Keywords: Dielectric, Electronic packaging, Electrografting, Isolation technology, Microfabrication, Polymer films, Stress measurement, Three-dimensional integrated, Through Silicon Vias, Keep Out Zone

Through-Silicon-Vias (TSV) are the key to 3D integrated microsystems. Their fabrication leads to reliability issues linked to the thermo-mechanical stress induced in the silicon around the vias. In this work, we propose to reduce the silicon residual stress in high aspect ratio copper TSVs (HAR TSV) using an electrografted polymer insulator, poly-4-vinylpyridine (P4VP), in replacement of the traditional silicon oxide layer. We use Raman spectroscopy to make the first investigation of the residual stress in the Si around P4VP insulated TSVs and compare it to SiO2 insulated TSVs. The results show that P4VP acts as a stress buffer layer because of its particular mechanical properties as the measured residual stress in Si is significantly reduced at room temperature around the polymer insulated HAR TSVs. The potential benefits of such a technology are not only a better thermo-mechanical reliability of 3D integrated microsystem, but also a greater integration density.

I. INTRODUCTION

HE development of 3D integration technologies constitutes a complementary approach to the constant effort realized on integrated circuit (IC) miniaturization. In 3D integration, dies are stacked and connected to each other using vertical interconnections, insulated from the silicon (Si) substrate, called through silicon vias (TSV). This technology allows better electrical performances, higher transistor density and advanced functionalities [START_REF] Knickerbocker | Three-dimensional silicon integration[END_REF], [START_REF] Garrou | Drivers for 3D Integration[END_REF].

The conductivity of a TSV depends on its filling material. Different materials have been used such as doped polysilicon T his work was performed at Université de Sherbrooke and INRS supported by grants from the Natural Sciences and Engineering Research Council of Canada (NSERC), PRIMA, MITACS and PROMPT, with the contributions Teledyne DALSA Semiconductor and aveni X. T. Le is with Teledyne Dalsa, Bromont, QC CANADA (e-mail: xuan.tuan.le@teledyne.com).

T. Dequivre, G. M. Brisard, S.A. Charlebois are with the Institut Interdisciplinaire d'Innovation Technologique (3IT), University of Sherbrooke, Sherbrooke, QC CANADA (e-mail: thomas.dequivre@ushserbrooke.ca).

G. Kolhatkar, A. Hadj Youssef and A. Ruediger are with the Institut National de la Recherche Scientifique (INRS), Varennes, QC CANADA (email: andreas.ruediger@emt.inrs.ca). [START_REF] Agarwal | Polysilicon interconnections (FEOL): Fabrication and characterization[END_REF], tungsten [START_REF] Pares | Through Silicon Via technology using tungsten metallization[END_REF] or copper [START_REF] Berube | Implementation Study of Single Photon Avalanche Diodes (SPAD) in 0.8 micrometre HV CMOS Technology[END_REF]. Because of their enhanced electrical performances, Cu TSVs are receiving enormous interest from the industry. The Cu TSV metallization normally includes the deposition of an insulator (to ensure the electrical insulation of the interconnection), the deposition of a barrier layer (to prevent the Cu migration from the interconnection to the Si substrate [START_REF] Ogawa | Electromigration reliability issues in dualdamascene Cu interconnections[END_REF]) and the electroplating of the Cu filling. A TSV therefore consists of a variety of materials with different mechanical properties deposited at different temperatures. This heterogeneous structure is a source of stress that affects the electrical and mechanical properties and reliability of the 3D interconnections during the fabrication or the operation of a microsystem. This induced stress can cause mobility shifts in the devices through piezoresistivity effects [START_REF] Ryu | Effect of thermal stresses on carrier mobility and keep-out zone around through-silicon vias for 3-D integration[END_REF], Cu via protrusion, cracks and material delamination [START_REF] Heryanto | Effect of Copper TSV Annealing on Via Protrusion for TSV Wafer Fabrication[END_REF] until degradation of the device's performance or mechanical failure of the Cu TSV. This is one of the main technological challenge one faces to integrate Cu TSVs in the microsystem fabrication process flow that impact significantly the layout optimization. Indeed, while transistor placement is forbidden within a significant keep-outzone (KOZ) around each TSV, the area above the TSV is also often forbidden to interconnections for mechanical integrity reasons. The KOZ defines the minimum distance between two TSV to ensure the mechanical reliability of the microsystem, and the minimum distance between a TSV and an active device to minimize the impact on the carrier mobility. Electrical KOZ is defined based on a 7% mobility change or an equivalent stress of 100 MPa [START_REF] Lu | Thermo-mechanical reliability of 3-D ICs containing through silicon vias[END_REF] (some state a 5 % mobility change [START_REF] Ryu | Effect of thermal stresses on carrier mobility and keep-out zone around through-silicon vias for 3-D integration[END_REF]). Moreover, the Si stress around a TSV is proportional to the square of the TSV radius [START_REF] Lu | Thermo-mechanical reliability of 3-D ICs containing through silicon vias[END_REF]. Thus, scaling down TSVs to few micrometers radius openings can significantly reduce the Si residual stress, along with a gain of real estate in the design.

T

However, the metallization of narrow structures in high aspect ratio (HAR) TSVs is challenging. A lot of efforts are dedicated by the community to mitigate the stress induced by TSV. Some study the residual stress coming from the electrodeposition process and annealing of copper itself or design strategies [START_REF] Lu | Thermo-mechanical reliability of 3-D ICs containing through silicon vias[END_REF]- [START_REF] Texier | Effect of TSV density on local stress concentration: Micro-Raman spectroscopy measurement and Finite Element Analysis[END_REF].

We have been studying an innovative technology for insulating TSVs based on the electrografting of an organic polymer, poly-4-vinylpyridine (P4VP) [START_REF] Truzzi | Wet-process deposition of TSV liner and metal films[END_REF]- [START_REF] Dequivre | Electrografted P4VP for High Aspect Ratio Copper TSV Insulation in Via-Last Process Flow[END_REF]. Indeed, the P4VP electrografting process has shown promising results to insulate high aspect ratio copper TSVs in Vial-Last for the 3D integration of an imaging system [START_REF] Dequivre | Electrografted P4VP for High Aspect Ratio Copper TSV Insulation in Via-Last Process Flow[END_REF]. This process is of interest as it provides an interesting alternative to SiO2 based materials for high aspect ratio copper TSVs insulation in 3D integration process flow. Copper TSVs cannot be metallized in Via-First as the Front End Of Line requires high temperature processes that cause the copper filling to diffuse into the silicon substrate [START_REF] Zhang | Characterization of copper-to-silicon diffusion for the application of 3D packaging with through silicon vias[END_REF]. The deposition of SiO2 based dielectric is thus limited to Chemical Vapor Deposition based methods, which are costly to implement to obtain a conformal and uniform layer in such deep and narrow structures. On the contrary in the same conditions of integration, the electrografted polymer film is highly conformal and uniform, easy to implement and shows similar dielectric film properties than that of SiO2 [START_REF]Integration of Electrografted Layers for the Metallization of Deep Through Silicon Vias[END_REF].

In this paper, we will review the basic phenomena leading to the induced mechanical stress and demonstrate the use of an organic insulator between the silicon substrate and the Cu TSV core to overcome the problem. We report the first measurements of the stress buffering effect of the polymer in HAR Cu TSV.

II. ORIGIN OF THE THERMOMECHANICAL STRESS

Let us recall the coefficient of thermal expansion (CTE), of Si (2.6 ppm/°C) and Cu (17.7 ppm/°C) [START_REF] Selvanayagam | Nonlinear thermal stress/strain analyses of copper filled TSV (Through Silicon Via) and their flip-chip microbumps[END_REF]. In the hypothesis that a Cu TSV is formed at a given temperature and that the deposition process produces no stress in the Cu core, then no stress will be present in the materials at that temperature (therefore dubbed zero-stress temperature). At any other temperature, a mechanical stress is induced in the Si substrate.

The thermomechanical behavior of the TSV is closely linked to the nature of its insulating material as the stress is transmitted to the surrounding Si through the insulator. From basic principles, one can show that the main dependences of the radial stress induced in the silicon by the CTE difference of materials follows:

𝜎 𝑖 ∝ (𝛼 ̅ 𝐶 ,𝐼𝑛 -𝛼 𝑖 )𝐸 ̅ ∆ , ( 1
)
𝛼 ̅ 𝐶 ,𝐼𝑛 = 𝛼 𝐶𝑢 𝐶𝑢 +𝛼 𝐼𝑛 𝐼𝑛 𝐶𝑢 + 𝐼𝑛 , ( 2
)
𝐸 ̅ 𝐶 ,𝐼𝑛 -∝ 𝐶𝑢 + 𝐼𝑛 { 𝐶𝑢 𝐸 𝐶𝑢 + 𝐼𝑛 𝐸 𝐼𝑛 }, (3)
𝐸 ̅ -∝ 𝐸 ̅ 𝐶𝑢,𝐼𝑛 + 𝐸 𝑆𝑖 , ( 4
)
where ∆ is the thermal excursion away from the zero-stress temperature, 𝛼 𝐶 , 𝛼 𝐼𝑛 and 𝐸 𝐶 , 𝐸 𝐼𝑛 the CTE and Young's modulus of the materials respectively, 𝐶 the radius of the copper core and 𝑡 𝐼𝑛 the thickness of the insulating layer. The radius of the hole in silicon is = 𝐶 + 𝑡 𝐼𝑛 . Three effective properties have also been defined: 𝛼 ̅ 𝐶 ,𝐼𝑛 and 𝐸 ̅ 𝐶 ,𝐼𝑛 are effective CTE and Young's modulus for the copper core and insulating layer pair, while 𝐸 ̅ is an effective Young's modulus for the three component system. The definition of the stresses and other parameters are illustrated in Fig. 1Fig. [START_REF] Knickerbocker | Three-dimensional silicon integration[END_REF].

In comparison to Cu or Si, the insulating SiO2 has a slightly smaller Young's modulus and smaller CTE. Though favorable according to Eq. 2 and 3, this barely reduces the induced stress. Then, considering that the SiO2 is very thin and that the Young's modulus of Cu is approximately that of Si, we recover the conventional limit:

𝜎 𝑖 → 𝛼 𝐶 -𝛼 𝑖 𝐸 𝑖 ∆ , (5)
However, using an insulator with a much reduced Young's modulus should lead to an important reduction of the stress. In the limit where 𝐸𝐼𝑛/𝑡𝐼𝑛 is very small, the second term would dominate in Eq. 3 leading to 𝐸 𝐶 ,𝐼𝑛 ~ 𝐸𝐼𝑛. The stress induced in silicon would tend to

𝜎 𝑖 → 𝛼 𝐶 -𝛼 𝑖 𝐸 𝐼𝑛 ∆ , (6)
Considering the specific case of P4VP that has a Young's modulus ~30 times smaller than Cu (see Table I), although its CTE is large, the use of P4VP lowers significantly the effective 𝐸 ̅ 𝐶 ,𝐼𝑛 leading to a stress reduced by half.

The previous equations are given for purely elastic materials which is not the case of P4VP. They nevertheless can be used as a simple approximation to compare the insulating material. The use of the electrografted P4VP as insulator should play a key role in the reliability of the interconnections as its mechanical properties are totally different from those of SiO2 based materials. Table I lists and compares the main properties.

The P4VP also undergoes a glass transition above which its Young's modulus is reduced by a factor of 1,000 to 100,000. In fact, above its glass transition temperature Tg, the polymer chains possess enough energy to slide against one another and reorganize, thus relaxing the stress. This phenomenon is known as "viscoelastic relaxation". For these reasons, the P4VP polymer could act as a buffer layer and relax the stress between the Cu filling and the surrounding Si, reducing the KOZ size and thus, increasing the integration density in microsystems.

Other studies have been realized to characterize polymer insulated Cu TSVs [START_REF] Lu | Thermo-mechanical reliability of 3-D ICs containing through silicon vias[END_REF], [START_REF] Chen | Thermo-mechanical characterization of copper filled and polymer filled tsvs considering nonlinear material behaviors[END_REF]. In these studies, the SiO2 dielectric layer was replaced by a parylene or benzocyclobutene (BCB) layer and finite element analysis (FEA) was used to simulate the thermo-mechanical behavior of such TSVs. It led to the conclusion that with the use of parylene to insulate TSVs, stress can be reduced by up to 50% in Cu filled TSVs.

In the following sections, we use Raman spectroscopy to investigate the stress in the Si surrounding P4VP insulated HAR Cu TSVs, and compare it to SiO2 insulated Cu TSVs.

III. RAMAN SPECTROSCOPY MEASUREMENT OF SILICON RESIDUAL STRESS

When a photon interacts with matter, an electron-hole pair can be created if the incident photon has a high enough energy. The electron will then relaxes to a lower energy state through radiative transition. In the case of Rayleigh scattering, this lower energy state is the same as the electron's initial state.

However, in some cases, a phonon can be generated (Stokes) or absorbed (anti-Stokes). This phonon is caused by lattice vibrations in the system. Raman spectroscopy measures these lattice vibrations, more specifically the optical modes [START_REF] Kittel | Introduction to Solid State Physics[END_REF], [START_REF] Ashcroft | Solid State Physics[END_REF].

In the case of Si, where the crystal structure is zinc-blend [START_REF] Temple | Multiphonon Raman spectrum of silicon[END_REF], the optical phonons are excited at the Γ point of the Brillouin zone. The typical Raman spectrum of Si at room temperature displays a first order peak at ~520 cm -1, and two second order peaks at 300 cm -1 and 950 cm -1 [START_REF] Parker | Raman scattering by silicon and germanium[END_REF]. The peak position varies with temperature at a rate between -0.02 to -0.03 cm -1 /°C in the 300-600°C range [START_REF] Tang | Subwavelength temperature probing in near-field laser heating by particles[END_REF]. The first order peak is caused by a single-phonon process and its full width at half maximum is ~5 meV. In the absence of stress, it is located at 520 cm -1 [START_REF] Iatsunskyi | One and two-phonon Raman scattering from nanostructured silicon[END_REF], [START_REF] Klemens | Anharmonic decay of optical phonon in diamond[END_REF] and corresponds to the energy of the optical phonon at Γ (~65 meV). The second order peaks at 300 cm -1 and 950 cm -1 are due to two-phonon processes [START_REF] Parker | Raman scattering by silicon and germanium[END_REF], [START_REF] Klemens | Anharmonic decay of optical phonon in diamond[END_REF], [START_REF] Weinstein | Two-phonon Raman spectra of Si and GaP[END_REF]. In this work, we focus on the first order peak. Stress, whether of mechanical or thermal origin, has an impact on the position of the Raman peaks.

There are two methods to extract information on the Si stress from Raman spectroscopy. The first method allows the full interpretation of the stress components of the experimental values, but it necessitates to calculate the strain tensor elements at various points in the Si using an analytical model or a finite element analysis and to compare the results to the experimental values [START_REF] Ryu | Micro-Raman spectroscopy and analysis of near-surface stresses in silicon around through-silicon vias for threedimensional interconnects[END_REF]- [START_REF] Vogel | Stress analyses of high spatial resolution on TSV and BEoL structures[END_REF]. The second method uses a simple and more straightforward method. Although it does not allow the interpretation of the stress components, it is the most often used method in the literature to study and compare the thermomechanical behavior of a TSV [START_REF] Wolf | Raman Spectroscopy Analysis Of Mechanical Stress Near Cu-TSVs[END_REF], [START_REF] Ghosh | Integration of Low-k Dielectric Liner in Through Silicon Via and Thermomechanical Stress Relief[END_REF]. In this method, for the TSV configuration, we assume a biaxial stress [START_REF] Wolf | Micro-Raman spectroscopy to study local mechanical stress in silicon integrated circuits[END_REF], [START_REF] Wolf | Raman Spectroscopy Analysis Of Mechanical Stress Near Cu-TSVs[END_REF], [START_REF] Yoo | Multiwavelength Raman characterization of silicon stress near through-silicon vias and its inline monitoring applications[END_REF]. It was shown that the axial stress σz is negligible near the surface of the silicon [START_REF] Ryu | Impact of Near-Surface Thermal Stresses on Interfacial Reliability of Through-Silicon Vias for 3-D Interconnects[END_REF]. As the 473nm excitation photons are absorbed in the first ~1 to 2 µm from the surface, it is justifiable to neglect the axial component resulting in the linear relation with the sum of the two in-plane stress normal components σr and σθ. In the silicon surrounding TSVs, these components are known to be of comparable magnitude but of opposite signs (exact in Lamé's equations for an infinite volume of silicon). The biaxial stress (σr + σθ) is thus expected to be small regardless of the magnitude of the individual components. Using the deformation potentials, the wavenumber difference between the Raman peek of the material with and without stress Δω is given by

∆𝜔 = 𝜔 0 [ + + ] 𝜎 𝑟 + 𝜎 𝜃 . ( 7
)
where p, q, S11 and S12 are material constants, and σr and σθ, the radial and tangential stress [START_REF] Wolf | Micro-Raman spectroscopy to study local mechanical stress in silicon integrated circuits[END_REF]. Hence, considering accepted values [START_REF] Yoo | Multiwavelength Raman characterization of silicon stress near through-silicon vias and its inline monitoring applications[END_REF] the biaxial stress is given by

𝜎 𝑟 + 𝜎 𝜃 = -5 × 6 ∆𝜔. ( 8
)
where ∆𝜔 is given in cm -1 and the stress in MPa. Note that the Raman peak shifts towards the higher wavenumbers in the case of compressive stress, and towards the lower wavenumbers in the case of tensile stress. Using Eq. 8, the Si stress surrounding the Cu TSV can be estimated using Raman spectroscopy.

IV. EXPERIMENT

A. TSVs Samples Fabrication

In order to compare SiO2 insulated TSVs to P4VP insulated TSVs, two series of samples were fabricated using 8-inch p-doped Si (100) wafers. Both of them followed the fabrication process flow described in Fig. 2.

First, a DRIE Bosch process was used to etch a periodic array of HAR TSVs [START_REF] Yu | Fabrication of high aspect ratio TSV and assembly with fine-pitch lowcost solder microbump for Si interposer technology with high-density interconnects[END_REF] of 5 µm diameter by 50 µm depths on both wafers. Then, prior to any metallization process, both wafers were prepared using the so-called RCA clean [START_REF] Garrou | Drivers for 3D Integration[END_REF]. First, a Standard Clean 1 (SC-1) (a mixture of H2O2, NH4OH and H2O) was performed to remove organic residues and particles from the Si substrate. Then, the sample was immersed in diluted HF to remove the native SiO2 of the exposed Si surface. Finally, we did a Standard Clean 2 (SC-2), (a mixture of HCl, H2O and H2O2) to remove the remaining metal contaminants. This process creates a thin SiO2 layer, up to ~10 Å thick [START_REF] Tsunoda | Spectroscopic characterization of naturally and chemically oxidized silicon surfaces[END_REF]. Between each cleaning steps, the wafers were rinsed using DI water.

The HAR TSVs of the first wafer were insulated with electrografted P4VP. The electrografting of P4VP was realized using aveni's eG 3D Isolation solutions [START_REF]Integration of Electrografted Layers for the Metallization of Deep Through Silicon Vias[END_REF]. This process was done in a dedicated production tool present at the MiQro Innovation Collaborative Center (C2MI). After the electrografting process, the sample was dried under N2 before annealing at 200°C also under N2. The HAR TSVs of the second wafers were insulated by a thermal SiO2 layer grown at 1,100°C. The resulting TSVs are shown in Fig. 3.

The TSVs were then metallized (Cu diffusion barrier layer and Cu filling) using identical process also in a dedicated production tool presents at C2MI. For both wafers, a conductive NiB barrier layer was deposited by chemical grafting using aveni's cG 3D Barrier chemical solutions [START_REF]Integration of Electrografted Layers for the Metallization of Deep Through Silicon Vias[END_REF]. This electroless process created strong chemical bonds between the dielectric (P4VP or SiO2) and the NiB layer with equivalent barrier properties than TiN [START_REF] Truzzi | Wet-process deposition of TSV liner and metal films[END_REF], [START_REF] Suhr | Through Silicon Via metallization: A novel approach for insulation/barrier/copper seed layer deposition based on wet electrografting and chemical grafting technologies[END_REF]. Finally, bottom-up Cu electrografting was realized using the conducting NiB barrier layer as seed layer with aveni's eG 3D fill chemical solutions [START_REF]Integration of Electrografted Layers for the Metallization of Deep Through Silicon Vias[END_REF]. A last annealing was realized at 250°C under H2/N2 for 20 minutes. At the end of the TSV metallization process, a Cu overburden layer was present on the surface. The Cu overburden was then removed from the Si surface by Chemical and Mechanical Planarization (CMP), using a 50 nm diameter colloidal silica with benzotriazole (BTA) as corrosion inhibitor and a R&D slurry.

B. Raman Spectroscopy Measurement

The Raman measurements were performed at room temperature using a Horiba system equipped with a modified Olympus BX-41 confocal microscope having a 0.9 NA objective 100×, a solid-state blue Cobolt 04-01 laser (473.2 nm excitation), and a Synapse Back-Illuminated DeepDepletion 1024 × 256 CCD detector. A focus spot of 0.6 µm in diameter was obtained, giving a spatial resolution of 0.3 µm. The laser power was varied between 10.5 mW and 1.08 mW for the laser power study. For the stress analysis, a value of 1.08 mW was used. The penetration depth of the laser radiation in the Si samples is ~800 nm. The scan lines on both types of sample were realized along the Si (100) crystallographic direction.

V. MEASUREMENTS AND RESULTS

Fig. 3 presents the two types of TSVs studied after the insulation process with either a) thermal SiO2 or b) and c) electrografted P4VP. The thermally grown SiO2 dielectric layer is highly conformal with a thickness ranging from ~110 to 130 nm. Fig. 3 b) andc) show that the walls of the P4VP insulated TSV are grafted with the polymer. The electrografted P4VP follows closely the textured walls of the TSV. This excellent uniformity is due to the electro-initiated mechanism [START_REF] Bélanger | Electrografting: a powerful method for surface modification[END_REF] of electrografting process. The top of the TSV wall is covered with ~100 nm of P4VP and the bottom of the TSV is covered with ~80 nm of P4VP, which corresponds to a an excellent step coverage of 80% (ratio between the thickness of the insulator on the bottom and the top of the TSV).

At the end of the metallization process (Fig. 4 a), the TSVs are filled without any visible default and a ~3 μm thick Cu overburden is present at the surface of the samples. The CMP process successfully removed the Cu overburden, leaving the top of the TSV with a smooth Cu surface (Fig. 4 b). A thin black region is visible between the Cu and the Si substrate corresponding to the insulating material. Small scratches are present on the Si surface, and are attributed to the SiO2 nanoparticles used during the CMP process.

This result attests to the capability of the aveni wet metallization process flow to fully metalize HAR TSVs of 5 µm diameter and 50 µm depths as described in the literature [START_REF]Integration of Electrografted Layers for the Metallization of Deep Through Silicon Vias[END_REF], [START_REF] Dequivre | Electrografted P4VP for High Aspect Ratio Copper TSV Insulation in Via-Last Process Flow[END_REF]. After the CMP process, Raman spectra were taken on the Si substrate to measure the stress in induced silicon.

The first step consisted of determining the laser power to use. The laser used for the Raman measurements can induce local heating and thus shift the Raman peak. This shift cannot be differentiated from the shift induced thermomechanically and must therefore be minimized. The effect of the laser power on the Si Raman peak shift was studied, as presented in Fig. 5.

We can see that as the laser power decreases from 9.02 to 0.101 mW, the Raman peak shifts towards the higher wavenumbers by 0.23 cm -1 corresponding to a temperature increase of ~10°C. There is no important shift between the two lower laser intensity Si peak positions. Nevertheless, as the laser intensity is decreasing, the Si peak position uncertainty is increasing. For these reasons, the measurements were performed with a laser power of ~1 mW, to limit the thermal contribution of the laser intensity and to limit the Si peak position uncertainty. A laser power of ~1 mW is a common power value used for Si stress investigation [START_REF] Texier | Effect of TSV density on local stress concentration: Micro-Raman spectroscopy measurement and Finite Element Analysis[END_REF], [START_REF] Texier | Investigation of local stress around TSVs by micro-Raman spectroscopy and finite element simulation[END_REF].

Fig. 6 presents a typical Si Raman spectrum centered at the expected first order Si peak (~520 cm -1 ). The measurement was taken on the bare Si substrate, far from the TSV, to serve as a reference. Its precise position was determined using a Lorentzian function fit, with a resolution of ~0.05 cm -1 inferring according to Eq. 8 a stress resolution of ~25 MPa. The Si peak wavenumber shift around the TSV was investigated using this method, for both SiO2 insulated TSVs and P4VP insulated TSVs. Raman spectra were measured from the edge of the TSVs to the bare Si substrate. For each analyzed TSV, the reference wavenumber ω0 was estimated from the average Si peak position far from the TSV where the stress is expected to be zero. Several devices were measured to ensure reproducibility. Fig. 7 shows that, in the case of the SiO2 insulated TSVs, the Si Raman peak shifts towards the lower wavenumbers towards the edges of the TSVs. This implies the presence of a tensile stress (Δω0 < 0) in the immediate vicinity of the TSV. According to equation ( 8), the corresponding stress was calculated for each position and is presented in Fig. 8. Note that for greater clarity the measurements of left side of the TSV are here folded onto those of the right side. To further the analysis, we also plot as grey area the 95% prediction interval considering the measurements as a sample of duplicate measurements of the zero-stress background. The interval was calculated using the data collected 3 µm away from the TSVs edge, where the silicon is supposed to be stress free. Thus, the distribution of the data is representative of the statistical variation of the zero-stress background.

As expected, these measurements show an increase in the tensile stress as we move closer to the TSVs up to a maximum value of 180 MPa at the edges, over the 95% prediction interval. The distance over which the stress in Si relaxes (~1.5 μm) is in the same range than that of HAR TSVs insulated using a polymeric compound, where the stress is estimated using the similar Eq. 8 [START_REF] Ghosh | Integration of Low-k Dielectric Liner in Through Silicon Via and Thermomechanical Stress Relief[END_REF]. As the spatial resolution of the system is ~0.3 μm, it is difficult to measure with precision the evolution of the stress within 1.5 μm. Moreover, the Raman signal becomes noisy closer to the Si/TSV interface due to the loss of the laser intensity in the Si. Interpretation of the stress values close to the TSV edge must be done with caution.

For SiO2 insulated TSVs, the residual Si stress could have three possible origins. At the end of the metallization process, the last annealing (250°C) expands the Cu resulting in a compressive stress in the Si due to their CTE mismatch. Above 200°C, Cu may develop plasticity reducing the stress in the TSV [START_REF] Ryu | Effect of thermal stresses on carrier mobility and keep-out zone around through-silicon vias for 3-D integration[END_REF]. When cooling down, and back at room temperature, the contraction of the Cu would cause a tensile stress in the surrounding Si. Another source of the residual Si stress could be the thermal oxidation process of the TSV hole that forms the insulator. It could be due to the difference of CTE between SiO2 and Si for the same reason as above. A last source is discussed in [START_REF] Zhu | On the origins of near-surface stresses in silicon around Cu-filled and CNT-filled through silicon vias[END_REF] where the authors gave clear evidence that residual Si stress existing around a metalized TSV is not only due to CTE mismatch between Cu and Si but also from a pre-existing stress present before the TSV Cu filling. They suspect this preexisting stress to be originating from the thermal growth technique of the oxide insulator.

As shown in Fig. 9, the case of P4VP is significantly different. Only few measurement points, mainly close to the TSVs, standout of what could be considered as background. We clearly see that all but 6 of the 65 measurements are inside the 95% prediction interval and thus, could be explained by the statistical dispersion of the measurements around their mean value. Moreover, the three highest measurements are located within 1.5 µm from to the TSV edge, it is likely that they show the presence of a small stress of approximately 40 MPa. For the reasons already discussed above, such low stress close to the TSV edge must be interpreted with caution.

VI. DISCUSSION

We conclude that, if existing, the stress close to the P4VP insulated TSV edge is considerably lower compared to SiO2 insulated TSVs. This observation is of great interest, as it clearly shows that with the use of the electrografted P4VP, the residual Si stress at the edges of the TSV is dramatically reduced compared to SiO2 insulated TSVs.

We will now discuss main phenomena leading to the reduction of the residual Si stress with the use of P4VP relating to two of its properties: 1) low Young's modulus and 2) viscoelastic relaxation.

We already mentioned that by considering the material properties of SiO2 and P4VP in Table I and the simple model of Eq. 1, we can show that the residual stress expected in P4VP insulated TSVs is reduced to half of SiO2 case. Furthermore, as the electrografting process is realized at room temperature, we expect the zero-stress temperature to be rather low (at least before the final anneal) and thus ΔT at room temperature. We leave the discussion on the impact of the final anneal at 250°C to the next paragraph. This case of thermal SiO2 formed above 400°C is quite opposite. A quantitative agreement is likely out of reach for many reasons. As mentioned, the mechanical properties of polymers are rather peculiar. Also, their properties in confined thin films (200 nm in this case) are certainly not those of the same bulk material. Their study and characterization in these conditions are very complex [START_REF] Isaacson | Fundamental limits of material toughening in molecularly confined polymers[END_REF]. Nevertheless, we believe that the large Young's modulus ratio between Si and polymers (typically more than one order of magnitude) is a dominant contribution.

The metallization process of the TSV includes a 250°C annealing step after the filling with Cu. This temperature is above the expected glass transition temperature of P4VP below 200°C, relatively close to its bulk value [START_REF] Zander | Thermal and FTIR Characterization of Poly ( 4vinylpyridine ) Crosslinked with Metal Salts[END_REF], [START_REF] Kuo | Thermal properties, interactions, morphologies, and conductivity behavior in blends of poly(vinylpyridine)s and zinc perchlorate[END_REF]. Our hypothesis is that Tg is reached during the final annealing of the TSV. Even if the polymer is confined between the Cu filling and the Si substrate, the polymer's chains can reconfigure between reticulation points: there is a viscoelastic relaxation of the stress through self-reorganization of the P4VP chains. The polymer would therefore act as a buffer layer and reconfigure itself to absorb the stress. After cool down, once below Tg, the mechanisms that lead to stress in TSV come into play again. This occurring at a lower temperature than that common to SiO2 process, we expect the residual stress to be lower and further reduced by the low Young's modulus of P4VP.

VII. CONCLUSION

We have fabricated two series of HAR TSV of identical geometry (5 µm diameter by 50 µm deep), one with polymer (P4VP) as insulator and the other with a thermal SiO2 as insulator to serve as reference. All process steps are identical except for those forming the insulator layer. We used Raman spectroscopy to make the first measurements of the residual stress in Si around P4VP insulated HAR TSVs. While we measured stress around SiO2 insulated TSVs (>100 MPa), we were not able to measure any significant stress around P4VP. We estimate that if present, the residual stress in the latter case is below 40 MPa. Complementary Finite Element Analysis will be realized in the further experiments to give a more accurate interpretation of the measurements results. We propose that this drastic reduction is due to the very low Young's modulus of P4VP and speculate on the possible role of a low temperature glass transition and viscoelastic properties to further reduce stress. So far, it has not been possible to properly measure the glass transition temperature of P4VP and to characterize its material properties above it.

We demonstrated that the use of a polymer as insulator, P4VP in this case, nearly eliminates the residual stress in Si around HAR TSVs. We further show that the full wet metallization process flow based on electrografting to form the P4VP insulator is perfectly suited for low stress HAR TSV.

These results are promising as the process could mitigate present technical limitations in terms of thermomechanical reliability and integration density of a microsystem. In particular, we believe that the KOZ around TSV due to the thermomechanical stress only could be strongly reduced.

Finally, the electrografted P4VP layer could be used in a wide range of applications confronted to thermo-mechanical issues and where highly conformal layer is needed.

Table:

Figures:

Fig.

1 .

 1 Fig. 1. Definition of the axes, stress components and geometrical parameters.

Fig. 2 .

 2 Fig. 2. Sample fabrication process flow prior to Raman spectroscopy measurements.

Fig. 3 .Fig. 5 .

 35 Fig. 3. SEM cross section of TSVs with insulator material: a) SiO2 and b-c) P4VP. Top b) and bottom c) of the TSV with P4VP.

ACKNOWLEDGMENT

The authors would like to thank, S. Ecoffey, C. Mireault and J. Plathier for their contribution. This work was performed at the Université de Sherbrooke, the Institut National de la Recherche Scientifique (INRS) and MiQro Innovation Collaborative Center (C2MI) supported by grants from the Natural Sciences and Engineering Research Council of Canada (NSERC), PRIMA, MITACS and PROMPT. We thank our industrial collaborators Teledyne DALSA Semiconductor and aveni. We finally thank the reviewers for their contributions at improving the completeness of this paper.


Gilles Pijaudier-Cabot

Mirvat Omar

Ahmed Loukili

Yann Le Pape

Creep-Damage interaction in concrete structures

come

INTRODUCTION

In common practice, it is usually assumed that concrete exhibits a linear visco-elastic response for low load levels and that the instantaneous mechanical response of concrete is elastic. For high load levels, deviation from linearity of the creep behaviour of concrete is expected. Under highsustained loads, cracks grow and interact with visco-elasticity. Some experimental and analytical results concerning non-linear creep can be found in the literature (see among others Bazant, 1988[START_REF] Gettu | Rate effects and load relaxation: static fracture of concrete[END_REF][START_REF] Mazzotti | Nonlinear creep damage model for concrete under uniaxial compression[END_REF][START_REF] Rüsch | Experimental Determination of the effect of the Duration of Loading on Strength and Deformation[END_REF][START_REF] Rüsch | Investigations on the strength of concrete under sustained load[END_REF]) but much remains to be learned.

Modelling non linear creep of concrete is of fundamental importance for severely loaded structures as creep may decrease the material strength with increasing time, for instance in the evaluation of the safety of structural components such as nuclear power plants. Actually, these structures may be exposed to high stresses for a long time, due to what damage develops. Therefore, it is of great interest to devise methods for evaluating the evolution of their safety margin with time. This problem has drawn the attention of some authors, among them [START_REF] Rüsch | Experimental Determination of the effect of the Duration of Loading on Strength and Deformation[END_REF]co-workers (1957, 1958), who have carried out the first experiments in order to determine the effect of continuous loading duration on the resistance and deflections of concrete specimens under compression tests. They found that the capacity of a specimen subjected to creep loads seems to be 70 to 80% of that observed in short time compression tests.

The main results of a set of experimental creep tests on concrete three points bend specimens at different load levels are presented first. The objective was to investigate the influence of the load level on the residual capacity of bending beams. For each creep test, a different load level has been applied, starting from low levels (36% of peak load) where linear visco-elasticity applies, to high levels (80% of peak load) where tertiary creep causing failure at a finite time can be observed. Size effect test data, before and after creep at 80% of the maximum load were also obtained.

The second part of the paper deals with a coupled creep -damage constitutive relation. It is based on the standard effective stress approach. The growth of damage is defined as a function of a total strain invariant, from which a portion of the creep strain is removed in order to achieve a coupling between creep and damage that is not too strong. The evolution of the size effect test data shows that the internal length in the damage model should decrease as the amount of creep strain increases. This effect is included in the constitutive relations too.

EXPERIMENTAL STUDY

Standard concrete, same as the one tested by [START_REF] Granger | Comportement différé du béton dans les enceintes de centrales nucléaires (analyse et modélisation[END_REF] has been used in this experimental program. The specimens were made with this mix which consisted of ordinary Portland cement CPA-CEMII 42.5, fine sand with a maximum size of 5 mm, crushed gravel of size 5 to 25 mm, a super-plasticiser agent (Glenium 21) and water. This mixture is characterised by a water-cement ratio of 0.56 and a slump of 4 cm.

The creep tests were performed on frames that apply a constant load on flexural beams. The frames have a capacity ranging from 5 to 50 KN and can accommodate geometrically similar specimens of three different sizes. The specimens tested have the same thickness of 10 cm. The smallest (D1) is 10 cm high and 35 cm long. The medium size (D2) is 20 cm high and 70 cm long and the largest (D3) is 40 cm high and 140 cm long. In the present study, we shall consider only basic creep test data. The curing conditions (3 months at 100% RH and 20°C) guaranteed to avoid early age autogeneous shrinkage so that basic creep could be measured only. Figure 1 compares the displacements due to basic creep for the smaller specimens (size D1). It shows the influence of the load level on the basis creep evolution versus the elapsed time. As expected, increasing the applied load increases the basic creep magnitude. Moreover, creep develops very fast in the first days of loading and stabilises after a few weeks for the two lower loading levels. For the specimen loaded at 80% of the peak load, basic creep increases rapidly.

In order to investigate the variation of the residual capacity due to creep, comparison specimens cast at the same time than those subjected to creep were kept under the same conditions of temperature and relative humidity. After 60 days of loading, beams subjected to 80% of the maximum load were removed from the creep frames and then immediately subjected to three point bending loading up to failure with a constant loading rate. The results were then processed according to Bazant's size effect law.

Figure 2 shows the size effect test data for the two sets of beams, with and without creep. We can observe a shift of the data points to the right when creep is applied prior to the fracture tests. Upon creep, the behaviour of the beams is closer to linear elastic fracture mechanics. Irwin's length decreases and the parameter d O in the size effect law decreases too. This shows that the size of the fracture process zone decreases.

CREEP-DAMAGE MODEL

Creep is modelled with visco-elasticity, in the form of a generalised Maxwell chain. The constitutive law is written with the help of a relaxation function R:

τ τ ε τ ε σ d t R t t t R t t t t ⋅ ⋅ ∫ + = ) ( ) , ( ) ( ). , ( ) ( 0 0 0 . ( 1
)
t σ is the effective stress applied to the Maxwell chain, t 0 is the age of loading. Note that the formulation is one-dimensional here. The relaxation function is:

∑ = - - + = n t E E t R 1 )) ( exp( ). ( ) ( 0 ) , ( µ τ µ τ τ µ τ τ .
(2) τ µ and E µ are the relaxation time and the modulus of elasticity of each branch µ in the generalised Maxwell model respectively. In general E µ (t 0 ) can be calculated by the method of least squares. As for the parameters τ µ , however, they cannot be calculated from measured creep data but must be suitably chosen in advance [START_REF] Bazant | Mathematical modeling of creep and shrinkage of concrete[END_REF].

In order to incorporate damage, which is a degradation of the overall elastic stiffness, into the time dependent creep functions, we use the effective stress approach [START_REF] Omar | Etude comparative du couplage endommagement -fluage[END_REF]

): ) 1 ( d t - = σ σ .
(3) It is a classical approach, in which we consider a time independent growth of damage. The damage evolution law follows the classical relationship by [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture de béton de structure[END_REF], coupled to an integral non local approach. Mechanical damage is controlled by the weighted average of the equivalent strain ε ~:

∫ Ω - = ∫ Ω - = ds s x x r V ds s s x x r V x ) ( ) ( with ) ( ) ( ) ( 1 ) ( ψ ε ψ ε , ( 4
)
and the weight function ψ (xs ) is:

          - - = - 2 2 4 exp ) ( c l s x s x ψ . ( 5
)
The equivalent strain is:

∑ + = 2 ' ~i ε ε , with
( ) i t i i C σ α α ε ε 1 ' . ) 1 .( - + - = , ( 6
)
where the subscript i denotes the principal strain components, C -1 is the inverse of the elastic (initial) stiffness of the material, and α is a model parameter. When it is equal to zero, damage is a function of the total strain (including the creep strain). For values of this parameter larger than zero, a portion of the creep strain is deducted from the total strain prior the equivalent strain is computed.

Time (days)

Figure 3: Creep deformation of a tensile bar.

Figure 3 shows a typical creep response of a one dimensional bar. Note that tertiary creep can be described, including failure after some finite load duration. Such a result cannot be obtained with a series arrangement between visco-elasticity and damage or plasticity. Figure 4 shows the influence of the parameter α for a monotonic loading that is displacement controlled. We have shown on this figure the rate effect due to creep too. When α is taken equal to 0, the strongest possible effect of creep on damage growth is observed. When α increases, the coupled effect becomes milder. The maximum load increases and the displacement at peak load increases too. Figure 4: Rate effect and influence of the parameter α for monotonic loading of bending beams. [START_REF] Zhou | Time-dependent fracture of concrete: testing and modelling[END_REF] have considered creep tests up to failure on notched bending beams. The beam were loaded to a constant level and then the time to failure was computed. Figure 5 shows how the present model can qualitatively reproduce this experiment. Again, the influence of the parameter α that controls the coupled effect between damage and creep is shown. For small values of α, creep failure is observed for loads that are bellow 70% of the maximum load (measured under quasistatic loading without creep effect). For α=0.65, creep failure does not occur for applied loads that are lower that 70% of the maximum one. This is more consistent with our experiments which does not exhibit creep failure at such low load levels.

CLOSURE

At present, calibration of the proposed constitutive relations and comparisons with the presented test data is in progress. Qualitatively, the model provides some consistent responses. The present computations, however, rely on a beam theory and size effect cannot be described. A full implementation into a 3D finite element code is required for that. This is necessary in order to The internal grows from l c0 to l c0 +β as the function f is in fact the local value of damage. A possible modification of this evolution of the internal length, which of course remains to be tested and analysed, could be:

0 ) ( . ~. ) ( c fl c l f l + - = ε ε ε ε β ε , ( 8
)
where fl ε ~is an invariant of the creep strain (total strain minus the strain computed according to the damage model without creep). Constructed like this, the growth of the creep strains would decrease the value of the internal length.

Figure 1 :

 1 Figure 1: Basic creep displacement of notched beams D1 for 36% , 60% , and 80% of peak load.

Figure 2 :

 2 Figure 2: Residual capacity of unotched beams of dimension D2.

  1,0E-05 2,0E-05 3,0E-05 4,0E-05 5,0E-05 6,0E-05

Figure 5 :

 5 Figure 5: Failure lifetimes for notched bending beams.

  on the size effect plot due to creep that the constitutive relation is capable of. Already, the variation of the internal length due to creep can be envisioned. We start with the non-local damage model with evolving internal length proposed by[START_REF] Pijaudier-Cabot | Non local damage model with evolving internal length[END_REF]. The value of the internal length is modified in the weight function:

Acknowledgements: Financial support from the European Commission through the MAECENAS project (FIS5-2001-00100) and from the partnership between Electricité de France and the R&DO group through the MECEN project are gratefully acknowledged.


Alain Rakotomamonjy

Abraham Traoré

Maxime Berar

Rémi Flamary

Nicolas Courty

Liva Ralaivola

Maxime Bérar

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

Most discriminative machine learning algorithms have focused on learning problems where inputs can be represented as feature vectors of fixed dimensions. This is the case of popular algorithms like support vector machines [START_REF] Schölkopf | Learning with kernels: support vector machines, regularization, optimization, and beyond[END_REF] or random forest [START_REF] Breiman | Random forests[END_REF]. However, there exists several practical situations where it makes more sense to consider input data as set of distributions or empirical distributions instead of a larger collection of single vector. As an example, multiple instance learning [START_REF] Dietterich | Solving the multiple instance problem with axis-parallel rectangles[END_REF] can be seen as learning of a bag of feature vectors and each bag can be interpreted as samples from an underlying unknown distribution. Applications related to political sciences [START_REF] Flaxman | Who supported Obama in 2012?: Ecological inference through distribution regression[END_REF] or astrophysics [START_REF] Ntampaka | A machine learning approach for dynamical mass measurements of galaxy clusters[END_REF] have also considered this learning from distribution point of view for solving some specific machine learning problems. This paper also addresses the problem of learning decision functions that discriminate distributions.

Traditional approaches for learning from distributions is to consider reproducing kernel Hilbert spaces (RKHS) and associated kernels on distributions. In this larger context, several kernels on distributions have been proposed in the literature such as the probability product kernel [START_REF] Jebara | Probability product kernels[END_REF], the Battarachya kernel [START_REF] Bhattacharyya | On a measure of divergence between two statistical populations defined by their probability distributions[END_REF] or the Hilbertian kernel on probability measures of [START_REF] Hein | Hilbertian metrics and positive definite kernels on probability measures[END_REF]. Another elegant approach for kernel-based distribution learning has been proposedby [START_REF] Muandet | Learning from distributions via support measure machines[END_REF]. It consists in defining an explicit embedding of a distribution as a mean embedding in a RKHS. Interestingly, if the kernel of the RKHS satisfies some mild conditions then all the information about the distribution is preserved by this mean embedding. Then owing to this RKHS embedding, all the machinery associated to kernel machines can be deployed for learning from these (embedded) distributions.

By leveraging on the flurry of distances between distributions [START_REF] Sriperumbudur | Non-parametric estimation of integral probability metrics[END_REF], it is also possible to build definite positive kernel by considering generalized radial basis function kernels of the form K(µ, µ ) = e -σd 2 (µ,µ ) ,

(1)

where µ and µ are two distributions, σ > 0 a parameter of the kernel and d(•, •) a distance between two distributions satisfying some appropriate properties so

x 1

x 2 Class 1 Class 2 Class 3

Figure 1: Illustrating the principle of the dissimilarity-based distribution embedding. We want to discriminate empirical normal distributions in R 2 ; their discriminative feature being the correlation between the two variables. An example of these normal distributions are given in the left panel. The proposed approach consists in computing a embedding based on the dissimilarity of all these empirical distributions (the blobs) to few of, them that serve as templates. Our theoretical results show that if we take enough templates and there is enough samples in each template then with high-probability, we can learn a linear separator that yields few errors. This is illustrated in the 3 other panels in which we represent each of the original distribution as a point after projection in an discriminant 2D space of the embeddings. From left to right, the dissimilarity embedding respectively considers 10, 45 and 90 templates and we can indeed visualize that using more templates improve separability.

as to make K definite positive [START_REF] Haasdonk | Learning with distance substitution kernels[END_REF]).

As we can see, most works in the literature address the question of discriminating distributions by considering either implicit or explicit kernel embeddings. However, is this really necessary? Our observation is that there are many advantages of directly using distances or even dissimilarities between distributions for learning. It would avoid the need for two-stage approaches, computing the distance and then the kernel, as proposed by [START_REF] Póczos | Distribution-Free Distribution Regression[END_REF] for distribution regression or estimating the distribution and computing the kernel as introduced by [START_REF] Sutherland | Kernels on sample sets via nonparametric divergence estimates[END_REF]. Using kernels limits the choice of distribution distances as the resulting kernel has to be definite positive. For instance, [START_REF] Póczos | Nonparametric kernel estimators for image classification[END_REF] used Reyni divergences for building generalized RBF kernel that turns out to be non-positive. For the same reason, the celebrated and widely used Kullback-Leibler divergence does not qualify for being used in a kernel. This work aims at showing that learning from distributions with distances or even dissimilarity is indeed possible. Among all available distances on distributions, we focus our analysis on Wasserstein distances which come with several relevant properties, that we will highlight later, compared to other ones (e.g Kullback-Leibler divergence).

Our contributions, depicted graphically in Figure 1, are the following : (a) We show that by following the underlooked works of [START_REF] Balcan | A theory of learning with similarity functions[END_REF], learning to discriminate population distributions with dissimilarity functions comes at no expense. While this might be considered a straightforward extension, we are not aware of any work making this connection. (b) Our key theoretical contribution is to show that Balcan's framework also holds for empirical distributions if the used dissimilarity function is endowed with nice convergence properties of the distance of the empirical distribution to the true ones. (c) While these convergence bounds have already been exhibited for distances such as the Wasserstein distance or MMD, we prove that this is also the case for the Bures-Wasserstein metric.(d) Empirically, we illustrate the benefits of using this Wasserstein-based dissimilarity functions compared to kernel or MMD distances in some simulated and real-world vision problem, including 3D point cloud classification task.

Framework

In this section, we introduce the global setting and present the theory of learning with dissimilarity functions of [START_REF] Balcan | A theory of learning with similarity functions[END_REF].

Setting

Define X as an non-empty subset of R d and let P denotes the set of all probability measures on a measurable space (X , A), where A is σ-algebra of subsets of X . Given a training set {µ i , y i } n i=1 , where µ i ∈ P and y i ∈ {-1, 1}, drawn i.i.d from a probability distribution P on P × {-1, 1}, our objective is to learn a decision function h : P → {-1, 1} that predicts the most accurately as possible the label associated to a novel measure µ. In summary, our goal is to learn to classify probability distributions from a supervised setting. While we focus on a binary classification, the framework we consider and analyze can be extended to multi-class classification.

Dissimilarity function

Most learning algorithms for distributions are based on reproducing kernel Hilbert spaces and leverage on kernel value k(µ, µ ) between two distributions where k(•, •) is the kernel of a given RKHS.

We depart from this approach and instead, we consider learning algorithms that are built from pairwise dissimilarity measures between distributions. Subsequent definitions and theorems are recalled from [START_REF] Balcan | A theory of learning with similarity functions[END_REF] and adapted so as to suit our definition of bounded dissimilarity. Definition 1. A dissimilarity function over P is any pairwise function D :

P × P → [0, M ].
While this definition emcompasses many functions, given two probability distributions µ and µ , we expect D(µ, µ ) to be large when the two distributions are "dissimilar" and to be equal to 0 when they are similar. As such any bounded distance over P fits into our notion of dissimilarity, eventually after rescaling. Note that unbounded distance which is clipped above M also fits this definition of dissimilarity. Now, we introduce the definition that characterizes dissimilarity function that allows one to learn a decision function producing low error for a given learning task.

Definition 2. [START_REF] Balcan | A theory of learning with similarity functions[END_REF] A dissimilarity function D is a ( , γ)-good dissimilarity function for a learning problem L if there exists a bounded weighting function w over P, with w(µ) ∈ [0, 1] for all µ ∈ P, such that a least 1 -probability mass of distribution examples µ satisfy :

E µ ∼P [w(µ )D(µ, µ )| (µ) = (µ )] + γ ≤ E µ ∼P [w(µ )D(µ, µ )| (µ) = (µ )] .
The function (µ) denotes the true labelling function that maps µ to its labels y.

In other words, this definition translates into: a dissimilarity function is "good" if with high-probability, the weighted average of the dissimilarity of one distribution to those of the same label is smaller with a margin γ to the dissimilarity of distributions from the other class.

As stated in a theorem of [START_REF] Balcan | A theory of learning with similarity functions[END_REF], such a good dissimilarity function can be used to define an explicit mapping of a distribution into a space. Interestingly, it can be shown that there exists in that space a linear separator that produces low errors.

Theorem 1. [START_REF] Balcan | A theory of learning with similarity functions[END_REF] if D is an ( , γ)good dissimilarity function, then if one draws a set

S from P containing n = ( 4M γ ) 2 log( 2 δ ) positive ex- amples S + = {ν 1 , • • • , ν n } and n negative exam- ples S -= {ζ 1 , • • • , ζ n }, then with probability 1 -δ, the mapping ρ S : P → R 2n defined as ρ S (µ) = (D(µ, ν 1 ), • • • , D(µ, ν n ), D(µ, ζ 1 ), • • • , D(µ, ζ n ))
has the property that the induced distribution ρ S (P) in R 2n has a separator of error at most + δ at margin at least γ/4.

The above described framework shows that under some mild conditions on a dissimilarity function and if we consider population distributions, then we can benefit from the mapping ρ S . However, in practice, we do have access only to empirical version of these distributions.

Our key theoretical contribution in Section 3 proves that if the number of distributions n is large enough and enough samples are obtained from each of these distributions, then this framework is applicable with theoretical guarantees to empirical distributions.

Learning with empirical distributions

In what follows, we formally show under which conditions an ( , γ)-good dissimilarity function for some learning problems, applied to empirical distributions also produces a mapping inducing low-error linear separator.

Suppose that we have at our disposal a dataset composed of {µ i , y i = 1} n i=1 where each µ i is a distribution. However, each µ i is not observed directly but instead we observe it empirical version μi =

1 Ni Ni j=1 δ xi,j with x i,1 , x i,2 , • • • x i,Ni i.i.d ∼ µ i .
For a sake of simplicity, we assume in the sequel that the number of samples for all distributions are equal to N . Suppose that we consider a dissimilarity D and that there exist a function g 1 such that D satisfies a property of the form P D(µ, μ) > ≤ g 1 (K, N, , d) then following theorem holds:

Theorem 2. For a given learning problem, if the dissimilarity D is an ( , γ)-good dissimilarity function on population distributions, with w(µ) = 1, ∀µ and K a parameter depending on this dissimilarity then, for a parameter λ ∈ (0, 1), if one draws a set

S from P containing n = 32M 2 γ 2 log( 2 δ 2 (1-λ) ) positive examples S + = {ν 1 , • • • , ν n } and n negative examples S -= {ζ 1 , • • • , ζ n }, and from each distribution ν i or ζ i , one draws N samples so that δ 2 λ ≥ N g 1 (K, N, 4 , d)
samples so as to build empirical distributions {ν i } or { ζi }, then with probability 1 -δ, the mapping ρS : P → R 2n defined as

ρS (μ) = 1 M (D(μ, ν1 ), • • • , D(μ, νn ), D(μ, ζ1 ), • • • , D(μ, ζn ))
has the property that the induced distribution ρ S (P)

in R 2n has a separator of error at most +δ and margin at least γ/4.

Let us point out some relevant insights from this theorem. At first, due to the use of empirical distributions instead of population one, the sample complexity of the learning problem increases for achieving similar error as in Theorem 1. Secondly, note that λ has a trade-off role on the number n of samples ν i and ζ i and the number of observations per distribution. Hence, for a fixed error + δ at margin γ/4, having less samples per distribution has to be paid by sampling more observations.

The proof of Theorem 1 has been postponed to the appendix. It takes advantage of the following key technical result on empirical distributions.

Lemma 1. Let D be a dissimilarity on P × P such that D is bounded by a constant M . Given a distribution µ ∈ P of class y and a set of independent distributions {ν j } n j=1 , randomly drawn from P, which have the same label y and denote as μ and {ν i } their empirical version composed of N observations. Let us assume that there exists a function g 1 and a constant K > 0 so that for any µ ∈ P, P D(µ, μ) > ≤ g 1 (K, N, , d), with typically g 1 tends towards 0 as N or goes to ∞. The following concentration inequality holds for any > 0 :

P 1 n n i=1 D(μ, νi ) -E ν∼P [D(µ, ν)| (µ) = (ν)] > ≤ N g 1 (K, N, 4 , d) + 2e -n 2 2M 2 .
This lemma tells us that, with high probability, the mean average of the dissimilarity between an empirical distribution and some other empirical distribution of the same class does not differ much from the expectation of this dissimilarity measured on population distributions. Interestingly, the bound on the probability is composed of two terms : the first one is related to the dissimilarity function between a distribution and its empirical version while the second one is due to the empirical version of the expectation (resulting thus from Hoeffding inequality). The detailed proof of this result is given in the supplementary material. Note that in order for the bound to be informative, we expect g 1 to have a negative exponential form in N . Another version of this lemma is proven in supplementary where the concentration inequality for the dissimilarity is on

|D(μ, ν) -D(µ, ν)|.
From a theoretical point of view, there is only one reason for choosing one ( , γ)-good dissimilarity function on population distributions from another. The rationale would be to consider the dissimilarity function with the fastest rate of convergence of the concentra-tion inequality Pr(D(µ, μ) > ), as this rate will impact the upper bound in Theorem 1.

From a more practical point of view, several factors may motivate the choice of a dissimilarity function : computational complexity of computing D(μ i , μj ), its empirical performance on a learning problem and adaptivity to different learning problems ( e.g without the need for carefully adapting its parameters to a new problem.)

( -γ) Good dissimilarity for distributions

We are interested now in characterizing convergence properties of some dissimilarities (or distance or divergence) on probability distributions so as to make them fit into the framework. Mostly, we will focus our attention on divergences that can be computed in a non-parametric way.

Optimal transport distances

Based on the theory of optimal transport, these distances offer means to compare data probability distributions. More formally, assume that X is endowed with a metric d X . Let p ∈ (0, ∞), and let µ ∈ P and ν ∈ P be two distributions with finite moments of order p (i.e. X d X (x, x 0 ) p dµ(x) < ∞ for all x 0 in X ). then, the p-Wasserstein distance is defined as:

W p (µ, ν) = inf π∈Π(µ,ν) X ×X d X (x, y) p dπ(x, y) 1 p
.

(2) Here, Π(µ, ν) is the set of probabilistic couplings π on (µ, ν). As such, for every Borel subsets A ⊆ X , we have that µ(A) = π(X × A) and ν(A) = π(A × X ). We refer to (Villani, 2009, Chaper 6) for a complete and mathematically rigorous introduction on the topic. Note when p = 1, the resulting distance belongs to the family of integral probability metrics [START_REF] Sriperumbudur | Non-parametric estimation of integral probability metrics[END_REF]. OT has found numerous applications in machine learning domain such as multi-label classification [START_REF] Frogner | Learning with a Wasserstein Loss[END_REF], domain adaptation [START_REF] Courty | Optimal transport for domain adaptation[END_REF] or generative models [START_REF] Arjovsky | Wasserstein Generative Adversarial Networks[END_REF]. Its efficiency comes from two major factors: i) it handles empirical data distributions without resorting first to parametric representations of the distributions ii) the geometry of the underlying space is leveraged through the embedding of the metric d X . In some very specific cases the solution of the infimum problem is analytic. For instance, in the case of two Gaussians µ ∼ N (m 1 , Σ 1 ) and ν ∼ N (m 2 , Σ 2 ) the Wasserstein distance with d X (x, y) = x -y 2 reduces to:

W 2 2 (µ, ν) = ||m 1 -m 2 || 2 2 + B(Σ 1 , Σ 2 ) 2 (3)
where B(, ) is the so-called Bures metric [START_REF] Bures | An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite σ-algebras[END_REF]:

B(Σ 1 , Σ 2 ) 2 = trace(Σ 1 +Σ 2 -2(Σ 1/2 1 Σ 2 Σ 1/2 1 ) 1/2 ). (4)
If we make no assumption on the form of the distributions, and distributions are observed through samples, the Wasserstein distance is estimated by solving a discrete version of Equation 2 which is a linear programming problem.

One of the necessary condition for this distance to be relevant in our setting is based on non-asymptotic deviation bound of the empirical distribution to the reference one. For our interest, [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF] have shown that for distributions with finite moments, the following concentration inequality holds

P(W p (µ, μ) > ) ≤ C exp (-KN d/p )
where C and K are constants that can be computed from moments of µ. This bound shows that the Wasserstein distance suffers the dimensionality and as such a Wasserstein distance embedding for distribution learning is not expected to be efficient especially in highdimension problems. However, a recent work of [START_REF] Weed | Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance[END_REF] has also proved that under some hypothesis related to singularity of µ better convergence rate can be obtained (some being independent of d).

Interestingly, we demonstrate in what follows that the estimated Wasserstein distance for Gaussians using Bures metric and plugin estimate of m and Σ has a better bound related to the dimension.

Lemma 2. Let µ be a d-dimensional Gaussian distribution and m and Σ the sample mean and covariance estimator of µ obtained from N samples. Assume that the true covariance matrix of µ satisfies Σ 2 ≤ C Σ the random vectors v used for computing these estimates are so that v 2 ≤ C v and The squared-Wasserstein distance between the empirical and true distribution satisfies the following deviation inequality:

P(W 2 (µ, μ)) > ) ≤ 2d exp - -N 2 /(8d 4 ) C v C Σ + 2C v /(3d 2 ) + exp - N 1/2 24 √ C v 2 -1 1/2
This novel deviation bound for the Gaussian 2-Wasserstein metric tells us that if the empirical data (approximately) follows a high-dimensional Gaussian distribution then it makes more sense to estimate the mean and covariance of the distribution and then to apply Bures-Wasserstein distance rather to apply directly a Wasserstein distance estimation based on the samples.

Kullback-Leibler divergence and MMD

Two of the most studied and analyzed divergences/distances on probability distribution are the Kullback-Leibler divergence and Maximum Mean Discrepancy. Several works have proposed non-parametric approaches for estimating these distances and have provided theoretical convergence analyses of these estimators.

For instance, [START_REF] Nguyen | Estimating divergence functionals and the likelihood ratio by convex risk minimization[END_REF] estimate the KL divergence between two distributions by solving a quadratic programming problem which aims a finding a specific function in a RKHS. They also proved that the convergence rate of such estimator in in O(N 1 2 ). MMD has originally been introduced by [START_REF] Gretton | A kernel method for the two-sample-problem[END_REF] as a mean for comparing two distributions based on a kernel embedding technique. It has been proved to be easily computed in a RKHS. In addition, its empirical version benefits from nice uniform bound. Indeed, given two distribution µ and ν and their empirical version based on N samples μ and ν, the following inequality holds [START_REF] Gretton | A kernel two-sample test[END_REF]:

P MMD 2 (μ, ν) -MMD 2 (µ, ν) > ≤ exp - 2 N 2 8K 2
where N 2 = N/2 , K is a bound on k(x, x ), ∀x, x , and k(•, •) is the reproducing kernel of the RKHS in which distributions have been embedded. We can note that this bound is independent of the underlying dimension of the data.

Owing to this property we can expect MMD to provide better estimation of distribution distance for highdimension problems than WD for instance. Note however that for MMD-based two sample test, [START_REF] Ramdas | On the decreasing power of kernel and distance based nonparametric hypothesis tests in high dimensions[END_REF] has provided contrary empirical evidence and have shown that for Gaussian distributions, as dimension increases MMD 2 (µ, ν) goes to 0 exponentially fast in d. Hence, we will postpone our conclusion on the advantage of one measure distance on another to our experimental analysis.

Discriminating normal distributions with the mean

( , γ)-goodness of a dissimilarity function is a property that depends on the learning problem. As such, it is difficult to characterize whether a dissimilarity will be good for all problems. In the sequel we characterize this property for these three dissimilarities, on a meanseparated Gaussian distribution problem. [START_REF] Muandet | Learning from distributions via support measure machines[END_REF] used the same problem as their numerical toy problem. We show that even in this simple case, MMD suffers high dimensionality more than the two other dissimilarities.

Consider a binary distribution classification problem where samples from both classes are defined by Gaussian distributions in R d . Means of these Gaussian distribution follow another Gaussian distribution which mean depends on the class while covariance are fixed. Hence, we have

µ i ∼ N (m i , Σ) with m i ∼ N (m -1 , Σ 0 ) if y i = -1 and m i ∼ N (m +1 , Σ 0 ) if y i = +1
where Σ and Σ 0 are some definite-positive covariance matrix. We suppose that both classes have same priors. We also denote

D = m -1 -m +1 2
2 which is a key component in the learnability of the problem. Intuitively, assuming that the volume of each µ i as defined by the determinant of Σ is smaller than the volume of Σ 0 , the larger D is the easier the problem should be. This idea appears formally in what follows.

Based on Wasserstein distance between two normal distributions with same covariance matrix, we have

W (µ i , µ j ) 2 = m i -m j 2 2 .
In addition, given a µ i with mean m i , regardless of its class, we have, with k ∈ {-1, +1}:

E µj :mj ∼N (m k ,Σ0) [ m i -m j 2 2 ] = m i -m k 2 2 +Tr(Σ 0 ) Given α ∈]0, 1], we define the subset of R d , E -1 = {m : (m -m -1 ) (m +1 -m -1 ) ≤ 1 -α 2 D
Informally, E -1 is an half-space containing of m -1 for which all points are nearer to m -1 than m +1 with a margin defined by 1-α 2 D . In the same way, we define E +1 as :

E +1 = {m : (m -m +1 ) (m -1 -m +1 ) ≤ 1 -α 2 D }
Based on these definition, we can state that

W (•, •) is a ( , γ) good dissimilarity function with γ = αD , = 1 2 R d \E-1 dN (m -1 , Σ 0 ) + 1 2 R d \E+1 dN (m +1 , Σ 0
) and w(µ) = 1, ∀µ. Indeed, it can be shown that for a given µ i with

y i = -1, if m i ∈ E -1 then m i -m -1 2 2 + α m -1 -m +1 2 2 γ ≤ m i -m +1 2 2
With a similar reasoning, we get an equivalent inequality for µ i of positive label. Hence, we have all the conditions given in Definition 2 for the Wasserstein distance to be an ( , γ) good dissimilarity function for this problem. Note that the γ and naturally depend on the distance between expected means. The larger this distance is, the larger the margin and the smaller are.

Following the same steps, we can also prove that for this specific problem of discriminating normal distribution, the Kullback-Leibler divergence is also a ( , γ) good dissimilarity function. Indeed, for µ 1 and µ 2 following two Normal distributions with same covariance matrix Σ 0 , we have

KL(µ 1 , µ 2 ) = m 2 -m 1 2 Σ -1 0
. And following exactly the same steps as above, but replacing inner product m m with m Σ -1 0 m leads to similar margin γ = α m -1 -m +1 2 Σ -1 0 and similar definition of .

While the above margins γ for KL and WD are valid for any Σ 0 , if we assume Σ 0 = σ 2 I, then according to [START_REF] Ramdas | On the decreasing power of kernel and distance based nonparametric hypothesis tests in high dimensions[END_REF], the following approximation holds for this problem

MMD 2 (µ 1 , µ 2 ) ≈ m 1 -m 2 2 2 σ 2 k (1 + 2σ 2 /σ 2 k ) d/2+1
where σ k is the bandwidth of the kernel embedding, leading to a ( -γ) distribution with margin

α m 1 -m 2 2 2 σ 2 k (1 + 2σ 2 /σ 2 k ) d/2+1
From these margin equations for all the dissimilarities, we can drive similar conclusions to those of [START_REF] Ramdas | On the decreasing power of kernel and distance based nonparametric hypothesis tests in high dimensions[END_REF] on test power. Regardless on the choice of the kernel embedding bandwidth, the margin of MMD is supposed to decrease with respect to the dimensionality either polynomially or exponentially fast. As such, even in this simple setting, MMD is theoretically expected to work worse than KL divergence or WD.

In practice, we need to compute these KL, WD or MMD distance from samples obtained i.i.d from the unknown distribution µ and ν. The problem of estimating in a non-parametric way some φ-divergence, especially the Kullback-Leibler divergence have been thoroughly studied by [START_REF] Nguyen | Nonparametric estimation of the likelihood ratio and divergence functionals[END_REF][START_REF] Nguyen | Estimating divergence functionals and the likelihood ratio by convex risk minimization[END_REF]. For KL divergence, this estimation is obtained by solving a quadratic programming problem. In a nutshell, compared to Kullback-Leibler divergence, Wasserstein distance benefits from a linear programming problem compared to a quadratic programming problem. In addition, unlike KL-divergence, Wasserstein distance takes into account the properties of X and as such it does not diverge for distributions that do not share support.

Numerical experiments

In this section, we have analyzed and compared the performances of Wasserstein distances based embedding for learning to classify distributions. Several toy problems, similar to those described in Section 4.3 have been considered as well as a computer-vision real-world problem.

Number

Competitors

Before describing the experiments we discuss the algorithms we have compared. We have considered two variants of our approach. The first one embeds the distributions based on ρS by using, unless specified, all distributions available in the training set. The Wasserstein distance is approximated using its entropic regularized version with λ = 0.01 for all problems. Then, we learn either a linear SVM or a Gaussian kernel classifier resulting in two methods dubbed in the sequel as WD+linear and WD+kernel. As discussed in section 3, we can use the closed-form Bures Wassrestein distance when we suppose that the distributions are Gaussian. Assuming that the samples come from a Normal distribution, plugging-in the empirical mean and covariance estimation into the Bures-Wasserstein distance 3 gives us a distance that we can use as an embedding. In the experiments, these approaches are named Bures+linear and Bures+kernel. In the family of integral probability metrics, we used the support measure machines of [START_REF] Muandet | Learning from distributions via support measure machines[END_REF], denoted as SMM. We have considered its non-linear version which used an Gaussian kernel on top of the MMD kernel. In SMM, we have thus two kernel hyperparameters. In order to evaluate the choice of the distance, we have also used the MMD distance in addition to the Wasserstein distance in our framework. These approaches are denoted as MMD+linear and MMD+kernel.

Note that we have not reported based on samples-based approaches such as SVM since [START_REF] Muandet | Learning from distributions via support measure machines[END_REF] have already reported that they hardly handle distributions.

Kullback-Leibler divergence can replace the Wasserstein distance in our framework. For instance, we have highlighted that for the problem in Section 4.3, KLdivergence is an ( , γ) good dissimilarity function. We have thus implemented the non-parametric estimation of the KL-divergence based on quadratic programming [START_REF] Nguyen | Estimating divergence functionals and the likelihood ratio by convex risk minimization[END_REF][START_REF] Nguyen | Nonparametric estimation of the likelihood ratio and divergence functionals[END_REF]. After few experiments on the toy problems, we finally decided to not report performance of the KL-divergence based approach due to its poor computational scalability as illustred in the supplementary material.

Simulated problem

These problems aim at studying the performances of our models in controlled setting. The toy problem corresponds to the one described in Section 4.3 but with 3 classes. For all classes, mean of a given distribution follows a normal distribution with mean m = [1, 1] and covariance matrix σI with σ = 5. For class i, the covariance matrix of the distribution is defined as σ i I + u i (I 1 + I -1 ) where I 1 and I -1 are respectively the super and sub diagonal matrices. The {σ i } are constant whereas u i follows a uniform distribution depending on the class. We have kept the number of empirical samples per distribution fixed at N = 30.

For these experiments, we have analyzed the effect of the number of training examples n (which is also the number of templates) and the dimensionality d of the distribution. Approaches are then evaluated on of 2000 test distributions. 20 trials have been considered for each n and dimension d. We define a trial as follows: we randomly sample the n number of distributions and compute all the embeddings and kernels. For learning, We have performed cross-validation on all parameters of all competitors. This involves all kernel and classifier parameters. Details of all parameters and hyperparameters are given in the appendix. Right panel shows the Impact of the dimensionality of the problem on the classification performance. We note that again MMD-based approaches do not perform as good as Wasserstein-based ones. Whereas MMD tops below 70%, our non-linear WDMM method achieves about 90% of classification rate across a large range of dimensionality.

Left plot in

Natural scene categorization

We have also compared the performance the different approaches on a computer vision problem. For this purpose, we have reproduced the experiments carried out by [START_REF] Muandet | Learning from distributions via support measure machines[END_REF]. Their idea is to consider an image of a scene as an histogram of codewords, where the codewords have been obtained by k-means clustering of 128-dim SIFT vector and thus to use this histogram as a discrete probability distribution for classifying the images. Details of the feature extraction pipeline can be found in the paper [START_REF] Muandet | Learning from distributions via support measure machines[END_REF]. The only difference our experimental set-up is that we have used an enriched version of the dataset1 they used. Similarly, we have used 100 images per class for training and the rest for testing. Again, all hyperparameters of all competing methods have been selected by cross-validation.

The averaged results over 10 trials are presented in Table 1. Again, the plot illustrates the benefit of Wasserstein-distance based approaches (through fully non-parametric distance estimation or through the estimated Bures-Wasserstein metric) compared to MMD based methods. We believe that the gain in performance for non-parametric methods is due to the ability of the Wasserstein distance to match samples of one distribution to only few samples of the other distribution. By doing so, we believe that it is able to capture in an elegant way complex interaction between samples of distributions.

3D point cloud classification

3D point cloud can be considered as samples from a distribution. As such, a natural tool for classifying them is to used metrics or kernels on distributions. In this experiment, we have benchmarked all competitors on a subset of the ModelNet10 dataset . 1, we report results based on original train and test sets and results using 50 -50 random splits and resamplings. Again, we highlight the benefit of using the Wasserstein distance as an embedding and contrarily to other experiments, the Bures-Wasserstein metric yields to poor performance as the object point clouds hardly fit a Normal distribution leading thus to model misspecification.

Conclusion

This paper introduces a method for learning to discriminate probability distributions based on dissimilarity functions. The algorithm consists in embedding the distributions into a space of dissimilarity to some template distributions and to learn a linear decision function in that space. From a theoretical point of view, when considering population distributions, our framework is an extension of the one of [START_REF] Balcan | A theory of learning with similarity functions[END_REF]. But we provide a theoretical analysis showing that for embeddings based on empirical distributions, given enough samples, we can still learn a linear decision functions with low error with high-probability with empirical Wasserstein distance. The experimental results illustrate the benefits of using empirical dissimilarity on distributions on toy problems and real-world data.

Futur works will be oriented toward analyzing a more general class of regularized optimal transport divergence, such as the Sinkhorn divergence [START_REF] Genevay | Learning Generative Models with Sinkhorn Divergences[END_REF] in the context of Wasserstein distance measure machines. Also, we will consider extensions of this framework to regression problems, for which a direct

Figure 2 :

 2 Figure 2: Comparing performances of Support Measure Machines and Wasserstein distance + classifier.

  Figure 2 represents the averaged classification accuracy with N = 30 samples per classes and d = 50 for increasing number n of empirical training distribution examples. Right plot represents the same but for fixed n = 250 and increasing dimensionality d.

  From the left panel, we note that MMD-based distance

	Method	Scenes	3DPC	3DPC-CV
	SMM	51.58 ± 2.46	92.79	92.99 ± 0.99
	MMD linear	24.83 ± 1.22	91.89	91.84 ± 1.13
	MMD kernel	27.02 ± 4.09	90.54	92.66 ± 1.02
	WD linear	61.58 ± 1.34	97.30 95.52 ± 0.89
	WD kernel	60.70 ± 2.49	96.86	94.89 ± 0.80
	BW linear	62.30 ± 1.32 64.86	63.52 ± 5.72
	BW kernel	62.06 ± 1.34	70.72	72.20 ± 1.51
	Table 1: Performances of competitors on real-world
	problems. 3DPC and 3DPC-CV columns report per-
	formances on original train/test split and for random
	splits. Bold denotes best test accuracy and under-
	line show statistically equivalent performance under
	Wilocoxon signrank test with p = 0.01.
	fails in achieving good performances regardless of how
	they are employed (kernel or distance based classifier).
	WDMM performs better than MMD especially as the
	number of training distrubtions increases. For n = 600,
	the difference in performance is almost 30% of accuracy
	when considering distance-based embeddings. We also
	remark that the Bures-Wasserstein metric naturally
	fits to this Normal distribution learning problem and
	achieves perfect performances for n ≥ 200.

  Among the 10 classes in that dataset, we have extracted the night stand, desk and bathtub classes which respectively have 400, 400 and 212 training examples and 172,172 and 100 test examples. Experiments and model selection have been run as in previous experiments. In Table

The dataset is available at http://www-cvr.ai.uiuc. edu/ponce_grp/data/