [START_REF] Kerkyacharian | Localized spherical deconvolution[END_REF]. A needlet is denoted ψ jη , βjη is the estimate of the scalar product between f and ψ jη , j is the resolution level and η is the quadrature point around which the corresponding needlet is almost exponentially localized. Each η belongs to a quadrature set Z j provides by HEALPix and which cardinality is equal to 12.2 2j . We have set J = 3. The estimator of f is given by:

f T = J j=0 η∈Zj βjη 1{| βjη | ≥ κt N |σ j |}ψ jη , 10 with t N = log N N , σ 2 j = A 2 j+1 l=2 j-1 n∈I l | m∈I l ψ * l jη,m (f * l ε ) -1 mn | 2 ,
with A ≥ f Z ∞ . The quantity σ 2 j constitutes an upper bound for the variance of the estimated coefficients βjη . Here, we have decided to estimate directly the variance of βjη and plug it in the expression of f T , like in [START_REF] Kerkyacharian | Localized spherical deconvolution[END_REF].

As for the tuning parameter κ, we set it to κ = 1 which gives the best results in terms of L 2 loss.

Let us give some comments to highlight the choice of tuning parameters in both methods. This latter value of κ = 1 was not easy to find and relies on the data at stake and the type of noise, in other words it is an ad hoc choice. Moreover, κ = 1 does not correspond to what the theory says. Theorems in Kerkyacharian Pham Ngoc and Picard [START_REF] Kerkyacharian | Localized spherical deconvolution[END_REF] states that for the L 2 loss for instance, κ should be taken greater than which is equal to 17 with our target density, not to mention that for real data f ∞ is unknown. This ad hoc choice of κ constitutes a real drawback especially when simulations are time consuming which is the case on the sphere. On the other hand, for the Lasso, once one has set γ = 1.01 which is the smallest value allowed by theoretical arguments, the Lasso offers a full-calibrated procedure and gives good results. Now, we present the graphics reconstructions, L 2 and L ∞ losses. The estimated losses are computed over 15 runs. The sup-norm loss is computed on an almost uniform grid of S 2 of 192 points provided by the software HEALPix. The analytic expression of our target density in spherical coordinates is:

f (θ, φ) = c 1 0.2(cos(1.7 × θ)) 2 sin(1.2 × φ)
+ exp -4(((sin(θ) × cos(φ) + 0.7071) 2 + (sin(θ) × sin(φ) + 0.7071) 2 + cos 2 (θ)))) + 0.2 , with c 1 a normalization constant and θ ∈ [0, π] and φ ∈ [0, 2π].

Concerning the graphics reconstructions, we present the target density, the Lasso estimate and the needlet thresholding one for various amounts of noise. Analyzing the graphic reconstructions, it appears that for the first case of noise with φ ∼ U [0, π 8 ], the Lasso selects only four coefficients whereas the needlet thresholding procedure keeps 31 coefficients. Consequently, the Lasso enhances much better the sparsity of the signal. As we increase the noise, with φ ∼ U [0, 3π 16 ], the Lasso keeps only one coefficient, whereas the needlet thresholding algorithm keeps 27 ones. Once again, the Lasso highlights the very sparse feature of the signal. Of course, for both methods, the intensity of the peaks decreases as the random rotations scatter the observations. We precise that on the graphics, the left extremity of the estimated density is the continuation of the right one because of the spherical symmetry.

At closer inspection, both methods manage to recover the principal mode even if the small fluctuations for the Lasso are slightly flattened on the top. Although the graphic reconstructions seem a bit better for the thresholding method, both procedures are capable to localize the main peak which constitutes the most important fact in directional problems. That said, as the Lasso only keeps very few coefficients, it is pretty normal that the reconstructions are visually a bit more distorted but on the other hand we stress that we gain sparsity and clearer interpretation of our signal.

Considering the L 2 loss and the sup-norm loss, the Lasso performs better in three of four cases and always better for the sup-norm which is a nice result.

Appendix

Proof of Proposition 1

Straightforward computations establish Proposition 1. We have:

E[C(λ)] = S 2 K k=1 λ k ϕ k (x) 2 dx - K k=1 λ k β k - K k=1 λ k β k = S 2 K k=1 λ k ϕ k (x) 2 dx - K k=1 λ k S 2 f (x)ϕ k (x)dx - K k=1 λ k S 2 f (x)ϕ k (x)dx = S 2 K k=1 λ k ϕ k (x) 2 dx - K k=1 λ k S 2 f (x)ϕ k (x)dx - K k=1 λ k S 2 f (x)ϕ k (x)dx = S 2 K k=1 λ k ϕ k (x) 2 dx - S 2 f (x) K k=1 λ k ϕ k (x) + λ k ϕ k (x) dx. But, K k=1 λ k ϕ k -f 2 2 = S 2 K k=1 λ k ϕ k (x) -f (x) K k=1 λ k ϕ k (x) -f (x) dx = S 2   K k=1 λ k ϕ k (x) 2 + f 2 (x) -f (x) K k=1 λ k ϕ k (x) -f (x) K k=1 λ k ϕ k (x)   dx = E[C(λ)] + f 2 2 ,
which proves the result.

Proof of Proposition 2

We set for any k, λ

k = (λ k ), λ (1)
k = (λ k ), ϕ (2)
k = (ϕ k ), ϕ (1)
k = (ϕ k ) and β(1) k = ( βk ), β (2)
k = ( βk ).

We now show that for any k, ∂C

= 2 ((Gλ) k -βk ), (λ) ∂λ (1) k
and

∂C(λ) ∂λ (2) k = 2 ( βk -(Gλ) k ). (25)
We have:

f λ 2 2 = K k=1 λ (1) k ϕ
(1)

k (x) -λ (2) k ϕ
(2)

k (x) + i(λ (2) k ϕ (1)
k (x) + λ (1) k ϕ
(2)

k (x)) × K k=1 λ (1) k ϕ
(1)

k (x) -λ (2) k ϕ
(2)

k (x) -i(λ (2) k ϕ (1)
k (x) + λ (1) k ϕ
(2)

k (x)) dx
Let us compute partial derivatives:

∂ f λ 2 2 λ (1) 1 = (ϕ (1) 1
(x) + iϕ (2) 1 (x))f λ (x) + f λ (x)(ϕ (1) 1
(x) -iϕ (2) 1 (x))dx = 2   ϕ 1 (x) K k=1 λ k ϕ k (x)   dx = 2 K k=1 λ k ϕ 1 (x)ϕ k (x)dx = 2 K k=1 λ k G 1k = 2 ((Gλ) 1 ).
Besides, we have

∂ f λ 2 2 λ (2) 1 = (-ϕ (2)
1 (x) + iϕ

(1)

1 (x))f λ (x) + f λ (x)(-ϕ (2)
1 (x) -iϕ

(1)

1 (x)) dx = iϕ 1 (x)f λ (x) + f λ (x)(-iϕ 1 (x)) dx = 2   iϕ 1 (x) K k=1 λ k ϕ k (x)   dx = 2 i K k=1 λ k ϕ 1 (x)ϕ k (x)dx = 2 (i(Gλ) 1 ) = -2 ((Gλ) 1 )
.

Finally, if we set A = 2 ( K k=1 λ k βk ), A = 2 K k=1 (λ (1)
k + iλ (2) k )( β(1) k + i β(2) k ) = 2 K k=1 (λ (1) k β(1) k -λ (2) k β(2) k ) hence we have ∂A λ (1) 1 = 2 ( β1 ) ∂A λ (2) 1 = -2 ( β1 ),
which completes the proofs of ( 24) and [START_REF] Massart | An 1 oracle inequality for the Lasso[END_REF]. KKT first-order conditions end the proof of Proposition 2.

Proof of Theorem 1

We only make the proof for the real part. Identical arguments hold for the imaginary part. First of all, let us establish that σ2 1,k is an unbiased estimator of the variance σ 2 1,k . We have

σ2 1,k = 1 2N (N -1) i =j ( 2 (φ k (Z i )) + 2 (φ k (Z j )) -2 (φ k (Z i )) (φ k (Z j ))).
So,

E(σ 2 1,k ) = N (N -1) N (N -1) E( 2 (φ k (Z 1 ))) - 1 N (N -1) i =j E( (φ k (Z i )))E( (φ k (Z j ))) = E( 2 (φ k (Z 1 ))) -(E( (φ k (Z 1 )))) 2 = σ 2 1,k . Now, we set W i = ( 1 N (φ k (Z i ) -β k )).
As

W i = 1 N φ k (Z i ) - S 2 φ k (x)f Z (x)dx
then the W i satisfy almost surely

|W i | ≤ 2 (φ k ) ∞ N .
Then, we apply Bernstein's Inequality (see [START_REF] Massart | Concentration inequalities and model selection[END_REF] on pages 24 and 26) with the variables W i and -W i : for any u > 0,

P   | ( βk -β k )| ≥ 2σ 2 1,k u N + 2u||φ k || ∞ 3N   ≤ 2e -u . (26)
Now, let us decompose σ2 1,k in two terms:

σ2 1,k = 1 2N (N -1) i =j ( (φ k (Z i ) -φ k (Z j ))) 2 = 1 2N N i=1 ( (φ k (Z i ) -β k )) 2 + 1 2N N j=1 ( (φ k (Z j ) -β k )) 2 - 2 N (N -1) N i=2 i-1 j=1 ( (φ k (X i ) -β k ))( (φ k (Z j ) -β k )) = s N - 2 N (N -1) u N with s N = 1 N N i=1 ( (φ k (Z i ) -β k )) 2 and u N = N i=2 i-1 j=1 ( (φ k (Z i ) -β k ))( (φ k (Z j ) -β k )). (27)
Let us first focus on s N that is the main term of σ2 1,k by applying again Bernstein's Inequality with

Y i = σ 2 1,k -( (φ k (Z i ) -β k )) 2 N which satisfies Y i ≤ σ 2 1,k N .
One has that for any u > 0

P σ 2 1,k ≥ s N + √ 2v k u + σ 2 1,k u 3N ≤ e -u with v k = 1 N E σ 2 1,k -( (φ k (Z i ) -β k )) 2 2 .
But we have

v k = 1 N σ 4 1,k + E (φ k (Z i ) -β k ) 4 -2σ 2 1,k E (φ k (Z i ) -β k ) 2 = 1 N E (φ k (Z i ) -β k ) 4 -σ 4 1,k ≤ σ 2 1,k N (|| (φ k )|| ∞ + | (β k )|) 2 ≤ 4σ 2 1,k N || (φ k )|| 2 ∞ .
Finally, with for any u > 0

S(u) = 2 √ 2σ 1,k || (φ k )|| ∞ u N + σ 2 1,k u 3N , we have P(σ 2 1,k ≥ s N + S(u)) ≤ e -u . (28)
The term u N is a degenerate U-statistics that satisfies for any u > 0

P(|u N | ≥ U (u)) ≤ 6e -u , (29)
with for any u > 0

U (u) = 4 3 Au 2 + 4 √ 2 + 2 3 Bu
where A, B, C, D and F are constants not depending on u that satisfy

A ≤ 4|| (φ k )|| 2 ∞ , B ≤ 2 √ N -1|| (φ k )|| 2 ∞ , C ≤ N (N -1) 2 σ 2 1,k , D ≤ N (N -1) 2 σ 2 1,k , and
F ≤ 2 √ 2|| (φ k )|| 2 ∞ (N -1) log(2N )
(see [START_REF] Reynaud-Bouret | On the influence of the support of functions for density estimation[END_REF]). Then, we have for any u > 0,

2 N (N -1) U (u) ≤ 32 3 || (φ k )|| 2 ∞ N (N -1) u 2 + 16 √ 2 + 8 3
|| (φ k )|| 2 ∞ N √ N -1 u 3 2 + 2 √ 2 σ 2 1,k N (N -1) + 8 √ 2 3 log(2N )|| (φ k )|| 2 ∞ N √ N -1 u + 4σ 2 1,k N (N -1) √ u.
Now, we take u that satisfies

u = o(N ) (30)
and

log(2N ) ≤ √ 2u. (31)
Therefore, for any ε 1 > 0, we have for N large enough,

2 N (N -1) U (u) ≤ ε 1 σ 2 1,k + 16 √ 2 + 8 || (φ k )|| 2 ∞ N √ N -1 u 3 2 + 32 3
|| (φ k )|| 2 ∞ N (N -1) u 2 .
So, for N large enough,

2 N (N -1) U (u) ≤ ε 1 σ 2 1,k + C 1 || (φ k )|| 2 ∞ u N 3 2 , (32)
where C 1 = 16 √ 2 + 19. Using Inequalities ( 28) and ( 29), we obtain

P σ 2 1,k ≥ σ2 1,k + S(u) + 2 N (N -1) U (u) = P σ 2 1,k ≥ s N - 2 N (N -1) u N + S(u) + 2 N (N -1) U (u) ≤ P σ 2 1,k ≥ s N + S(u) + P (u N ≥ U (u)) ≤ 7e -u .
Now, using [START_REF] Van De Geer | High dimensional generalized linear models and the Lasso[END_REF], for any 0 < ε 2 < 1, we have for n large enough,

σ2 1,k + S(u) + 2 N (N -1) U (u) = σ2 1,k + 2 √ 2σ 0,m || (φ k )|| ∞ u N + σ 2 1,k u 3N + 2 N (N -1) U (u) ≤ σ2 1,k + 2 √ 2σ 1,k || (φ k )|| ∞ u N + σ 2 1,k u 3N + ε 1 σ 2 1,k + C 1 || (φ k )|| 2 ∞ u N 3 2 ≤ σ2 1,k + 2 √ 2σ 1,k || (φ k )|| ∞ u N + ε 2 σ 2 1,k + C 1 || (φ k )|| 2 ∞ u N 3 2 .
Therefore,

P (1 -ε 2 )σ 2 1,k ≥ σ2 1,k + 2 √ 2σ 1,k || (φ k )|| ∞ u N + C 1 || (φ k )|| 2 ∞ u N 3 2 ≤ 7e -u . (33)
Now, let us set

a = 1 -ε 2 , b = √ 2|| (φ k )|| ∞ u N , c = σ2 1,k + C 1 || (φ k )|| 2 ∞ u N 3 2
and consider the polynomial

P (x) = ax 2 -2bx -c, with roots b± √ b 2 +ac a
. So, we have

P (σ 1,k ) ≥ 0 ⇐⇒ σ 1,k ≥ b + √ b 2 + ac a ⇐⇒ σ 2 1,k ≥ c a + 2b 2 a 2 + 2b √ b 2 + ac a 2 .
It yields

P σ 2 1,k ≥ c a + 2b 2 a 2 + 2b √ b 2 + ac a 2 ≤ 7e -u , so, P σ 2 1,k ≥ c a + 4b 2 a 2 + 2b √ c a √ a ≤ 7e -u ,
which means that for any 0 < ε 3 < 1, we have for N large enough,

P σ 2 1,k ≥ (1 + ε3) σ2 1,k + C1|| (φ k )|| 2 ∞ u N 3 2 + 8|| (φ k )|| 2 ∞ u N + 2 √ 2|| (φ k ||)∞ u N σ2 1,k + C1|| (φ k )|| 2 ∞ u N 3 2 ≤ 7e -u .
Finally, we can claim that for any 0 < ε 4 < 1, we have for N large enough,

P σ 2 1,k ≥ (1 + ε 4 ) σ2 1,k + 8|| (φ k )|| 2 ∞ u N + 2(|| (φ k )|| ∞ 2σ 2 1,k u N ≤ 7e -u .
Now, we take u = γ log K. Under Assumptions of Theorem 1, Conditions (30) and ( 31) are satisfied. The previous concentration inequality means that [START_REF] Meinshausen | High dimensional graphs and variable selection with the Lasso[END_REF], we have for N large enough,

P σ 2 1,k ≥ (1 + ε 4 )σ 2 1,k ≤ 7K -γ . Now, using
P | (β k -βk )| ≥ η 1,k = P   | (β k -βk )| ≥ 2σ 2 1,k γ log K N + 2|| (φ k )|| ∞ γ log K 3N , σ 2 1,k < (1 + ε 4 )σ 2 1,k   + P | (β k -βk )| ≥ η 1,k , σ 2 1,k ≥ (1 + ε 4 )σ 2 1,k ≤ P   |β k -βk | ≥ 2σ 2 1,k γ(1 + ε 4 ) -1 log K N + 2|| (φ k )|| ∞ γ(1 + ε 4 ) -1 log K 3N   + P σ 2 1,k ≥ (1 + ε 4 )σ 2 1,k ≤ 2K -γ(1+ε4) -1 + 7K -γ .
Then, the first part of Theorem 1 is proved: for any ε > 0,

P | (β k -βk )| ≥ η 1,k ≤ C 1 (ε, δ, γ)K -γ 1+ε ,
where C 1 (ε, δ, γ) is a constant that depends on ε, δ and γ.

Proof of Theorem 2

We first state the following lemma.

Lemma 1. Let J 0 ⊂ {1, . . . , K} with cardinality |J 0 | = s and ∆ ∈ C K . We have:

||f ∆ || 2 ≥ ξ min (2s)||∆ J0 || 2 - µ s √ s ||∆ J c 0 || 1 , with µ s = θ s,2s
ξ min (2s) .

Proof. We denote by J 1 the subset of {1, . . . , K} corresponding to the s largest coordinates of ∆ (in modulus) outside J 0 and we set J 01 = J 0 ∪ J 1 . We denote by P J01 the projector on the linear space spanned by (ϕ k ) k∈J01 . For k > 1, we denote by J k the indices corresponding to the coordinates of ∆ outside J 0 whose absolute values are between the ((k -1) × s + 1)-th and the (k × s)-th largest ones (in absolute value). Note that this definition is consistent with the definition of J 1 . Using this notation, we have

||P J01 f ∆ || 2 ≥ ||P J01 f ∆ J 01 || 2 -|| k≥2 P J01 f ∆ J k || 2 ≥ ||f ∆ J 01 || 2 - k≥2 ||P J01 f ∆ J k || 2 .
Since J 01 has 2s elements, we have

||f ∆ J 01 || 2 ≥ ξ min (2s)||∆ J01 || 2 .
Note that P J01 f ∆ J k = f C J 01 for some vector C ∈ C K . Since,

P J01 f ∆ J k -f ∆ J k , P J01 f ∆ J k = 0,
one obtains that

||P J01 f ∆ J k || 2 2 = f ∆ J k , f C J 01
and thus Now, let λ ∈ C K and J ⊂ {1, . . . , K} such that |J| = s. We set ∆ = λ -λ where λ stands for λL . The 1 -norm of ∆ satisfies the inequality stated in the following lemma.

||P J01 f ∆ J k ||
Lemma 2. Using assumptions of Theorem 2, we have:

||∆|| 1 ≤ 2 |J| κ s ||f ∆ || 2 + 2 λ J c 1 1 + 2µ s κ s . Proof. Since || λ|| 1 ≤ ||λ|| 1 ,
we have

||∆ J -λ J || 1 + ||∆ J c -λ J c || 1 ≤ ||λ J || 1 + ||λ J c || 1 ,
and thus

||λ J || 1 -||∆ J || 1 + ||∆ J c || 1 -||λ J c || 1 ≤ ||λ J || 1 + ||λ J c || 1 .
So, we have

||∆ J c || 1 -||∆ J || 1 ≤ 2||λ J c || 1 . (34)
Using Lemma 1 with J 0 = J, we obtain that

||∆ J || 2 ≤ 1 κ s ||f ∆ || 2 + 2 µ s |J|κ s λ J c 1 .
By using again (34), we deduce then

||∆|| 1 ≤ 2||∆ J || 1 + 2 λ J c 1 . ≤ 2 |J|||∆ J || 2 + 2 λ J c 1 ≤ 2 |J| κ s ||f ∆ || 2 + 2 λ J c 1 1 + 2µ s κ s ,
which ends the proof of the lemma.

Now, let us focus on the proof of Theorem 2. We have:

||f λ -f || 2 2 = (f λ (x) -f (x)) f λ (x) -f (x) dx = ||f λ -f λ|| 2 2 + ||f -f λ|| 2 2 + 2 f λ (x) -f λ(x) f λ(x) -f (x) dx = ||f ∆ || 2 2 + ||f -f λ|| 2 2 + 2 K k=1 ∆ k ϕ k (x) × K k =1 λk ϕ k (x) -f (x) dx .
So, using Proposition 2, on Ω,

||f λ -f || 2 2 = ||f λ -f || 2 2 -||f ∆ || 2 2 -2 K k=1 ∆ k (G λ) k -βk + βk -β k ≤ ||f λ -f || 2 2 -||f ∆ || 2 2 + 2 K k=1 |∆ k | × 2 η 2 1,k + η 2 2,k ≤ ||f λ -f || 2 2 -||f ∆ || 2 2 + 4||η|| ∞ ||∆|| 1 . (35)

x∈S 2 g

 2 (x)h(x)dx, and ||.|| 2 is the associated norm. |.| will denote the modulus, and the real and the imaginary part of a complex number respectively.

Figure 1 :

 1 Figure 1: The exact density, the Lasso, the needlet thresholding, φ ∼ U [0, π 16 ].

Figure 2 :

 2 Figure2: The exact density, the Lasso, the needlet thresholding, φ ∼ U [0,3π 16 ].

  2 2 ≤ θ s,2s ||∆ J k || 2 ||C J01 || 2 ≤ θ s,2s ||∆ J k || 2 ||f C J 01 || 2 ||∆ J k || 2 ||P J01 f ∆ J k || 2 . ||∆ J k || 2 = µ s ||∆ J k || 2 . Now using that ||∆ J k+1 || 2 ≤ ||∆ J k || 1 / √ s, we obtain k≥2 ||P J01 f ∆ J k || 2 ≤ µ s √ s ||∆ J c 0 || 1 and ||P J01 f ∆ || 2 ≥ ξ min (2s)||∆ J01 || 2 -µ s √ s ||∆ J c 0 || 1 ,which finally leads to ||f ∆ || 2 ≥ ξ min (2s)||∆ J0 || 2 -

		µ s √ s	||∆ J c 0 || 1 .
			ξ min (2s)
	≤	θ s,2s ξ min (2s)
	This implies that
	||P J01 f ∆ J k || 2 ≤	θ s,2s ξ min (2s)

  ||f ∆ || 2 ≥ ξ min (2s)||∆ J || 2 -µ s |J| (||∆ J || 1 + 2 λ J c 1 ).Using ||∆ J || 1 ≤ |J|||∆ J || 2 , we deduce that ||f ∆ || 2 ≥ ξ min (2s) -µ s ||∆ J || 2 -2µ

s |J| λ J c 1 ≥ κ s ||∆ J || 2 -2µ s |J| λ J c 1 ,

and thus

  Now, we use Lemma 2 to obtain4||η|| ∞ ||∆|| 1 ≤ 8 |J| κ s ||η|| ∞ ||f ∆ || 2 + 8 λ J c 1 1 + 2µ s κ s ||η|| ∞ Since || f L -f || 2 ≤ ||f λL -f || 2 , (35) and (36) yield the result.

	≤	16|J| κ 2 s	||η|| 2 ∞ + ||f ∆ || 2 2 + 8 λ J c 1 1 +	2µ s κ s	||η|| ∞ ,
	so, we have for any α > 0,
	4||η|| ∞ ||∆|| 1 -||f ∆ || 2 2 ≤ 16|J|	1 α	+	1 κ 2 s	||η|| 2 ∞ + α	λ J c 2 1 |J|	1 +	2µ s κ s	2	.	(36)

√3π f ∞

+ 2D + 2

F u + 2 √ 2C √ u,