Bruno Lanson
email: bruno.lanson@obs.ujf-grenoble.fr

Matthew A Marcus

Sirine Fakra

Frédéric Panfili

Nicolas Geoffroy

Alain Manceau

Formation of Zn-Ca Phyllomanganate Nanoparticles in Grass Roots

It is now well established that a number of terrestrial and aquatic microorganisms have the capacity to oxidize and precipitate Mn as phyllomanganate.

However, this biomineralization has never been shown to occur in plant tissues, nor has the structure of a natural Mn(IV) biooxide been characterized in detail. We show that the graminaceous plant Festuca rubra (red fescue) produces a Zn-rich phyllomanganate with constant Zn:Mn and Ca:Mn ratios (0.46 and 0.38, respectively) when grown on a contaminated sediment. This new phase is so far the Zn-richest manganate known to form in nature (chalcophanite has a Zn:Mn ratio of 0.33) and has no synthetic equivalent. Visual examination of root fragments under a microscope shows black precipitates about ten to several tens of microns in size, and their imaging with backscattered and secondary electrons demonstrates that they are located in the root epidermis. In situ measurements by Mn and Zn K-edge extended X-ray absorption fine structure (EXAFS) spectroscopy and X-ray diffraction (XRD) with a micro-focused beam can be quantitatively described by a single-phase model consisting of Mn(IV) octahedral layers with 22% vacant sites capped with tetrahedral and octahedral Zn in proportions of 3:1. The layer charge deficit is also partly balanced by interlayer Mn and Ca. Individual crystallites have a domain radius of 33 Å in the ab plane and contain only 1.2 layers (~8.6 Å) on average. Since this biogenic Mn oxide consists mostly of isolated layers, basal 00l reflections are essentially absent despite its lamellar structure.

Individual Mn layers are probably held together in the Mn-Zn precipitates by stabilizing organic molecules. Zinc biomineralization by plants likely is a defense mechanism against toxicity induced by excess concentrations of this metal in the rhizosphere.

INTRODUCTION

Graminaceous plants, like other so-called metal-tolerant plants, mostly sequester metals in roots to protect reproductive and photosynthetic tissues [START_REF] Li | Response of four turfgrass cultivars to limestone and biosolids-compost amendment of a zinc and cadmium contaminated soil at Palmerton, Pennsylvania[END_REF][START_REF] Simon | Stabilization of metals in acidic mine spoil with amendments and red fescue (Festuca rubra L.) growth[END_REF]. The ability to store metals in underground tissues is used in phytoremediation to reinstall a vegetation cover on heavily contaminated areas and limit the propagation of metals into the food chain [START_REF] Smith | Stabilization of toxic mine wastes by the use of tolerant plant populations[END_REF][START_REF] Cunningham | Phytoremediation of contaminated soils[END_REF][START_REF] Vangronsveld | Reclamation of a bare industrial area contaminated by non-ferrous metals: In situ metal immobilization and revegetation[END_REF][START_REF] Ma | The isolation and characterization of Type 1 metallothionein (MT) cDNA from a heavymetal-tolerant plant, Festuca rubra cv[END_REF][START_REF] Mench | Progress in remediation and revegetation of the barren Jales gold mine spoil after in situ treatments[END_REF][START_REF] Krämer | Phytoremediation: novel approaches to cleaning up polluted soils[END_REF]. [START_REF] Panfili | The effect of phytostabilization on Zn speciation in a dredged contaminated sediment using scanning electron microscopy, X-ray fluorescence, EXAFS spectroscopy, and principal components analysis[END_REF] showed that the grass species Festuca rubra (red fescue) and Agrostis tenuis (colonial bentgrass) accelerate the weathering of zinc sulfide when grown on contaminated dredged sediment, thus increasing Zn bioavailability in the rhizosphere. After two years of plant growth, micrometersized Mn-Zn black precipitates were observed at the surface of Festuca rubra roots, but not characterized [START_REF] Panfili | Etude de l'évolution de la spéciation du zinc dans la phase solide d'un sédiment de curage contaminé induite par phytostabilisation[END_REF]. Zinc precipitation may be a bioactive tolerance mechanism in response to metal toxicity, or a passive mineralization at the soil-root interface [START_REF] Cotter-Howells | Mineralogy of Pb-P grains in the roots of Agrostis capillaris L. by ATEM and EXAFS[END_REF]. Clarifying this issue and determining the mineralogy and structure of this natural precipitate is important to enhance the effectiveness of using graminaceous plants in phytoremediation. These questions are addressed here with electron microscopy and synchrotron-based microanalytical tools, including X-ray fluorescence (μ-XRF), extended Xray absorption fine structure (μ-EXAFS) spectroscopy and X-ray diffraction (μ-XRD) (Manceau et al., 2002b). Micro-XRD was employed to determine the nanocrystalline structure of the Mn-Zn precipitates and the nature of defects (layer stacking faults, cationic vacancies and occupancies, Mn, Zn, and Ca site configuration, stoichiometry) through modeling of their scattering properties [START_REF] Villalobos | Structural model for the biogenic Mn oxide produced by Pseudomonas putida[END_REF][START_REF] Drits | Birnessite polytype systematics and identification by powder X-ray diffraction[END_REF]. We show that the root precipitates are present in the root epidermis (the outermost layer of root cells) and consist of a poorly crystallized phyllomanganate with a constant Zn:Mn ratio higher than reported so far for any natural and synthetic manganate. A structure model is proposed for this new biomineral.

MATERIALS AND METHODS

Materials

The composition in major elements of the dredged sediment was 68.3 % SiO 2 , 6.9 % CaO, 4.8 % Al 2 O 3 , 2.4 % Fe 2 O 3 , 0.7 % P 2 O 5 , and 7.2% organic carbon, and the composition in a trace metals was 4700 mg.kg -1 Zn, 700 mg.kg -1 Pb, and ~270 mg.kg -1 Mn. Seeds of F. rubra were sown in plastic pots filled with 40 kg of either the untreated sediment, the sediment amended with 3 wt. % hydroxylapatite, or the sediment amended with 5 wt. % Thomas basic slag. The pots were placed in a greenhouse without artificial lighting and daily irrigated with tap water in an amount similar to the mean rainfall in northern France. After two years of culture, the pots were dismantled to collect samples. The texture and color of the sediment in areas colonized by the roots (upper 30 cm of the pots) were similar to a brown silty soil, whereas the initial sediment was black and compact. Roots of F. rubra from the treated and untreated pots were washed meticulously with distilled water to remove soil particles from the surface and then freeze-dried.

The speciation of zinc in the initial sediment and in the rhizosphere of F. rubra after the two years of vegetation was described previously [START_REF] Panfili | The effect of phytostabilization on Zn speciation in a dredged contaminated sediment using scanning electron microscopy, X-ray fluorescence, EXAFS spectroscopy, and principal components analysis[END_REF]. Briefly, in the untreated and unvegetated sediment, Zn was distributed as ~50% (mol ratio of total Zn) sphalerite, ~40% Zn-ferrihydrite, and ~10 to 20% (Zn-Al)-hydrotalcite plus Zn-phyllosilicate.

In the presence of plants, ZnS was almost completely dissolved, and the released Zn bound to phosphate (~40-60%) and to Zn phyllosilicate plus (Zn,Al)-hydrotalcite (~20-40%). The coaddition of mineral amendment did not affect the Zn speciation in the vegetated sediment.

Methods

Electron microscopy

Regions of the roots rich in black precipitates were carbon-coated and examined in secondary and backscattered electron modes by high-resolution scanning electron microscopy (JEOL JSM-6320F with a field emission gun) and analyzed with energy dispersive X-ray spectroscopy (EDS, Tracor analyzer).

X-ray fluorescence, diffraction, and absorption spectroscopy

The Zn:Mn and Ca:Mn ratios were measured with an Eagle III µ-XRF spectrometer (Röntgenanalytik Messtechnik GmbH) equipped with a Rh anode and a 40 µm poly-capillary.

The spectrometer was operated under vacuum at 20 kV and 400 µA, and fluorescence was measured for 300 s per point. Micro XRF, XRD and EXAFS data were collected on beamline 10.3.2 at the Advanced Light Source (ALS, Berkeley -Marcus et al., 2004a). Short root fragments were attached to the tips of glass capillaries and cooled down to 110-150 K (Oxford CryoSystems Cryo-Stream) to minimize radiation damage (Manceau et al., 2002b).

X-ray fluorescence maps were taken at 10 keV incident energy, with a beam size ranging from 5×5 µm to 16×7 µm (H×V). Fluorescence counts were collected for K, Ca, Mn, Fe and Zn with a seven-element Ge solid-state detector and a counting time of 100 ms per pixel. For µ-EXAFS measurements, the vertical beam size ranged from 5 to 7 µm. A maximum of two spectra per precipitate were taken at either the Mn or the Zn K-edge to prevent the reduction of tetravalent to divalent Mn and the increase of structural disorder under the beam (Manceau et al., 2002b). Diffraction data were collected with a CCD camera (Bruker SMART6000, SMART software) at 17 keV (λ = 0.729 Å) and exposure times of 120-240 seconds. At this energy, the incident flux and absorption cross-sections are low enough to make radiation damage during an exposure negligible even at room temperature. A background pattern was recorded next to each precipitate to subtract the scattering contribution from the root so as to obtain the precipitate pattern. Diffraction patterns collected on different precipitates were all statistically identical, and thus summed up to optimize data quality. Calibration of the energy and camera distance were obtained using an Al 2 O 3 standard and Fit2D software [START_REF] Hammersley | FIT2D V9.129 -Reference manual V3[END_REF]. This software was also used to calculate the one-dimensional XRD traces from the radial integration of the two-dimensional patterns.

Data processing

The EXAFS data were analyzed according to standard procedure [START_REF] Teo | EXAFS: basic principles and data analysis[END_REF]Marcus et al., 2004b). The µ-XRD patterns were simulated following the trial-and-error approach developed by [START_REF] Drits | X-ray diffraction by disordered lamellar structures: Theory and applications to microdivided silicates and carbons[END_REF], and applied previously to natural and synthetic phyllomanganates [START_REF] Chukhrov | Crystal structure of birnessite from the Pacific ocean[END_REF][START_REF] Manceau | Structural mechanism of Co 2+ oxidation by the phyllomanganate buserite[END_REF][START_REF] Drits | Substructure and superstructure of four-layer Ca-exchanged birnessite[END_REF]Lanson et al., 2000Lanson et al., , 2002a, b;, b;[START_REF] Gaillot | Structure of synthetic K-rich birnessite obtained by high-temperature decomposition of KMnO 4 . I. Two-layer polytype from 800°C experiment[END_REF][START_REF] Gaillot | Structure of birnessite obtained from decomposition of permanganate under soft hydrothermal conditions. 1. Chemical and structural evolution as a function of temperature[END_REF][START_REF] Gaillot | Structure of the synthetic K-rich phyllomanganate birnessite obtained by high-temperature decomposition of KMnO4 -Substructures of K-rich birnessite from 1000°C experiment[END_REF][START_REF] Villalobos | Structural model for the biogenic Mn oxide produced by Pseudomonas putida[END_REF]. Details on the program and fitting procedure can be found in the articles by [START_REF] Drits | Substructure and superstructure of four-layer Ca-exchanged birnessite[END_REF][START_REF] Plançon | CALCIPOW-a program for calculating the diffraction by disordered lamellar structures[END_REF]. The scattering background was considered to be linear in the 0.35-0.80 Å -1 1/d interval (2.86-1.25 Å). The fit quality was evaluated over this interval using the conventional R WP and R Exp values [START_REF] Howard | Profile fitting of powder diffraction patterns[END_REF].

RESULTS AND INTERPRETATION

Optical and electron microscopy

Under the optical microscope, the Mn-Zn precipitates appear as black stains about ten to several tens of micrometers in size on the root surface (Fig. 1a, EA-1). They are also observed in backscattered electron microscopy (Fig. 1b) due to the presence of high-Z elements (Mn, Zn, Ca, and minor Pb), but always are hardly noticeable in secondary electron imaging mode (Fig. 1c). This suggests that the precipitates are engulfed in the root epidermis [START_REF] Cotter-Howells | Mineralogy of Pb-P grains in the roots of Agrostis capillaris L. by ATEM and EXAFS[END_REF] and do not coat the root surface as iron and manganese plaques do [START_REF] Otte | Iron plaque on roots of Aster tripolium L.: Interaction with zinc uptake[END_REF][START_REF] St-Cyr | Metals (Fe, Mn, Zn) in root plaque of submerged aquatic plant collected in situ: relations with metal concentrations in adjacent sediments and in the root tissue[END_REF]. No differences were observed among precipitates from plants grown in the untreated and mineral amended sediments. This result, together with the compartmentation of the precipitates inside the roots, suggests a biological origin. This interpretation is supported also by the absence of Zn-rich phyllomanganates in the surrounding soil matrix [START_REF] Panfili | The effect of phytostabilization on Zn speciation in a dredged contaminated sediment using scanning electron microscopy, X-ray fluorescence, EXAFS spectroscopy, and principal components analysis[END_REF].

Micro-XRF

Elemental mapping of F. rubra roots shows that Zn is associated with Mn in localized spots, and uniformly distributed without manganese in the vascular cylinder as expected for this nutritive element (Fig. 2 - [START_REF] Rout | Effect of metal toxicity on plant growth and metabolism: I[END_REF]. All roots have Zn in their central stele, but not all are speckled with Mn-Zn precipitates. Some root fragments are partly covered by Zn-free Fe-rich plaques (Fig. EA-2). These plaques are made of ferric oxyhydroxides, as indicated by their optical rusty color (Fig. 1a top right). In Zn-Mn-Ca tricolor representation all Mn-Zn precipitates generally have the same color (Fig. 2), even among different roots (Fig. EA-2), meaning that the relative proportions of Zn, Mn and Ca are about the same. The correlation coefficient between Zn and Mn counts for the precipitates is 0.8, with P-value < 0.0001 for the Anova F-test (Fig. 2). The Zn:Mn atomic ratio was calculated from the relative absorption jumps measured at the Mn and Zn K-edges on four particles. For each particle, a pre-absorption edge background was removed first, and then a linear fit to the post-edge region was extrapolated back to the edge to measure edge jumps.

The ratio of the Zn to Mn edge jumps is 0.310(7), which translates into a Zn:Mn ratio of 0.46(1) when taking into account the atomic absorptions of the two elements. A consistent 0.44 value was obtained independently with the Eagle III spectrometer. This analysis also confirmed that root precipitates have a constant Ca:Mn ratio. An atomic ratio of 0.41 was calculated after correction of the Ca-fluorescence from the root.

Micro-EXAFS spectroscopy

The Mn and Zn μ-EXAFS spectra for all precipitates were indistinguishable from each other, and thus averaged to improve the signal-to-noise ratio. If the precipitates consisted of an assemblage of distinct Zn and Mn species, the proportions of these species would most likely vary among the analyzed grains, and this variability would be detected by μ-EXAFS [START_REF] Panfili | The effect of phytostabilization on Zn speciation in a dredged contaminated sediment using scanning electron microscopy, X-ray fluorescence, EXAFS spectroscopy, and principal components analysis[END_REF]. However, this was not the case, suggesting that all root precipitates consist of a single species, hereafter referred to as "Mn-Zn precipitate", in agreement with μ-XRF data.

Mn-EXAFS in Mn-Zn precipitate

Fig. 3 compares the EXAFS spectrum of Mn-Zn precipitate with those of reference compounds. The references used are hollandite (Hol) and todorokite (Todo), two tectomanganates with 2×2 (Hol) and 3×3 (Todo) tunnel structures, TcBi, a triclinic birnessite with 31% Mn 3+ in the layers, [Na + 0.31 [START_REF] Villalobos | Structural model for the biogenic Mn oxide produced by Pseudomonas putida[END_REF]]. TcBi and Lit have a similar content of layer Mn 3+ , but different Mn 4+ -Mn 3+ cation ordering [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF][START_REF] Manceau | Natural speciation of Mn, Ni, and Zn at the micrometer scale in a clayey paddy soil using X-ray fluorescence, absorption, and diffraction[END_REF].

Mn-Zn precipitate is not a tectomanganate because i) the second EXAFS oscillation of tectomanganates is split or has a shoulder at 6.5 Å -1 , depending on the tunnel size [START_REF] Manceau | Structure of Mn and Fe oxides and hydroxides: A topological approach by EXAFS[END_REF]Mckeown and Post, 2001;Manceau et al., 2007b), and ii) their [7.3-9.5 Å -1 ] indicator region (Marcus et al., 2004b) does not match the data (Figs. 3a-b). In the indicator region, Mn-Zn precipitate has a single maximum at 8.0-8.1 Å -1 like dBi, HBi, and Chalco. The shape and position of this maximum is diagnostic of Mn 4+ -rich manganate layers with hexagonal symmetry [START_REF] Gaillot | Structure of the synthetic K-rich phyllomanganate birnessite obtained by high-temperature decomposition of KMnO4 -Substructures of K-rich birnessite from 1000°C experiment[END_REF]. For example, this maximum is shifted to 7.9 Å -1 (i.e., consistent with longer distances) in Lit [START_REF] Manceau | Natural speciation of Zn at the micrometer scale in a clayey soil using X-ray fluorescence, absorption, and diffraction[END_REF] due to the large amount of Mn 3+ in the hexagonal layer. TcBi has a distinctive double peak with a minimum at 7.9-8.0 Å -1 as a result of the split of the Mn-Mn distances induced by the linear ordering of Mn 3+ and Mn 4+ cations in the layer [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF]Lanson et al., 2002a;[START_REF] Gaillot | Structure of synthetic K-rich birnessite obtained by high-temperature decomposition of KMnO 4 . I. Two-layer polytype from 800°C experiment[END_REF][START_REF] Gaillot | Structure of the synthetic K-rich phyllomanganate birnessite obtained by high-temperature decomposition of KMnO4 -Substructures of K-rich birnessite from 1000°C experiment[END_REF][START_REF] Manceau | Natural speciation of Mn, Ni, and Zn at the micrometer scale in a clayey paddy soil using X-ray fluorescence, absorption, and diffraction[END_REF]. This cation ordering lowers the layer symmetry from hexagonal to orthogonal. Thus, the Mn-Zn precipitate is a phyllomanganate having hexagonal layer symmetry and little layer Mn 3+ . The subtle differences between Mn-Zn precipitate and some phyllomanganate references are more obvious when the data are Fourier transformed (FT, Fig. 4). The phase of the Mn1 E peak matches those of dBi, Chalco, and HBi, and is shifted to a shorter distance relative to Lit. In phyllomanganates, the Mn1 E peak represents Mn atoms in the first-neighbor layer octahedra [START_REF] Manceau | Structure of Mn and Fe oxides and hydroxides: A topological approach by EXAFS[END_REF]. Therefore, its phase depends on the nearest Mn-Mn distance (Mn-Mn1 pair), and is sensitive to the amount of Mn 3+ in the layer similarly to the phase of the O peak. However, the sensitivity of the phase of this peak to the presence of Mn 3+ is not high enough to distinguish HBi (11% layer Mn 3+ per octahedral site) from dBi and Chalco (0% -Fig. EA-3). The Mn3 E peak is twice as sensitive as the Mn1 E peak because it arises from the third-neighbor octahedra at twice the Mn-Mn1 distance (see for example Fig. 11 in [START_REF] Manceau | Natural speciation of Mn, Ni, and Zn at the micrometer scale in a clayey paddy soil using X-ray fluorescence, absorption, and diffraction[END_REF]. The sensitivity is now high enough to differentiate HBi from dBi and Chalco, as shown in Fig. 4. Thus, using the phase of the Mn1 E and Mn3 E peaks as chemical probes to the layer composition, we conclude that Mn-Zn precipitate has less Mn 3+ in the octahedral layers than HBi (11%), if any.

Confirming evidence is found in the comparison of the frequency of EXAFS spectra. Fig. 3 shows that the overall EXAFS frequency, and hence interatomic distances, increase from dBi/Chalco/Mn-Zn precipitate, to HBi, to Lit/TcBi, as a result of the increasing amount of layer Mn 3+ [START_REF] Gaillot | Structure of the synthetic K-rich phyllomanganate birnessite obtained by high-temperature decomposition of KMnO4 -Substructures of K-rich birnessite from 1000°C experiment[END_REF].

In contrast to the Mn1 peak, the phase of the O peak (i.e., average Mn-O distance in the layer and interlayer) matches those of the two Mn 3+ -free references (dBi, Chalco) and is slightly shifted to lower distance relative to both HBi and Lit. This shift is barely visible in Fig. 4, but clearly apparent when the [1-3 Å] R+ΔR interval is expanded (Fig. EA-3). Thus, Mn-Zn precipitate has no detectable Mn 3+ in the layer similarly to dBi and Chalco, nor in the interlayer, in contrast to HBi.

The FT of Mn-Zn precipitate also differs from that of HBi by the absence of the Mn1 TC peak from Mn layer -Me pairs at ~3.5 Å (R + ΔR ~ 3.1 Å), where Me is an interlayer metal cation, such as Mn, Zn, or Pb, in triple-corner sharing position above or below vacant layer sites (TC linkage - Manceau et al., 2002a). When there are interlayer Mn atoms, this peak is intense because each interlayer Mn is surrounded by as many as six Mn layer neighbors.

In contrast, Mn layer atoms near an octahedral vacancy have fewer interlayer Mn neighbors, their exact number depending on the density of layer vacancies and composition of the interlayer. Here, the lack of Mn1 Tc peak in Mn-Zn precipitate suggests that it has no detectable Me cations on either side of the layer vacancies. The Mn5 E peak is also absent in the FT of Mn-Zn precipitate, as in dBi, but not HBi nor Chalco. This peak arises from the 5 th Mn shell at about 2.90 × 3 = 8.7 Å, and is enhanced by forward scattering from two intervening Mn layer [START_REF] Manceau | Natural speciation of Mn, Ni, and Zn at the micrometer scale in a clayey paddy soil using X-ray fluorescence, absorption, and diffraction[END_REF]. The absence of this peak in Mn-Zn precipitate indicates that the octahedral layers have a small lateral dimension. This peak is also absent in Lit, but in this case because the Mn-Mn5 shell is split as a result of the Mn 4+ -Mn 3+ ordering in the layer [START_REF] Manceau | Natural speciation of Mn, Ni, and Zn at the micrometer scale in a clayey paddy soil using X-ray fluorescence, absorption, and diffraction[END_REF].

Zn-EXAFS in Mn-Zn precipitate

The best spectral match of Mn-Zn precipitate to our Zn species database was obtained with tetrahedrally coordinated Zn ( IV Zn) sorbed above/below octahedral vacancies of a phyllomanganate ( IV TC site -Fig. 5a). This Zn complex is common in nature and its local structure has been described previously (Manceau et al., 2000b(Manceau et al., , 2002a;;Marcus et al., 2004b;[START_REF] Isaure | Zinc mobility and speciation in soil covered by contaminated dredged sediment using micrometer-scale and bulk-averaging X-ray fluorescence, absorption and diffraction techniques[END_REF][START_REF] Toner | Zinc sorption to biogenic hexagonal-birnessite particles within a hydrated bacterial biofilm[END_REF]. Although the IV Zn-sorbed phyllomanganate reference provides a good approximation of Zn local structure in Mn-Zn precipitate, there are significant differences between the two spectra, particularly in the phase mismatches between 6.5 and 10 Å -1 , and the symmetry of the first and second oscillations. These differences have been documented previously and result from the mixed occupancy of the TC site by tetrahedral and octahedral Zn (Manceau et al., 2002a(Manceau et al., , 2007b)). Adding the chalcophanite or VI/IV Zn-sorbed dBi reference (Fig. 5b - Toner et al., 2005a[START_REF] Toner | Zinc sorption to biogenic hexagonal-birnessite particles within a hydrated bacterial biofilm[END_REF] in a two-component linear fit to the data compensated for the phase shift and asymmetry of the first oscillation, but not entirely for that of the second (Fig. 5c). All attempts to suppress this residual by adding a third component to the linear fit failed. Therefore, although there is some uncertainty on the exact configuration of the minor VI Zn site, the IV TC + VI TC model is the best description of the data we can offer. Consideration of an edge-sharing complex ( VI TE) at layer edges did not reproduce the phase as well as the VI TC surface species. The VI TC species may occur also at the edge of the Mn-Zn precipitate layer, instead of the basal surface, in which case it has a slightly different binding environment from that of dBi. Thus, the dBi and chalcophanite standards may not be the best proxies for the VI TC species. The fraction of each Zn species derived from the two-component least-squares fit is 87 ± 10% IV TC and 27 ± 10% VI TC.

Zn-EXAFS in Zn-only precipitate

A few spots in the µ-XRF maps contained Zn and little else detectable. EXAFS spectroscopy identified zincite (ZnO) and sphalerite (ZnS), two mineral species originally present in the untreated sediment [START_REF] Isaure | Quantitative Zn speciation in a contaminated dredged sediment by µ-PIXE, µ-SXRF, EXAFS spectroscopy and principal component analysis[END_REF]. These grains are likely residual slag material stuck to the root surface that were not removed by washing. None of the Zn species formed in the rhizosphere of F. rubra after the two years of experiment were detected in or at the root surface.

Zn-EXAFS in the root vascular cylinder

Micro-EXAFS spectra were recorded in the vascular cylinder of four distinct roots, at spots containing little Mn. All spectra were indistinguishable, indicating that Zn speciation is uniform, and thus averaged. The resulting Root spectrum has the same frequency as the Mn-Zn precipitate spectrum, which suggests that Zn is also mostly tetrahedral in the roots (Fig. 5d). However, in contrast to Mn-Zn precipitate, the second and third oscillations of the Root spectrum are not split, indicative of "light" backscatters from second-shell contributions.

Consistently, the best spectral match to our organic and inorganic database of the Root spectrum was provided with Zn in a biofilm (Zncell7, Fig. 5e). This reference has 80 ± 10%

Zn complexed to phosphoryl groups and 20 ±10% to carboxyl groups (Toner et al., 2005b).

Consistent with this other study, consideration of carboxyl (citrate, Fig. 5f) and phosphate (phytate, Fig. 5g; Zn-sorbed Penicillium chrysogenum, Fig. 5h - Sarret et al., 1998a) ligands alone, did not yield an optimal fit to the data. Zinc preferential binding to phosphate groups has been reported also in the roots of Arabidopsis halleri and A. lyrata grown hydroponically, on bacterial and fungi cells, and in biofilms (Sarret et al., 1998b[START_REF] Sarret | Forms of zinc accumulated in the hyperaccumulator Arabidopsis halleri[END_REF][START_REF] Fein | Metal adsorption onto bacterial surfaces: Development of a predictive approach[END_REF].

These studies have shown that Zn has a higher affinity for phosphate than for carboxyl groups, which is consistent with the predominance of the phosphate species in F. rubra roots.

Micro-XRD

Qualitative description of the data

According to µ-XRF and µ-EXAFS, Mn-Zn precipitate is a phyllomanganate with hexagonal layer symmetry, little to no layer Mn 3+ , less interlayer Mn 3+ than HBi, and as much as 0.46 interlayer Zn per total Mn, of which 87 ± 10% is tetrahedral and 27 ± 10% octahedral.

Despite its lamellar structure, the XRD pattern of Mn-Zn precipitate shows no patent basal reflections (Fig. 6). The XRD trace is dominated by two reflections at ~2.45 and ~1.42 Å, the first one being asymmetric towards higher 1/d values. This profile is characteristic of lamellar compounds with turbostratic stacking, i.e., lacking well-defined displacement/rotation between successive layers [START_REF] Warren | X-ray diffraction in random layer lattices[END_REF][START_REF] Biscoe | An X-ray study of carbon black[END_REF][START_REF] Brindley | Order-disorder in clay mineral structures[END_REF]. The positions, profiles, and relative intensities of the two peaks match the [20,11] and [02,31] scattering bands of turbostratic birnessite, choosing a C-centered cell for their crystallographic assignment [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF][START_REF] Drits | Birnessite polytype systematics and identification by powder X-ray diffraction[END_REF][START_REF] Villalobos | Structural model for the biogenic Mn oxide produced by Pseudomonas putida[END_REF]. Their d-spacings are in the ratio of 1.73, close to the √3 value for hexagonal symmetry. Their profiles are controlled by the structure factor and can thus be used for structural determinations [START_REF] Villalobos | Structural model for the biogenic Mn oxide produced by Pseudomonas putida[END_REF][START_REF] Drits | Birnessite polytype systematics and identification by powder X-ray diffraction[END_REF], as shown below. ). The atomic coordinates for the IV TC and VI TC sites were considered to be the same as those in Zn-sorbed birnessite (Table 1 - Lanson et al., 2002b). The Ca position was assumed to be close to that in Ca-rich birnessite [START_REF] Drits | Substructure and superstructure of four-layer Ca-exchanged birnessite[END_REF], that is in the mid-plane of the interlayer almost above a tetrahedral surface site (TE position -Fig. EA-5). Thus, parameters optimized in the simulations are the amount of vacant layer sites, the amount, position, and coordination of interlayer Zn 2+ and Mn 2+,3+ , the xy coordinates of Ca 2+ , the position of water molecules, and the size of the coherent scattering domains (CSDs) in the ab plane. The CSDs were considered to have a disk-like shape, whose average radius was constrained by fitting the maximum at ~2.45 Å [START_REF] Villalobos | Structural model for the biogenic Mn oxide produced by Pseudomonas putida[END_REF].

Since the phyllomanganate has 0.46 interlayer Zn per total Mn, at least 0. 2]. Overall, only the Mn TE position is new, all others have been described previously for other birnessite varieties.

Sensitivity of calculated XRD patterns to structural parameters

As the optimal fit to data was obtained using a trial-and-error approach, the sensitivity of the XRD simulations to key structural parameters needs to be assessed. A key parameter for birnessite's ability to sorb trace metals is the origin of the layer charge. In the present model, the layer charge arises from layer vacancies only, not at all from substitution of Mn 3+ for Mn 4+ in the layer, because the hexagonal layer symmetry and small b unit-cell dimension (2.85 Å) are incompatible with appreciable amount of Mn 3+ in the layer [START_REF] Gaillot | Structure of birnessite obtained from decomposition of permanganate under soft hydrothermal conditions. 1. Chemical and structural evolution as a function of temperature[END_REF](Gaillot et al., , 2007;;[START_REF] Manceau | Natural speciation of Mn, Ni, and Zn at the micrometer scale in a clayey paddy soil using X-ray fluorescence, absorption, and diffraction[END_REF]. When birnessite contains ~25% layer Mn 3+ per octahedral site with the long Mn-O bonds oriented at random, b = 2.895 Å [START_REF] Gaillot | Structure of the synthetic K-rich phyllomanganate birnessite obtained by high-temperature decomposition of KMnO4 -Substructures of K-rich birnessite from 1000°C experiment[END_REF] and the [20,11] and [02,31] bands are shifted to lower 1/d values relative to data (R WP = 7.15%; Fig. 7c). Mn-Zn precipitate has the same b value as HBi (2.850 Å) which contains 11% layer Mn 3+ per octahedral site [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]. Based on EXAFS results, this amount is regarded as an upper limit. Also, if the layer contained a high proportion of Mn 3+ cations ordered in rows, as in triclinic birnessite, a similar b value (2.84-2.85 Å) would be measured but the [02,31] band would be a doublet due to the departure from hexagonal symmetry [START_REF] Drits | Birnessite polytype systematics and identification by powder X-ray diffraction[END_REF][START_REF] Gaillot | Structure of the synthetic K-rich phyllomanganate birnessite obtained by high-temperature decomposition of KMnO4 -Substructures of K-rich birnessite from 1000°C experiment[END_REF]. Calculations show that the band splitting decreases with the layer dimension, but that it should still be observed for a CSD dimension of 33 Å, the optimal CSD value (Fig. 7d -R WP = 6.79%).

The sensitivity of XRD profiles to the amount, coordination, and position of high-Z interlayer scatterers, and to the number of vacant layer sites is illustrated next. If interlayer Mn atoms are located only in VI TC sites (0.056 per octahedron) rather than distributed over TE and VI TC sites (0.046 and 0.010, respectively), the dip at ~0.52 Å -1 is deeper and the intensity of the broad hump at ~0.60 Å -1 is increased (Fig. 7e -R WP = 3.60%). When the number of interlayer Ca 2+ is decreased from 0.318 to 0.240, there is a deficit of intensity in the 0.43-0.45 Å -1 region (Fig.

Simulation of the 0.05-0.35 Å -1 1/d interval

Despite the lamellar structure of Mn-Zn precipitate, its XRD pattern lacks welldefined 00l reflections. To assess the possible origin of this oddity, the 001 and 002 reflections were calculated for crystallites with extremely small CSD dimension along the c* axis (Fig. 9). Calculations show that at least two layers are required to modulate the scattering profile at low 1/d values, otherwise the X-ray intensity steadily decreases with increasing 1/d (Fig. 9). The best fit to data was obtained with a mixture of crystallites containing 1, 2, and 3 layers in the ratio of 20:3:1, respectively, leading to an average CSD dimension along the c* axis of 1.2 layers (~8.6 Å).

DISCUSSION

Structure model

Amount of vacant sites and Zn loading

Except for one sample obtained by metal sorption on poorly crystalline Mn oxides [START_REF] Nelson | Production of biogenic Mn oxides by Leprothrix discophora SS-1 in a chemically defined growth medium and evaluation of their Pb adsorption characteristics[END_REF], the new Zn-rich phyllomanganate contains higher amounts of vacant layer sites (0.22 per layer octahedron) and transition elements (0.44 total Mn and Zn per layer octahedron) than any other natural and synthetic variety described so far (0.167 and 0.22, respectively; [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]Lanson et al., , 2002b;;[START_REF] Villalobos | Structural model for the biogenic Mn oxide produced by Pseudomonas putida[END_REF]. The constant Zn:Mn ratio of Mn-Zn precipitate suggests that Zn co-precipitated with Mn by a yet unknown mechanism to form a chemically well-defined phase as natural solids formed by metal sorption on preexisting mineral surfaces are chemically heterogeneous (see for example Manceau et al., 2007b).

Presence of heavy scatterers above and below vacant layers sites

According to XRD and Zn-EXAFS, Mn-Zn precipitate has 0.300 IV Zn and 0.088 VI Zn per octahedron in TC position (Table 1). The Zn-Mn pairs are detected at the Zn K-edge (Fig. EA-7) but not at the Mn K-edge for two possible reasons. In a vacancy-free layer, each

Mn has six nearest-neighbor Mn atoms. In Mn-Zn precipitate, this number is 4.25 if two vacancies cannot be adjacent which is always the case in layered structures (Manceau et al., 2000a). Thus, each Mn octahedron is surrounded statistically by 1.75 vacant sites. With 80% of these vacant sites capped by Me cations on both sides, and 20% on one side only, Mn layer atoms are coordinated on average to 1.75 × (0.2 + (0.8 × 2)) = 3.15 Me in TC sites. This number is about half the value for Zn, as each Zn is linked to six Mn layer . Structural disorder also can be invoked to explain the absence of Mn-Zn pairs in Mn-EXAFS. The TC position is occupied by at least three types of complex: 0.300 IV Zn, 0.088 VI Zn, and 0.010 VI Mn, and the TE position by 0.046 Mn. Each Mn layer - (TC,TE) Me pair has a different distance, and this broad distribution results in an apparent loss of coordination.

Intensity of basal reflections

Birnessite and vernadite minerals were given different names because the basal reflections of birnessite at 7.2-7.0 Å (001) and 3.6-3.5 Å (002) were not observed originally in the diffraction pattern of vernadite (Chukhrov and Gorshkov, 1980;Chukhrov et al., 1980).

However, recent studies have shown that natural vernadite and its biogenic and chemical (δ-MnO 2 ) analogs most often display a 001 reflection when their XRD pattern is recorded on modern diffractometers [START_REF] Mandernack | Manganese mineral formation by bacterial spores of the marine Bacillus, strain SG-1: Evidence for the direct oxidation of Mn(II) to Mn(IV)[END_REF][START_REF] Villalobos | Characterization of the manganese oxide produced by Pseudomonas putida strain MnB1[END_REF][START_REF] Villalobos | Mechanisms of Pb(II) sorption on a biogenic manganese oxide[END_REF][START_REF] Jurgensen | The structure of the manganese oxide on the sheath of the bacterium Leptothrix discophora: An XAFS study[END_REF]Webb et al., 2005b;[START_REF] Bodeï | Formation of todorokite from vernadite in Ni-rich hemipelagic sediments[END_REF]Manceau et al., 2007a, b), thus confirming the view of [START_REF] Arrhenius | Counterions in marine manganates. In La genèse des nodules de manganèse[END_REF] and [START_REF] Giovanoli | Vernadite is random-stacked birnessite[END_REF] that this mineral is a c-disordered variety of birnessite. [START_REF] Villalobos | Structural model for the biogenic Mn oxide produced by Pseudomonas putida[END_REF] showed that basal reflections are present when the diffracting crystallites have only 2-3 layers, on average. Here, this number is as low as 1.2 layers, meaning that Mn-Zn precipitate is essentially an assemblage of isolated layers.

Measurements of the Mn edge jumps on different Mn-Zn precipitates provide an inkling of how the constitutive nanoparticles are joined at particle or so-called grain boundaries. The Mn edge jump was typically between 0.2 and 0.3 for aggregates ~15-25 µm in diameter, which indicates that the phyllomanganate represents only a small fraction (~20 volume %) of the black precipitates, thus revealing a high micro-porosity. This porosity is possibly filled, at least partly, by organics that may help disrupt the parallelism of the layers, but also to tie them together.

Biologically induced oxidation of manganese

Many microorganisms have the capacity to oxidize and precipitate Mn as manganate (Tebo et al., 2004, and references therein). Since biological oxidation of Mn is generally faster than abiotic oxidation, most natural Mn oxides are considered to be biogenic.

Pseudomonas putida [START_REF] Villalobos | Characterization of the manganese oxide produced by Pseudomonas putida strain MnB1[END_REF][START_REF] Villalobos | Structural model for the biogenic Mn oxide produced by Pseudomonas putida[END_REF]Toner et al., 2005a), Leptothrix discophora [START_REF] Nelson | Production of biogenic Mn oxides by Leprothrix discophora SS-1 in a chemically defined growth medium and evaluation of their Pb adsorption characteristics[END_REF][START_REF] Nelson | Effect of oxide formation mechanisms on lead adsorption by biogenic manganese (hydr)oxides, iron (hydr)oxides, and their mixtures[END_REF]Nelson and Lion, 2003;[START_REF] Jurgensen | The structure of the manganese oxide on the sheath of the bacterium Leptothrix discophora: An XAFS study[END_REF][START_REF] Saratovsky | Manganese oxides: Parallels between abiotic and biotic structures[END_REF], and Bacillus sp. strain SG-1 [START_REF] Mandernack | Manganese mineral formation by bacterial spores of the marine Bacillus, strain SG-1: Evidence for the direct oxidation of Mn(II) to Mn(IV)[END_REF][START_REF] Tebo | Manganese oxidation by spores of the marine Bacillus sp. strain SG-1 -Application for the bioremediation of metal pollution[END_REF]Webb et al., 2005a, b) are the three fairly-well characterized bacterial model systems for the oxidation of manganese in the environment. Manganese oxidation and the subsequent precipitation of Mn(III,IV) bioxides by microscopic fungi is also well documented [START_REF] Krumbein | Biogenic rock varnishes of the Negev Desert (Israel) an ecological study of iron and manganese transformation by cyanobacteria and fungi[END_REF][START_REF] Emerson | Formation of Metallogenium-like structures by a manganese-oxiding fungus[END_REF][START_REF] Schulze | Manganese Oxidation-States in Gaeumannomyces-Infested Wheat Rhizospheres Probed by Micro-Xanes Spectroscopy[END_REF]Tani et al., 2003[START_REF] Tani | Interaction of inorganic arsenic with biogenic manganese oxide produced by a Mnoxidizing fungus, strain KR21-2[END_REF]Miyata et al., 2004Miyata et al., , 2006a, b;, b;[START_REF] Thompson | Fungal manganese oxidation in a reduced soil[END_REF]. Here, we showed that Mn can be biomineralized also in higher living organisms, such as plants. Except for its atypical high Zn content and the structural consequences thereof, this new manganese biomineral is no exception to the intrinsic nanocrystalline nature of biogenic phyllomanganates.

Although the mechanism of Mn(II) to Mn(IV) oxidation is presently unknown, the constant Zn:Mn ratio of the new Mn biooxide suggests the existence of a well-defined bioactive process, likely in response to metal toxicity. The occurrence of Zn-Mn precipitate only in the root epidermis and the absence in the roots of any Zn-rich species from the soil matrix [START_REF] Panfili | The effect of phytostabilization on Zn speciation in a dredged contaminated sediment using scanning electron microscopy, X-ray fluorescence, EXAFS spectroscopy, and principal components analysis[END_REF] suggest that Mn oxidation did not occur in the rhizosphere, and thus does not result from bacterial activity or abiotic reaction. Divalent manganese may have been complexed and transported to the roots by phytosiderophores [START_REF] Römheld | The role of phytosiderophores in acquisition of iron and other micronutrients in graminaceaous species: an ecological approach[END_REF], and then oxidized by the plant itself or by endomycorrhizal fungi, as shown for wheat and soybean [START_REF] Schulze | Manganese Oxidation-States in Gaeumannomyces-Infested Wheat Rhizospheres Probed by Micro-Xanes Spectroscopy[END_REF][START_REF] Thompson | Fungal manganese oxidation in a reduced soil[END_REF][START_REF] Thompson | Evidence of a multicopper oxidase in Mn oxidation by Gaeumannomyces graminis var. tritici[END_REF].

Knowing how to stimulate the formation of this new phase in biological systems, or how to synthesize it abiotically, would be a significant progress towards Zn immobilization in contaminated environments and their (phyto-)remediation. Formation of this new phase could in particular facilitate the growth of plants in highly contaminated environments in lowering the concentration of bio-available Zn in the rhizosphere. and8, Table 1). b) The Mn layer -TE Mn inter. distance can be increased by ~0.1 Å by shifting Mn inter in the ab plane from its ideal TE position towards the nearest vacant site (see Fig. EA-5). This shift occurs along the a ±120° directions with equal probabilities resulting in an increased Debye-Waller B factor without significant alteration of the fit quality. (b) O1 coordinated to 3 Mn 4+ in Mn1 (Table 1).

FIGURE CAPTIONS

(c) O1 coordinated to 2 Mn 4+ in Mn1 and 1 Mn 3+ in Mn2 or Mn3.

(d) O1 coordinated to 2 Mn 4+ in Mn1 and 1 Zn 2+ in Zn1.

(e) O1 coordinated to 2 Mn 4+ in Mn1 and 1 Zn 2+ in Zn2.

(f) O5-H-O1 H-bond.

(g) Depending on whether this O1 receives additional valence from H + through H-bond. Intensities (red lines) were calculated for the optimal structure model (Table 1) with different occurrence probabilities of random stacking faults (W r ). Arrows indicate the positions of hkl reflections. (b) Model with IV Zn (Zn2) in (0, 0, 1.97 Å), instead of (0, 0, 1.77 Å) in the optimal fit, so as to increase the Zn-O bond length from 1.82 À (Table 2) to 1.91 Å. Coordinated H 2 O molecules (O4) were moved from (0, 0, 3.70 Å) to (0, 0, 3.90 Å); R wp = 3.92%. (c) Model with VI Zn (Zn1) in (0, 0, 2.30 Å), instead of (0, 0, 2.20 Å) in the optimal fit so as to decrease the sum valence of Zn1 from 2.3 (Table EA-1) to 2.0. Coordinated H 2 O molecules (O2) were moved from (-0.333, 0, 3.45 Å) to (-0.333, 0, 3.65 Å); R wp = 3.55%. (d) Model with Ca in (-0.333, 0, 3.60 Å), instead of (-0.410, 0, 3.60 Å) and equivalent positions in the optimal fit; R wp = 3.51%. Unless specified, all parameters used in calculations are those of the optimal model.

3. 4 . 2

 42 Simulation of the 0.35-0.80 Å -1 1/d interval XRD simulations were performed with the C-centered unit-cell parameters b = 2.850 Å, a = b√3 = 4.936 Å, γ = 90° derived from the position of the [20,11] and [02,31] bands, d (001) = 7.20 Å, and a random layer stacking (Wr = 100%, Fig. EA-4

  186 vacant layer sites per octahedron are needed to accommodate all Zn [(0.186 × 2) / (1.00 -0.186) = 0.46]. In this case, every vacant site is capped by one Zn on either side. No satisfactory agreement between theory and experiment could be obtained with this model, regardless of the amount and position of Ca 2+ . In particular, the high 1/d tail of the [20,11] band at ~0.43-0.45 Å -1 was poorly reproduced (Fig. 7a -R WP = 3.57%; R Exp = 1.57%). To reproduce this feature it was necessary to increase the density of vacant sites to 0.22, and thus to introduce interlayer Mn in TC and TE sites to keep the Zn:Mn ratio constant. Our best theoretical model to the data is shown in Fig. 7b (R WP = 3.49%; R Exp = 1.57%), and structural parameters 300 Ca 0.318 (H 2 O) 0.972 ], and the structure model is schematized in Figs. 8 and EA-5. The refined model contains 0.38 Ca per Mn, in agreement with μ-XRF data (0.41), at a position [(-0.410, 0, ½) and symmetric positions] slightly shifted from the ideal TE position [(-0.333, 0, ½) -Fig. EA-5]. Interlayer H 2 Omolecules that are not bound to Zn and interlayer Mn are located in (0.220, 0, ½) and symmetric positions. This position is similar to ordered(Lanson et al., 2002a) and disordered[START_REF] Villalobos | Structural model for the biogenic Mn oxide produced by Pseudomonas putida[END_REF] Na-exchanged birnessite varieties. It allows for the formation of strong H-bonds with O layer atoms [d(O layer -H 2 O inter. ) = 2.66 Å, Table

  7f -R WP = 3.54%). Similarly, decreasing the number of interlayer Zn from 0.388 to 0.244, and that of vacant layer sites from 0.22 to 0.15 broadens the high 1/d tail of the[20,11] band and shifts the hump at ~0.60 Å -1 to lower 1/d values (Fig.7g-R WP = 4.92%). Increasing the proportion of octahedrally coordinated interlayer Zn from 23% to 50% also broadens the high 1/d tail of the[20,11] band and increases the intensity of the broad hump at ~0.60 Å -1 (Fig.7h-R WP = 3.99%). Finally, the sensitivity of XRD profiles to IV Zn, VI Zn, and Ca coordinates is shown in Figure

Fig. 1 .

 1 Fig. 1. Roots of Festuca rubra grown on a Zn-contaminated sediment. Optical

Fig. 2 .

 2 Fig. 2. Tricolor (RGB) µ-XRF map of a root with Mn-Zn precipitates. Red codes for Ca,

Fig. 3 .

 3 Fig. 3. Manganese K-edge EXAFS spectra of Mn-Zn precipitate (dotted lines) and the

Fig. 4 .Fig. 5 .

 45 Fig. 4. Fourier transforms (FTs) of the k-weighted Mn-EXAFS spectra for the Mn-Zn precipitate (dotted lines) and synthetic VI Zn-sorbed δ-MnO 2 (dBi), chalcophanite

Fig. 6 .

 6 Fig. 6. XRD pattern of Mn-Zn precipitate after subtraction of the scattering from the root.

Fig. 7 .

 7 Fig. 7. Simulations of the [20,11] and [02,31] X-ray scattering bands (C-centered layer cell).

Fig. 9 .

 9 Fig. 9. Calculation of the 001 and 002 reflections for crystallites composed of one (dashed

  : b = 2.850 Å, a = b√3 = 4.936 Å, γ = 90°, and d (001) = 7.20 Å. x and y coordinates are expressed as fractions of the a and b parameters, respectively. Coordinates along the c* axis, ζ, are expressed in Å to point out the thickness of layer and interlayer polyhedra.The average dimension of the coherent scattering domains (CSDs) along the c* axis is 1.2 layers (8.6 Å). The average radius of the disk-like CSDs in the ab plane is 33 Å. This value was calculated by fitting the[20,11] reflection. Un-refined thermal B factors are 0.5 Å 2 for Mn layer , 1.0 Å 2 for O layer , interlayer Mn, and interlayer Zn, and 2.0 Å 2 for Ca 2+ and H 2 O.

  The IV Zn inter. -O layer distance should be increased by ~0.1 Å to provide more realistic Zn-O bond valence (Table EA-1). This cannot be achieved by changing the z coordinate of IV Zn (Fig. EA-6b). Alternatively, moving apart the three oxygen atoms delimiting the underlying vacancy by 0.15 Å in the [110], [100 ], and [0 10 ] directions increases the IV Zn inter. -O layer distance from 1.82 Å to 1.95 Å, as discussed in Manceau et al. (2002a).

(

  Fig. EA-1: Roots of Festuca rubra grown on a Zn-contaminated sediment. (a) Close-up photograph from Fig. 1a. (b) Lower magnification of Fig. 1c.

Fig. EA- 5 :

 5 Fig. EA-5: Idealized structure of Mn-Zn precipitate. (a) Projection on the ab plane. The upper surface of the layer is shown as light shaded triangles, and the atomic notations are the same as in Table 1. (b) Projection along the b axis. Open and solid symbols indicate atoms at y = 0 and y = ±½, respectively. Squares represent vacant layer octahedra.The Mn 3 and Zn 2 atoms can be shifted from their positions as indicated by the dashed arrows to obtain more realistic interatomic distances (seeTable EA-1).

  Fig. EA-7: Fourier transform of the EXAFS spectra for the Mn-Zn precipitate at the Mn and Zn K-edges. The average Mn-O and Zn-O EXAFS distances (R values) are 1.90 Å and 2.00 Å, respectively.

Table 1

 1 Mn = 0.46) in the optimal fit. (h) Model with 50% tetrahedral and 50% octahedral Zn 2+ , instead of 77% and 23% in the optimal fit. Unless specified, all parameters used in calculations are those of the optimal model.

	; Figs. 8 and EA-5). (c) Model with a =

Table 1 .

 1 Structural parameters of Mn-Zn precipitate derived from the simulation of XRD data.

	Atom	x	y	ζ	Occ.	x	y	ζ	Occ.
	Mn layer (Mn1)	0	0	0	0.780	-	-	-	-
	O layer (O1)	0.333	0	1.00	1.00 -0.333	0	-1.00	1.00
	TC Mn inter (Mn2)	0	0	2.20	0.005	0	0	-2.20 0.005
	H 2 O inter. (O2)	-0.333	0	3.45	0.015 0.333	0	-3.45 0.015
	TE Mn inter (Mn3)	-0.333	0	2.20	0.023 0.333	0	-2.20 0.023
	H 2 O inter. (O3)	0	0	3.45	0.069	0	0	-3.45 0.069
	VI Zn inter (Zn1)	0	0	2.20	0.044	0	0	-2.20 0.044
	H 2 O inter. (O2)	-0.333	0	3.45	0.132 0.333	0	-3.45 0.132
	IV Zn inter (Zn2)	0	0	1.77	0.150	0	0	-1.77 0.150
	H 2 O inter. (O4)	0	0	3.70	0.150	0	0	-3.70 0.150
	Ca inter (Ca1)	-0.410	0	3.60	0.053 0.410	0	-3.60 0.053
	Ca inter (Ca1)	-0.295 0.115	3.60	0.053 0.295 0.115 -3.60 0.053
	Ca inter (Ca1)	-0.295 -0.115 3.60	0.053 0.295 -0.115 -3.60 0.053
	H 2 O inter. (O5)	0.220	0	3.60	0.04 -0.220	0	-3.60	0.04
	H 2 O inter. (O5)	-0.110 0.330	3.60	0.04 -0.110 -0.330 -3.60	0.04
	H 2 O inter. (O5)	-0.110 -0.330 3.60	0.04	0.110 -0.330 -3.

Table 2 .

 2 Selected interatomic distances for the optimal structure model.

	Atomic pair	Distance (Å)	Atomic pair	Distance (Å)
	Mn layer -O layer	1.925
	TC,TE Mn inter. -O layer	2.04	TC,TE Mn inter. -H 2 O	2.07
	VI Zn inter. -O layer	2.04	VI Zn inter. -H 2 O	2.07
	IV Zn inter. -O layer	1.82 a	IV Zn inter. -H 2 O	1.93
	Ca inter. -O layer	2.89	H 2 O inter -O layer	2.66
	Mn layer -VI Zn inter	3.60	Mn layer -IV Zn inter	3.35
	Mn layer -TC Mn inter.	3.60	Mn layer -

TE Mn inter.

Table EA -

 EA 1. Empirical bond-valences for Mn-Zn precipitate a .

		O1 b	O1 c	O1 d	O1 e	O2, O3	O3	O4	Σ	Formal
										valence
		0.628 ×6→							3.8	4
	Mn1	×3↓	0.628 ×2↓	0.628 ×2↓	0.628 ×2↓
	Mn2, Mn3		0.469 ×3→			0.433 ×3→			2.7	3
	Zn1			0.403 ×3→			0.372 ×3→		2.3	2
	Zn2				0.731 ×3→			0.543	2.7	2
	H +	0.11 f
	Σ	1.9 -2.0 g	1.7	1.7	2.0

(a)

Bond valences in valence unit (v.u.) were calculated using the Valence for Dos program (v. 2.0http://www.ccp14.ac.uk/solution/bond_valence/index.html) and the parameters from

[START_REF] Brese | Bond-valence parameters for solids[END_REF]

.

  . Simulations of the[20,11] and [02,31] X-ray scattering bands (C-centered layer cell, black crosses). Small but significant misfits between experimental and calculated patterns are pointed out with arrows. Intensities (red lines) were calculated with a turbostratic layer stacking (no interlayer correlation). (a) Optimal model (Table1; Figs. 8 and EA-5); R wp = 3.49%.

	a	O 1	Mn 1				O 3	O 2		O 2 Zn 1	Zn 2 ,O 4
	O 1						O 2
					O 1
		O 2				O 2	O 3	Mn 3	O 3	O 2	a
				Mn 2
					O 2	O 5	TE		TE	Ca Ca Ca
					O 5	O 5					O 2	a
												b
								O 3	a
								vac.			b
							O 3	Mn 3	O 3
								b			c
	b						Mn 3				O 4	d
	Mn 1 0.35	O 2	O 2 0.40	Mn 2 ,Zn 1 0.45	O 3	O 3 0.50		0.55	0.60	0.65	Zn 2 0.70	0.75
	O 1									1/d (Å -1 )
	O 1 Fig. EA-6:
		O 2			O 2							a
			Mn 2 ,Zn 1			Mn 2 ,Zn 1

ACKNOWLEDGMENTS

The following persons are thanked for their contribution on various aspects of this work: Fabienne Marseille and Bertrand Girondelot (CNRSSP, Douai) for the setup of pot experiments, Clément Falk (LGIT, Grenoble) for optical microscopy on Mn-Zn precipitate, Serge Nitsche, Damien Chaudanson, and Alain Baronnet (CRMCN, Marseilles) for SEM analysis, Manuel Muñoz (LGCA, Grenoble) for XRF measurements on the Eagle spectrometer, Didier Arnal, Nicole Paoletti, and Gérard Souche (R&S, Montpellier) for growing Festuca rubra in hydroponic conditions, and Benoît Jaillard and Philippe Hinsinger (R&S, Montpellier) for fruitful discussions about metal uptake by plants. This research was supported by the ACI/FNS Ecco (EcoDyn) program. The ALS is supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the U.S. Department of Energy, under Contract No. DE-AC02-05CH11231.


Antoine De Mathelin
email: antoine.de-mathelin-de-papigny@michelin.com

Guillaume Richard
email: guillaume.richard@ens-paris-saclay.fr

François Deheeger
email: francois.deheeger@michelin.com

Mathilde Mougeot
email: mathilde.mougeot@ens-paris-saclay.fr

Nicolas Vayatis
email: nicolas.vayatis@ens-paris-saclay.fr

Adversarial Weighting for Domain Adaptation in Regression

published or not. The documents may come

I. INTRODUCTION

Many applications require to learn a regression task, for instance, estimation of manufactured products performance, forecasting of supply and demand or prediction of the time spent by a patient in a hospital. In most of these applications, we observe groups of products or patients defining several domains characterized by different distributions. Acquiring a sufficient amount of labeled data to provide a model performing well on all of these domains is known to be difficult and expensive. In practical cases, no or only a few labeled data are available for the target domain of interest whereas a large amount of labeled data are available from another source domain. One then seeks to leverage information from the source domain to learn efficiently the task on the target one. This problem is referred as domain adaptation [START_REF] Ben-David | Analysis of representations for domain adaptation[END_REF].

The domain adaptation framework is characterized by the presence of a source domain (Q, f Q ) and a target domain (P, f P ) where Q, P define two distributions on the input space and f Q , f P are two labeling functions returning the labels for each domain: y = f Q (x) for any x ∼ Q and y = f P (x) for any x ∼ P . The main assumption for domain adaptation is that the source and target distributions differ: P = Q. From there, two common assumptions are considered: the covariate shift which states that the source and target domains share the same labeling functions, i.e. f = f P = f Q [START_REF] Bickel | Discriminative learning under covariate shift[END_REF] and the assumption that the labeling functions match under a specific feature transformation [START_REF] Courty | Optimal transport for domain adaptation[END_REF]. For domain adaptation with a regression task, the covariate shift assumption is often considered [START_REF] Cortes | Domain adaptation and sample bias correction theory and algorithm for regression[END_REF], [START_REF] Huang | Correcting sample selection bias by unlabeled data[END_REF], [START_REF] Sugiyama | Direct importance estimation with model selection and its application to covariate shift adaptation[END_REF]. In this case, instance-based approaches are mainly used to perform a reweighting of source instances which corrects the shift between input distributions. On the other hand, the second assumption is essentially considered for feature-based approaches which perform a transformation of the input features in order to match the source distribution to the target distribution [START_REF] Courty | Optimal transport for domain adaptation[END_REF].

In recent years, research in domain adaptation has mainly focused on feature-based approaches due to the success of adversarial neural networks [START_REF] Ganin | Domain-adversarial training of neural networks[END_REF], [START_REF] Tzeng | Adversarial discriminative domain adaptation[END_REF]. Adversarial domain adaptation methods were originally introduced to learn an encoding space in which the source and target data can not be distinguished by any domain classifier, with the encoder and domain classifier trained with opposing objectives [START_REF] Ganin | Domain-adversarial training of neural networks[END_REF]. Since then, adversarial training has been used as a powerful tool to minimize complex losses defined as maxima on functional spaces [START_REF] Richard | Unsupervised multi-source domain adaptation for regression[END_REF], [START_REF] Saito | Maximum classifier discrepancy for unsupervised domain adaptation[END_REF], [START_REF] Shen | Wasserstein distance guided representation learning for domain adaptation[END_REF], [START_REF] Zhang | Bridging theory and algorithm for domain adaptation[END_REF]. The success of adversarial methods lies in their computational speed and ability to scale to large data sets through the use of stochastic gradient descent-ascent algorithms. Moreover, the efficiency of adversarial methods leads to significant improvements for many tasks as image classification [START_REF] Zhang | Bridging theory and algorithm for domain adaptation[END_REF], image segmentation [START_REF] Saito | Maximum classifier discrepancy for unsupervised domain adaptation[END_REF] or sentiment analysis [START_REF] Richard | Unsupervised multi-source domain adaptation for regression[END_REF].

However, the developed adversarial methods are exclusively designed to learn a feature transformation through an encoder network. Thus, existing methods are generally not applicable to domain adaptation issues where the covariate shift assumption holds. Indeed, when the source and target labeling functions are the same, the projection of the source instances onto the target instances is likely to impair the learning of the task. We, indeed, observe negative transfer in our numerical experiments when applying feature-based methods on covariate shift domain adaptation issues. On the contrary, instance-based methods often succeed in adapting in this context. However, most of the developed instance-based methods are based on positive semi-definite kernels and their weighting strategy often involves solving a quadratic problem [START_REF] Cortes | Domain adaptation and sample bias correction theory and algorithm for regression[END_REF], [START_REF] Cortes | Adaptation based on generalized discrepancy[END_REF], [START_REF] Huang | Correcting sample selection bias by unlabeled data[END_REF] which presents a computational burden when the number of data is large.

Our present work aims to address this issue of regression domain adaptation under covariate shift. In this context, the difficulties of negative transfer with unsupervised featurebased methods and the computational burden of instance-based approaches must be overcome. Based on our experience, the scenario where a few labeled target data are available is the most encountered in practice [START_REF] Cortes | Adaptation based on generalized discrepancy[END_REF]. In this case, supervised methods can greatly improve the performance of domain adaptation [START_REF] Motiian | Few-shot adversarial domain adaptation[END_REF]. These reasons motivate our choice to use a purely supervised metric between source and target distributions: the Y-discrepancy. Furthermore, discrepancy metrics are suited for regression task as they provide theoretical adaptation guarantees on the target risk for common regression losses (as the L p losses) [START_REF] Cortes | Adaptation based on generalized discrepancy[END_REF], [START_REF] Mansour | Domain adaptation: Learning bounds and algorithms[END_REF], [START_REF] Munoz | Learning Theory and Algorithms for Auctioning and Adaptation Problems[END_REF], [START_REF] Richard | Unsupervised multi-source domain adaptation for regression[END_REF]. We propose an original algorithm called Weighting Adversarial Neural Network (WANN) which relies on an adversarial weighting approach to minimize the Y-discrepancy. The WANN algorithm learns the optimal source weights through a weighting network fitted with adversarial training. Thus WANN takes the best of both worlds by combining the computation efficiency of adversarial neural networks with the ability of instance-based approaches to handle regression domain adaptation under covariate shift.

The organization of the paper is as follows. First, the learning scenario with the definition and the main properties of the Ydiscrepancy are introduced (Sections II-A and II-B). Then, the optimization formulation of the WANN algorithm is presented (Section II-C). A presentation of related works and discussions are given in Section III. Finally, numerical experiments are presented on both synthetic and real-world data sets (Section IV).

II. WEIGHTING ADVERSARIAL NEURAL NETWORK A. Learning scenario

Let X and Y ⊂ R denote the respective input and output spaces. A domain is defined as a pair, composed of a distribution over X and a labeling function mapping from X to Y. The source and target domains are respectively written (Q, f Q ) and (P, f P ) with Q, P the source and target distributions over X and f Q , f P : X → Y their respective labeling functions.

We consider the supervised domain adaptation scenario inspired from [START_REF] Cortes | Adaptation based on generalized discrepancy[END_REF], where the learner has access to a labeled source sample of size m, S = {(x 1 , y 1 ), ..., (x m , y m )} ∈ (X × Y) m where S X = {x 1 , ..., x m } is supposed to be drawn iid according to distribution Q and y i = f Q (x i ) for all i ∈ [|1, m|]; and to a labeled target sample of size n, T = {(x 1 , y 1 ), ..., (x n , y n )} ∈ (X × Y) n where T X = {x 1 , ..., x n } is assumed drawn iid according to P and y i = f P (x i ) for all i ∈ [|1, n|]. We denote by Q and P the empirical distributions of the respective samples S X and T X . In the supervised domain adaptation setting, n is typically smaller than m (n << m). We highlight that, similarly to [START_REF] Pardoe | Boosting for regression transfer[END_REF], we do not introduce an unlabeled target sample as the proposed method does not require unlabeled data. We consider in the following that the covariate shift assumption holds, i.e.

f = f Q = f P .
We consider a loss function L : Y × Y → R + and a hypothesis set H of functions mapping from X to Y. We denote by L D (h, h ) = E x∼D [L(h(x), h (x))] the average loss (or risk) for any distribution D over X between two functions h, h . As we consider the instance-based setup where the source losses can be reweighted, we extend the definition of the average loss to functions with a fixed support: L q (h, h ) = x∈X q(x)L(h(x), h (x)) with q ∈ F and F a functional space of functions mapping from X → R + .

The learning problem consists in finding a hypothesis h over H with a small target risk L P (h, f P ).

B. Y-discrepancy

The Y-discrepancy, first introduced in [START_REF] Mohri | New analysis and algorithm for learning with drifting distributions[END_REF], is defined as the maximal difference between source and target risk over a set of hypotheses. We recall below its definition for any weighting function q ∈ F. Definition 1. Let (Q, f Q ) and (P, f P ) be respectively the source and target domains and q ∈ F a weighting function. We define the Y-discrepancy between P and q as:

Y-disc H (P, q) = max h ∈H |L P (h , f P ) -L q (h , f Q )| (1)
The Y-discrepancy characterizes how differently a hypothesis from H can behave between source and target domains. The intuition behind this metric is that the adaptation is successful if the loss obtained on the reweighted sources with any hypothesis h ∈ H is the same as the h loss obtained on the targets. The Y-discrepancy measures then the worst-case loss gap that a hypothesis h ∈ H can achieve. Notice that, with the covariate shift assumption, the Y-discrepancy is null when q = P .

It can be easily shown [START_REF] Munoz | Learning Theory and Algorithms for Auctioning and Adaptation Problems[END_REF] that the Y-discrepancy provides the following generalization bounds of the target risk for any h ∈ H and any q ∈ F:

L P (h, f P ) ≤ L q (h, f Q ) + Y-disc H (P, q) (2)

C. Optimization formulation

From the inequality (2), we infer that finding a reweighting function q ∈ F and a hypothesis h ∈ H minimizing L q (h, f Q ) + Y-disc H (P, q) allows one to learn an efficient hypothesis on the target domain. Since the Y-discrepancy is defined as a maximum on a functional space, we propose to use adversarial training to learn the function q that minimizes it. To this end, we introduce an adversarial hypothesis h ∈ H to approximate the maximal hypothesis returning the Y-discrepancy. We thus derive this original optimization formulation:

min h∈H, q∈F max h ∈H L q (h, f Q ) + L P (h , f P ) -L q (h , f Q ) (3)
The above optimization is a minimization under h and q of the upper bound from equation (2); except that absolute values were removed because they were not needed to obtain the inequality. Furthermore, by removing them, the optimization objective becomes differentiable and the problem can be solved through stochastic gradient descent-ascent algorithm. Notice that the target loss is computed on the empirical distribution P of the target set T X . This can be done as a few labels are available on the target domain. Also, similar to [START_REF] Cortes | Adaptation based on generalized discrepancy[END_REF], the reweighting function q is applied on the training set of size m + n obtained by combining S and T .

Neural networks are used as base hypotheses for the functional spaces H and F because of their ability to learn non-linear functions and their fast optimization with stochastic gradient algorithms. A clipping weight regularization, parameterized by the constants C h , C q , is added to the networks from H and F respectively. By using a regularized neural network as weighting function q ∈ F, instead of optimizing individual weights for each training instances, the smoothness of the weighting map can be controlled (by the clipping constant C q ). With low value of C q , two nearby training points in X will tend to have similar weights. The algorithm then avoids degenerate weighting schemes where all but a few of the training weights become zero. Adequate value of C q can be selected through cross-validation on the few available labeled target data.

The architecture of the neural networks from H as well as the clipping parameter C h are chosen based on the performances of the hypotheses on the source data. As the covariate shift assumption holds, a good architecture for learning the labeling function on the source domain will likely be adequate for learning this same function on the target domain. The architecture of q ∈ F could be optimized by cross-validation on the few target data. In practice, we set F = H for simplicity. However, a ReLU activation is added at end of q ∈ F to ensure positive weights. In addition, to initialize all training weights to 1/(m + n), the network q is first fitted on training data with 1/(m + n) as output labels.

III. RELATED WORK AND DISCUSSION

A. Adversarial domain adaptation

Adversarial training of neural networks have been first used for domain adaptation with DANN [START_REF] Ganin | Domain-adversarial training of neural networks[END_REF]. This method focuses on finding a new representation of the input features where source and target instances cannot be distinguished by any domain classifier. This process aims to minimize the H-divergence [START_REF] Ben-David | Analysis of representations for domain adaptation[END_REF]. Since this seminal work, many domain adaptation methods based on adversarial networks have been proposed with other metrics as the Wasserstein's [START_REF] Shen | Wasserstein distance guided representation learning for domain adaptation[END_REF], the discrepancy [START_REF] Saito | Maximum classifier discrepancy for unsupervised domain adaptation[END_REF], the disparity discrepancy [START_REF] Zhang | Bridging theory and algorithm for domain adaptation[END_REF] and the hdiscrepancy [START_REF] Richard | Unsupervised multi-source domain adaptation for regression[END_REF]. The latter work highlights the advantage of using discrepancy to handle regression tasks over the Hdivergence.

The previously mentioned methods consider the unsupervised domain adaptation setting and use then unsupervised distances between distributions. Even though labeled target data can be added to the source data set as [START_REF] Ganin | Domain-adversarial training of neural networks[END_REF] suggests, some adversarial method focus specifically on the use of labeled samples from the target domain: [START_REF] Motiian | Few-shot adversarial domain adaptation[END_REF], [START_REF] Saito | Semi-supervised domain adaptation via minimax entropy[END_REF]. These methods, however, focus only on classification tasks and can hardly be extended to regression.

B. Adversarial domain adaptation under target shift

Previous works on adversarial domain adaptation mainly focus on feature-based approaches. However, recent works introduce importance weighting in adversarial methods especially for partial domain adaptation (PDA) which considers the target shift issue. Target shift occurs when the output distribution differ between the source and target domain [START_REF] Zhang | Domain adaptation under target and conditional shift[END_REF]. Some PDA methods propose to reweight the source instances during the training process to "discard" the ones unrelated to the target domain. For instance, [START_REF] Chen | Re-weighted adversarial adaptation network for unsupervised domain adaptation[END_REF], [START_REF] Li | On target shift in adversarial domain adaptation[END_REF] propose for classification task, to give an importance weight to each class in the source domain. Other approaches propose to weight specifically each source instance [START_REF] Cao | Learning to transfer examples for partial domain adaptation[END_REF], [START_REF] Zhang | Importance weighted adversarial nets for partial domain adaptation[END_REF], the proposed weighting scheme is obtained by using the output of one domain classifier. Notice that these methods learn at the end a new representation of the input features with adversarial training.

A work closely related to ours [START_REF] Binkowski | Batch weight for domain adaptation with mass shift[END_REF] explicitly introduces a weighting network to reweight batches during a GAN training. However, their framework, in line with cycle-GAN, differs from ours as they focus on the divergence given by a domain discriminator (H-divergence) instead of the Y-discrepancy.

C. Instance-based domain adaptation and curriculum learning

Instance-based domain adaptation methods propose to correct the difference between source and target distributions by reweighting the source instances. Most of them propose to minimize a specific distance between distributions as for instance the KL divergence [START_REF] Sugiyama | Direct importance estimation with model selection and its application to covariate shift adaptation[END_REF], the MMD [START_REF] Huang | Correcting sample selection bias by unlabeled data[END_REF], the discrepancy and other related distances [START_REF] Cortes | Domain adaptation and sample bias correction theory and algorithm for regression[END_REF], [START_REF] Cortes | Adaptation based on generalized discrepancy[END_REF], [START_REF] Mansour | Domain adaptation: Learning bounds and algorithms[END_REF], [START_REF] Mohri | New analysis and algorithm for learning with drifting distributions[END_REF]. Other methods consider boosting iterations as [START_REF] Pardoe | Boosting for regression transfer[END_REF] for example. Most of these methods consider positive semi-definite kernels and propose to solve a quadratic problem [START_REF] Cortes | Domain adaptation and sample bias correction theory and algorithm for regression[END_REF], [START_REF] Cortes | Adaptation based on generalized discrepancy[END_REF], [START_REF] Huang | Correcting sample selection bias by unlabeled data[END_REF] which hardly scales to large samples.

A work on Bayesian instance-based transfer learning [START_REF] Ruder | Learning to select data for transfer learning with bayesian optimization[END_REF] highlights the links between instance-based domain adaptation and curriculum learning [START_REF] Bengio | Curriculum learning[END_REF]. Indeed curriculum learning methods propose to reweight the training instances. Some of these methods have also been used for domain adaptation [START_REF] Zhang | Curriculum domain adaptation for semantic segmentation of urban scenes[END_REF], [START_REF] Zou | Unsupervised domain adaptation for semantic segmentation via classbalanced self-training[END_REF]. However, as mentioned in [START_REF] Ruder | Learning to select data for transfer learning with bayesian optimization[END_REF], curriculum learning aims at "ordering" the training data whereas instance-based domain adaptation aims at "selecting" the best source sample for adaptation.

D. Discussion

Most domain adaptation methods focus on the feature-based framework, where the goal is to learn a transformation of the feature space φ correcting the domain shift between Q and P .

The assumption commonly made in this framework is that the labeling functions of the two domains are matching after the transformation, i.e. f P (x) = f Q (φ(x)) ∀ x ∈ X in the asymmetric setting [START_REF] Courty | Optimal transport for domain adaptation[END_REF], [START_REF] Tzeng | Adversarial discriminative domain adaptation[END_REF], or f P (φ(x)) = f Q (φ(x)) ∀ x ∈ X in the symmetric one [START_REF] Ganin | Domain-adversarial training of neural networks[END_REF], [START_REF] Shen | Wasserstein distance guided representation learning for domain adaptation[END_REF]. This assumption is usually motivated by the existence of domain-invariant features [START_REF] Ganin | Domain-adversarial training of neural networks[END_REF] or by a change in the acquisition conditions between the two domains [START_REF] Courty | Optimal transport for domain adaptation[END_REF] (noise, sensor drift...). The covariate shift assumption states that the labeling functions match for φ = Id. Applying, in this context, another transformation φ as for instance, optimal transport [START_REF] Courty | Optimal transport for domain adaptation[END_REF] or encoding networks trained to minimized the H-divergence [START_REF] Ganin | Domain-adversarial training of neural networks[END_REF], will probably produce negative transfer.

In an industrial design scenario for instance, the learner aims at building an accurate predictive model for new products based on an historical labeled data set from previous products. In this kind of regression problems, the shift between the input distribution P and Q is likely to be informative and looking for a perfect matching of the two distributions has no sense. For a better understanding of this problem, we propose the following illustrative example: considering Q as a uniform distribution between [0, 2], i.e Q = U[0, 2] and P = U[1, 3] and f Q = f P = Id, in this case, matching Q and P with optimal transport [START_REF] Courty | Optimal transport for domain adaptation[END_REF] or with domain adversarial neural networks [START_REF] Tzeng | Adversarial discriminative domain adaptation[END_REF], will lead to negative transfer as it is observed in Figure 1. Notice that, in this illustrative example, an instance-based method will not perform better than a standard approach but will avoid negative transfer.

IV. EXPERIMENTS

In this section, we report the results of WANN algorithm on several experiments. The first set of experiments is based on a synthetic data set proposed by [START_REF] Cortes | Domain adaptation and sample bias correction theory and algorithm for regression[END_REF] on which we compare for several input dimensions the WANN algorithm to the naive approaches: Uniform Weighting and Target Only.

The second set of experiments are conducted on two realworld public data sets: CityCam [START_REF] Zhang | Understanding traffic density from large-scale web camera data[END_REF] and Amazon reviews [START_REF] Blitzer | Biographies, bollywood, boom-boxes and blenders: Domain adaptation for sentiment classification[END_REF] which consist respectively in a counting task on images and a sentiment analysis on text. The WANN algorithm is compared to the most representative domain adaptation methods from both feature-based and instance-based strategies. The selected feature-based methods are the following:

• DANN [START_REF] Ganin | Domain-adversarial training of neural networks[END_REF] is used here for regression tasks by considering the mean squared error as task loss instead of the binary cross-entropy proposed in the original algorithm.

In the following DANN uses all labeled data to learn the task and all available training target data (including unlabeled ones) to train the domain classifier.

• ADDA [START_REF] Tzeng | Adversarial discriminative domain adaptation[END_REF] performs a DANN algorithm in two-stage: it first learns a source encoder and a task hypothesis using labeled data and then learns the target encoder with adversarial training. • Deep-CORAL [START_REF] Sun | Deep coral: Correlation alignment for deep domain adaptation[END_REF] learns a new feature representation by minimizing the difference between the correlation matrix of encoded source and target data.

• MDD [START_REF] Zhang | Bridging theory and algorithm for domain adaptation[END_REF] learns a new feature representation by minimizing the disparity discrepancy between the encoded source and target domains. The selected instance-based competitors are the following:

• Uniform Weighting assigns a uniform weight to every instance. • Target Only uses only labeled target data.

• TrAdaBoostR2 [START_REF] Pardoe | Boosting for regression transfer[END_REF] is based on a reverse-boosting principle where the weight of source instances poorly predicted are decreased at each boosting iteration. 10 boosting iterations are used.

• KLIEP [START_REF] Sugiyama | Direct importance estimation with model selection and its application to covariate shift adaptation[END_REF] is a kernel-based sample bias correction method minimizing the KL-divergence between a reweighted source and target distributions. The default number of maximal centers: 100 is used.

• KMM [START_REF] Huang | Correcting sample selection bias by unlabeled data[END_REF] reweights source instances in order to minimize the MMD between domains. A gaussian kernel and the default parameters B = 1000 and = ( √ m -1)/ √ m are used. The source code of the selected domain adaptation methods is provided in the python library ADAPT1 [START_REF] De Mathelin | Adapt: Awesome domain adaptation python toolbox[END_REF], the source code of WANN and the scripts used to obtain the presented results are available on GitHub2 . The presented results of this section are computed on a (2.7 GHz, 16 G RAM) computer. For all neural network optimizations, the Tensorflow library is used [START_REF] Abadi | TensorFlow: Large-scale machine learning on heterogeneous systems[END_REF] as well as the Adam optimizer [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF]. If nothing else is specified, the optimization parameters used in the presented experiments are lr = 0.001, β 1 = 0.9 and β 2 = 0.999 and the loss function is the mean squared error (MSE).

A. Synthetic data set

The first set of experiments aim to test the efficiency of the WANN algorithm in a controlled environment. We consider the regression domain adaptation issue proposed by [START_REF] Cortes | Domain adaptation and sample bias correction theory and algorithm for regression[END_REF] which satisfies the covariate shift assumption (the source and target domains share the same labeling function). In the proposed setup, the source input distribution Q is a mixture of N gaussians with bandwidth equal to 1 and centered in the hypercube [-1, 1] N . The target distribution is a single gaussian also with bandwidth equal to 1 and centered in [-1, 1] N . The source distribution contains 20% of instances drawn according to the target gaussian distribution, these instances are then considered as target labeled instances in the WANN algorithm. The labeling function f is defined as the mean of absolute values of the input vector's components and is shared by the two domains:

f (x) = 1 N N 1 |x i | with x = (x 1 , ..., x N ) ∈ R N .
The base hypothesis used to learn the task is a neural network with two hidden fully-connected layers of 100 neurons each, ReLU activation functions and weights clipping C h = 1; 300 epochs with a batch size of 128 are performed.

We consider a number of source instance m = 1000 from which around 200 instances are drawn according to the target distribution, we also consider 1000 target instances used as validation data. Following [START_REF] Cortes | Domain adaptation and sample bias correction theory and algorithm for regression[END_REF], we vary the dimension N of the input space N ∈ {32, 64, 128, 246}. We report on Figure 3 the MSE evolution computed on the target validation data through the 300 epochs for the three algorithms: WANN, Uniform Weighting and Target Only. The results are averaged on 10 runs to provide confidence intervals (in light color).

We observe that the difficulty of the task is increasing with the number of dimension N as the target MSE of Uniform Weighting and Target Only increase with the dimension. It appears clearly that WANN outperforms the two other approaches, especially when the task difficulty increases. Indeed, we observe that WANN MSEs on target data decrease faster and lower than the MSEs of the two naive methods. These results highlight the efficiency of WANN to handle regression domain adaptation issue under covariate shift. The distribution of importance weights returned by the weighting network of the WANN algorithm (cf Figure 2) presents two distinct modes corresponding to the 20% of source instances drawn according to the target gaussian distribution (in orange) and the 80% other source instances (in blue). We observe that the weighting network assigns higher weights to the first group of instances (around 3) while keeping a reasonable weighting on other instances (around 0.5).

B. Real-world data sets

For the experiments with real-world data sets, we compare the results obtained with the WANN algorithm to the ones obtained with the previously described competitors. To compare only the adaptation effect of each method, we use, for all of them, the same class of hypotheses H to learn the task which is the class of fully-connected neural networks with ReLU activation functions and a static architecture. All networks from H implement a weight clipping regularization of parameter C h . For WANN algorithm, the two networks h, h are chosen in the specified class H. The weighting network q is taken in the functional space F, if nothing is specified we choose F = H with the same clipping parameter C q = C h . Notice, however, that we add a ReLU activation function at the end of q to ensure positive source weights. For feature-based methods, we split the networks h ∈ H into an encoder part φ and a task part g such that h = g • φ. Thus the learning hypothesis for the task has the same complexity for each method.

The present work focuses on the covariate shift domain adaptation setting in which the source and target labeling functions match. To test whether this assumption holds for the selected real-world data sets we conduct the following experiment: we compute cross-validation scores on the source and target domain for different training data sets: those composed of the respective instances of each domain and those containing the instances of both domains. We assume that if the source and target cross-validation scores obtained with models trained on both domain are as good as those obtained with models trained on single domains, it means that the source and target labeling functions are close. The obtained cross-validation scores for CityCam are reported in Table I. We observe that the scores are close and therefore consider the covariate shift to be a reasonable assumption for these experiments. A similar observation is made for experiments on Amazon reviews. 1) CityCam vehicle counting data set: For the first realworld experiment, we consider the CityCam vehicle counting data set [START_REF] Zhang | Understanding traffic density from large-scale web camera data[END_REF]. This data set is composed of images of city camera videos coming from several cameras distributed in different city locations. The data set is annotated, including for each image the box location of each vehicle and the total number of vehicles. We focus on predicting the number of vehicles. This task is useful, for instance, for understanding the traffic density or for detecting traffic jams. This data set has been previously used for regression domain adaptation in the work of [START_REF] Zhao | Adversarial multiple source domain adaptation[END_REF] considering the images from each camera as belonging to separated domains. To demonstrate the effectiveness of WANN in this setting, we select images coming from 4 cameras: two located on a highway and the two others located at an intersection. The images from one of the intersection camera are selected as target data (∼ 5000 data) and the images from the three other cameras as source data (∼ 10000 data). We use, as input features, the outputs from ResNet50 [START_REF] He | Deep residual learning for image recognition[END_REF] to which we further apply a standard scaling using the source input data. We also rescale the counting number of vehicles using the source output labels. We select the best trade-off parameter λ for DANN, Deep-CORAL and MDD and the best kernel bandwidth for KMM and KLIEP among several tested ones.

We conduct each experiment 10 times and report the mean and standard deviation of mean absolute errors (MAEs) for each method in Table II. We randomly pick n target data as labeled target data set. We vary n ∈ {20, 50, 100, 200}. The other target instances are used as unlabeled target data by feature-based methods and KMM and KLIEP; they are also used to compute the MAEs. We observe on Table II that WANN, KMM and KLIEP provide the best performances, which underlines that instancebased approaches are more suited than feature-based ones for adaptation under the covariate shift assumption. In fact, none of the feature-based approaches provide a better performance than the two naive strategies: Uniform Weighting (Unif.) and Target Only (Tgt O.) and thus perform negative transfer. We also observe that the gap between WANN results and the ones of KMM and KLIEP progressively increases when the number of labeled target data in the training set increases. This can be explained by the fact that WANN, in the contrary of KMM and KLIEP, is based on the Y-discrepancy which is a purely supervised domain adaptation metric, thus the WANN performance improves as the estimation of L P (h , f P ) become more accurate.

We report on Figure 4 the two first components of the PCA in the encoded space for the feature-based methods and in the input space for instance-based methods. We observe that feature-based methods manage to make source and target data look close in the encoded space. However, this is detrimental for their performances on target data. We observe indeed that Deep-CORAL performs the best match between source and target distribution but also presents the higher target MAE. On the contrary, instance-based methods provide reasonable target MAEs. We observe that these methods assign higher weights to the target labeled data and to source instances close from the target domain. We also observe that the weighting scheme of WANN is more conservative than the one of KMM and KLIEP. Figure 5 provides the detail of the WANN importance weights according to each source, we observe that the instances of the source intersection camera (source 1) have in average higher weights than other source instances. It is interesting to notice that source 1 importance weights present two modes because the images from this source come from two different views, the higher mode corresponding to the intersection view. This highlights the ability of WANN algorithm to catch useful source instances for the learning on the target domain.

2) Amazon reviews sentiment analysis: We conduct the second experiment on the cross-domain sentiment analysis data set of Amazon reviews [START_REF] Blitzer | Biographies, bollywood, boom-boxes and blenders: Domain adaptation for sentiment classification[END_REF] where reviews from four domains: dvd, kitchen, electronics and books are rated in [|1, 5|]. The task consists to predict the rating given one review. The data set has been used for regression domain adaptation experiments in [START_REF] Cortes | Adaptation based on generalized discrepancy[END_REF] considering one domain as source and another as target, defining then 12 experiments. We follow the settings from [START_REF] Cortes | Adaptation based on generalized discrepancy[END_REF] using the top 1000 uni-grams and bi-grams as input features, 700 randomly picked labeled source and unlabeled target data and 50 labeled target data. The results are computed on 1000 target data. Each of the 12 experiment is conducted 10 times to get standard deviation indices. The base network h ∈ H used to learn the task is composed of 2 layers of respectively 100 and 10 neurons with parameter C h = 1. As we encounter over-fitting when training the network on source instances, dropouts is added at the end of each layer with rates (0.8, 0.5). These rates are chosen based on validation scores computed on the source domain. We use cross-validation on the 50 labeled A summary of the results is reported on Table III. For these experiments, it appears that WANN provides the more interesting performances although it should be notice that the advantage of using WANN is less significant than for previous experiments. We compute, however, the probabilities that one algorithm is better than the others according to an extension for regression task of the pairwise Poisson binomial test [START_REF] Ganin | Domain-adversarial training of neural networks[END_REF]. We observe that WANN and TrAdaBoost are better than other algorithms with probability 0.67 against Unif, 0.74 against Deep-CORAL and more than 0.9 against the others. WANN and TrAdaBoost provide similar performances, but TrAdaBoost is an ensemble method which takes advantage of averaging the predictions over several models. To make a fair comparison, we conduct the same experiments with a bagging of WANN models and observe that WANN has then a significantly better performance than TrAdaBoost with probability 0.62.

V. CONCLUSION

This work presents a novel instance-based approach for regression tasks in the context of supervised domain adaptation under the covariate shift assumption. We show that the training weights can be optimally adjusted with a neural network in order to efficiently learn the target task. This is achieved by the WANN algorithm which provides, on various experiments, results which outperform the domain adaptation baselines. We show that feature-based methods are not suited to handle regression domain adaptation when the covariate shift assumption holds. Our work also highlights the efficiency of using a supervised metric: the Y-discrepancy when a few labeled target data are available. Future work will focus on the extension of WANN to the unsupervised and semi-supervised domain adaptation framework as well as the ability of the weighting network to provide useful information on the next target labels to query from an active learning perspective.

Fig. 1 .

 1 Fig. 1. Negative transfer caused by feature-based methods. Matching Q and P is detrimental here, as the shift between the two distributions is informative with respect to the task.

Fig. 2 .

 2 Fig. 2. Distribution of training weights returned by the WANN weighting network for N = 256. The weights are normalized so that the mean is equal to one.

Fig. 3 .

 3 Fig. 3. Synthetic experiments results: target MSE evolution across epochs for four values of N .

Fig. 4 .

 4 Fig. 4. Two first PCA components of the encoded space for the four plots at the top and of the input space for the four plots at the bottom. For instance-based methods, the sizes of the points are proportional to the importance weights.

Fig. 5 .

 5 Fig. 5. Distribution of training weights returned by the weighting network of WANN in the CityCam experiment with n = 50. The weights are normalized and log scaled for a better visualization.

TABLE I CITYCAM

 I CROSS-VALIDATION SCORES FOR TESTING COVARIATE SHIFT.

	Train set	Source	Target	Both
	Source	1.58 (0.04)	/	1.59 (0.02)
	Target	/	1.39 (0.08)	1.39 (0.04)

TABLE II MAE

 II BETWEEN THE TRUE VEHICLE COUNT AND THE ESTIMATED COUNT ON THE CITYCAM DATA SET.

	n	20	50	100	200
	Tgt O. 3.05 (0.18)	2.69 (0.10)	2.43 (0.07)	2.16 (0.11)
	Unif.	3.20 (0.23)	2.91 (0.16)	2.63 (0.10)	2.24 (0.06)
	DANN 4.92 (1.25)	4.01 (1.59)	3.26 (0.53)	2.71 (0.21)
	ADDA 3.21 (0.17)	3.78 (0.79)	3.75 (1.24)	3.71 (0.98)
	CORAL 5.37 (0.27)	4.52 (0.50)	3.86 (0.41)	2.87 (0.21)
	MDD	3.35 (0.41)	3.03 (0.28)	2.78 (0.23)	2.42 (0.15)
	TrAdaB. 3.30 (0.18)	2.96 (0.13)	2.52 (0.16)	2.28 (0.11)
	KLIEP 3.05 (0.48)	2.60 (0.15)	2.34 (0.1)	2.07 (0.07)
	KMM 2.79 (0.16) 2.52 (0.10)	2.32 (0.07)	2.06 (0.09)
	WANN 2.79 (0.22) 2.48 (0.08) 2.26 (0.04) 1.98 (0.07)

TABLE III AMAZON

 III REVIEWS SENTIMENT ANALYSIS RESULTS SUMMARY (MSE ×1000). AVERAGES ARE DONE OVER THE 12 EXPERIMENTS.

		Tgt O.	Unif.	WANN	TrAdaB.	KMM	KLIEP	DANN	ADDA	CORAL	MDD
	Avg MSE	1066 (41)	999 (39)	988 (40)	984 (38)	1031 (48)	1059 (64)	1071 (61)	1116 (52)	1008 (44)	1126 (153)
	Median	1068 (38)	1009 (40)	995 (43)	1003 (34)	1039 (47)	1068 (65)	1064 (63)	1118 (47)	1012 (47)	1109 (121)
	Avg rank	6.83	3.33	1.83	2.0	5.0	6.92	7.17	9.0	4.08	8.83
	1 st rank	0	0	7	5	0	0	0	0	0	0
	Top 3	2	7	11	10	2	0	0	0	4	0

https://github.com/adapt-python/adapt

https://github.com/anonymousaccount0/wann