Yann Goueffon

Grégory Aldebert

Catherine Mabru
email: catherine.mabru@isae.fr

Laurent Arurault

Claire Tonon

Pascale Guigue

Flaking of black anodic films in space environment: Ageing and numerical simulation

Keywords:

Black coatings can be used for managing passive thermal control on spacecraft and for avoiding stray light in optical equipment. Inorganic black anodizing of aluminium alloys is a solution to obtain films with a low outgassing and a sufficient thermal stability for this kind of application. Flaking of these coatings has sometimes been observed after thermal cycling on 2XXX and 7XXX aluminium alloys. This phenomenon could generate particulate contamination on satellites optics and may affect mission lifetime. In this work, the influence of thermal cycling on the adhesion of the films was investigated thanks to both a qualitative method (peel-test) and a quantitative method (scratch-test). It has been shown that differential dilatations and dehydration of the film result in stresses inside the film. A finite element model was developed to identify the mechanisms of crack propagation and bifurcation inside the film and thus explain the reasons of flaking. Especially, it has been shown that cracks can bifurcate and propagate in the film along the interface under thermal solicitation resulting in a mixed opening/shearing mode loading at the crack tip.

Introduction

Under space vacuum, thermal exchanges between a satellite and its environment are only radiative. The spacecraft temperature is mainly depending on the interactions with three main external sources (the Earth, the sun and the deep cold space). The Earth is emitting an infrared flux heating the satellite while, on the contrary, the deep cold space is equivalent to a source at 3 K and allows evacuation of heat. Those two heat fluxes can be considered as approximately constant during revolutions of the satellite. Nevertheless, the main heating source is the sun (direct and albedo fluxes). Successive passages in the shadow of the Earth result in thousands thermal cycles suffered by the satellite during its lifetime. The temperature of exposed parts like solar arrays can typically vary between À140 and +140 °C.

The radiative exchanges of a part are driven by its thermo-optical properties (e emissivity and a s solar absorptance) and the equilibrium temperature is directly linked to the ratio a s /e [START_REF] Agrawal | Design of Geosynchronous Spacecraft[END_REF]. The required specification directly depends on the equipment needs. Cold parts can be obtained for a s /e < 1 [START_REF] Siva Kumar | Studies on white anodizing on aluminium alloy for space applications[END_REF] while a ratio a s /e > 1 results in a high equilibrium temperature [START_REF] Saxena | Studies on ultra high solar absorber black electroless nickel coatings on aluminum alloys for space application[END_REF]. Warm parts are obtained for a ratio close to one. In the case of aluminium alloys, largely used in spacecraft design, different surface treatments are available to provide such thermo-optical properties: black paints [START_REF] Mccroskey | Aeroglaze Z306 black paint for cryogenic telescope use: outgassing and water vapor regain[END_REF], micro-arc oxidation [START_REF] Shrestha | Microstructural and thermooptical properties of black Keronite PEO coating on aluminium alloy AA7075 for spacecraft materials applications[END_REF][START_REF] Wu | The influence of the time on the optical properties of the ceramic thermal control coating prepared by micro-plasma oxidation[END_REF], electro-deposition [START_REF] Magdy | Black nickel electrodeposition from a modified Watts bath[END_REF] or black anodizing (Goueffon et al., 2009a;[START_REF] Levesque | Black anodize as a thermal control coating for space station freedom[END_REF][START_REF] Shashikala | Solar selective black nickel-cobalt coatings on aluminum alloys[END_REF].

The black inorganic anodizing of aluminium alloys is used to provide specific thermo-optical properties to surfaces and then manage passive thermal control. Actually, the process allows the increase of the normal emissivity to 0.90 and the solar absorptance to 0.93 (ECSS-Q-ST-70-03C, 2008). In addition, the high absorptance avoids stray lights reflections that could be harmful especially near optical instruments while the use of inorganic dyes limits the outgassing in vacuum. Nevertheless cases of flaking of black anodic films were sometimes observed on aluminium from 2XXX and 7XXX series after thermal cycling performed to simulate the space environment (ESA Alert, 2005;Goueffon et al., 2009a). Such particles could pollute sensitive equipment of the satellite and shorten the lifetime of the mission.

The aim of this work is to study the environmental conditions leading to flaking of the anodic film. The influence of ageing on the adhesion of the film was measured thanks to three different methods: the peel test, the scratch test and four-point bending test. Flaking was then reproduced after experimental thermal cycling and numerical simulation was used to understand the mechanisms involved.

Experimental procedure

The aluminium substrate was AA 7175 T7351 often used in the space industry. Some cases of flaking have been observed on this alloy. Two sizes of samples were used: 40 Â 20 Â 3 mm for scratch and peel tests and 160 Â 20 Â 10 mm for bending tests.

Process of black anodizing

The black anodizing process followed the ESA standard (ECSS-Q-ST-70-03C, 2008) for spacecraft design. It consists in four main steps: pretreatments, anodizing, inorganic colouring and sealing.

The aluminium sheets were degreased with ethanol, etched in an aqueous mixed solution of Na 2 CO 3 /Na 3 PO 4 for 5 min at 93 °C and neutralised 3 min at room temperature in HNO 3 (50% v/v). The samples were rinsed with distiled water at the end of each step.

The anodizing was performed in sulphuric acid during 60 min with a current density of 1.25 A/dm 2 . It results in the growth of a porous oxide film with a typical thickness of 20 lm (Goueffon et al., 2010a). In this study, the electrolyte concentration (C) and the anodizing temperature (T) were adjusted to control the film's initial porosities. Thus, two types of selected and prepared anodic films are 40% (C 1 = 150 g/L; T 1 = 20 °C) and 50% (C 1 = 150 g/L; T 1 = 25 °C) of porosities.

The anodized parts were then coloured by successive immersions in two different baths. Firstly a solution of cobalt acetate (200 g/L) at 43 °C was used to fill in the pores during 15 min. Secondly, an immersion in ammonium sulphide (30 g/L) at room temperature during 10 min resulted in the precipitation of black dyes (CoS) in the pores.

The final step was a sealing in an aqueous solution of nickel acetate (5 g/L) and boric acid (5 g/L) at 98 ± 2 °C. The hydration of the oxide was used to close pores and protect dyes.

A previous study (Goueffon et al., 2009b) showed that films with initial porosities of 40% and 50% were crazed after colouring and sealing steps. Actually, the differential thermal dilatations in the sealing bath, associated to a poor limit tensile stress of highly porous films, were proved to lead to formation of cracks perpendicular to the interface during the process. In both cases residual stresses from the process were relaxed by the crazing.

Ageing

To simulate the space environment, the ESA Standard (ECSS-Q-ST-70-04C, 2008) recommends performing 100 cycles between À100 and +100 °C under vacuum (10 À5 Pa) with a dwell time of at least 5 min and a slope of 10 °C per minute. These conditions were defined for the general case and all kinds of materials (from polymers to metals). The number cycles' impact, the pressure and the temperature range on the adhesion will be evaluated in this paper. A Sun Electronic System EC11 environmental chamber was used to perform thermal cycles. Dwells times of 5 min and warming/cooling speeds of 10 °C/min under nitrogenous atmosphere were used. Nevertheless, this speed was 1 °C/min for cycles performed under vacuum (10 À6 Pa) because of the warming system.

Adhesion measurements

Peel tests

Adhesion was evaluated qualitatively using 90°peel tests with tape strengths of 250 and 670 g/cm. The removal speed was 500 mm/min controlled with tensile testing device (Instron). Three identical samples were tested for each configuration with each tape to observe an average behaviour.

Scratch test

The adhesion was quantitatively evaluated using a scratch-test device (CSM Revetest instrument) with a diamond stylus (Rockwell, 200 lm radius tip). Scratch tests were configured with an increasing normal load from 1 to 60 N, a loading speed of 30 N/min and an advance speed of 5 mm/min. The mechanism of degradation was the lateral flaking of the film on both sides of the scratch [START_REF] Bull | Failure mode maps in the thin film scratch adhesion test[END_REF]. The resulting scratch-print of about 10 mm length was then observed by Scanning Electron Microscopy (SEM -Philips XL30 ESEM) to determine the corresponding normal load. The given values are averages of six different scratches on each sample.

Four-point bending test

Four-point bending tests with acoustic emission were also performed to evaluate the evolution of adhesion of the film [START_REF] Delmas | Characterization of adherence and cracking within coated materials by an acoustic emission method: application to a WC-Co coating on a steel substrate[END_REF][START_REF] Ollendorf | A comparative study of adhesion test methods for hard coatings[END_REF][START_REF] Richard | Fourpoint bending tests of thermally produced WC-Co coatings[END_REF]. Tests were done on a tensile testing device (Adamel DY 26 of 100kN) with a distance between supports of 40 mm for the face under compression and 120 mm for the face under tension. Samples (10 Â 20 Â 160 mm) were anodized and black coloured only on the compressive face between the supports. All the anodic film is then submitted to a uniform strain field. The tests were coupled with acoustic emission measurements to determine the damaging loads. The detection threshold was set at 50 dB to filter noise. In this configuration, no acoustic hit is detected during the bending of an unanodized aluminium sheet. Samples were observed by SEM after loading to determine if flaking occurred. Three samples were tested to obtain an average value of bending critical load.

Evaluation of the dehydration

Thermo-gravimetric analysis (TGA) were carried out to evaluate the kinetic of water desorption during thermal cycles. The anodic film was isolated from the substrate by mechanical compression. Actually, samples were fourpoint bent until delaminating of the anodic film on the compressive face. The film was thus collected under powder form. A speed of 10 °C/min, dwells times of 5 min, and a nitrogenous atmosphere were used. A test at constant temperature under a pressure of 10 2 Pa was also performed. All measurements were realised with a Setaram B24 thermo-balance.

Numerical simulation

As the anodic films elaborated at 20 and 25 °C (40% and 50% initial porosity) are crazed after the whole black anodizing process (Goueffon et al., 2009b), a finite element simulation was used to evaluate mechanical behaviour of a crazed thin film on its substrate during thermal cycling. The software SAMCEF was used to perform isotropic thermo-elastic calculations with 2-dimensional plane strain hypotheses. The geometry used was a substrate (thickness of 3 mm) with an anodic film of 20 lm thick. Mechanical characteristics of the substrate used for the calculations can be found in the literature [START_REF] Davis | ASM Specialty Handbook: Aluminum and Aluminum Alloys[END_REF] and were the following: Young modulus of 72 GPa, Poisson's ratio of 0.33 and Coefficient of Thermal Expansion (CTE) of 23.6 Â 10 À6 K À1 . Concerning the film, previous studies (Goueffon et al., 2010a) proved that the Young modulus is linked to the porosity and those different values were used. Poisson's ratio was chosen to 0.28 (Goueffon et al., 2010b) and the CTE value (13 Â 10 À6 K À1 ) was previously obtained by beam bending analysis (Goueffon et al., 2010b). Cracks with various geometries (1 micron width and variable length) were added through the film. Boundary conditions were defined in order to permit free deformations of the system under thermal loadings. Applied loadings were only thermal, constant and uniform. The temperature was chosen between À140 and +140 °C while the reference of the system is 20 °C. All stresses in the model are then due to differential thermal dilatations between the film and the substrate. A contact criterion has been implemented between the two lips of the crack. The friction coefficient was set up at 0.3. A study on the sensibility of this parameter in the range from 0.1 to 0.5 has shown that its influence on the calculated stress intensity factors is negligible.

Meshing of the whole model and of the crack tip has been validated by comparing the stress intensity factor K I in the case of a straight crack perpendicular to the surface with theoretical values [START_REF] Anderson | Fracture Mechanics: Fundamentals and Applications[END_REF] on a mono-material (same mechanical properties for substrate and film) and with numerical values for a bi-material [START_REF] Kaddouri | Finite element analysis of crack perpendicular to bi-material interface: case of couple ceramic-metal[END_REF] at room temperature.

Thus, the potential evolution of a crack through the coating was evaluated when thermally loaded by considering the stress intensity factors. These stress intensity factors were calculated thanks to the Equivalent Domain Integral (EDI) Method [START_REF] Bittencourt | Quasi-automatic simulation of crack propagation for 2D LEFM problems[END_REF] and strain energy density criterion was used to evaluate the bifurcation angle of the crack [START_REF] Sih | Some basic problems in fracture mechanics and new concepts[END_REF]. The crack will propagate if K > K C [START_REF] Irwin | Analysis of stresses and strains near the end of a crack traversing a plate[END_REF] where K C is the toughness of the film. However, in the present study, toughness of the anodic film has not been determined. Consequently, results from the numerical simulation are not able to quantitatively predict conditions of cracking. Nevertheless, comparative analyses of the K values can give information on the cases that are potentially the more detrimental and that can favour crack propagation.

Experimental results and discussion

Influence of the pressure on the adhesion of the film

Both types of black anodic films (40% and 50% of initial porosities) were cycled ten times between À80 and +80 °C. The influence of the atmosphere (inert gas or vacuum) used during thermal cycling on the adhesion of anodic films was particularly studied.

The evolution of the scratch-test critical load was observed after thermal cycling (Fig. 1). After cycles under nitrogenous atmosphere, the measured critical load is affected only for the anodic films with the highest initial porosity (from 20 N to 5 N). However, cycles performed under vacuum are detrimental for both types of films, with the same effect than inert gas for film with 50% of initial porosity while a slight decrease of the scratch critical load of the film with 40% of porosity is observed.

Four-point bending tests were also realised after the same ageing. Before thermal cycling, no acoustic signal is detected before a critical load which depends on the porosity of the film (Fig. 2). The bending critical load corresponds to plastic strains of the aluminium substrate. Those acoustic hits are here associated to the degradation of the film under compressive stresses. After thermal ageing under vacuum, both types of samples have a modified behaviour during bending. Acoustic events are detected at low loads: they can be attributed to propagations of cracks created during the sealing step in the anodic film. After thermal ageing under nitrogenous atmosphere, very few acoustic hits are detected at low loads in the case of samples with a porosity of 50%. No acoustic event is detected for the samples with a porosity of 40%.

Those results are consistent with peel tests. Before or after thermal ageing under nitrogen of samples with 40% of porosity, no particle of the coating is detached from the substrate, whatever the tape used. After cycles under vacuum, the tape test results in several particles removed with the 250 g/cm tape while numerous are detached with the 670 g/cm one (Fig. 3). For samples with 50% of porosity, cycles under inert gas or vacuum are both leading to flaking.

The decrease of the critical load in scratch-test, bending-test or peel-test after thermal cycling is attributed to crack propagations in the film during ageing. Actually, during thermal cycling, the differential thermal dilatations between the film and the substrate results in tensile stresses in the film during heating and compressive stresses for temperatures lower than the ambient. These stresses can cause crack propagation. However, as observed for films with 40% of initial porosity, temperature range is not the only parameter affecting the adhesion: temperatures between À80 and +80 °C under inert gas are not modifying the adhesion. Nevertheless, thermal cycling at the same temperatures under vacuum has a detrimental effect on the adhesion.

Fig. 4 shows the mass loss of an anodic film during the thermal cycles between À80 and +80 °C under a nitrogen flux at ambient pressure and under a low pressure (10 2 Pa). The mass loss is more important at low pressure than with an inert gas. Those observations are even confirmed at ambient temperature (Fig. 5). So the dehydration of black anodic films occurring during thermal cycles is amplified by the vacuum. This phenomenon results in additional tensile stresses in the film (Goueffon et al., 2009b) favourable to cracks development. After several days in a vacuum of 10 À6 Pa at ambient temperature, the cracks width increases (Fig. 6) what is a consequence of the dehydration [START_REF] Rayhani | Physical modeling of desiccation cracking in plastic soils[END_REF]. Nevertheless, seven days in a vacuum atmosphere without thermal stresses have no influence on any adhesion measurement performed. Thus, the loss of adhesion during thermal cycling is a consequence of combined thermal and dehydration stresses. Films with an initial porosity of 40% are not damaged even for a high number of thermal cycles. This result was confirmed by four-point bending tests and peel tests. Actually, when degradation occurs, it is during the first thermal cycles as we can see for the sample showing initial 50% porosity. This observation is confirmed under vacuum for both types of porosities (Fig. 8).

Influence of the number of cycles on the adhesion

These results must be related with the fast dehydration that mainly occurs during the first cycles, as seen in Fig. 4. In addition, they are consistent with the brittleness of the anodic film: the mechanism of degradation is not cyclic loading but results from the combined effect of thermal and dehydration stresses that damage the film quasiimmediately.

Effects of the temperature range

The influence of temperature range has been investigated for both types of anodic films after ten thermal cycles under nitrogen atmosphere (Fig. 9). The adhesion of anodic films with an initial porosity of 50% was already degraded with thermal cycles between À80 and +80 °C. The scratch-test critical load measured after ageing was around 5 N. Ageing with larger temperature range is also leading to the same results (Fig. 9). Nevertheless, if samples with a lower initial porosity (40%) are not affected by thermal cycles À80/+80 °C, the increase of the temperature range (À110/+110 °C and À140/+140 °C) is leading to a progressive decrease of the critical load.

For all types of samples four-point bending is not leading to significant results. This can be attributed to the fact that the dehydration is not complete at 140 °C at ambient pressure under nitrogenous atmosphere. After thermal cycling under nitrogen, any eventual fracture in the hydrated anodic film with 50% of initial porosity during four-point bending could lead to acoustic emission with amplitudes lower than the detection threshold (50 dB).

For 50% of initial porosity, peel-tests are always leading to particles detachment after ageing, whatever the temperature of cycling. Films with the lower porosity are not damaged by peeling during cycling between À80 and +80 °C. For the two larger temperature ranges tested, some particles are pulled up with both tapes used.

The detrimental influence of increasing temperature range is due to two phenomena. Firstly, the increase of the temperature range is resulting in higher thermal stresses in the anodic films because the stresses induced by differential dilatations are proportional to the temperature (Goueffon et al., 2009b). Secondly, the dehydration and then associated tensile stresses in the film are amplified when the maximum temperature reached increases (Fig. 10). The cracks are then more solicited and may propagate in the film.

To understand the mechanisms leading to the collapse of adhesion during ageing, it is now necessary to dissociate the influence of the cold temperatures (<20 °C) and hot ones (>20 °C) during cycling.

Dissociation cold/hot parts of thermal cycles

The contributions of each part of the thermal cycling to the loss of adhesion were studied for 10 cycles between À140 °C and +140 °C under nitrogen. During cycling between À140 and 20 °C, no dehydration may occur (Fig. 11). The film is then only solicited with compressive stresses due to differential dilatations. On the contrary, between 20 and 140 °C, the dehydration and then tensile stresses in the film may be equivalent than during cycles between À140 and 140 °C. Fig. 11 shows that the decrease of critical load for samples with an initial porosity of 40% is less important for cycles with amplitudes of 20/140 °C or À140/20 °C than for À140/+140 °C.

Tensile and compressive stresses level reached during cycling are separately responsible of a small loss of adhesion of the film. However, successions of hot and cold loadings are more damaging for black anodic films.

Influence of the initial porosity of the anodic film

As seen on all the results presented in the previous sections, the loss of adhesion due to ageing is different for the films with 40% and 50% of initial porosity. In particular, some ageing conditions can seriously affect the adhesion of the film with the highest porosity while they do not modify the behaviour of the film showing 40% initial porosity. In addition when they are both affected for a given ageing condition, the loss of adhesion (see quantitative results obtained by scratch tests), is more important for high porosity films. Thus, films with 50% of initial porosity seem to be more sensitive to thermal ageing. The difference of anodizing temperature between films with 40% and 50% porosity is only 5 °C. So an insufficient control of the bath temperature could explain the apparent non reproducibility of measurements performed after ageing classically encountered on black anodic films.

Analysis of the flaking

The black anodic films were observed by SEM after ageing and adhesion evaluation. These observations were performed after four-point bending, scratch test and peel test, especially where particles were pulled-off (Fig. 12). The sulphur detected by additional energy-dispersive X-ray spectroscopy (EDX) analyses, performed at the bottom of the print let by the particle, have shown that the fracture is cohesive in the film. Thus, the cracks propagate only in the film and then bifurcate before the interface film/substrate as schematically presented Fig. 13.

A finite element model was developed to better understand the types of solicitations leading to these crack bifurcation and propagation along the interface.

Numerical results and discussion

Crack propagation through the thickness of the film

The stress state near the crack when the sample is thermally loaded was evaluated thanks to the finite element model. Effects of dehydration are not modelled here, only thermal stresses due to the dilatations mismatch are calculated. At 80 °C, thermal tensile stresses are released on both sides of the crack; nevertheless a stress concentration is observed near the tip (Fig. 14). The Young modulus of the film decreases when its initial porosity increases (Goueffon et al., 2010a). It can vary from approximately 90 GPa (10% of initial porosity) to less than 20 GPa (50% of initial porosity). Thus, the stress intensity factors were calculated for different young modulus of the film as a function of the temperature applied (Fig. 14). Whatever the modulus of the film considered, the crack is only solicited in mode I (K II = 0 MPa m 1/2 ). When the temperature is lower than 20 °C, the film is under compressive thermal stresses, thus the crack is closed and the tip is not solicited. For higher temperatures, the stress intensity factor corresponding to the opening mode K I is linearly increasing with the temperature as expected. For a given temperature and a given crack length, stress intensity factor increases when Young modulus of the film increases: the crack propagation will be favored in films of higher Young modulus.

Bifurcation of the crack

In the previous conditions, there is symmetry of the geometry and loadings: the propagation angle is always equal to zero. It is necessary to break this symmetry to cause the bifurcation of the crack. The first possibility is the initial deviation of the crack. Actually, on experimental samples, cracks are not always exactly perpendicular to the interface. It can be explained by the roughness of the film (Ra % 1.6 lm) and the presence of defects at its surface (Goueffon et al., 2009a). The crack was then designed with an initial angle w which can vary between 0°and 10°( Fig. 15).

At À80 or 120 °C, K I is not significantly modified by the presence of a small initial angle (Fig. 16). Nevertheless, we can observe in both cases the apparition of K II (shearing mode) when w increases.

At hot temperatures, the crack is then under a mixed mode (I + II) but K I >> K II then it can be assimilated as pure mode I. The bifurcation angle is then tending to redress the crack perpendicularly to the interface (Fig. 16). At cold temperatures the solicitations are purely in the mode II resulting in a bifurcation angle of approximately 70°. Nevertheless, the value of

K II is low (K 120 C II < K À80 C II << K 120 C I
), so the temperature has to be cold enough to generate crack propagation.

A second possibility to break the symmetry is to insert neighbour cracks in the model (Fig. 17). Two cracks (17 lm length) were added to the initial model at a distance of 100 lm on both sides of the first one. This corresponds to the average size of islets observed on samples with scanning electron microscopy (Fig. 3). The central crack was then moved off centre at a distance ''e''. When e increases, the central crack closers to its neighbour and enters in its stress field. The symmetry of the stress field is then broken resulting in the simultaneous apparition of a mode II and release of the mode I. This phenomenon is amplified when the cracks become closer (i.e. when e increases) (Fig. 18). The apparition of this mixed mode is generating a crack bifurcation. The corresponding angle is actually increasing with e but is somewhat dependant on the central crack length (Fig. 18).

Two major causes of crack bifurcation have been highlighted. First, a geometrical asymmetry can lead to a pure mode II loading of the crack at temperatures lower than 20 °C. The temperature has to be cold enough to generate the propagation of the crack with a bifurcation angle of approximately 70°. Secondly, a close crack (or local defect) unbalances the stress field around the crack which is then solicited in mixed mode (I + II). The bifurcation angle is then strongly depending on the ratio K I /K II and thus on the distance to the neighbour crack.

Propagation along the interface film/substrate

The shape of the initial model has been modified to study the crack propagation along the interface. A kinked crack was designed with depth ''a'', a global width ''W'' and a radius of curvature ''r'' (Fig. 19).

At 80 °C, the substrate strains generate tensile stresses in the film. Stress concentration between the crack and the interface is observed. However, strains are not transmitted to the upper part of the film. A released area and even local compressive stresses near the crack tip appear (Fig. 20), creating local shear loading (mode II).

At hot temperatures (T > 20 °C), Fig. 21 shows that the crack is also solicited in opening mode (mode I). At cold temperatures (T < 20 °C), K I = 0 MPa m 1/2 , thus the crack is solicited in pure mode II. In both cases, the intensity of the solicitations is proportional to the applied temperature.

It is worthwhile to notice that the choice of the radius r does not have any significant effect on the values of calculated stress intensity factors. However, when the crack depth increases, these factors (mode I and II) are more important (Fig. 22).

Thus, the propagation along the interface will be favoured if the initial crack is deep enough.

Global discussion on the degradation mechanisms of black anodic films

The decrease of adhesion detected on several black anodic films after ageing was attributed to the propagation of cracks initiated during the process. Adhesion measurements have shown an important difference of behaviour between films with 40% and 50% of initial porosities. Flaking occurs much easier on films with the highest initial porosity.

Yet, the Young modulus of these highly porous films is lower than for other films (Goueffon et al., 2010a). According to the Fig. 14, the stress intensity factors are then lower in films at 50% of initial porosity than in 40% initial porosity films at a given temperature. The difference between the two samples can then be explained by two hypotheses. Firstly, it can be attributed to the fact that cracks resulting from the colouring and sealing processes are thicker and then probably deeper in the highest porosity film as noted in Goueffon et al. (2010a) and [START_REF] Liu | The cracking behavior of anodic films on cast aluminum alloy after heating in the temperature range up to 300 °C[END_REF]. So, according to numerical results (Fig. 22), this could result in slightly higher values of stress intensity factors for the 50% of initial porosity films. Secondly, and with probably a more important influence, this could be due to a decrease of the toughness K c with the increasing porosity as [START_REF] Hashimoto | Effect of Dy211 content on fracture toughness of Dy123 bulks[END_REF] have observed on ceramics.

With these hypothesis, stresses (thermal + dehydration) encountered during thermal cycling between À80 and +80 °C are not high enough to cause crack propagation in films with 40% of initial porosity. The toughness of the films with 50% of initial porosity being lower, crack propagation can occur resulting in the loss of adhesion detected by peel-tests and scratch-tests.

The thermal cycling range enlargement implies an increase of stress intensity factors as noted by numerical analysis (see for instance Fig. 14). Crack propagation occurs between À110 and +110 °C (and then À140/+140 °C) for films with 40% of initial porosity and an adhesion loss is detected. The vacuum amplifies tensile stresses in the films, especially at hot temperatures. Actually, the release of water favours crack propagation what explains the lower critical loads measured by scratch-tests after ageing under vacuum than under nitrogenous atmosphere.

The numerical model has shown that a bifurcation of the crack can be induced by the presence of a neighbour crack, a defect, or cold temperatures. The crack can then propagate along the interface either under mixed mode I + II at hot temperatures or under pure mode II at cold temperatures, justifying the respective detrimental effect of only cold temperatures cycling and only hot temperatures cycling on the adhesion, as noted experimentally (Fig. 11). When this adhesion loss occurs, it is detected after the first thermal cycles. The fast propagation is consistent with the brittle behaviour and the instable character of crack propagation. The loss is more important in vacuum due to additional effects of dehydration. When the range of thermal cycles increases, the stresses, due to differential dilatations or dehydration, are amplified arising risks of crack propagations.

Conclusion

Fig. 1 .

 1 Fig. 1. Evolution of the scratch critical load measured after 10 thermal cycles between À80 and +80 °C under N 2 or vacuum (10 À6 Pa) for samples with two different initial porosities (40% and 50%).

Fig. 2 .

 2 Fig. 2. Cumulated hits detected by acoustic emission measurements as a function of the load of four-point bending: samples with 40% and 50% of initial porosity and different ageing.

Fig. 3 .

 3 Fig. 3. Surface SEM observation of an anodic film (initial porosity 40%) after 10 thermal cycles under vacuum between À80 and +80 °C and peel test (670 g/cm).

Fig. 4 .

 4 Fig. 4. Thermo-gravimetric analysis of anodic films with initial porosity of 40% during cyclic warming at +80 °C.

Fig. 5 .

 5 Fig. 5. Thermo-gravimetric analysis of anodic films with initial porosity of 40% at ambient temperature.

Fig. 7

 7 Fig. 7 presents the evolution of the scratch critical load as a function of the number of thermal cycles performed between À80 and +80 °C under nitrogenous atmosphere. Films with an initial porosity of 40% are not damaged even for a high number of thermal cycles. This result was confirmed by four-point bending tests and peel tests. Actually, when degradation occurs, it is during the first thermal cycles as we can see for the sample showing initial 50% porosity. This observation is confirmed under vacuum for both types of porosities (Fig.8).These results must be related with the fast dehydration that mainly occurs during the first cycles, as seen in Fig.4. In addition, they are consistent with the brittleness of the anodic film: the mechanism of degradation is not cyclic loading but results from the combined effect of thermal and dehydration stresses that damage the film quasiimmediately.

Fig. 6 .

 6 Fig. 6. Evolution of the cracks width on a 40% porosity film as a function of the time of storage under vacuum (10 À6 Pa) at ambient temperature.

Fig. 7 .

 7 Fig. 7. Evolution of the scratch-test critical load as a function of the number of thermal cycles between À80 and +80 °C under nitrogenous atmosphere.

Fig. 8 .

 8 Fig. 8. Evolution of the scratch-test critical load as a function of the number of thermal cycles between À80 and +80 °C under vacuum (10 À6 Pa).

Fig. 9 .

 9 Fig. 9. Evolution of the scratch-test critical load as a function of the temperature of thermal cycles under nitrogenous atmosphere.

Fig. 10 .

 10 Fig. 10. Thermo-gravimetric analysis of anodic films with initial porosity of 40% under inert atmosphere for different temperatures of cycling.

Fig. 11 .

 11 Fig. 11. Evolution of the scratch-test critical load with dissociating cold parts and hot parts of thermal cycles (À140/+140 °C) under nitrogenous atmosphere.

Fig. 12 .

 12 Fig. 12. Surface SEM observation of black anodic film (initial porosity 50%) after 10 cycles between À80 and +80 °C and peel test with a tape strength of 250 g/cm.

Fig. 13 .

 13 Fig. 13. Schema of crack propagation in an anodic film leading to flaking.

Fig. 14 .

 14 Fig. 14. Evolution of the stress intensity factor K I of a 17 lm length crack as a function of the applied temperature and for different Young modulus of the film. The stress state around a crack tip is presented on the upper part of the figure.

Fig. 15 .

 15 Fig. 15. Schema of the crack geometry with an initial angle (on the left) and comparison with SEM cross-section view (on the right).

Fig. 16 .

 16 Fig. 16. Evolution of the stress intensity factors and the bifurcation angle for an initial angle from 0°to 10°(E f = 32 GPa).

Fig. 17 .

 17 Fig. 17. Schema of the film with three cracks (on the left) and comparison with SEM cross-section view (on the right).

Fig. 18 .

 18 Fig. 18. Stress intensity factors and bifurcation angle of the central crack as a function of its eccentricity (a = 9 lm or a = 17 lm, T = 120 °C and E film = 32 GPa).

Fig. 19 .

 19 Fig. 19. Schema of the numerical model with a kinked crack (on the left) and comparison with SEM cross-section view (on the right).

Fig. 20 .

 20 Fig. 20. Stress field (r xx ) around the crack tip at 80 °C (r = 5 lm, W = 10 lm, a = 9 lm, E film = 32 GPa).

Fig. 21 .

 21 Fig. 21. Stress intensity factors as a function of the thermal loading applied (r = 5 lm, W = 10 lm, a = 9 lm, E film = 32 GPa).

  On the three methods used to evaluate the adhesion of black anodic films on their substrates, two of them (scratch test and peel test) have clearly spotted an ageing effect during thermal cycling. A flaking of the film can occur and the associated fracture is cohesive in the film. Numerical simulation has shown that thermo-mechanical solicitations (differential dilatations) related to both warm and cold parts of the thermal cycling can cause crack propagations according to different mechanisms, what could result in a loss of the measured adhesion. Samples with the highest initial porosity are the more sensitive to flaking. It can be explained by their low toughness favoring crack propagation.

Acknowledgements

The authors thank Rafal Pijewski and Muguette Monjauze for their participations to this study.


Zoltán Szabó
email: zoltan.szabo@polytechnique.edu

Francis Bach

Characteristic and Universal Tensor Product Kernels

Keywords: tensor product kernel, kernel mean embedding, characteristic kernel, Icharacteristic kernel, universality, maximum mean discrepancy, Hilbert-Schmidt independence criterion

Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. Thanks to their kernel-based foundations, MMD and HSIC are applicable on a wide variety of domains. Despite their tremendous success, quite little is known about when HSIC characterizes independence and when MMD with tensor product kernel can discriminate probability distributions. In this paper, we answer these questions by studying various notions of characteristic property of the tensor product kernel.

Introduction

Kernel methods [START_REF] Schölkopf | Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond[END_REF] are among the most flexible and influential tools in machine learning and statistics, with superior performance demonstrated in a large number of areas and applications. The key idea in these methods is to map the data samples into a possibly infinite-dimensional feature space-precisely, a reproducing kernel Hilbert space (RKHS; [START_REF] Aronszajn | Theory of reproducing kernels[END_REF]-and apply linear methods in the feature space, without the explicit need to compute the map. A generalization of this idea to probability measures, i.e., mapping probability measures into an RKHS (Berlinet and Thomas-Agnan, 2004, Chapter 4;[START_REF] Smola | A Hilbert space embedding for distributions[END_REF] has found novel applications in nonparametric statistics and machine learning. Formally, given a probability measure P defined on a measurable space X and an RKHS H k with k : X×X → R as the reproducing kernel (which is symmetric and positive definite), P is embedded into H k as

P → X k(•, x) dP(x) =: µ k (P), (1)
where µ k (P) is called the mean element or kernel mean embedding of P. The mean embedding of P has lead to a new generation of solutions in two-sample testing [START_REF] Baringhaus | On a new multivariate two-sample test[END_REF]Székely andRizzo, 2004, 2005;[START_REF] Borgwardt | Integrating structured biological data by kernel maximum mean discrepancy[END_REF][START_REF] Harchaoui | Testing for homogeneity with kernel Fisher discriminant analysis[END_REF][START_REF] Gretton | A kernel twosample test[END_REF], goodness-of-fit testing [START_REF] Chwialkowski | A kernel test of goodness of fit[END_REF][START_REF] Liu | A kernelized Stein discrepancy for goodness-of-fit tests[END_REF]Jitkrittum et al., 2017b;[START_REF] Balasubramanian | On the optimality of kernel-embedding based goodness-of-fit tests[END_REF], domain adaptation (Zhang et al., 2013) and generalization [START_REF] Blanchard | Domain generalization by marginal transfer learning[END_REF], kernel belief propagation [START_REF] Song | Kernel belief propagation[END_REF], kernel Bayes' rule [START_REF] Fukumizu | Kernel Bayes' rule: Bayesian inference with positive definite kernels[END_REF], model criticism [START_REF] Lloyd | Automatic construction and natural-language description of nonparametric regression models[END_REF][START_REF] Kim | Examples are not enough, learn to criticize! criticism for interpretability[END_REF], approximate Bayesian computation [START_REF] Park | K2-ABC: Approximate Bayesian computation with kernel embeddings[END_REF], probabilistic programming [START_REF] Schölkopf | Computing functions of random variables via reproducing kernel Hilbert space representations[END_REF], distribution classification [START_REF] Muandet | Learning from distributions via support measure machines[END_REF]Zaheer et al., 2017), distribution regression [START_REF] Szabó | Learning theory for distribution regression[END_REF][START_REF] Law | Bayesian approaches to distribution regression[END_REF] and topological data analysis [START_REF] Kusano | Persistence weighted Gaussian kernel for topological data analysis[END_REF]. A recent survey on the topic is provided by [START_REF] Muandet | Kernel mean embedding of distributions: A review and beyond[END_REF].

Crucial to the success of the mean embedding based representation is whether it encodes all the information about the distribution, in other words whether the map in (1) is injective in which case the kernel is referred to as characteristic [START_REF] Fukumizu | Kernel measures of conditional dependence[END_REF][START_REF] Sriperumbudur | Hilbert space embeddings and metrics on probability measures[END_REF]. Various characterizations for the characteristic property of k is known in the literature [START_REF] Fukumizu | Kernel measures of conditional dependence[END_REF][START_REF] Fukumizu | Kernel dimension reduction in regression[END_REF][START_REF] Sriperumbudur | Hilbert space embeddings and metrics on probability measures[END_REF][START_REF] Gretton | A kernel twosample test[END_REF] using which the popular kernels on R d such as Gaussian, Laplacian, B-spline, inverse multiquadrics, and the Matérn class are shown to be characteristic. The characteristic property is closely related to the notion of universality [START_REF] Steinwart | On the influence of the kernel on the consistency of support vector machines[END_REF][START_REF] Micchelli | Universal kernels[END_REF][START_REF] Carmeli | Vector valued reproducing kernel Hilbert spaces and universality[END_REF][START_REF] Sriperumbudur | Universality, characteristic kernels and RKHS embedding of measures[END_REF]-k is said to be universal if the corresponding RKHS H k is dense in a certain target function class, for example, the class of continuous functions on compact domains-and the relation between these notions has recently been explored by [START_REF] Sriperumbudur | Universality, characteristic kernels and RKHS embedding of measures[END_REF]; Simon-Gabriel and Schölkopf (2016).

Based on the mean embedding in (1), [START_REF] Smola | A Hilbert space embedding for distributions[END_REF] and [START_REF] Gretton | A kernel twosample test[END_REF] defined a semi-metric, called the maximum mean discrepancy (MMD) on the space of probability measures:

MMD k (P, Q) := µ k (P) -µ k (Q) H k ,
which is a metric iff k is characteristic. A fundamental application of MMD is in nonparametric hypothesis testing that includes two-sample [START_REF] Gretton | A kernel twosample test[END_REF] and independence tests [START_REF] Gretton | A kernel statistical test of independence[END_REF]. Particularly in independence testing, as a measure of independence, MMD measures the distance between the joint distribution P XY and the product of marginals P X ⊗ P Y of two random variables X and Y which are respectively defined on measurable spaces X and Y, with the kernel k being defined on X × Y. As aforementioned, if k is characteristic, then MMD k (P XY , P X ⊗ P Y ) = 0 implies P XY = P X ⊗ P Y , i.e., X and Y are independent. A simple way to define a kernel on X × Y is through the tensor product of kernels k X and k Y defined on X and Y respectively:

k = k X ⊗ k Y , i.e., k ((x, y) , (x , y )) = k X (x, x )k Y (y, y ), x, x ∈ X, y, y ∈ Y, with the corresponding RKHS H k = H k X ⊗ H k Y being the tensor product space generated by H k X and H k Y . This means, when k = k X ⊗ k Y , MMD k (P XY , P X ⊗ P Y ) = µ k X ⊗k Y (P XY ) -µ k X ⊗k Y (P X ⊗ P Y ) H k X ⊗H k Y . (2)
In addition to the simplicity of defining a joint kernel k on X × Y, the tensor product kernel offers a principled way of combining inner products (k X and k Y ) on domains that can correspond to different modalities (say images, texts, audio). By exploiting the isomorphism between tensor product Hilbert spaces and the space of Hilbert-Schmidt operators 1 , it follows from (2) that

MMD k (P XY , P X ⊗ P Y ) = C XY HS =: HSIC k (P XY ), (3)
which is the Hilbert-Schmidt norm of the cross-covariance operator

C XY := µ k X ⊗k Y (P XY )- µ k X (P X ) ⊗ µ k Y (P Y )
and is known as the Hilbert-Schmidt independence criterion (HSIC) (Gretton et al., 2005a). HSIC has enjoyed tremendous success in a variety of applications such as independent component analysis (Gretton et al., 2005a), feature selection [START_REF] Song | Feature selection via dependence maximization[END_REF], independence testing [START_REF] Gretton | A kernel statistical test of independence[END_REF]Jitkrittum et al., 2017a), post selection inference (Yamada et al., 2018) and causal detection [START_REF] Mooij | Distinguishing cause from effect using observational data: Methods and benchmarks[END_REF][START_REF] Pfister | Kernel-based tests for joint independence[END_REF][START_REF] Strobl | Approximate kernel-based conditional independence tests for fast non-parametric causal discovery[END_REF]. Recently, MMD and HSIC (as defined in (3) for two components) have been shown by Sejdinovic et al. (2013b) to be equivalent to other popular statistical measures such as the energy distance [START_REF] Baringhaus | On a new multivariate two-sample test[END_REF]Székely andRizzo, 2004, 2005)-also known as N-distance (Zinger et al., 1992;[START_REF] Klebanov | N-Distances and Their Applications[END_REF]-and distance covariance [START_REF] Székely | Measuring and testing dependence by correlation of distances[END_REF][START_REF] Székely | Brownian distance covariance[END_REF][START_REF] Lyons | Distance covariance in metric spaces[END_REF] respectively. HSIC has been generalized to M ≥ 2 components [START_REF] Quadrianto | Kernelized sorting[END_REF]Sejdinovic et al., 2013a) to measure the joint independence of M random variables

HSIC k (P) = µ ⊗ M m=1 km (P) -⊗ M m=1 µ km (P m ) ⊗ M m=1 H km , (4)
where P is a joint measure on the product space X := × M m=1 X m and (P m ) M m=1 are the marginal measures of P defined on (X m ) M m=1 respectively. The extended HSIC measure has recently been analyzed in the context of independence testing [START_REF] Pfister | Kernel-based tests for joint independence[END_REF]. In addition to testing, the extended HSIC measure is also useful in the problem of independent subspace analysis (ISA; [START_REF] Cardoso | Multidimensional independent component analysis[END_REF], wherein the latent sources are separated by maximizing the degree of independence among them. In all the applications of HSIC, the key requirement is that k = ⊗ M m=1 k m captures the joint independence of M random variables (with joint distribution P)-we call this property as I-characteristic-, which is guaranteed if k is characteristic. Since k is defined in terms of (k m ) M m=1 , it is of fundamental importance to understand the characteristic and I-characteristic properties of k in terms of the characteristic property of (k m ) M m=1 , which is one of the main goals of this work. For M = 2, the characterization of independence, i.e., the I-characteristic property of k, is studied by [START_REF] Blanchard | Generalizing from several related classification tasks to a new unlabeled sample[END_REF] and [START_REF] Gretton | A simpler condition for consistency of a kernel independence test[END_REF] where it has been shown that if k 1 and k 2 are universal, then k is universal 2 and therefore HSIC captures independence. A stronger version of this result can be obtained by combining (Lyons, 2013, Theorem 3.11) and (Sejdinovic et al., 2013b, Proposition 29): if k 1 and k 2 are characteristic, then the HSIC associated with k = k 1 ⊗ k 2 characterizes independence. Apart from these results, not much is known about the characteristic/I-characteristic/universality properties of k in terms of the individual kernels. Our goal is to resolve this question and understand the characteristic, I-characteristic and universal property of the product kernel (⊗ M m=1 k m ) in terms of the kernel components ((k m ) M m=1 ) for M ≥ 2. Because of the relatedness of MMD and HSIC to energy distance and distance covariance, our results also contribute to the better understanding of these other measures that are popular in the statistical literature.

Specifically, our results shed light on the following surprising phenomena of the Icharacteristic property of ⊗ M m=1 k m for M ≥ 3: 1. characteristic property of (k m ) M m=1 is not sufficient but necessary for ⊗ M m=1 k m to be I-characteristic; 2. universality of (k m ) M m=1 is sufficient for ⊗ M m=1 k m to be I-characteristic, and 3. if at least one of (k m ) M m=1 is only characteristic and not universal, then ⊗ M m=1 k m need not be I-characteristic. The paper is organized as follows. In Section 3, we conduct a comprehensive analysis about the above mentioned properties of k and (k m ) M m=1 for any positive integer M . To this end, we define various notions of characteristic property on the product space X (see Definition 1 and Figure 2(a) in Section 3) and explore the relation between them. In order to keep our presentation in this section to be non-technical, we relegate the problem formulation to Section 3, with the main results of the paper being presented in Section 4. A summary of the results is captured in Figure 1 while the proofs are provided in Section 5. Various definitions and notation that are used throughout the paper are collected in Section 2.

Definitions and Notation

M m=1 A m = {(a 1 , . . . , a M ) : a m ∈ A m m ∈ [M ]
} is the Descartes product of sets (A m ) M m=1 . P(X) denotes the power set of a set X, i.e., all subsets of X (including the empty set and X). The Kronecker delta is defined as δ a,b = 1 if a = b, and zero otherwise. χ A is the indicator function of set A: χ

A (x) = 1 if x ∈ A and χ A (x) = 0 otherwise. R d 1 ×...×d M is the set of d 1 × . . . × d M -sized tensors.
For a topological space (X, τ X ), B(X) := B(τ X ) is the Borel sigma-algebra on X induced by the topology τ X . Probability and finite signed measures in the paper are meant w.r.t. the measurable space (X, B(X)). Given {(X i , τ i )} i∈I topological spaces, their product × i∈I X i is enriched with the product topology; it is the coarsest topology for which the canonical projections π i : × i∈I X i → (X i , τ i ) are continuous for all i ∈ I. A topological space (X, τ X ) is called second-countable if τ X has a countable basis.3 C(X) denotes the space of continuous functions on X. C 0 (X) denotes the class of real-valued functions vanishing at infinity on a locally compact Hausdorff (LCH) space4 X, i.e., for any > 0, the set {x ∈ is compact. C 0 (X) is endowed with the uniform norm f ∞ = sup x∈X |f (x)|. M b (X) and M + 1 (X) are the space of finite signed measures and probability measures on X, respectively. For P m ∈ M + 1 (X m ), ⊗ M m=1 P m denotes the product probability measure on the product space

X : |f (x)| ≥ } ⊗ 0 -char ⊗-char char c 0 -universal I-char (k m ) M m=1 char (k m ) M m=1 c 0 -universal (
× M m=1 X m , i.e., ⊗ M m=1 P m ∈ M + 1 (× M m=1 X m ). δ x is the Dirac measure supported on x ∈ X. For F ∈ M b × M
m=1 X m , the finite signed measure F m denotes its marginal on X m . H km is the reproducing kernel Hilbert space (RKHS) associated with the reproducing kernel k m : X m × X m → R, which in this paper is assumed to be measurable and bounded. The tensor product of (k m ) M m=1 is a kernel, defined as (Berlinet and Thomas-Agnan, 2004, Theorem 13), where the r.h.s. is the tensor product of RKHSs (H km

⊗ M m=1 k m (x 1 , . . . , x M ) , x 1 , . . . , x M = M m=1 k m x m , x m , x m , x m ∈ X m , whose associated RKHS is denoted as H ⊗ M m=1 km = ⊗ M m=1 H km
) M m=1 . For h m ∈ H m , m ∈ [M ], the multi-linear operator ⊗ M m=1 h m ∈ ⊗ M m=1 H m is defined as ⊗ M m=1 h m (v 1 , . . . , v M ) = M m=1 h m , v m Hm , v m ∈ H m . A kernel k : X × X → R defined on a LCH space X is called a c 0 -kernel if k(•, x) ∈ C 0 (X) for all x ∈ X. k : R d × R d → R is said to be a translation invariant kernel on R d if k(x, y) = ψ(x -y), x, y ∈ R d for a positive definite function ψ : R d → R. µ k (F) denotes the kernel mean embedding of F ∈ M b (X) to H k which is defined as µ k (F) = X k(•, x) dF(x),
where the integral is meant in the Bochner sense.

Problem Formulation

In this section, we formally introduce the goal of the paper. To this end, we start with a definition. For simplicity, throughout the paper, we assume that all kernels are bounded.

The definition is based on the observation (Sriperumbudur et al., 2010, Lemma 8) that a bounded kernel k on a topological space (X, τ X ) is characteristic if and only if

X X k(x, x ) dF(x) dF(x ) > 0, ∀ F ∈ M b (X)\{0} such that F(X) = 0.
In other words, characteristic kernels are integrally strictly positive definite (ispd; see [START_REF] Sriperumbudur | Hilbert space embeddings and metrics on probability measures[END_REF]Sriperumbudur et al., , p. 1523) ) w.r.t. the class of finite signed measures that assign zero measure to X. The following definition extends this observation to tensor product kernels on product spaces.

Definition 1 (F-ispd tensor product kernel) Suppose k m : X m × X m → R is a bounded kernel on a topological space (X m , τ Xm ) , m ∈ [M ]. Let F ⊆ M b (X) be such that 0 ∈ F where X := × M m=1 X m . k := ⊗ M m=1 k m is said to be F-ispd if µ k (F) = 0 ⇒ F = 0 (F ∈ F), or equivalently µ k (F) 2 H k = × M m=1 Xm × M m=1 Xm ⊗ M m=1 k m x, x dF(x) dF(x ) > 0, ∀ F ∈ F\{0}. (5)
Specifically,

• if k m -s are c 0 -kernels on locally compact Polish (LCP)5 spaces X m -s and

F = M b (X), then k is called c 0 -universal. • if F = [M b (X)] 0 := {F ∈ M b (X) : F(X) = 0} , F = ⊗ M m=1 M b (X m ) 0 := F ∈ ⊗ M m=1 M b (X m ) , F(X) = 0 , F = I := P -⊗ M m=1 P m : P ∈ M + 1 × M m=1 X m , (M ≥ 2) F = ⊗ M m=1 M 0 b (X m ) := F = ⊗ M m=1 F m : F m ∈ M b (X m ) , F m (X m ) = 0, ∀ m ∈ [M ] ,
then k is called characteristic, ⊗-characteristic, I-characteristic and ⊗ 0 -characteristic, respectively.

In Definition 1, k being characteristic matches the usual notion of characteristic kernels on a product space, i.e., there are no two distinct probability measures on X = × M m=1 X m such that the MMD between them is zero. The other notions such as ⊗-characteristic, I-characteristic and ⊗ 0 -characteristic are typically weaker than the usual characteristic property since

⊗ M m=1 M 0 b (X m ) ⊆ ⊗ M m=1 M b (X m ) 0 ⊆ M b × M m=1 X m 0 ⊆ M b × M m=1 X m I ⊆ . (6)
Below we provide further intuition on the F measure classes enlisted in Definition 1.

Remark 2 (i) F = M b (X) : If k m -s are c 0 -kernels on LCH spaces X m for all m ∈ [M ],
then k is also a c 0 -kernel on LCH space X implying that if k satisfies (5), then k is c 0universal (Sriperumbudur et al., 2010, Proposition 2). It is well known [START_REF] Sriperumbudur | Hilbert space embeddings and metrics on probability measures[END_REF]) that c 0 -universality reduces to c-universality (i.e., the notion of universality proposed by [START_REF] Steinwart | On the influence of the kernel on the consistency of support vector machines[END_REF]

if X is compact which is guaranteed if and only if each X m , m ∈ [M ] is compact.
(ii) F = I : This family is useful to describe the joint independence of M random variableshence the name I-characteristic-defined on kernel-endowed domains (X m ) M m=1 : If P denotes the joint distribution of random variables (X m ) M m=1 and (P m ) M m=1 are the associated marginals on

(X m ) M m=1 , then by definition k = ⊗ M m=1 k m is I-characteristic iff HSIC k (P) = 0 ⇐⇒ P = ⊗ M m=1 P m .
In other words, HSIC captures joint independence exactly with I-characteristic kernels.

Similarly, the I-characteristic property ensures that COCO (constrained covariance; Gretton et al., 2005b) is a joint independence measure as COCO is defined by replacing the Hilbert-Schmidt norm of the cross-covariance operator (see (3) and (4)) with its spectral norm.

(iii) F = ⊗ M m=1 M 0 b (X m ) :
In this case F is chosen to be the product of finite signed measures on X such that each marginal measure F m assigns zero to the corresponding space X m . This choice is relevant as the characteristic property of individual kernels

(k m ) M m=1 need not imply the characteristic property of ⊗ M m=1 k m , but is equiva- lent to the ⊗ 0 -characteristic property of ⊗ M m=1 k m . The equivalence holds for bounded kernels k m : X m × X m → R on topological spaces X m (m ∈ [M ]) since for any F = ⊗ M m=1 F m ∈ ⊗ M m=1 M b (X m ), F m (X m ) = 0 (∀ m ∈ [M ]) µ k (F) 2 H ⊗ M m=1 km = M m=1 µ km (F m ) 2 H km , (7)
and the l.h.s. is positive iff each term on the r.h.s. is positive.

(iv) F = ⊗ M m=1 M b (X m ) 0 :
This class is similar to the one discussed in (iii) abovei.e., class of product measures-with the slight difference that the joint measure F is restricted to assign zero measure to X without requiring all the marginal measures F m c 0 -univ.

characteristic

I-char. ⊗-char. ⊗ 0 -char. (a) Example 1 M b (X) [M b (X)] 0 I ⊗ m M 0 b (X m ) [⊗ m M b (X m )] 0 (b) Figure 2: (a) F-ispd ⊗ M m=1 k m kernels (see (8)); (b) F ⊆ M b (X), X = × M m=1 X m . Exam- ple 1: ⊗ M m=1 k m is ⊗ 0 -characteristic but not ⊗-characteristic and therefore not characteristic.
to assign zero measure to the corresponding space X m . While the need for considering such a measure class may not be clear at this juncture, however, based on (7), it turns out that this choice of F has quite surprising connections to the characteristic property and c 0 -universality of the product kernel; for details see Remark 7.

(v) F-ispd relations: Given the relations in (6), it immediately follows that k

= ⊗ M m=1 k m satisfies ⊗ 0 -characteristic ⊗-characteristic ⇐= characteristic ⇐= ⇐ c 0 -universal ⇐= I-characteristic (8)
when X m for all m ∈ [M ] are LCP. A visual illustration of (6) and (8) is provided in Figure 2.

(vi) ⊗ M m=1 M b (X m ) 0 ∩I = {0} : While it is clear that ⊗ M m=1 M b (X m ) 0 and I are subsets of M b (× M m=1 X m ) 0 , it is interesting to note that ⊗ M m=1 M b (X m )
0 and I have a trivial intersection with 0 being the measure common to each of them, assuming that X m -s are second-countable for all m ∈ [M ]; see Section 5.1.

Having defined the F-ispd property, our goal is to investigate whether the characteristic or c 0 -universal property of k m -s (m ∈ [M ]) imply different F-ispd properties of ⊗ M m=1 k m , and vice versa.

Main Results

In this section, we present our main results related to the F-ispd property of tensor product kernels, which are summarized in Figure 1. The results in this section will deal with various assumptions on X m , such as second-countability, Hausdorff, locally compact Hausdorff (LCH) and locally compact Polish (LCP), so that they are presented in more generality. However, for simplicity, all these assumptions can be unified by simply assuming a stronger condition that X m 's are LCP.

Our first example illustrates that the characteristic property of k m -s does not imply the characteristic property of the tensor product kernel. In light of Remark 2(iv) of Section 3, it follows that the class of ⊗ 0 -characteristic tensor product kernels form a strictly larger class than characteristic tensor product kernels; see also Figure 2.

Example 1 Let X 1 = X 2 = {1, 2}, τ X 1 = τ X 2 = P({1, 2}), k 1 (x, x ) = k 2 (x, x ) = 2δ x,x -1.
It is easy to verify that k 1 and k 2 are characteristic. However, it can be proved that k 1 ⊗ k 2 is not ⊗-characteristic and therefore not characteristic. On the hand, interestingly, k 1 ⊗ k 2 is I-characteristic. We refer the reader to Section 5.2 for details.

In the above example, we showed that the tensor product of k 1 and k 2 (which are characteristic kernels) is I-characteristic. The following result generalizes this behavior for any bounded characteristic kernels. In addition, under a mild assumption, it shows the converse to be true for any M .

Theorem 3 Let k m : X m × X m → R be bounded kernels on topological spaces X m for all m ∈ [M ], M ≥ 2. Then the following holds. (i) Suppose X m is second-countable for all m ∈ [M ] with M = 2. If k 1 and k 2 are characteristic, then k 1 ⊗ k 2 is I-characteristic. (ii) Suppose X m is Hausdorff and |X m | ≥ 2 for all m ∈ [M ]. If ⊗ M m=1 k m is I-characteristic, then k 1 , . . . , k M are characteristic.
Lyons (2013) has showed an analogous result to Theorem 3(i) for distance covariances (M = 2) on metric spaces of negative type (Lyons, 2013, Theorem 3.11), which by Sejdinovic et al. (2013b, Proposition 29) holds for HSIC yielding the I-characteristic property of k 1 ⊗k 2 . Recently, [START_REF] Gretton | A simpler condition for consistency of a kernel independence test[END_REF] presented a direct proof showing that HSIC corresponding to k 1 ⊗ k 2 captures independence if k 1 and k 2 are translation invariant characteristic kernels on R d (which is equivalent to c 0 -universality). [START_REF] Blanchard | Generalizing from several related classification tasks to a new unlabeled sample[END_REF] proved a result similar to Theorem 3(i) assuming that X m 's are compact and k 1 , k 2 being c-universal. In contrast, Theorem 3(i) establishes the result for bounded kernels on general secondcountable topological spaces. In fact, the results of [START_REF] Gretton | A simpler condition for consistency of a kernel independence test[END_REF]; [START_REF] Blanchard | Generalizing from several related classification tasks to a new unlabeled sample[END_REF] are special cases of Theorems 4 and 5 below. Theorem 3(i) raises a pertinent question:

whether ⊗ M m=1 k m is I-characteristic if k m -s are characteristic for all m ∈ [M ]
where M > 2?

The following example provides a negative answer to this question. On a positive side, however, we will see in Theorem 5 that the I-characteristic property of ⊗ M m=1 k m can be guaranteed for any M ≥ 2 if a stronger condition is imposed on k m -s (and X m -s). Theorem 3(ii) generalizes Proposition 3.15 of [START_REF] Lyons | Distance covariance in metric spaces[END_REF] for any M > 2, which states that every kernel k m , m ∈ [M ] being characteristic is necessary for the tensor kernel ⊗ M m=1 k m to be I-characteristic.

Example 2 Let M = 3 and X m := {1, 2}, τ Xm = P(X m ), k m (x, x ) = 2δ x,x -1 (m = 1, 2, 3). As mentioned in Example 1, (k m ) 3 m=1 are characteristic. However, it can be shown that ⊗ 3 m=1 k m is not I-characteristic. See Section 5.4 for details.

In Remark 2(iii) and Example 1, we showed that in general, only the ⊗ 0 -characteristic property of ⊗ M m=1 k m is equivalent to the characteristic property of k m -s. Our next result shows that all the various notions of characteristic property of ⊗ M m=1 k m coincide if k m -s are translation-invariant, continuous bounded kernels on R d .

Theorem 4 Suppose k m : R dm × R dm → R are continuous, bounded and translationinvariant kernels for all m ∈ [M ]. Then the following statements are equivalent:

(i) k m -s are characteristic for all m ∈ [M ]; (ii) ⊗ M m=1 k m is ⊗ 0 -characteristic; (iii) ⊗ M m=1 k m is ⊗-characteristic; (iv) ⊗ M m=1 k m is I-characteristic; (v) ⊗ M m=1 k m is characteristic.
The following result shows that on LCP spaces, the tensor product of M ≥ 2 c 0 -universal kernels is also c 0 -universal, and vice versa.

Theorem 5 Suppose k m : X m × X m → R are c 0 -kernels on LCP spaces X m (m ∈ [M ]). Then ⊗ M m=1 k m is c 0 -universal iff k m -s are c 0 -universal for all m ∈ [M ].
Remark 6 (i) A special case of Theorem 5 for M = 2 is proved by Lyons (2013, Lemma 3.8) in the context of distance covariance which reduces to Theorem 5 through the equivalence established by Sejdinovic et al. (2013b). Another special case of Theorem 5 is proved by Blanchard et al. (2011, Lemma 5.2) for c-universality with M = 2 using the Stone-Weierstrass theorem: if k 1 and k 2 are c-universal then k 1 ⊗ k 2 is c-universal.

(ii) Since the notions of c 0 -universality and characteristic property are equivalent for translation invariant c 0 -kernels on R d [START_REF] Carmeli | Vector valued reproducing kernel Hilbert spaces and universality[END_REF], Prop. 5.16, Sriperumbudur et al., 2010, Theorem 9), Theorem 4 can be considered as a special case of Theorem 5. In other words, requiring (k m ) M m=1 to be also c 0 -kernels in Theorem 4(i)-(iv) is equivalent to

(v) k m -s are c 0 -universal for all m ∈ [M ]; (vi) ⊗ M m=1 k m is c 0 -universal. (iii) Since the c 0 -universality of ⊗ M
m=1 k m implies its I-characteristic property (see (8)), Theorem 5 also provides a generalization of Theorem 3(i) to M ≥ 2 under additional assumptions on k m -s, while constraining X m -s to LCP-s instead of second-countable topological spaces.

In Example 2 and Theorem 5, we showed that for M ≥ 3 components while the characteristic property of (k m ) M m=1 is not sufficient, their universality is enough to guarantee the I-characteristic property of ⊗ M m=1 k m . The next example demonstrates that these results are tight: If at least one k m is not universal but only characteristic, then ⊗ M m=1 k m might not be I-characteristic.

Example 3 Let M = 3 and X m := {1, 2}, τ Xm = P(X m ), for all m ∈ [3], k 1 (x, x ) = 2δ x,x -1, and k m (x, x ) = δ x,x (m = 2, 3). k 1 is characteristic (Example 1), k 2 and k 3 are universal since the associated Gram matrix G = [k m (x, x )] x,x ∈Xm is an identity matrix, ⊗-char. =characteristic =c 0 -universal I-char. ⊗ 0 -char. which is strictly positive definite (m = 2, 3). However, ⊗ 3 m=1 k m is not I-characteristic. See Section 5.7 for details.

Remark 7 Note that the l.h.s. in (7) is positive if and only if each term on the r.h.s. is positive, i.e., if k = ⊗ M m=1 k m is ⊗-characteristic with k m -s being c 0 -kernels on LCP X m -s, then all k m -s are c 0 -universal. A similar result was also proved by Steinwart and Ziegel (2017, Lemma 3.4). Combining this with Theorem 5 yields that for tensor product c 0kernels, the notions of ⊗-characteristic, characteristic and c 0 -universality are equivalent, which is quite surprising as for a joint kernel k (that is not of product type), these notions need not necessarily coincide. In light of this discussion, Figure 2(a) can be simplified to Figure 3.

Proofs

In this section, we provide the proofs of our results presented in Section 4.

Proof of Remark 2(iv)

By the second-countability of X m -s, B × M m=1 X m = ⊗ M m=1 B(X m )
, where the r.h.s. is defined as the σ-field generated by the cylinder sets A m × n =m X n where m ∈ [M ] and A m ∈ B(X m ). Suppose there exists

F ∈ ⊗ M m=1 M b (X m ) 0 ∩ I such that F = 0. This means there exists P ∈ M + 1 × M m=1 X m with (P m ) M m=1 being the marginals of P such that F = ⊗ M m=1 F m = P -⊗ M m=1 P m . Since F = 0 there exists A m × n =m X n for some m ∈ [M ] and A m ∈ B(X m ) such that 0 = F(A m × n =m X n ) = F m (A m ) n =m F n (X n ) = P (A m × n =m X n )- P m (A m ) n =m P n (X n ) = P m (A m ) -P m (A m ) = 0, leading to a contradiction.

Proof of Example 1

The proof is structured as follows.

1. First we show that k := k 1 = k 2 is a kernel and it is characteristic.

Next it is proved that k

1 ⊗ k 2 is not ⊗-characteristic, which implies k 1 ⊗ k 2 is not characteristic. 3. Finally, the I-characteristic property of k 1 ⊗ k 2 is established.
The individual steps are as follows:

k is a kernel. Assume w.l.o.g. that x 1 = . . . = x N = 1, x N +1 = . . . = x n = 2. Then it is easy to verify that the Gram matrix G = [k(x i , x j )] n i,j=1 = aa where a := 1 N , -1 n-N and a is the transpose of a. Clearly G is positive semidefinite and so k is a kernel. k is characteristic. We will show that k satisfies (5). On X = {1, 2} a finite signed measure F takes the form F = a 1 δ 1 + a 2 δ 2 for some a 1 , a 2 ∈ R. Thus,

F ∈ M b (X)\{0} ⇔ (a 1 , a 2 ) = 0 and F(X) = 0 ⇔ a 1 + a 2 = 0. (9) Consider X X k(x, x ) dF(x) dF(x ) = a 2 1 k(1, 1) + a 2 2 k(2, 2) + 2a 1 a 2 k(1, 2) = a 2 1 + a 2 2 -2a 1 a 2 = (a 1 -a 2 ) 2 = 4a 2 1 > 0, ( 10
)
where we used ( 9) and the facts that

k(1, 1) = k(2, 2) = 1, k(1, 2) = -1. k 1 ⊗ k 2 is not ⊗-characteristic. We construct a witness F = F 1 ⊗ F 2 ∈ ⊗ 2 m=1 M b (X m )\{0} such that F(X 1 × X 2 ) = F 1 (X 1 )F 2 (X 2 ) = 0, ( 11
) and 0 = X 1 ×X 2 X 1 ×X 2 (k 1 ⊗ k 2 )((i 1 , i 2 ), (i 1 , i 2 )) k 1 (i 1 ,i 1 )k 2 (i 2 ,i 2 ) dF(i 1 , i 2 ) dF(i 1 , i 2 ) = 2 m=1 Xm Xm k m (i m , i m ) dF m (i m ) dF m (i m ). ( 12
)
Finite signed measures on {1, 2} take the form

F 1 = F 1 (a) = a 1 δ 1 + a 2 δ 2 , F 2 = F 2 (b) = b 1 δ 1 + b 2 δ 2 form, where a = (a 1 , a 2 ) ∈ R 2 , b = (b 1 , b 2 ) ∈ R 2 .
With these notations, ( 11) and ( 12) can be rewritten as

0 = (a 1 + a 2 )(b 1 + b 2 ), 0 =   2 i,i =1 k 1 (i, i )a i a i     2 j,j =1 k 2 (j, j )b j b j   = (a 1 -a 2 ) 2 (b 1 -b 2 ) 2 .
Keeping the solutions where neither a nor b is the zero vector, there are 2 (symmetric) possibilities: (i)

a 1 + a 2 = 0, b 1 = b 2 and (ii) a 1 = a 2 , b 1 + b 2 = 0.
⊗ 2 m=1 M b (X m ) 0 -ispd property of k 1 ⊗ k 2 . k 1 ⊗ k 2 is I-characteristic. Our goal is to show that k 1 ⊗ k 2 is I-characteristic, i.e., for any P ∈ M + 1 (X 1 × X 2 ), µ k 1 ⊗k 2 (F) = 0 implies F = 0
, where F = P -P 1 ⊗ P 2 . We divide the proof into two parts:

1. First we derive the equations of F(X 1 × X 2 ) = 0 and

(X 1 ×X 2 ) 2 (k 1 ⊗ k 2 ) ((i, j), (r, s)) dF(i, j) dF(r, s) = 0 (13)
for general finite signed measures F = 2 i,j=1 a ij δ (i,j) on X 1 × X 2 . 2. Then, we apply the F = P-P 1 ⊗P 2 parameterization and solve for P that satisfies (13) to conclude that P = P 1 ⊗ P 2 , i.e., F = 0. Note that in the chosen parametrization for F, F(X 1 × X 2 ) = 0 holds automatically.

The details are as follows.

Step 1. Solving ( 14) and ( 15) yields a 11 + a 22 = 0 and a 12 + a 21 = 0. ( 16)

0 = F(X 1 × X 2 ) ⇔ 0 = a 11 + a 12 + a 21 + a 22 , ( 14
) 0 = X 1 ×X 2 X 1 ×X 2 (k 1 ⊗ k 2 ) ((i, j), (r, s)) k 1 (i,r)k 2 (j,s) dF(i, j) dF (r, s) = 2 i,j=1 2 r,s=1 k 1 (i, r)k 2 (j, s)a ij a rs = 2 i,r=1 k 1 (i, r) 2 j,s=1 k 2 (j, s)a ij a rs = k 1 (1, 1) [k 2 (1,
Step 2. Any P ∈ M + 1 (X 1 × X 2 ) can be parametrized as

P = 2 i,j=1 p ij δ (i,j) , p ij ≥ 0, ∀ (i, j) and 2 i,j=1 p ij = 1. ( 17
)
Let F = P-P 1 ⊗P 2 = 2 i,j=1 a ij δ (i,j) ; for illustration see Table 1. It follows from step 1 that F satisfying ( 16) is equivalent to satisfying (13). Therefore, for the choice of F := P-P 1: Joint (P), joint minus product of the marginals (P -P 1 ⊗ P 2 ).

P: y\x 1 2 P 2 1 p 11 = a[1-(a+b)] a+b p 21 = a q 1 = a a+b 2 p 12 = b[1-(a+b)] a+b p 22 = b q 2 = b a+b P 1 p 1 = 1 -(a + b) p 2 = a + b
p 11 = a[1 -(a + b)] a + b , p 12 = b[1 -(a + b)] a + b , p 21 = a and p 22 = b, with 0 ≤ a, b ≤ 1, a + b ≤ 1 and (a, b) = 0.
The resulting distribution family with its marginals is summarized in Table 2. It can be seen that each member of this family (any a, b in the constraint set) factorizes: P = P 1 ⊗ P 2 . In other words,

F = P -P 1 ⊗ P 2 = 0; hence k 1 ⊗ k 2 is I-characteristic.
Remark. We would like to mention that while k 1 and k 2 are characteristic, they are not universal. Since X is finite, the usual notion of universality (also called c-universality) matches with c 0 -universality. Therefore, from (10), we have X X k(x, x ) dF(x) dF(x) = (a 1 -a 2 ) 2 where F = a 1 δ 1 + a 2 δ 2 for some a 1 , a 2 ∈ R\{0}. Clearly, the choice of a 1 = a 2 establishes that there exists F ∈ M b (X)\{0} such that X X k(x, x ) dF(x) dF(x) = 0. Hence k is not universal. Note that the constraint in (9), which is needed to verify the characteristic property of k is not needed to verify its universality.

Proof of Theorem 3

Define H m := H km .

(i) Suppose k 1 and k 2 are characteristic and that for some

F = P -P 1 ⊗ P 2 ∈ I, H 1 ⊗ H 2 X 1 ×X 2 (k 1 ⊗ k 2 ) (•, x) dF(x) = X 1 ×X 2 k 1 (•, x 1 ) ⊗ k 2 (•, x 2 ) dF(x) = 0, (20)
where x = (x 1 , x 2 ). We want to show that F = 0. By the second-countability of X m -s, the product σ-field, i.e., ⊗ 2 m=1 B(X m ) generated by the cylinder sets B 1 × X 2 and X 1 × B 2 (B m ∈ B(X m ), m = 1, 2), coincides with the Borel σ-field B(X 1 × X 2 ) on the product space (Dudley, 2004, Lemma 4.1.7):

⊗ 2 m=1 B (X m ) = B (X 1 × X 2 ) .
Hence, it is sufficient to prove that

F (B 1 × B 2 ) = 0, ∀ B m ∈ B(X m ), m = 1, 2.
To this end, it follows from ( 20) that for all

h 2 ∈ H 2 , H 1 X 1 ×X 2 k 1 (•, x 1 )h 2 (x 2 ) dF(x) = X 1 k 1 (•, x 1 ) dν(x 1 ) = 0, (21)
where

ν(B 1 ) := ν h 2 (B 1 ) = X 1 ×X 2 χ B 1 (x 1 )h 2 (x 2 ) dF(x), B 1 ∈ B(X 1 ).
Since k 1 is characteristic, ( 21) implies ν = 0, provided that |ν|(X 1 ) < ∞ and ν(X 1 ) = 0. These two requirements hold:

ν(X 1 ) = X 1 ×X 2 h 2 (x 2 ) dF(x) = X 2 h 2 (x 2 ) d[P 2 -P 2 ](x 2 ) = 0, |ν|(X 1 ) ≤ X 1 ×X 2 |h 2 (x 2 )| h 2 ,k 2 (•,x 2 ) H 2 d[P + P 1 ⊗ P 2 ](x 1 , x 2 ) ≤ h 2 H 2 X 1 ×X 2 k 2 (x 2 , x 2 ) d[P + P 1 ⊗ P 2 ](x 1 , x 2 ) ≤ 2 h 2 H 2 X 2 k 2 (x 2 , x 2 ) dP 2 (x 2 ) < ∞,
where the last inequality follows from the boundedness of k 2 . The established ν = 0 implies that for

∀ B 1 ∈ B(X 1 ) and ∀ h 2 ∈ H 2 , 0 = ν(B 1 ) = h 2 , X 1 ×X 2 χ B 1 (x 1 )k 2 (•, x 2 ) dF(x) H 2
, and hence 0 =

X 1 ×X 2 χ B 1 (x 1 )k 2 (•, x 2 ) dF(x) = X 2 k 2 (•, x 2 ) dθ B 1 (x 2 ), (22)
where

θ B 1 (B 2 ) = X 1 ×X 2 χ B 1 (x 1 )χ B 2 (x 2 ) dF(x), B 2 ∈ B(X 2 ).
Using the characteristic property of k 2 , it follows from ( 22) that θ B 1 = 0 for ∀B 1 ∈ B(X 1 ), i.e.,

0 = θ B 1 (B 2 ) = F(B 1 × B 2 ), ∀ B 1 ∈ B(X 1 ), ∀ B 2 ∈ B(X 2 ) provided that θ B 1 (X 2 ) = 0 and |θ B 1 |(X 2 ) < ∞.
Indeed, both these conditions hold:

θ B 1 (X 2 ) = X 1 ×X 2 χ B 1 (x 1 ) dF(x) = X 1 χ B 1 (x 1 ) d[P 1 -P 1 ](x 1 ) = 0, |θ B 1 |(X 2 ) ≤ X 1 ×X 2 d[P + P 1 ⊗ P 2 ](x) = 2.
(ii) Assume w.l.o.g. that k 1 is not characteristic. This means there exists

P 1 = P 1 ∈ M + 1 (X 1 ) such that µ k 1 (P 1 ) = µ k 1 (P 1 ). Our goal is to construct an F ∈ M + 1 × M m=1 X m such that µ ⊗ M m=1 km F -⊗ M m=1 F m = × M m=1 ⊗ M m=1 k m (•, x m ) d F -⊗ M m=1 F m = 0, but F = ⊗ M m=1 F m . Define I := F -⊗ M m=1 F m ∈ I.
In other words we want to get a witness

I ∈ I proving that ⊗ M m=1 k m is not I-characteristic. Let us take z = z ∈ X 2 , which is possible since |X 2 | ≥ 2. Let us define F as 6 F = P 1 ⊗ δ z ⊗ (⊗ M m=3 Q m ) + P 1 ⊗ δ z ⊗ (⊗ M m=3 Q m ) 2 ∈ M + 1 × M m=1 X m .
It is easy to verify that

F 1 = P 1 + P 1 2 , F 2 = δ z + δ z 2 and F m = Q m (m = 3, . . . , M ),
where Q 3 , . . . , Q M are arbitrary probability measures on X 3 , . . . , X M , respectively. First we check that I = 0. Indeed it is the case since

• z = z and X 2 is a Hausdorff space, there exists B 2 ∈ B(X 2 ) such that z ∈ B 2 , z ∈ B 2 .

• P 1 = P 1 , P 1 (B 1 ) = P 1 (B 1 ) for some B 1 ∈ B(X 1 ).

Let S = B 1 × B 2 × × M m=3 X m , and compare its measure under F and ⊗ M m=1 F m :

F(S) = P 1 (B 1 ) =1 (z∈B 2 ) δ z (B 2 ) M m=3 =1 Q m (X m ) +P 1 (B 1 ) =0 (z ∈B 2 ) δ z (B 2 ) M m=3 =1 Q m (X m ) 2 = P 1 (B 1 ) 2 , ⊗ M m=1 F m (S) = M m=1 F m (B m ) = P 1 (B 1 ) + P 1 (B 1 ) 2 =1 δ z (B 2 ) + =0 δ z (B 2 ) 2 M m=3 =1 Q m (X m )
6. The F construction specializes to that of Lyons (2013, Proposition 3.15) in the M = 2 case; Lyons used it for distance covariances, which is known to be equivalent to HSIC (Sejdinovic et al., 2013b).

= P 1 (B 1 ) + P 1 (B 1 ) 4 = P 1 (B 1 ) 2 ,
where the last equality holds since P 1 (B 1 ) = P 1 (B 1 ). This shows that I = F -⊗ M m=1 F m = 0 since I(S) = 0. Next we prove that µ ⊗ M m=1 km F -⊗ M m=1 F m = 0. Indeed,

µ ⊗ M m=1 km (I) = × M m=1 Xm ⊗ M m=1 k m (•, x m ) d F -⊗ M m=1 F m (x 1 , . . . , x M ) = × M m=1 Xm ⊗ M m=1 k m (•, x m ) d P 1 ⊗ δ z + P 1 ⊗ δ z 2 - P 1 + P 1 2 ⊗ δ z + δ z 2 ⊗ ⊗ M m=3 Q m (x 1 , . . . , x M ) = × M m=1 Xm ⊗ M m=1 k m (•, x m ) d P 1 (x 1 ) ⊗ δ z (x 2 ) + P 1 (x 1 ) ⊗ δ z (x 2 ) 2 - P 1 (x 1 ) ⊗ δ z (x 2 ) + P 1 (x 1 ) ⊗ δ z (x 2 ) 4 - P 1 (x 1 ) ⊗ δ z (x 2 ) + P 1 (x 1 ) ⊗ δ z (x 2 ) 4 ⊗ (⊗ M m=3 Q m (x m )) ( * ) = µ k 1 (P 1 ) ⊗ k 2 (•, z) + µ k 1 (P 1 ) ⊗ k 2 (•, z ) 2 - µ k 1 (P 1 ) ⊗ k 2 (•, z) + µ k 1 (P 1 ) ⊗ k 2 (•, z ) 4 - µ k 1 (P 1 ) ⊗ k 2 (•, z) + µ k 1 (P 1 ) ⊗ k 2 (•, z ) 4 ⊗ ⊗ M m=3 µ km (Q m ) = 0 ∈ H k 1 ⊗k 2 ⊗ ⊗ M m=3 µ km (Q m ) = 0,
where we used µ k 1 (P 1 ) = µ k 1 (P 1 ) in ( * ).

Proof of Example 2

Let M = 3, × M m=1 X m = {(i 1 , i 2 , i 3 ) : i m ∈ {1, 2}, m ∈ [3]}, k m (x, x ) = 2δ x,x -1. Our goal is to show that ⊗ 3 m=1 k m is not I-characteristic.
The structure of the proof is as follows: 1. First we describe the equations of the non-characteristic property of ⊗ 3 m=1 k m with a general finite signed measure

F = 2 i 1 ,i 2 ,i 3 =1 a i 1 ,i 2 ,i 3 δ (i 1 ,i 2 ,i 3 ) on × 3 m=1 X m where a i 1 ,i 2 ,i 3 ∈ R (∀ i 1 , i 2 , i 3 ).
2. Next, we apply the F = P -⊗ 3 m=1 P m parameterization and show that there exists P that satisfies the equations of step 1 to conclude that ⊗ 3 m=1 k m is not I-characteristic. The details are as follows.

Step 1. The equations of non-characteristic property in terms of A

= [a i 1 ,i 2 ,i 3 ] (im) 3 m=1 ∈[2] 3 ∈ R 2×2×2 are F ∈ M b × 3 m=1 X m \{0} ⇔ A = 0, 0 = F(× 3 m=1 X m ) ⇔ 0 = 2 i 1 ,i 2 ,i 3 =1 a i 1 ,i 2 ,i 3 , (23) 0
= × 3 m=1 Xm × 3 m=1 Xm (⊗ 3 m=1 k m ) (i 1 , i 2 , i 3 ), (i 1 , i 2 , i 3 ) 3 m=1 km(im,i m ) dF(i 1 , i 2 , i 3 ) dF(i 1 , i 2 , i 3 ) = 2 i 1 ,i 2 ,i 3 =1 2 i 1 ,i 2 ,i 3 =1 3 m=1 k m (i m , i m )a i 1 ,i 2 ,i 3 a i 1 ,i 2 ,i 3 . (24)
Solving ( 23) and ( 24) yields

a 1,1,1 + a 1,2,2 + a 2,1,2 + a 2,2,1 = 0 and a 1,1,2 + a 1,2,1 + a 2,1,1 + a 2,2,2 = 0.
Step 2. The equations of non I-characteristic property can be obtained from step 1 by choosing F = P -⊗ M m=1 P m , where

P = 2 i 1 ,i 2 ,i 3 =1 p i 1 ,i 2 ,i 3 δ (i 1 ,i 2 ,i 3 ) and P = [p i 1 ,i 2 ,i 3 ] (im) 3 m=1 ∈[2] 3 ∈ R 2×2×2 .
In other words, it is sufficient to obtain a P that solves the following system of equations for which A = A(P) = 0:

2 i 1 ,i 2 ,i 3 =1 p i 1 ,i 2 ,i 3 = 1, ( 25
)
p i 1 ,i 2 ,i 3 ≥ 0, ∀ (i 1 , i 2 , i 3 ) ∈ [2] 3 , (26)
a 1,1,1 + a 1,2,2 + a 2,1,2 + a 2,2,1 = 0, ( 27)

a 1,1,2 + a 1,2,1 + a 2,1,1 + a 2,2,2 = 0, (28)
where

a i 1 ,i 2 ,i 3 = p i 1 ,i 2 ,i 3 -p 1,i 1 p 2,i 2 p 3,i 3 , (29)
and

p 1,i 1 = 2 i 2 ,i 3 =1 p i 1 ,i 2 ,i 3 , p 2,i 2 = 2 i 1 ,i 3 =1 p i 1 ,i 2 ,i 3 , p 3,i 3 = 2 i 1 ,i 2 =1 p i 1 ,i 2 ,i 3 . ( 30
)
One can get an analytical description for the solution of ( 25)-( 30), where the solution P(z) is parameterized by z = (z 0 , . . . , z 5 ) ∈ R 6 . For explicit expressions, we refer the reader to Appendix A. In the following, we present two examples of P that satisfy (25)-( 30) such that A = 0, thereby establishing the non I-characteristic property of ⊗ 3 m=1 k m . 1. P: andA:

p 1,1,1 = 1 5 , p 1,1,2 = 1 10 , p 1,2,1 = 1 10 , p 1,2,2 = 1 10 , p 2,1,1 = 1 5 , p 2,1,2 = 1 10 , p 2,2,1 = 1 10 , p 2,2,2 = 1 10 ,
a 1,1,1 = 1 50 , a 1,1,2 = - 1 50 , a 1,2,1 = - 1 50 , a 1,2,2 = 1 50 , (31)
a 2,1,1 = 1 50 , a 2,1,2 = - 1 50 , a 2,2,1 = - 1 50 , a 2,2,2 = 1 50 . (32)
2. P:

p 1,1,1 = 0, p 1,1,2 = 1 10 , p 1,2,1 = 1 10 , p 1,2,2 = 1 10 , p 2,1,1 = 1 10 , p 2,1,2 = 1 10 , p 2,2,1 = 3 10 , p 2,2,2 = 1 5 ,
and A:

a 1,1,1 = - 9 200 , a 1,1,2 = 11 200 , a 1,2,1 = - 1 200 , a 1,2,2 = - 1 200 , a 2,1,1 = - 1 200 , a 2,1,2 = - 1 200 , a 2,2,1 = 11 200 , a 2,2,2 = - 9 200 .
In fact these examples are obtained with the choices z = ( 1 10 , 1 10 , 1 10 , 1 10 , 1 10 , 1 10 ) and z = ( 3 10 , 1 10 , 1 10 , 1 10 , 1 10 , 2 10 ) respectively. See Appendix A for details.

Proof of Theorem 4

It follows from (8) and Remark 2(iii) that (v) ⇒ (iii) ⇒ (ii) ⇔ (i). It also follows from (8) and Theorem 3(ii) that (v) ⇒ (iv) ⇒ (i). We now show that (i) ⇒ (v) which establishes the equivalence of (i)-(v). Suppose (i) holds. Then by Bochner's theorem (Wendland, 2005, Theorem 6.6), we have that for all m ∈ [M ],

k m (x m , y m ) = R dm e - √ -1 ωm,xm-ym dΛ m (ω m ), x m , y m ∈ R dm ,
where (Λ m ) M m=1 are finite non-negative Borel measures on (R dm ) M m=1 respectively. This implies Sriperumbudur et al. (2010, Theorem 9) showed that k m is characteristic iff supp (Λ m ) = R dm , where supp(•) denotes the support of its argument. Since supp

⊗ M m=1 k m (x m , y m ) = ⊗ M m=1 R dm e - √ -1 ωm,xm-ym dΛ m (ω m ) = R d e - √ -1 ω,x-y dΛ(ω), where x = (x 1 , . . . , x M ) ∈ R d , y = (y 1 , . . . , y M ) ∈ R d , ω = (ω 1 , . . . , ω M ) ∈ R d , d = M m=1 d m and Λ := ⊗ M m=1 Λ m .
(Λ) = supp ⊗ M m=1 Λ m = × M m=1 supp (Λ m ) = × M m=1 R dm = R d , it follows that ⊗ M m=1 k m is characteristic.

Proof of Theorem 5

The c 0 -kernel property of k m -s (m = 1, . . . , M ) implies that of ⊗ M m=1 k m . Moreover, X m -s are LCP spaces, hence × M m=1 X m is also LCP.

(⇐) Assume that ⊗ M m=1 k m is c 0 -universal. Since ⊗ M m=1 M b (X m ) ⊆ M b × M m=1 X m , we have that for all F = ⊗ M m=1 F m ∈ ⊗ M m=1 M b (X m )\{0}, 0 < × M m=1 Xm × M m=1 Xm ⊗ M m=1 k m (x, x ) M m=1 km(xm,x m ) dF(x) dF(x ) = M m=1 Xm×Xm k m (x m , x m ) dF m (x m ) dF m x m ,
where x = (x 1 , . . . , x M ) and x = (x 1 , . . . , x M ). The above inequality implies

Xm×Xm k m (x m , x m ) dF m (x m ) dF m x m > 0, ∀ m ∈ [M ]. Since F ∈ ⊗ M m=1 M b (X m ) \{0} iff F m ∈ M b (X m )\{0} for all m ∈ [M ], the result follows. (⇒) Assume that k m -s are c 0 -universal. By the note above ⊗ M m=1 k m is c 0 -kernel; its c 0 - universality is equivalent to the injectivity of µ = µ ⊗ M m=1 km on M b × M m=1 X m .
In other words, we want to prove that µ(F) = 0 implies F = 0, where F ∈ M b × M m=1 X m . We will use the shorthand H m = H km below.

Suppose there exists (Steinwart and Christmann, 2008, page 480). Hence, in order to get F = 0 it is sufficient to prove that

F ∈ M b × M m=1 X m such that µ F = × M m=1 Xm ⊗ M m=1 k m (•, x) ⊗ M m=1 km(•,xm) dF(x) = 0 (∈ ⊗ M m=1 H m ). ( 33
) Since X m -s are LCP, ⊗ M m=1 B (X m ) = B × M m=1 X m
F × M m=1 B m = 0, ∀ B m ∈ B(X m ), m ∈ [M ].
We will prove by induction that for m = 0, . . . , M From the above, it is clear that (34) holds for m = 0. Assuming (34) holds for some m, we now prove that it holds for m + 1. To this end, it follows from (34) that ∀ h m+2 ∈ H m+2 , . By the c 0 -universality of k m+1 ,

⊗ M j=m+1 H j 0 = × M j=1 X j m j=1 χ B j (x j ) ⊗ M j=m+1 k j (•, x j ) dF(x) =: o(B 1 , . . . , B m , k m+1 , . . . , k M ), ∀ B j ∈ B(X j ), j ∈ [m], (34)
ν = 0 for ∀ h m+2 ∈ H m+2 , . . . , ∀ h M ∈ H M (35)
provided that ν ∈ M b (X m+1 ), in other words if |ν|(X m+1 ) < ∞. This condition is met:

|ν|(X m+1 ) ≤ × M j=1 X j M j=m+2 h j , k j (•, x j ) H j ≤ h j H j √ k j (x j ,x j ) d|F|(x) ≤ |F| × M m=1 X m M j=m+2 h j H j sup x∈X j ,x ∈X j k j (x, x ) < ∞,
where we used the boundedness of k m -s in the last inequality. (35) implies that for ∀ B 1 ∈ B(X 1 ), . . . , ∀ B m+1 ∈ B(X m+1 ) and ∀ h m+2 ∈ H m+2 , . . . , for ∀ B 1 ∈ B(X 1 ), . . . , ∀ B m+1 ∈ B(X m+1 ), i.e., (34) holds for m+1. Therefore, by induction, (34) holds for m = M and the result follows from ( †). To justify the convention in ( †), consider the case of m = M -1 in which case (34) can be written as

∀ h M ∈ H M 0 = ν(B m+1 ) = × M j=1 X j   m+1 j=1 χ B j (x j )     M j=m+2 h j (x j )   dF(x) = ⊗ M j=m+2 h j , × M j=1 X j   m+1 j=1 χ B j (x j )   ⊗ M j=m+2 k j (•, x j ) dF(x) ⊗ M j=m+2 H j
X M k M (•, x M ) dν(x M ) = 0,
where

ν(B) = × M j=1 X j   M -1 j=1 χ B j (x j )   χ B (x M ) dF(x), B ∈ B(X M ).
Then by the c 0 -universal property of k M , since

|ν|(X M ) ≤ × M j=1 X j 1 d|F|(x) = |F| × M j=1 X j < ∞ we obtain × M j=1 X j M j=1
χ B j (x j ) dF(x) = F × M j=1 B j = 0, ∀ B 1 ∈ B(X 1 ), . . . , ∀ B M ∈ B(X M ).

Proof of Example 3

The proof follows by a simple modification of that of Example 2 (Section 5.4). The equations of a witness A = [a i 1 ,i 2 ,i 3 ] (im) 3 m=1 ∈[2] 3 (and corresponding P = [p i 1 ,i 2 ,i 3 ] (im) 3 m=1 ∈[2] 3 ) for the non-I-characteristic property of ⊗ 3 m=1 k m take the form:

A = 0, 0 = 2 i 1 ,i 2 ,i 3 =1 a i 1 ,i 2 ,i 3 , (36) 0
= 2 i 1 ,i 2 ,i 3 =1 2 i 1 ,i 2 ,i 3 =1 3 m=1
k m (i m , i m )a i 1 ,i 2 ,i 3 a i 1 ,i 2 ,i 3 = (a 1,1,1 -a 2,1,1 ) 2 + (a 1,1,2 -a 2,1,2 ) 2 + (a 1,2,1 -a 2,2,1 ) 2 + (a 1,2,2 -a 2,2,2 ) 2 , (37)

where ( 36) and ( 37) are equivalent to 0 = 2 i 1 ,i 2 ,i 3 =1 a i 1 ,i 2 ,i 3 , a 1,1,1 = a 2,1,1 , a 1,1,2 = a 2,1,2 , a 1,2,1 = a 2,2,1 , a 1,2,2 = a 2,2,2 . (38)

While ( 38) is more restrictive than ( 27) and (28) (hence its solution set might even be empty), one can immediately see that the example of A = 0 given in ( 31) and ( 32) fulfills (38) proving the non-I-characteristic property of ⊗ 3 m=1 k m . +2z 0 z 2 3 + 2z 2 0 z 3 + 2z 0 z 2 4 + 2z 1 z 2 3 + 2z 2 0 z 4 + 4z 0 z 2 5 + 2z 2 0 z 5 + 2z 1 z 2 5 + 2z 2 z 2 5 +2z 3 z 2 4 + 2z 2 3 z 4 + 4z 3 z 2 5 + 2z 2 3 z 5 + 4z 4 z 2 5 + 2z 2 4 z 5 -z 2 0 -z 2 3 -z 2 4 -3z 2 5 + 2z 3 5 +2z 0 z 1 z 2 + 2z 0 z 1 z 3 + 2z 0 z 1 z 4 + 2z 0 z 2 z 3 + 2z 0 z 1 z 5 + 2z 0 z 2 z 4 + 2z 1 z 2 z 3 +4z 0 z 2 z 5 + 4z 0 z 3 z 4 + 6z 0 z 3 z 5 + 2z 1 z 2 z 5 + 2z 1 z 3 z 4 + 6z 0 z 4 z 5 + 4z 1 z 3 z 5 +2z 2 z 3 z 4 + 2z 1 z 4 z 5 + 2z 2 z 3 z 5 + 2z 2 z 4 z 5 + 6z 3 z 4 z 5 ) ≤ 0, and 0 ≤ z 0 , z 1 , z 2 , z 3 , z 4 , z 5 ≤ 1.

The above analytic solution to ( 25)-( 30) is obtained by symbolic math programming in MATLAB.

N

  := {1, 2, . . .} and R denotes the set of natural numbers and real numbers respectively. For M ∈ N, [M ] := {1, . . . , M }. 1 d := (1, 1, . . . , 1) ∈ R d and 0 denotes the matrix of zeros. For a := (a 1 , . . . , a d ) ∈ R d and b := (b 1 , . . . , b d ) ∈ R d , a, b = d i=1 a i b i is the Euclidean inner product. For sets A and B, A\B = {a ∈ A : a / ∈ B} is their difference, |A| is the cardinality of A and ×

Figure 1 :

 1 Figure 1: Summary of results: "char" denotes characteristic. In addition to the usual characteristic property, three new notions ⊗ 0 -characteristic, ⊗-characteristic and I-characteristic are introduced in Definition 1 which along with c 0 -universal (in the top right corner) correspond to the property of the tensor product kernel ⊗ M m=1 k m , while the bottom part of the picture corresponds to the individual kernels (k m ) M m=1 being characteristic or c 0 -universal. If (k m ) M m=1 -s are continuous, bounded and translation invariant kernels on R dm , m ∈ [M ], all the notions are equivalent (see Theorem 4).

Figure 3 :

 3 Figure 3: Simplification of the F-ispd property of tensor product kernels; see Remark 7.

  In other words, for any a, b = 0, the possibilities are (i) a = (a, -a), b = (b, b) and (ii) a = (a, a), b = (b, -b). This establishes the non-

  reduces to (33) when m = 0 by defining0 j=1 χ B j (x j ) := 1; ( †) for m = M , ⊗ M m=M +1 H m is defined to be equal to R and ⊗ M m=M +1 k m (•, x m ) := 1, in which case o(B 1 , . . . , B M ) = F × M j=1 B j = 0 ⇒ F = 0, the result we want to prove.

k

  . . , ∀ h M ∈ H M , (H m+1 ) 0 = o(B 1 , . . . , B m , k m+1 , . . . , k M ) (h m+2 , . . . , h M ) m+1 (•, x m+1 ) dν(x m+1 ),where ν(B) := ν B 1 ,...,Bm,h m+2 ,...,h M (x), B ∈ B(X m+1 ).

χ

  1 , . . . , B m+1 , k m+2 , . . . , k M ) = B j (x j )   ⊗ M j=m+2 k(•, x j ) dF(x) = 0 ∈ ⊗ M j=m+2 H j

  1 ⊗P 2 , we obtain p 11 -(p 11 + p 12 )(p 11 + p 21 ) + p 22 -(p 21 + p 22 )(p 12 + p 22 ) = 0, 11 -(p 11 + p 12 )(p 11 + p 21 ) a 21 = p 21 -(p 21 + p 22 )(p 11 + p 21 ) 2 a 12 = p 12 -(p 11 + p 12 )(p 12 + p 22 ) a 22 = p 22 -(p 21 + p 22 )(p 12 + p 22 )

	P: y\x	1	2	P 2
	1 2	p 11 p 12	p 21 p 22	q 1 = p 11 + p 21 q 2 = p 12 + p 22	⇒
	P 1	p 1 = p 11 + p 12 p 2 = p 21 + p 22
	F := P -P 1 ⊗ P 2	1			2
	1 a 11 = p Table
					(18)

Table 2 :

 2 Family of probability distributions solving (17)-(19).

p 12 -(p 11 + p 12 )(p 12 + p 22 ) + p 21 -(p 21 + p 22 )(p 11 + p 21 ) = 0, (19) where (p ij ) i,j∈[2] satisfy (17). Solving (17)-(19), we obtain

Second-countability implies separability; in metric spaces the two notions coincide(Dudley, 2004, Proposition 2.1.4). By the Urysohn's theorem, a topological space is separable and metrizable if and only if it is regular, Hausdorff and second-countable. Any uncountable discrete space is not second-countable.

4. LCH spaces include R d , discrete spaces, and topological manifolds. Open or closed subsets, finite products of LCH spaces are LCH. Infinite-dimensional Hilbert spaces are not LCH.

A topological space is called Polish if it is complete, separable and metrizable. For example, R d and countable discrete spaces are Polish. Open and closed subsets, products and disjoint unions of countably many Polish spaces are Polish. Every second-countable LCH space is Polish.

Acknowledgments

The authors profusely thank Ingo Steinwart for fascinating discussions on topics related to the paper and for contributing to Remark 7. The authors also thank the anonymous reviewers for their constructive comments that improved the manuscript. A part of the work was carried out while BKS was visiting ZSz at CMAP, École Polytechnique. BKS is supported by NSF-DMS-1713011 and also thanks CMAP and DSI for their generous support. ZSz is highly grateful for the Greek hospitality around the Aegean Sea; it greatly contributed to the development of the induction arguments.

Appendix A. Analytical Solution to (25)-( 30) in Example 2

The solution of ( 25)-( 30) takes the form p 1,1,1 = -