Dr Adriano H Braga

Dr Natália J S Costa

Dr Karine Phillipot

Renato V Gonçalves

János Szanyi

Liane M Rossi
email: lrossi@iq.usp.br

Prof M Liane

Rossi

Prof V Renato

Gonçalves

Structure and activity of supported bimetallic NiPd nanoparticles: influence of preparation method on CO2 reduction

Keywords: bimetallic, CO2, segregation, alloy, activity

Bimetallic Ni-Pd and Ni, Pd reference catalysts were prepared by decomposing organometallic precursors, Ni(cod)2 and Pd2(dba)3, leading to nanoparticles with sizes ranging from 3 to 6 nm. Two different synthesis procedures were followed: i) solution synthesis using capping ligand (hexadecylamine) followed by impregnation of pre-formed nanoparticles on SiO2, called Sol-immobilization (SI); and 2) direct precursor decomposition onto SiO2, without stabilizer, called Direct Decomposition (DD). Samples prepared by SI procedure are alloyed bimetallic nanoparticles, whereas samples obtained by DD one show phase segregation. Interestingly, DD samples show better activity for CO2 hydrogenation into CO (reverse water-gas shift reaction -RWGS), than SI ones. The 1Ni:1Pd DD sample, that has the lowest activation energy for RWGS, shows high concentration of surface undercoordinated sites. These sites chemisorb CO weakly, thus improving activity in RWGS, in opposition to other samples that chemisorb CO strongly, in multibond configuration, having lower activity and selectivity. The results indicate that different decomposition rates drive the formation of smaller and more active Ni clusters, when in presence of Pd. However, the knowledge acquired here on the influence of synthesis conditions on the catalytic properties of Ni-Pd catalysts should help in finding better catalysts for CO2 transformation.

Introduction

Carbon dioxide, generated in several industrial chemical processes is a greenhouse gas which made the scientific community to boost efforts in its recycling using catalytic chemical processes, in order to produce value-added chemicals. The reverse water-gas shift (RWGS) reaction converts CO2 into CO (∆H 0 =41 kJ mol -1 ) and is among the possible ways to reuse CO2 by means of an integrated RWGS/Fischer-Tropsch synthesis. One main drawback of RWGS is the selectivity towards CO at low temperature, since the hydrogenation of CO2 to CH4 is a thermodynamically favorable side-reaction (∆H 0 =-164 kJ mol -1 ). In principle, RWGS reaction can be seen as intermediary step to the methanation process, since the CO formed can be hydrogenated to methane, [1] and also for other hydrocarbon processing like oxidative dehydrogenation of alkanes [2] and reforming reactions. [3,4] RWGS is also an important model reaction to study structure and activity correlations. The H2/CO2 molar ratio and the metal particle size in the catalysts are mostly reported as factors that influence the catalytic selectivity in RWGS. [5,6] A high H2 concentration is known to enhance the hydrogenation to methane. [7] Small nanoclusters generally improve the selectivity to CO, [6] which might be due to the higher density of low-coordinated atoms on top of small nanoclusters, [8] while larger particles size and high metal loadings have been associated to CH4 production, [9,10] although there is still discussion about the particle size effects on pathways of CO2 reduction reactions. [11] Nickel is a classical hydrogenation catalyst that is typically used for the hydrogenation of CO2 to methane. [12,13] This metal is of interest due to its relatively low-price, high abundance and elevated activity. [14,15] It is believed that a support-metal interaction can influence CO2 activation. [16,17] H2 dissociation occurs on Ni surface, which provides H atoms able to reduce CO2 to CO or directly to CH4, or also the hydrogenation of the formed CO to CH4. The reaction pathways depend strongly on the structure of Ni sites. But Ni usually suffers deactivation with time on stream due to sintering, coke formation, and sometimes oxidation. Bimetallic compounds are a frequently used strategy to improve stability and selectivity. [18] Pd, like Ni, is a metal that can easily dissociate H2, and is also known as a catalyst for hydrocarbon transformation and oxidation reactions. [8,19,20] However, the high cost of Pd prohibits its large-scale use as a promoter, in favor to more abundant and low-cost metals. Theoretical calculations on different metal surfaces demonstrated that Ni has a d-band center energy closer to the Fermi level than Pd; [21][22][23] also the O binding energy on Ni is higher than on Pd, and the activation barrier for O bonding to the metal surface is lower on Ni compared to Pd. [8] In terms of CO chemisorption energy, Ni values (BECO= 2.17 eV) are slightly higher than Pd (BECO= 2.36 eV). [8,22] Hence, Ni appears to be more reactive towards *CO, *O and *H, and more oxophilic than Pd. The alloying of Pd with Ni has a lower effect on disturbing Pd electronic properties, but the addition of Pd to Ni surfaces could slightly decrease the reactivity of Ni, thus decreasing the CO bonding energy to the catalyst surface. This, in turn, can help CO desorption to gas phase. Additionally, except for tetrahedron and cubic shaped nanoparticles, NiPd nanoalloys are found to be relatively stable in a broad range of compositions. [24] The organometallic complex Ni(cod)2 has been described as a good precursor to Ni clusters, [25,26] being easily decomposed under H2 atmosphere at low temperature that leads to well-defined particle size distribution and existence of Ni(0) species if caution is taken. [27] The direct preparation of Ni 0 avoids an additional thermal procedure usually necessary to reduce Ni 2+ species to metallic state. [28] We previously reported the direct deposition of Ni 0 nanoparticles (Ni NPs) by the in-situ decomposition of the organometallic precursor Ni(cod)2 over silica-coated magnetite nanoparticles, [25] which resulted in a very active nickel catalyst for hydrogenation of cyclohexene under mild conditions. On the contrary, sol-immobilization of pre-formed Ni NPs prepared by decomposition of Ni(cod)2 in hexadecylamine media produced poorly active Ni catalyst. The organometallic complex Pd2(dba)3 is also known as a good precursor for Pd NPs, being also easily decomposed in mild reaction conditions. [26] Interestingly, we have also reported the controlled synthesis of well-defined Pd and NixPdy NPs using hexadecylamine (HDA) as a stabilizer, which were successfully deposited onto silica-coated Fe3O4 nanoparticles [26] by the sol-immobilization method for application in hydrogenation catalysis. In this previous work, excess HDA was removed during a postimpregnation washing step with pentane, but HDA was still present at the metal surface. To avoid the interference of HDA, in the present work we are investigating the synthesis of NiPd NPs by direct co-decomposition of parent organometallic precursors over the support without adding a stabilizer.

Herein, the advantages of using Ni(cod)2 and Pd2(dba)3 organometallic complexes were explored for the synthesis of well-defined Ni and NiPd NPs. Different Ni:Pd molar ratios have been applied in order to modify the electronic structure of Ni in a controlled way and to investigate the composition/structure effect on RWGS. Colloidal NiPd NPs were synthesized using HDA as a stabilizer and then deposited onto a support by taking advantage of our previous knowledge to achieve well-defined alloyed bimetallic systems, in order to rationalize the effect of Ni/Pd composition on RWGS. Also, in order to obtain more realistic catalysts, a direct co-decomposition of the two organometallic precursors over the support was performed without capping ligand. To accurately study the compositional and size effects, SiO2 nanospheres were used as a support. These catalysts were evaluated in RWGS reaction at low H2 partial pressure in order to limit the CH4 formation, by decreasing the equilibrium constant for the hydrogenation reaction pathway.

Results and Discussion

Metal composition, particle size distribution and NiPd configuration in bimetallic catalysts

The metal content and the bulk Pd concentration, determined by ICP-OES, and the surface Pd concentration, determined by XPS, for the two series of catalysts are summarized in Table 1.

The catalysts prepared by the direct decomposition of organometallic precursors on functionalized silica support (DD) present a concentration of NiPd metal at least 4 times larger compared with preformed colloidal nanoparticles on the same support. The presence of capping ligand (hexadecylamine, HDA) with long carbonic chain may explain this difference, as the result of a steric hindrance around the NPs that limits their incorporation onto the amino-functionalized silica support. [26] The bulk and surface Pd molar fractions calculated from experimental results (ICP-OES and XPS) were found to be close to the nominal values.

These results suggest that the Ni and Pd elements are well-distributed in the catalysts; however, the characterization of bimetallic nanoparticles is a difficult task owing to different possible structures, such as core-shell, ordered or disordered alloys, or even segregated phases. More information was obtained by morphological analysis (vide infra).

In order to compare the two preparation methods, a structural analysis of the Ni, Pd and NiPd catalysts was done. The characterization of the catalysts prepared by solimmobilized method (SI) was reported in a previous work. [26] The TEM images of the catalysts prepared by the direct decomposition method (DD) are shown in Supplementary Information (Figure S1) and the histograms of particle size distributions are given in the (Figure S2). With mean sizes in the range of 3.2 to 4.2 nm, the nanoparticle sizes of DD samples are quite lower than those for SI samples where mean sizes in the range of 4.2 to 6.3 nm were observed. However, size distributions are comparable for both methodologies.

A comparison of elemental content, morphology and structure in supported catalysts prepared by DD and SI methodologies was investigated by high-resolution STEM with EDS elemental mapping for the representative NiPd (1:1 molar ratio) (Figure 1). The results for Ni5Pd5 DD catalyst (Figure 1a-d) show segregation of small Ni clusters and larger Pd NPs.

Metal segregation usually derive from the different decomposition kinetics of Ni(cod)2 and Pd2(dba)3 precursors, which may induce a successive metal deposition on the support instead of allowing the kinetic control necessary for alloy formation. In contrast, for the SI samples, the mapping results (Figure 1e-h) revealed homogeneous metal distribution attesting of the formation of alloyed nanoparticles. This can be explained by the better conditions offered by the solution synthesis method applied for elaborating the SI catalysts.

X-ray diffraction patterns were recorded for NiPd DD samples (Figure S3). Due to their low metal content, no XRD peaks were detected for SI samples. The obtained XRD patterns for NiPd DD catalysts show an amorphous halo from SiO2 and diffraction peaks assigned to the Fe3O4 core, at ca. 30.5, 35.5, 43.4, 54.0, 57.4 and 63.0 o mainly. Additional peaks can be attributed to metallic Ni (ca. 44.5° at Ni DD) and metallic Pd (ca. 40.2° and 47.1° at Pd DD). The Ni5Pd5 DD sample did not show reflections that could be attributed to pure Ni, but this can be expected if very small clusters have been obtained. Also, the Pd XRD peaks did not show any shift to higher 2θ values as can be expected when alloying with Ni. [29] These results are in agreement with the STEM/EDS analysis showing very small Ni clusters segregated from Pd nanoparticles (Figure 2b-d).

CO2 activation and conversion through RWGS

After reduction treatment at 400 °C for 30 min, the catalysts were tested in the hydrogenation of CO2 reaction using 1:1 (CO2:H2) gas composition in a temperature range from 150 to 700 o C. The curves of conversion obtained as function of temperature for all NixPdy catalysts are shown in Figure 2. CO2 conversion starts at about 250 °C for the sample of pure Ni whatever the preparation method. Pure Pd catalysts (Pd DD) sample showed an onset temperature for CO2 conversion at around 450 °C and at temperatures higher than 600 °C for the Pd SI sample.

The addition of Pd to Ni shifts the CO2 conversion to higher temperatures for the SI samples (Figure 3b). Among the DD samples, the Ni5Pd5 DD starts the reaction at the same temperature as pure Ni1 DD while the Ni1Pd9 DD sample becomes active at ca. 350 °C (Figure 3a). Regarding the selectivity, the CO and CH4 formation rates as function of temperature are shown in Figure 3. All the samples displayed low formation of methane (quantities in the range of 2% in molar fraction at the reactor outlet) but that of CO as the main product, thus showing a high selectivity towards RWGS. The overall rate of methane formation is given in Table 2 for all catalysts.

Pure Ni and Ni-rich samples presented higher methane formation rates, with Ni SI sample showing the highest rate of CH4 formation at 400 o C (Figure 3d). Interestingly, for DD samples, the methane formation was almost the same for Ni DD and Ni5Pd5 DD (Figure 3c).

The temperature of maximum methane formation follows well the conversion curves, i.e. the methane starting to be formed when CO2 starts to be converted. Ni1 DD and Ni5Pd5 DD catalysts showed a maximum methane formation temperature of 450°C, whereas for Ni1Pd9 DD and Pd DD it was 550°C. For the SI samples, the rate of methane formation decreased at higher Pd loading, being Pd SI the sample that yielded the lowest amount of CH4 (Figure 3d).

Regarding the temperatures for maximum CH4 formation, it was at 400 °C for Ni SI, and 550 °C for the Ni5Pd5, Ni9Pd1 and Pd1 SI samples.

The apparent activation energy (Ea app ; Table 2) was estimated at conversion rates lower than 25% from Arrhenius plots (Figure S4). For the SI catalysts the Ni1 SI sample provided the lowest value, namely 79 kJ mol -1 ; the addition of Pd led to an almost monotonically increase of the Ea app , achieving 122 kJ mol -1 for Pd1 SI sample. For the DD samples, surprisingly, the lowest value was observed for Ni5Pd5 DD sample, namely 51 kJ mol -1 . Pd1 DD showed the highest value 86 kJ mol -1 and the other catalysts presented activation energies in the range of 50-70 kJ mol -1 . These results show that the apparent activation energy is higher for pure Pd than with pure Ni. This can be explained by the electronic configuration of Pd, which has a lower d-band energy than Ni and, consequently, a lower reactivity. [21] Concerning the bimetallic NiPd DD samples, there is no clear correlation between metallic composition and Ea app . But, increasing the Pd content up to 1Ni:1Pd led to a decrease in the energy barrier values; nevertheless, on Pd rich samples the values start to increase again, agreeing with the expected trend based on reactivity of both metals. Regarding CH4 formation rate (rCH4, Table 2) in general, higher Pd content led to a decrease of the rCH4 for the DD and SI samples, except for pure Pd. A good compromise between activity for CO2 activation and selectivity towards RWGS was achieved with catalyst Ni5Pd5 DD, with Ea app remarkably lower and a rCH4 lower than those of pure Ni catalyst.

Ni-Pd surface interaction and its influence on CO chemisorption

XPS results are summarized in Table 3 and spectra presented on Figure S4. For the Ni1 DD sample, there is a main Ni 2p3/2 peak centered at 853.4 eV, which is characteristic of NiO species. The small XPS peak at 852.5 eV is assigned to metallic Ni. For the Ni5Pd5 DD catalyst, a major peak is observed at ca. 855.0 eV, with a well-defined satellite at ca. 860 eV that can be assigned to Ni(OH)2 species. The very low intense spectra of Ni1Pd9 DD sample show a main peak at 856.3 eV with a satellite at ca. 861.6 eV, again assigned to Ni(OH)2 species, shifted to higher BEs, probably due to interaction with Pd. The Ni1 and Ni5Pd5 SI samples both show a main peak at ca. 853 eV, characteristic of NiO species. Ni1Pd9 SI sample also show a main peak shifted to higher BE values at 854.6 eV that can be assigned to NiO species, interacting with Pd.

At Pd 3d level, the DD samples showed contribution of two well-defined peaks typical of metallic Pd and PdO species: namely at ca. 335 and 336.6 eV for Pd1 DD, 335 and 337 eV for Ni1Pd9 DD, and 334 and 335.8 eV for Ni5Pd5 DD, which might be shifted to lower BE values due to some interaction with Ni. For all SI samples, a higher concentration of metallic Pd was observed indicating that the HDA capping ligand limits surface oxidation of the NPs in some extent. For the Pd1 SI and Ni1Pd9 SI samples, main peaks that are typical of metallic Pd and PdO species were found at ca. 335 and 337 eV, respectively. For the Ni5Pd5 SI two main peaks were observed at 335.6 and 337.8 eV, with a higher intensity for the last contribution, assigned to PdO.

The XPS data collected for the two sets of catalysts showed that in general the nanoparticles are in metallic state, with surface oxidation, in a core-shell like structure. The spectra of the parent metal catalysts (Ni1 DD, Pd1 DD, Ni1 SI and Pd1 SI) resemble to what can be expected for small nanoparticles of the respective elements. Importantly, for SI samples, the total metal content is lower than on DD ones, in good agreement with the metal content determined by ICP OES analysis. The bimetallic samples presented more complex spectra, which can be attributed to interaction between the Ni and Pd. At Ni 2p level, the presence of Ni(OH)2 species detected for Ni5Pd5 and Ni1Pd9 DD samples indicates that segregation of Ni probably occurs, and that this Ni might be in close interaction with SiO2 support that naturally has surface silicon hydroxyl species. For the bimetallic samples with higher Pd thus evidencing a contribution of the less oxophilic Pd metal that withdraws electron density from Ni. Likewise, at Pd 3d level, the spectrum of Ni5Pd5 DD sample is shifted to lower from charge transfer from Ni to Pd. [30] If during the preparation of DD samples, there is formation of segregated Ni particles, however, the bimetallic NiPd structure might still be present. Due to the absence of capping ligands detrimental interdiffusion process can occur and lead to element reconfiguration in the bimetallic nanoparticles [31] . Strong chemical bonds between Ni and hydrogen-containing adsorbates (such as surface hydroxyls) can also drive alloy segregation. [32] On the contrary, for the SI samples, the Pd showed lower PdO contribution, and the BE values agree better with that expected for Pd, except for Ni5Pd5 SI sample, that shows a higher contribution of PdO. The Ni 2p spectra showed mainly NiO species, with some charge transfer occurring on the Ni1Pd9 that can result from interaction with Pd, and lead to shifting of the spectrum to higher BE values.

The DRIFTS spectra recorded after CO chemisorption on the surface of reduced catalysts are shown in Figure 4. The high frequency (HF) bands in the region of 2080-2020 cm -1 are assigned to linear carbonyls bonded to metallic Ni, Pd sites. The low frequency (LF) bands at 2000-1850 cm -1 range are assigned to CO in multibond (bidentate, bridge and 3-fold configuration). The HF bands are associated with CO in weak bonding configuration to the metal surfaces (*COw), while, the LF bands correspond to CO strongly chemisorbed to the metal surface (*COs). [33] On pure Ni samples, bands at ca. 2020 cm -1 can be attributed to linear carbonyl species on planar crystal planes. [34] Linear carbonyls at ca. 2050 cm -1 visible on Ni samples (Figure 4), can be attributed to CO adsorbed on top of low-coordinated Ni, [34,35] a characteristic of highly dispersed Ni. [36] Comparing the two synthesis methodologies, on the Ni1 DD, the 2050 cm -1 band is more prominent than that Ni1 SI sample, which suggesting i) the presence of smaller NPs in the DD system or ii) the partial coverage of these lowcoordinate sites by remaining HDA. The multibond species observed at ca. 1900 cm -1 can also give information about nanoparticle size and dispersion, with higher intensity bands being expected on larger NPs. [37] This can be observed on Ni1 SI sample, where the band at 1910 cm -1 is significantly more intense than that in the linear region. On the Pd samples (Figure 4d), the multibond CO configurations are significantly larger than the linear ones. For DD samples, two bands are observed at ca. 1980 and 1955 cm -1 , as well as low intensity HF (*COw) bands at ca. 2055 and 2075 cm -1 . On the other hand, for the SI, a very intense band at ca. 1945 is visible (*COs); also, the HF bands are very low intense. CO chemisorption probed by FTIR is a powerful tool in order to investigate metal nanoparticles characteristics, such as particle size, electron density on surface, exposed facets, support interactions, nature and population of different active sites. [38] Nevertheless, the independent investigation of these features is difficult. The higher *COs intensities compared to those of *COw for pure Pd catalysts, is reported in the literature to occur at lower CO coverages (symbolized as ). [39] The linear modes appear as the coverage increases (=0.75).

In terms of nanoparticle size, different interpretations are reported in the literature. Szanyi and Goodman [40] describe that on small nanoparticles of ca. 10-50 atoms, linear Pd carbonyls dominate in opposition of bridging CO. However, with Pd nanoparticles such as those presented here, and the DRIFTS configuration, with high CO partial pressures, the very low intensity of *COw bands observed (Figure 5d, right) can be explained by the presence of remaining hexadecylamine capping ligand more strongly bonded to low coordinated Pd sites on the Pd SI sample. This is supported by the fact that the presence of HDA was observed by N 1s XPS spectra in significant surface concentration, even higher than Ni+Pd ones (Table 3).

In the Ni case, the higher predominance of linear CO modes (Figure 4a), is indicative of a more heterogeneous surface, with terrace planar sites and edge sites in similar proportion; it also indicates a higher reactivity of the Ni sites, due to higher density of Ni in lowcoordination configuration.

The higher intensity of linear carbonyls observed on Ni5Pd5 DD compared to other samples can be explained by 1) changing the NiPd alloy metal bond length in terms that it can hinder the CO in multibond configuration and 2) increase of low coordinated Ni,Pd sites, that are more prone to adsorb linear carbonyls, in the form of Ni,Pd(CO)4; as these sites are more reactive, the CO bonding strength is lower and can be desorbed easily. For more clarity, the Ea app and the ratio of CO weakly to CO strongly (*COw/*COs) bonded to surface are plotted as function of Pd molar fraction (Figure 5). The higher presence of low-coordinated sites is the consequence of smaller nanoparticle sizes, like that encountered in Ni5Pd5 DD sample (Table 1, Figure 5). Nevertheless, a compensation of the amount of such low coordinated sites (particle size) and the nature of the metal dictates the reactivity. As showed by Dietz et al., the binding energy of CO to Pd surfaces is slightly lower than to Ni surfaces, [22] whereas the O binding energy to Ni (-5.69 eV) is significantly lower than on Pd (-4.75 eV). It is worth noting that binding energy is defined as the total energy of slab with chemisorbed molecule minus the sum of the total energies of clean slab and free molecule, thus a more negative value indicates a more exothermic chemisorption. [22,23] Therefore, Ni sites are more reactive towards O and less stable CO are formed on Ni compared to Pd; chemisorbed CO can be more stable on less reactive Pd surface. These literature data are in good accordance with our experimental observations, where the Pd-rich samples are in general less active than Ni-rich ones, with conversion degrees significantly lower at a given temperature. On the SI samples, the weak HDA ligand can bond preferentially to the most reactive low coordinated metallic sites, which are the ones expected to adsorb CO linearly, and deliver gas phase CO. This is in good agreement with both the Ea app values and the low *COw/*COs ratio values encountered for this set of samples. In terms of selectivity, in general the Pd-rich samples are less prone to CO2, CO hydrogenation to CH4. The CO-DRIFTS and reactivity results show that if the Pd NPs were active in the same temperature range than Ni, CH4 formation was supposed to be significantly higher than on Ni NPs, taking into account the lower binding energy of CO on Pd surfaces compared to Ni. [22,23] However, at the temperature ranges where Pd is active, in this case, CH4 formation from Sabatier (Equation 1) reaction is not favored.

(1)

These findings are pointing out to the different nanoparticle self-assembling process. Without SiO2 in the medium but with capping ligands, the decomposition of organometallic precursors follows the adequate kinetic control to produce Ni-Pd nuclei. Without the capping ligands, the assembling of the Ni and Pd nuclei was more difficult, and even more due to the presence of SiO2, which address the fast metal deposition on its surface, two parameters that do not allow the formation of bimetallic alloy. The higher Ni(COD)2 reactivity drives the formation of small nanoclusters, interacting strongly with the surface, helped by the -OH terminations on the surface. Pd2(dba)3 precursor, on the contrary, can produce nanoparticles easily in such conditions. These different arrangements are directly related to the catalytic activity in a model reaction such as RWGS.

Conclusion

Two synthesis methodologies were explored in order to produce well-defined bimetallic NiPd nanoparticles starting from organometallic complexes at the metal source, focusing on the catalytic activity for CO2 conversion. Nanoparticles with average nanoparticle sizes in the range of 3 to 6 nm were pre-stabilized with HDA and then deposited onto SiO2 by simple impregnation method which led to metal loadings ranging from 0.6 to 1.0 wt% (SI samples).

XPS and TEM analyses showed the formation of nanoalloys whatever the Ni/Pd ratio. When the decomposition of the organometallic precursors was performed directly onto SiO2 and in the absence of capping ligands, the decomposition and nucleation process were boosted, which did not allow the formation of nanoalloys. These DD samples showed a total metal content in the range 2 to 3 wt% that contained nanoparticles of mean sizes ranging from 3 to 4 nm. The XPS and TEM investigation showed that segregated Pd nanoparticles and very small Ni nanoclusters were produced in that case.

The obtained two sets of catalysts were then applied in the CO2 transformation into CO focusing on RWGS model reaction. SI samples showed a typical behavior of the conversion rate shifting to higher temperatures with increasing of Pd content. Selectivity to CO improved following the same trend, which can be due to the CO2 conversion in temperature regimes were CH4 is unfavorable. On the contrary, the DD samples appeared active for CO2 activation at lower temperatures compared to their SI counterparts, which can be explained by the absence of capping ligands and the general lower nanoparticle sizes, especially the Nicontaining catalysts, for which the results obtained led us to conclude that more active isolated Ni clusters are present. With Ni:Pd ratio of 1:1, surprisingly, the conversion degree and selectivity to CH4 and CO is similar to pure Ni, but with a lower energy barrier for CO2 conversion; the bimetallic Pd-rich sample presented a behavior more close to pure Pd. This finding was consistent with characterization results that showed the very small Ni nanoclusters segregated from Pd nanoparticles, which in turn are more active than the pure Ni samples themselves.

The FTIR analysis of chemisorbed CO showed that CO was preferentially chemisorbed strongly in bidentate and 3-fold configurations on Pd-rich samples while on Ni-rich ones the linear CO configuration arose. This helped to understand why the Ni-rich samples were more active: the CO weakly bonded to Ni surface is more available to desorb as gas phase CO and to be hydrogenated to CH4 under reaction conditions. These findings helped to understand how the synthesis protocol changed the structure of bimetallic nanoparticles and the consequent behavior under reaction conditions.

Experimental Section

Catalyst Preparation

Magnetic core-shell Fe3O4@SiO2 support material was used for the preparation of the catalysts in order to facilitate their separation being air sensitive synthesis systems. The magnetic support was prepared via a reverse microemulsion route, and further functionalized by a surface reaction with 3-aminopropiltrietoxysilane (APTES), in order to facilitate the impregnation of the nanoparticles, as described in a previous work. [41] No magnetite particles were found outside the SiO2 shell.

The preparation of the catalysts was performed following two different methods: 1) immobilization of preformed colloidal nanoparticles on the functionalized silica support by simple impregnation, samples identified as SI (sol-immobilization method) and 2) direct codecomposition of the organometallic precursors over the support without capping ligand, samples identified as DD (direct decomposition). The metal precursors were used as received and handled in a glove box under Ar atmosphere; the solvents were dried and degassed using standard air sensitive procedures.

The set of SI samples were prepared by a procedure previously described: [26] briefly, Ni and Pd precursors, Ni(cod)2 and Pd2(dba)3, respectively, in desired amounts to give nominal proportions of Ni, 1Ni:1Pd, 1Ni:9Pd and Pd (total 0.9 mmol) were introduced in a Fisher-Porter reactor under inert atmosphere and placed in a cryogenic bath. Then, a toluene solution (30 mL) of hexadecylamine (0.3 mol L -1 ) was added. The reactor was then evacuated (0.2 torr) and pressurized under H2 (3 bar), and the cold bath removed before heating to 110 o C (silicon oil bath). Vigorous stirring was maintained up to 20 h, leading to a black colloidal dispersion containing metal nanoparticles. Afterwards, the NP immobilization was performed as follows: functionalized SiO2 (1 g) support (outgassed for 1 h under vacuum) was introduced into a Schlenk flask followed by the colloidal dispersion (6 mL); the mixture was stirred at R.T. under Ar atmosphere for 2 h. The obtained grey solid was magnetically recovered from the dispersion by simple application of a magnet, and the supernatant was removed with a syringe. The final solid was then dried under vacuum for 1 h (0.2 torr) and stored under inert conditions until use in characterization and catalytic testes.

The set of DD samples was prepared as follows: the functionalized SiO2 (1 g) support was introduced into a Fisher-Porter reactor, followed by air evacuation (0.2 torr) and filling with Ar. Toluene solution (25 mL) of Ni and Pd precursors (0.45 mmol mL -1 ; at desired Ni/Pd ratios) was prepared in a Schlenk flask, under Ar atmosphere. This solution was then transferred to the SiO2 containing reactor, followed by outgassing under vacuum and pressurization with H2 (3 bar). The mixture was then stirred for 1 h at R.T. Afterwards, the solid was washed with dried toluene 5 times, under Ar atmosphere. The final solid was magnetically recovered by simple application of a magnet and dried under vacuum (0.2 torr) for 1 h and stored under air conditions.

The catalysts were labeled as NixPdy DD for the direct decomposition method and NixPdy SI for the sol-immobilization method, with the x and y numbers referring to the nominal Ni:Pd molar ratio (1:0, 1:1, 1:9, 0:1).

Characterization

Ni and Pd contents in the catalysts were determined by ICP-OES. The samples (10 mg) were digested in aqua regia solution (10 mL) at 100 o C for 3 h and analyzed in a Spectro Ciros equipment.

The surface composition of the catalysts was studied by X-ray photoelectron spectroscopy (XPS), using a conventional XPS spectrometer (ScientaOmicron ESCA+) with a high-performance hemispheric analyzer (EAC2000) with monochromatic Al Kα (hν = 1486.6 eV) radiation as the excitation source. The as prepared catalysts were deposited on a carbon tape and loaded to the XPS equipment. The base vacuum was 10 -9 torr. The XPS high-resolution spectra were recorded at a constant pass energy of 20 eV with an energy step of 0.05 eV. The fitting of the XPS components was performed using a Gaussian/Lorentzian function (30/70), and binding energy was referred to the C 1s level of adventitious carbon (284.6 eV).

Transmission electron microscopy was used to evaluate the particle size distribution and morphology. The images were obtained at a Jeol JEM 2100F microscope, with a field emission gun accelerated at 200 kV. The as prepared catalysts were dispersed in isopropyl proceeded up to 700°C, with reaction data acquisition on each 50°C step, for 30 minutes, to reach steady state conditions. The product formation was evaluated using a calibrated mass spectrometer (Hiden Analytical). The gas composition applied for the catalysis tests was 2% CO2 / 2% H2 / Ar balance and a total gas flow of 100 mL min -1 . The reaction rate was evaluated following the Equation 2, where rCO2 is the experimental reaction rate (in mol s -1 gmetal -1 ), X is the conversion degree, FCO2 is the CO2 inlet flow (mol s -1 ) and Wmetal the metal mass (in g). Arrhenius plots were obtained from the reaction rates calculated at maximum conversions of 25%, and the apparent activation energy values (Ea app , in kJ mol -1 ) were obtained from fitting of the experimental data.

Figure 1 .

 1 Figure 1. STEM-HAADF image of the Ni5Pd5 DD (a) and EDS elemental analysis of Ni (b), Pd (c) with color mix image (d); and STEM-HAADF image of Ni5Pd5 SI (e) with the corresponding Ni (f), Pd (g) and color mix image (h) elemental mappings.

Figure 2 .

 2 Figure 2. CO2 conversion as function of temperature for the series of Ni-Pd catalysts prepared from Direct Deposition, DD, (a) and Sol-Immobilization, SI, (b) methodologies. Samples were pre-treated under H2 at 400 o C for 30 min; Wmetal=0.5 mg of total metal (Ni+Pd); 1 CO2:4 H2:45 Ar.

Figure 3 :

 3 Figure 3: Rates of CO (a,b) and CH4 (c,d) formation from RWGS as function of temperature for the series of NiPd catalysts prepared from Direct Deposition, DD, (a,c) and Sol Immobilization, SI, (b,d) methodologies. Samples were pre-treated under H2 at 400 o C for 30 min; Wmetal=0.5 mg of total metal (Ni+Pd); 1 CO2:4 H2:45 Ar.

Figure 4 :

 4 Figure 4: DRIFTS spectra of adsorbed CO at room temperature onto the reduced catalysts at 400 o C, for 1h. (left) NixPdy DD samples; (right) NixPdy SI samples. (a) Ni1, (b) Ni5Pd5, (c) Ni1Pd9, (d) Pd1.

Figure 5 :

 5 Figure 5: Correlation between the Apparent Activation Energies (Ea app ) and weakly to strongly adsorbed CO ratio (*COw/*COs) with Pd content.

Figure S4 :

 S4 Figure S4: XPS spectra of the fresh Ni-Pd samples prepared from (a) Direct Decomposition, DD, and (b) Sol-Immobilization, SI.

Figure S5 :

 S5 Figure S5: Arrhenius plot for RWGS in Ni-Pd/SiO2 prepared following DD procedure (left) and SI procedure (right). Samples were pre-treated under H2 at 400 o C for 30 min; Wmetal=0.5 mg of total metal (Ni+Pd); 1 CO2:4 H2:45 Ar.

Table 1 .

 1 Metal content, bulk Pd concentration, surface Pd concentration and average particle size for the as prepared NiPd catalysts. Metal composition determined by ICP OES; b) Bulk Pd/(Ni+Pd) ratio calculated from ICP OES results; c) Surface Pd/(Ni+Pd) ratio calculated from surface composition determined by XPS; d) Average particle size measured by TEM.

	Catalysts	Metal content [%] a) Ni Pd	Bulk Pd/(Pd+Ni) ratio [%] b)	Surface Pd/(Pd+Ni) ratio [%] c)	Particle size [nm] d)
	Ni1 DD	2.64		-	-	4.4±1.5
	Ni5Pd5 DD	1.33	2.01	60.2	59.9	3.7±0.7
	Ni1Pd9 DD	0.30	3.69	92.5	95.0	3.2±0.1
	Pd1 DD		3.29	100	100	3.5±0.8
	Ni1 SI	0.64		-	-	6.3±1.3
	Ni5Pd5 SI	0.43	0.63	59.4	61.3	4.2±1.0
	Ni1Pd9 SI	0.05	0.68	93.2	96.5	5.3±1.0
	Pd1 SI		0.80	100	100	5.8±1.4
	a)

Acknowledgements

We gratefully acknowledge support of the RCGI -Research Centre for Gas Innovation, hosted by the University of São Paulo (USP) and sponsored by FAPESP -São Paulo Research Foundation (2014/50279-4) and Shell Brasil. Acknowledgements are given to Brazilian Nanotechnology Laboratory, LNNano, for use of TEM facilities (proposal #TEM-24456). All authors contributed equally.

Received: ((will be filled in by the editorial staff))

Revised: ((will be filled in by the editorial staff))

alcohol with sonication, and the dispersion was dripped on a lacey carbon coated cupper grid.

The images were acquired in conventional TEM and scanning TEM mode. The particle size distribution was estimated by fitting of histograms with a log-normal function. Highresolution STEM images and EDS mapping were acquired on aberration corrected FEI Titan Themis 3 equipment, with acceleration voltage of 300 kV; compositional EDS mapping was obtained with an Espirit system. DRIFTS of adsorbed CO was performed in order to probe the differences in electronic densities and catalytic sites of the catalysts. The experiments were performed in a commercial Harrick high-temperature cell and spectra were acquired on a Thermo iS50R spectrometer, equipped with an MCT detector (resolution 4 cm -1 and 64 scans). The as prepared samples were loaded into the cell with a subsequent thermal treatment under 5% H2/He mixture, at 400°C for 1 h. Then, the samples were cooled down to R.T. under He flow, when the background spectrum was acquired. The CO adsorption was performed by flowing 10 mL/min of a 5% CO/He diluted in 10 mL/min of pure He; the CO flow passed through the sample for 5 min. After, the flow was switched to He to desorb the gas phase CO, with spectra acquisition.

Catalytic evaluation

The RWGS catalytic tests were performed using a fixed bed quartz reactor, at atmospheric pressure, from 200 to 700C, in steps of 50C, in a CatLab PCS unit (Hiden Analytical). Due to different metal loading achieved during synthesis, the catalyst mass was adjusted in order to have 0.5 mg of metal in total (Ni+Pd). Thus, total mass in the range of 10-20 mg was used for the series of catalysts prepared by direct decomposition (DD); and of 15-40 mg for the catalysts prepared by sol-immobilization (SI). All the catalysts were previously thermally treated at 400°C, for 30 min, under flow of a 5% H2/Ar mixture, in order to recover a fully metallic surface. Afterwards, the catalysts were cooled down to 100°C, and the reaction

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

Supporting Information


Navonil Chatterjee
email: chatterjee.navonil@univ-ubs.fr

Hemanta Kumar Mondal
email: hemanta.mondal@ece.nitdgp.ac.in

Rodrigo Cataldo
email: rodrigo.cataldo@acad.pucrs.br

Jean-Philippe Diguet
email: jean-philippe.diguet@univ-ubs.fr

CDMA-based Multiple Multicast communications on WiNOC for eicient parallel computing

Keywords: CCS CONCEPTS, Networks → Network on chip, • Computer systems organization → Interconnection architectures, • Computing methodologies → Parallel computing methodologies Wired/Wireless Network-on-Chip, Broadcast and Multicast operation, Parallel computing, Code Division Multiple Access (CDMA)

In this work, we introduce an hybrid WiNoC, which judicially uses the wired and wireless interconnects for broadcasting/multicasting of packets. A code division multiple access (CDMA) method is used to support multiple broadcast operations originating from multiple applications executed on the multiprocessor platform. The CDMAbased WiNoC is compared in terms of network latency and power consumption with wired-broadcast/multicast NoC.

INTRODUCTION

WiNoC has emerged as a viable solution for the implementation of broadcast [START_REF] Abadal | OrthoNoC: A Broadcast-Oriented Dual-Plane Wireless Network-on-Chip Architecture[END_REF] communications on large manycores. Also, multiple applications can be executed simultaneously on a given WiNoC platform designed for large manycore architectures. In such a case, broadcast operation in one application may interfere with another one running in the same WiNoC, which might lead to contention and increase in power consumption as a single wireless channel is used by multiple application for broadcast communication.

In this work, we propose an original strategy to mitigate the aforementioned problems. We present a WiNoC topology which is divided into uniform clusters where each cluster is associated with one wireless interface. As, all the PEs are not associated with wireless interface, we use hybrid mechanism where both wired and wireless links are used for broadcast and multicast operations. For unicast packet transmission, only wired NoC is used. We have used code division multiple access (CDMA) based wireless packet Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for prot or commercial advantage and that copies bear this notice and the full citation on the rst page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specic permission and/or a fee. Request permissions from permissions@acm.org. NOCS '19, October 17-18, 2019, New York, NY, USA © 2019 Association for Computing Machinery. ACM ISBN 978-1-4503-6700-4/19/10. . . $15.00 https://doi.org/10.1145/3313231.3352371 transmission to support parallel broadcast communication in two or more applications executing in a WiNoC based multiprocessor platform. The proposed method helps to improve the overall system performance by providing ecient broadcast/multicast communication for single and multiple applications.

SYSTEM ARCHITECTURE

We consider a 2D Mesh based Wireless NoC which is divided into equal sized clusters as shown in Figure 1a. Each cluster consists of 16 routers which are arranged in 4 ⇥ 4 fashion. Every cluster has a centrally placed wireless hub with a wireless interface (WI) providing inter-cluster wireless links. The wireless hub is connected to one router in the cluster, which is known as hub connection (HC) router. Thus, all packets communicated using the wireless hub, ow though the HC router. For unicast communication, we have used table based routing while broadcast packets are communicated using WHIRL routing algorithm [START_REF] Krishna | Towards the Ideal On-chip Fabric for 1-to-many and Many-to-1 Communication[END_REF].

The Wireless Interface (WI) connects the router with the wireless medium. The WI consists of 3 parts: (1) Antenna, (2) Analog domain and (3) Digital domain, as shown in Figure 1b. The digital domain consists of Serializer/ Deserializer, CDMA encoder/decoder and channel access controller. The main component of the analog domain is the On-O Keying (OOK) transceiver module. zigzag metal antenna is used for signal transmission and reception. A MAC mechanism is used to share wireless channels among multiple users. We based our choice on a contention free MAC protocol as used in OrthoNoC [START_REF] Abadal | OrthoNoC: A Broadcast-Oriented Dual-Plane Wireless Network-on-Chip Architecture[END_REF] and modied it to suit our requirements.

BROADCAST & MULTICAST OPERATIONS

In this work, we have assumed that a single application or multiple applications may be executed in parallel on the multicore WiNoC. In case of single application, during broadcast operation only one hub works as the source and all other hubs as the destination. There is no conict with another broadcast, if two of them do not start at the same time. Thus no code is required as all the hubs receive the message. However, in case of multiple applications are executed in parallel, a conict may occur if both the applications launch a broadcast operation at the same time. In such as case, dierent CMDA codes will be assigned to the applications. In this work, we have used two type of CMDA based code-channel: (i) two codechannels and (ii) four code-channels, where maximum of two and four applications are executed in parallel, respectively.

RESULTS & DISCUSSION

To analyze the performance of the proposed CDMA-based multicast, we have used the Noxim [START_REF] Catania | Improving Energy Eciency in Wireless Network-on-Chip Architectures[END_REF] network-on-chip simulator. We To evaluate of the multicast operations, synthetic applications are generated and combined to form 4 dierent scenarios. We have compared our proposed technique (WiNoC) with wired broadcast/multicast scheme denoted as NoC+WHIRL [START_REF] Krishna | Towards the Ideal On-chip Fabric for 1-to-many and Many-to-1 Communication[END_REF]. To reduce the power consumption of the hubs, power gating is used in LNA and PA when hubs are not used as described in [START_REF] Mondal | Broadcast-and Power-Aware Wireless NoC for Barrier Synchronization in Parallel Computing[END_REF].

The network latency is presented in Fig 2(a). Compared to wired broadcast (NoC+WHIRL), we observe that our proposed method provides 29.11% , 19.81% and 28.48%, improvement in network latency for Bodytrack(med), Bodytrack(lar) and Streamcluster(med), respectively. Figure 2

CONCLUSION

In this paper, we present a solution to eciently implement broadcast and multicast operations on future large manycore architecture with the proposed Wireless NoC. We have implemented a CDMAbased transmission protocol for multicast operations in parallel applications. The proposed WiNoC supports maximum of four code-channels which allows execution of four parallel applications.

ACKNOWLEDGEMENT

This work is a part of BBC project supported by Labex CominLABS.

Figure 1 :

 1 Figure 1: Overview of the system Modelhave used PARSEC benchmark applications, bodytrack and streamcluster to analyze broadcast operations in WiNoC. To evaluate of the multicast operations, synthetic applications are generated and combined to form 4 dierent scenarios. We have compared our proposed technique (WiNoC) with wired broadcast/multicast scheme denoted as NoC+WHIRL[START_REF] Krishna | Towards the Ideal On-chip Fabric for 1-to-many and Many-to-1 Communication[END_REF]. To reduce the power consumption of the hubs, power gating is used in LNA and PA when hubs are not used as described in[START_REF] Mondal | Broadcast-and Power-Aware Wireless NoC for Barrier Synchronization in Parallel Computing[END_REF].The network latency is presented in Fig2(a). Compared to wired broadcast (NoC+WHIRL), we observe that our proposed method provides 29.11% , 19.81% and 28.48%, improvement in network latency for Bodytrack(med), Bodytrack(lar) and Streamcluster(med), respectively. Figure2(b) depicts the power delay product (PDP) for NoC+WHIRL and the proposed WiNoC for dierent benchmark applications.When compared with NoC+WHIRL, we observe that the propose method achieves 38. 8%, 30.7%, and 38.2% improvement in PDP for Bodytrack(med), Bodytrack(lar) and Streamcluster(med).

  (b) depicts the power delay product (PDP) for NoC+WHIRL and the proposed WiNoC for dierent benchmark applications.When compared with NoC+WHIRL, we observe that the propose method achieves 38. 8%, 30.7%, and 38.2% improvement in PDP for Bodytrack(med), Bodytrack(lar) and Streamcluster(med).

Figure 2 :Figure 3 :

 23 Figure 2: Analysis of (a) Network Latency and (b) PDP for broadcast operation using dierent benchmark applicationsFigure 3(a) reports the latency for dierent scenarios. Compared to NoC+WHIRL, CDMA-based WiNoC shows 21.01% , 20.8%, 19% and 5% improvement in network latency for scenario 1, 2, 3 and 4 respectively. The power delay product for NoC+WHIRL and proposed WiNoC for dierent scenarios is presented in Figure 3(b).Comparison with NoC+WHIRL, there is 4.25%, 4.12%, and 2.65%


Ahmet Iscen

Giorgos Tolias

Yannis Avrithis

Ondřej Chum

Cordelia Schmid

Graph convolutional networks for learning with few clean and many noisy labels

In this work we consider the problem of learning a classifier from noisy labels when a few clean labeled examples are given. The structure of clean and noisy data is modeled by a graph per class and Graph Convolutional Networks (GCN) are used to predict class relevance of noisy examples. For each class, the GCN is treated as a binary classifier, which learns to discriminate clean from noisy examples using a weighted binary cross-entropy loss function. The GCN-inferred "clean" probability is then exploited as a relevance measure. Each noisy example is weighted by its relevance when learning a classifier for the end task. We evaluate our method on an extended version of a few-shot learning problem, where the few clean examples of novel classes are supplemented with additional noisy data. Experimental results show that our GCNbased cleaning process significantly improves the classification accuracy over not cleaning the noisy data, as well as standard few-shot classification where only few clean examples are used.

Introduction

State-of-the-art deep learning methods require a large amount of manually labeled data. The need for supervision may be reduced by decoupling representation learning from the end task and/or using additional training data that is unlabeled, weakly labeled (with noisy labels), or belong to different domains or classes. Example approaches are transfer learning [START_REF] Wang | Unsupervised learning of visual representations using videos[END_REF], unsupervised representation learning [START_REF] Wang | Unsupervised learning of visual representations using videos[END_REF], semi-supervised learning [START_REF] Weston | Deep learning via semi-supervised embedding[END_REF], learning from noisy labels [START_REF] Joulin | Learning visual features from large weakly supervised data[END_REF] and few-shot learning [START_REF] Snell | Prototypical networks for few-shot learning[END_REF].

However, for several classes, only very few or even no clean labeled examples might be available at the representation learning stage. Few-shot learning severely limits the number of labeled samples on the end task, while the representation is learned on a large training set of different classes [START_REF] Hariharan | Low-shot visual recognition by shrinking and hallucinating features[END_REF][START_REF] Snell | Prototypical networks for few-shot learning[END_REF][START_REF] Vinyals | Matching networks for one shot learning[END_REF]. Nevertheless, in many situations, more data with noisy labels can be acquired or is readily available for the end task.

One interesting mix of few-shot learning with additional large-scale data is the work of Douze et al. [START_REF] Douze | Low-shot learning with large-scale diffusion[END_REF], where labels are propagated from few clean labeled examples to a large-scale collection. This collection is unlabeled and actually Query by admiral Fig. 1. Overview of our cleaning approach for 1-shot learning with noisy examples. We use the class name admiral to crawl noisy images from the web and create an adjacency graph based on visual similarity. We then assign a relevance score to each noisy example with a graph convolutional network (GCN). Relevance scores are displayed next to the images.

contains data for many more classes than the end task. Their method overall improves the classification accuracy, but at an additional computational cost.

It is a transductive method, i.e., instead of learning a parametric classifier, the large-scale collection is still necessary at inference. In this work, we learn a classifier from a few clean labeled examples and additional weakly labeled data, while the representation is learned on different classes, similarly to few-shot learning. We assume that the class names are known, and we use them to search an existing large collection of images with textual description. The result is a set of images with novel class labels, but potentially incorrect (noisy). As shown in Figure 1, we clean this data using a graph convolutional network (GCN) [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF], which learns to predict a class relevance score per image based on connections to clean images in the graph. Both the clean and the noisy images are then used to learn a classifier, where the noisy examples weighted by relevance. Unlike most existing work, our method operates independently per class and applies when clean labeled examples are few or even only one per class.

We make the following contributions: 1. We learn a classifier on a large-scale weakly-labeled collection jointly with only a few clean labeled examples. 2. To our knowledge, we are the first to use a GCN to clean noisy data: we cast a GCN as a binary classifier which learns to discriminate clean from noisy data, and we use its inferred "clean" probabilities as a relevance score per example. 3. We apply our method to two few-shot learning benchmarks and show significant improvement in accuracy, outperforming the method by Douze et al. [START_REF] Douze | Low-shot learning with large-scale diffusion[END_REF] using the same large-scale collection of data without labels.

Related work

Learning with noisy labels is often concerned with estimating or learning a transition matrix [START_REF] Natarajan | Learning with noisy labels[END_REF][START_REF] Patrini | Making deep neural networks robust to label noise: A loss correction approach[END_REF][START_REF] Sukhbaatar | Training convolutional networks with noisy labels[END_REF] or knowledge graph [START_REF] Li | Learning from noisy labels with distillation[END_REF] between labels and correcting the loss function, which does not apply in our case since the true classes in the noisy data is unknown. Most recent work on learning from large-scale weakly-labeled data focuses on learning the representation through metric learning [START_REF] Lee | CleanNet: Transfer learning for scalable image classifier training with label noise[END_REF][START_REF] Wang | Iterative learning with open-set noisy labels[END_REF], bootstrapping [START_REF] Reed | Training deep neural networks on noisy labels with bootstrapping[END_REF], or distillation [START_REF] Li | Learning from noisy labels with distillation[END_REF]. In our case however, since the clean labeled examples are scarce, we need to keep the representation mostly fixed.

Dealing with the noise by thresholding [START_REF] Lee | CleanNet: Transfer learning for scalable image classifier training with label noise[END_REF], outlier detection [START_REF] Wang | Iterative learning with open-set noisy labels[END_REF] or reweighting [START_REF] Liu | Classification with noisy labels by importance reweighting[END_REF], is applicable while the representation is learned, based on the gradient of the loss [START_REF] Ren | Learning to reweight examples for robust deep learning[END_REF]. In contrast, the relatively-shallow GCN that we propose effectively decouples reweighting from both representation learning and classifier learning. Learning to clean the noisy labels [START_REF] Veit | Learning from noisy large-scale datasets with minimal supervision[END_REF] typically assumes adequate human verified labels for training, which again is not the case in this work.

Few-shot learning. Meta-learning [START_REF] Vilalta | A perspective view and survey of meta-learning[END_REF] refers to learning at two levels, where generic knowledge is acquired before adapting to more specific tasks. In fewshot learning, this translates to learning on a set of base classes how to learn from few examples on a distinct set of novel classes without overfitting. For instance, optimization meta-learning [START_REF] Finn | Model-agnostic meta-learning for fast adaptation of deep networks[END_REF][START_REF] Finn | Probabilistic model-agnostic meta-learning[END_REF][START_REF] Ravi | Optimization as a model for few-shot learning[END_REF] amounts to learning a model that is easy to fine-tune in few steps. In our work, we study an extension of fewshot learning where more novel class instances are available, reducing the risk of overfitting when fine-tuning the model. Metric learning approaches learn how to compare queries for instance to few examples [START_REF] Vinyals | Matching networks for one shot learning[END_REF] or to the corresponding class prototypes [START_REF] Snell | Prototypical networks for few-shot learning[END_REF]. Hariharan and Girshick [START_REF] Hariharan | Low-shot visual recognition by shrinking and hallucinating features[END_REF] and Wang et al. [START_REF] Wang | Low-shot learning from imaginary data[END_REF] learn how to generate novel-class examples, which is not needed when more data is actually available.

Gidaris and Komodakis [START_REF] Gidaris | Dynamic few-shot visual learning without forgetting[END_REF] learn on base classes a simpler cosine similaritybased parametric classifier, or simply cosine classifier, without meta-learning. The same classifier has been introduced independently by Qi et al. [START_REF] Qi | Low-shot learning with imprinted weights[END_REF], who further fine-tune the network, assuming access to the base class training set. A recent survey [START_REF] Chen | A closer look at few-shot classification[END_REF] confirms the superiority of the cosine classifier to previous work including meta-learning [START_REF] Finn | Model-agnostic meta-learning for fast adaptation of deep networks[END_REF]. We use the cosine classifier in this work, both for base and novel classes.

Making use of unlabeled data has been little explored in few-shot learning until recently. Ren et al. [START_REF] Ren | Meta-learning for semi-supervised few-shot classification[END_REF] introduce a semi-supervised few-shot classification task, where some labels are unknown. Liu et al. [START_REF] Liu | Learning to propagate labels: Transductive propagation network for few-shot learning[END_REF] follow the same semisupervised setup, but use graph-based label propagation (LP) [START_REF] Zhou | Learning with local and global consistency[END_REF] for classification and consider all test images jointly. These methods assume a meta-learning scenario, where only small-scale data is available at each training episode; arguably, such a small amount of data limits the representation adaptation and generalization to unseen data. Similarly, Rohrbach et al. [START_REF] Rohrbach | Transfer learning in a transductive setting[END_REF] use label propagation in a transductive setting, but at a larger scale assuming that all examples come from a set of known classes. Douze et al. [START_REF] Douze | Low-shot learning with large-scale diffusion[END_REF] extend to an even larger scale, leveraging 100M unlabeled images in a graph without using additional text information. We focus on the latter large-scale scenario using the same 100M dataset. However, we filter by text to obtain noisy data and follow an inductive approach by training a classifier for novel classes, such that the 100M collection is not needed at inference.

Graph neural networks are generalizations of convolutional networks to non-Euclidean spaces [START_REF] Bronstein | Geometric deep learning: going beyond euclidean data[END_REF]. Early spectral methods [START_REF] Bruna | Spectral networks and locally connected networks on graphs[END_REF][START_REF] Henaff | Deep convolutional networks on graph-structured data[END_REF] have been succeeded by Chebyshev polynomial approximations [START_REF] Defferrard | Convolutional neural networks on graphs with fast localized spectral filtering[END_REF], which avoid the high computational cost of computing eigenvectors. Graph convolutional networks (GCN) [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF] provide a further simplification by a first-order approximation of graph filtering and are applied to semi-supervised [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF] and subsequently few-shot learning [START_REF] Garcia | Few-shot learning with graph neural networks[END_REF]. Kipf and Welling [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF] apply the loss function to labeled examples to make predictions on unlabeled ones. Similarly, Garcia and Bruna [START_REF] Garcia | Few-shot learning with graph neural networks[END_REF] use GCNs to make predictions on novel class examples. Gidaris and Komodakis [START_REF] Gidaris | Generating classification weights with gnn denoising autoencoders for few-shot learning[END_REF] use Graph Neural Networks as denoising autoencoders to generate class weights for novel classes. By contrast, we cast GCNs as binary classifiers discriminating clean from noisy examples: we apply a loss function to all examples, and then use the inferred probabilities as a class relevance measure, effectively cleaning the data.

Our counter-intuitive objective of treating all noisy examples as negative can be compared to treating each example as a different class in instance-level discrimination [START_REF] Wu | Unsupervised feature learning via nonparametric instance-level discrimination[END_REF]. In fact, our loss function is similar to noise-contrastive estimation (NCE) [START_REF] Gutmann | Noise-contrastive estimation: A new estimation principle for unnormalized statistical models[END_REF]. Our experiments show that our GCN-based classifier outperforms classical LP [START_REF] Zhou | Learning with local and global consistency[END_REF] used for a similar purpose by [START_REF] Rohrbach | Transfer learning in a transductive setting[END_REF].

Problem formulation

X c E = X c C ∪ X c N .
Examples or sets of examples having clean (noisy) labels are referred to as clean (noisy) as well. The goal is to train a K-way classifier, using the additional noisy set in order to improve the accuracy compared to only using the small clean set.

We assume that we are given a feature extractor g θ : X → R d , mapping an example to a d-dimensional vector. For instance, when examples are images, the feature extractor is typically a convolutional neural network (CNN) and θ are the parameters of all layers.

In this work, we assume that the noisy set X N is collected via web crawling. Examples are images accompanied by free-form text descriptions and/or user tags originating from community photo collections. To make use of text data, we assume that the names of the classes in C are given. An example in X N is given a label in class c ∈ C if its textual information contains the name of class c; it may then have none, one or more labels. In this way, we automatically infer labels for X N without human effort, which are however noisy.

Cleaning with graph convolutional networks

We perform cleaning by predicting a class relevance measure for each noisy example in X c N , independently for each class c ∈ C. To simplify the notation, we drop superscript c where possible in this subsection and we denote

X c E by {x 1 , . . . , x k , x k+1 , . . . , x N }, where X c C = {x 1 , . . . , x k } and X c N = {x k+1 , . . . , x N }.

The features of these examples are similarly represented by matrix

V = [v 1 , . . . , v k , v k+1 , . . . , v N ] ∈ R d×N , where v i = g θ (x i ) for i = 1, . . . , N .
We construct an affinity matrix

A ∈ R N ×N with elements a ij = [v i v j ] + if examples v i and v j are reciprocal nearest neighbors in X c
E and 0 otherwise. Matrix A has zero diagonal, but self-connections are added before A is normalized as à = D -1 (A + I) with D = diag((A + I)1) being the degree matrix of A + I and 1 the all-ones vector.

Graph convolutional networks (GCNs) [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF] are formed by a sequence of layers. Each layer is a function

f Θ : R N ×N × R l×N → R n×N of the form f Θ ( Ã, Z) = h(Θ Z Ã), (1)
where Z ∈ R l×N represents the input features, Θ ∈ R l×n holds the parameters of the layer to be learned, and h is a nonlinear activation function. Function f Θ maps l-dimensional input features to n-dimensional output features.

In this work we consider a two-layer GCN with a scalar output per example. This network is a function

F Θ : R N ×N × R d×N → R N given by F Θ ( Ã, V ) = σ(Θ 2 [Θ 1 V Ã] + Ã), (2)
where

Θ = {Θ 1 , Θ 2 }, Θ 1 ∈ R d×m , Θ 2 ∈ R m×1 , [•]
+ is the positive part or ReLU function [START_REF] Nair | Rectified linear units improve restricted boltzmann machines[END_REF] and σ(a) = (1 + e -a ) -1 for a ∈ R is the sigmoid function. Function F Θ performs feature propagation through the affinity matrix in an analogy to classical graph-based propagation methods for classification [START_REF] Zhou | Learning with local and global consistency[END_REF] or search [START_REF] Zhou | Ranking on data manifolds[END_REF]. The output F Θ ( Ã, V ) is a vector of length N , with element F Θ ( Ã, V ) i in [0, 1] representing a relevance value of example x i for class c. To learn the parameters Θ, we treat the GCN as a binary classifier where target output 1 corresponds to clean examples and 0 to noisy. In particular, we minimize the loss function

L G (V, Ã; Θ) = - 1 k k i=1 log F Θ ( Ã, V ) i - λ N -k N i=k+1 log 1 -F Θ ( Ã, V ) i . (3)
This is a binary cross-entropy loss function where noisy examples are given an importance weight λ. Given the propagation on the nearest neighbor graph, and depending on the relative importance λ of the second term, noisy examples that are strongly connected to clean ones are still expected to receive high class relevance, while noisy examples that are not relevant to the current class are expected to get a class relevance near zero.

The impact of parameter λ is validated in Section 6, where we show that the fewer the available clean images are (smaller k) the smaller the importance weight should be. As is standard practice for GCNs in classification [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF], training is performed in batches of size N , that is the entire set of features.

Figure 2 shows examples of clean images, corresponding noisy ones and the predicted relevance. Using the visual similarity to the clean image, we can use relevance to resolve cases of polysemy, e.g. black widow (spider) vs. black widow (superhero), or cases like pineapple vs. pineapple juice. Discussion. Loss function ( 3) is similar to noise-contrastive estimation (NCE) [START_REF] Gutmann | Noise-contrastive estimation: A new estimation principle for unnormalized statistical models[END_REF] as used by We et al. [START_REF] Wu | Unsupervised feature learning via nonparametric instance-level discrimination[END_REF] for instance-level discrimination, whereas we discriminate clean from noisy examples. The semi-supervised learning setup of GCNs [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF] uses a loss function that applies only to the labeled examples, and makes discrete predictions on unlabeled examples. In our case, all examples contribute to the loss but with different importance, as we infer real-valued class relevance for the noisy examples, to be used for subsequent learning.

Function F Θ in (2) reduces to a Multi-Layer Perceptron (MLP) when the affinity matrix A is zero, in which case all examples are disconnected. Using an MLP to perform cleaning would take each example into account independently of the others, while the GCN considers the collection of examples as a whole. MLP training is performed identically to GCN by minimizing [START_REF] Chen | A closer look at few-shot classification[END_REF]. We compare the two alternatives in our experiments.

Learning a classifier with few clean and many noisy examples

Our cleaning process applies when the clean labeled examples are few, but assumes a feature extractor5 g θ . That is, representation learning, label cleaning and classifier learning are decoupled. We perform GCN-based cleaning as described in Section 4, and learn a classifier by weighting examples according to class relevance. The process of training the classifier is described below.

Cosine-similarity based classifier

We use a cosine-similarity based classifier [START_REF] Gidaris | Dynamic few-shot visual learning without forgetting[END_REF][START_REF] Qi | Low-shot learning with imprinted weights[END_REF], or cosine classifier for short. Each class c ∈ C is represented by a learnable parameter

w c ∈ R d . The prediction of example x ∈ X is the class c of maximum cosine similarity ŵ c ĝθ (x) 6 π θ,W (x) = arg max c ŵ c ĝθ (x), (4)
where W = [w 1 , . . . , w K ] ∈ R d×K .

Classifier learning

The goal is to learn a K-way classifier for unseen data in X . Unlike the typical few-shot learning task, each class contains a few clean and many noisy examples.

Prior to learning classifiers, training examples x i ∈ X c E are weighted by their relevance r(x i ) to class c. For a noisy example x i ∈ X c N , we define r(x i ) = F Θ ( Ã, V ) i where F Θ ( Ã, V ) is the output vector of the GCN, while for a clean example x i ∈ X c C we fix r(x i ) = 1. Note that optimizing (3) does not guarantee

F Θ ( Ã, V ) i = 1 for clean examples x i ∈ X c
C . We define r(X) = x∈X r(x) for any set X ⊂ X .

We first assume that the given feature extractor is fixed and consider two different classifiers, namely class prototypes and cosine-similarity based classifier.

Then, this assumption is dropped and the classifier and feature representation are learned jointly by fine-tuning the entire network.

Class prototypes. For each class c ∈ C, we define prototype w c by

w c = 1 r(X c E ) x∈X c E r(x)g θ (x). (5)
Prototypes are fixed vectors, not learnable parameters. Collecting them into matrix W = [w 1 , . . . , w K ] ∈ R d×K , K-way prediction is made by classifier π θ,W (4).

Cosine classifier learning.

Given examples X E , we learn a parametric cosine classifier with parameters W = [w 1 , . . . , w K ] ∈ R d×K by minimizing the weighted cross entropy loss L(C, X E , θ; W ) over W , given by

L(C, X E , θ; W ) = - c∈C 1 r(X c E ) x∈X c E r(x) log(σ(s Ŵ ĝθ (x)) c ), (6)
where σ : R K → R K is the softmax function with σ(a) c = e ac / j∈C e aj for a ∈ R K , s is a scale parameter and Ŵ = [ ŵ1 , . . . , ŵK ]. The parameters θ of the feature extractor are fixed. The scale parameter s is also fixed according to the training of the feature extractor. Prediction is made as in the previous case.

Deep network fine-tuning. An alternative is to drop the assumption that the feature extractor is fixed. In this case, we jointly learn the parameters θ of the feature extractor and W of the K-way cosine classifier by minimizing the righthand side of ( 6). This requires access to examples X E , while for the previous two classifiers, access to features g θ (x) is enough. Note that, due to over-fitting on the few available examples, such learning is typically avoided in a few-shot learning setup.

6 Experiments

Experimental setup

Datasets and task setup. We extend the Low-Shot ImageNet benchmark [START_REF] Hariharan | Low-shot visual recognition by shrinking and hallucinating features[END_REF] by assuming many noisy examples in addition to the few clean ones. In this benchmark, the 1000 ImageNet classes [START_REF] Russakovsky | Imagenet large scale visual recognition challenge[END_REF] are split into 389 base classes and 611 novel classes. The validation set contains 193 base and 300 novel classes, and the test set the remaining 196 base and 311 novel classes. The base classes are used to learn the feature extractor (see supplementary material), while the novel classes form the set of classes C on which we apply the cleaning and learn the classifier. We only assume noisy examples for the novel classes, not for the base ones. Additionally, we apply a similar setup to the Places365 dataset [START_REF] Zhou | Places: A 10 million image database for scene recognition[END_REF]. We randomly choose 183 test and 182 validation classes. We use the model learned on the base classes of Low-Shot ImageNet benchmark as the feature extractor. Therefore, all classes in Places365 dataset are considered novel. We refer to this setup as Low-Shot Places365 benchmark.

The standard benchmark includes k-shot classification, i.e. classification on k clean examples per class, with k ∈ {1, 2, 5, 10, 20}. We extend it to k clean and many noisy examples per class. Similar to the work of Hariharan and Girshick [START_REF] Hariharan | Low-shot visual recognition by shrinking and hallucinating features[END_REF], we perform 5 episodes, each drawing a subset of k clean examples per class. We report the average top-5 accuracy over the 5 episodes on novel classes of the test set. Accuracy over all classes (base and novel) is reported in supplementary material.

Noisy data and statistics. We use the YFCC100M dataset [START_REF] Thomee | Yfcc100m: The new data in multimedia research[END_REF] as a source of additional data with noisy labels. It contains approximatively 100M images collected from Flickr. Each image comes with a text description obtained from the user title and caption. We use the text description to obtain images with noisy labels, as discussed in Section 3. Figure 3 (bottom) presents the same statistics for Low-Shot Places365 benchmark. The trends are similar to those from Figure 3, except that there are more than 20 classes without any additional noisy data in this task (b). Nevertheless, there is a more consistent improvement in accuracy for classes that do have sufficient noisy data(c).

Representation and classifier learning. In most experiments, we use ResNet-10 [START_REF] He | Deep residual learning for image recognition[END_REF] as feature extractor as in [START_REF] Gidaris | Dynamic few-shot visual learning without forgetting[END_REF]. Classification for novel classes is performed with class prototypes (5), cosine classifier learning [START_REF] Finn | Model-agnostic meta-learning for fast adaptation of deep networks[END_REF] or deep network finetuning. Hyper-parameters, such as batch size and number of epochs, are tuned on the validation set. We cross-validate the possible values of 512, 1024, 2048, 4096, and 8192 for batchsize and 10, 20, 30 and 50 for number of epochs. The learning rate starts from 0.1 and is reduced to 0.001 at the end of training with cosine annealing [START_REF] Loshchilov | Sgdr: Stochastic gradient descent with warm restarts[END_REF]. We handle the imbalance of the noisy set by normalizing by r(X c ) in [START_REF] Finn | Model-agnostic meta-learning for fast adaptation of deep networks[END_REF]. Prototypes [START_REF] Douze | Low-shot learning with large-scale diffusion[END_REF] are used to initialize the weights W of the cosine classifier in [START_REF] Finn | Model-agnostic meta-learning for fast adaptation of deep networks[END_REF]. We ignore examples x i with relevance r(x i ) < 0.1 to reduce the complexity when fine-tuning the network. We also report results with ResNet-50 as feature extractor, using the model trained on base classes by [START_REF] Hariharan | Low-shot visual recognition by shrinking and hallucinating features[END_REF]. Following [START_REF] Douze | Low-shot learning with large-scale diffusion[END_REF], we apply PCA to the features to reduce their dimensionality to 256. Base classes are represented by class prototypes (5) in this case.

GCN training is performed with the Adam optimizer and a learning rate of 0.1 for 100 iterations. We use dropout with probability 0.5. The dimensionality of the input descriptors is d = 512 for ResNet-10 and d = 256 for ResNet-50 (after PCA). Dimensionality of the internal representation in (1) is m = 16. The affinity matrix is constructed with reciprocal top-50 nearest neighbors.

Baseline methods. We implement and evaluate several baseline methods:

1. β-cleaning assigns r(x i ) = β to all additional examples. We report results for β = 1.0 (unit relevance score) and β * , the optimal β for all k obtained on the validation set. 2. Similarity uses the scaled cosine similarity as the relevance weight, i.e. r( 4. Label Propagation (LP) [START_REF] Zhou | Learning with local and global consistency[END_REF] propagates information by a linear operation. It solves the linear system (I -αD -1/2 AD -1/2 )r c = y c [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] for each class c, where D is the degree matrix of A, α = 0.9 and y c ∈ R N is a k-hot binary vector indicating the clean (labeled) examples of class c. Relevance r(x i ) is the i-th element (r c ) i of the solution. 5. MLP, discussed in Section 4, learns a nonlinear mapping to assign relevance weights, but does not propagate over the graph. It is trained using (3) and therefore includes part of our contribution.

x i ) = (1+v i x)/2,

Experimental results

The impact of the importance weight λ is measured on the validation set and the best performing value is used on the test set for each value of k. Results on Low-shot ImageNet benchmark are shown in Figure 4. The larger the value of λ, the more the loss encourages noisy examples to be classified as negatives. As a consequence, large (small) λ results in smaller (larger) relevance, on average, for noisy examples. The optimal λ per value of k suggests that as the number of clean examples decreases, the need for additional noisy ones increases.

Comparison with baselines using additional data on Low-shot Imagenet benchmark is presented in Table 1. Qualitative results are presented in Figure 2. The use of additional data is mostly harmful for β-weighting except for 1 and 2-shot. MLP offers improvements in most cases, implying that it manages to appropriately down-weight irrelevant examples. The consistent improvement of GCN compared to MLP, especially large for small k, suggests that it is beneficial to incorporate relations, with the affinity matrix A modeling the structure of the feature space. LP is a classic approach that also uses A but is a linear operation with no parameters, and is inferior to our method. The gain of cleaning (β = 1 vs. ours) ranges from 11% to 20%.

Comparison with the state of the art on Low-shot Imagenet benchmark is presented in Table 2. We significantly improve the performance by using additional data and cleaning compared to a number of different approaches, including the work by Gidaris and Komodakis [START_REF] Gidaris | Dynamic few-shot visual learning without forgetting[END_REF], which is our starting point. As expected, the gain is more pronounced for small k, reaching more than 20% improvement for 1-shot novel accuracy. Closest to ours is the work by Douze et al. [START_REF] Douze | Low-shot learning with large-scale diffusion[END_REF], who use the same experimental setup and the same additional data, but without filtering by text or using noisy labels. We outperform this approach in all cases, while requiring much less computation: offline, we construct a separate small graph per class rather than a single graph over the entire 100M collection; online, we perform inference by cosine similarity to one prototype per class or a learned classifier rather than iterative diffusion on the entire collection. By ignoring examples that are not given any noisy label, we are only using a tiny fraction of the 100M collection: in particular, only 3,744,994 images for the 311-class test split of the Low-shot Im-ageNet benchmark. In contrast to [START_REF] Douze | Low-shot learning with large-scale diffusion[END_REF], additional data brings improvement even at 20-shot with classifier learning or network fine-tuning. Most importantly, our approach does not require the entire 100M collection at inference. Analysis of relevance weights. We manually label all the noisy examples from 20 classes in order to quantitatively measure the accuracy of the assigned relevance. We measure the noise ratio per class, i.e. the ratio of irrelevant (negative) noisy images to all noisy (positive and negative) images. Positive and negative images are defined according to the manual labels. The 20 classes are selected such that 10 of them have the highest 1-shot accuracy, and the rest have the lowest. This allows us to examine success and failure cases.

In the case of 1-shot classification (k = 1), the average relevance weight is 0.71 for positive and 0.40 for negative examples. A success case is the "bee eater" class with noise ratio equal to 0.52. Our method achieves 98.4% accuracy for 1shot classification, compared to 68.8% without any additional data. The average relevance weight is 0.99 for positive examples and 0.25 for negative examples of this class. One failure case is the "muzzle" class; it corresponds to the muzzle of an animal. The noise ratio is high; 94% of the 980 collected images are not relevant with most being animals without a muzzle or a firearm. The 1-shot classification accuracy without noisy data is 4%. Our method offers only a small increase to 8%. This can be explained by inaccurate relevance weights, which are on average 0.18 for positive and 0.30 for negative examples.

Experiments in Low-Shot Places365 are reported in Table 3. Our results indicate that our method consistently outperforms the baselines on this benchmark as well. Note that MLP, which is also competitive for this task, is trained with our proposed loss function 3. This is our contribution as well as the use of GCN. These methods significantly improve over existing methods, such as Label Propagation [START_REF] Zhou | Learning with local and global consistency[END_REF]. Further improvements are brought by cosine classifier learning [START_REF] Finn | Model-agnostic meta-learning for fast adaptation of deep networks[END_REF] and deep network fine-tuning.

We also present qualitative results on Low-Shot Places365 in Figure 5. The first example at the top shows that top-ranked images by relevance depict different views of cafeterias for the cafeteria class, while bottom-ranked images depict food served in a cafeteria, which are irrelevant to our task. Similarly, our method assigns high relevance to images of soccer stadiums and low relevance to soccer players for the soccer class. Finally, our method finds similar images to the clean example for the ruin class. In general, top-ranking images exhibit diversity and are not just near-duplicates.

Conclusions

In this paper we have introduced a new method for assigning class relevance to noisy images obtained by textual queries with class names. Our approach leverages one or a few labeled images per class and relies on a graph convolutional network (GCN) to propagate visual information from the labeled images to the noisy ones. The GCN is trained as a binary classifier discriminating clean from noisy examples using a weighted binary cross-entropy loss function and inferring "clean" probability as a relevance score for that class. Experimental results show that using noisy images weighted by this relevance score significantly improves the classification accuracy.

  We consider a space X of examples. We are given a set X C ⊂ X of examples, each having a clean (manually verified) label in a set C of classes with |C| = K. We assume that the number |X c C | of examples 4 labeled in each class c ∈ C is only k, typically in {1, 2, 5, 10, 20}. We are also given an additional set X N of examples, each with a set of noisy labels in C. The extended set of examples for class c is now

6 ( 2 (Fig. 2 .

 622 Fig. 2. Examples of clean images from the Low-Shot ImageNet Benchmark (left) for 1-shot classification, cumulative histogram of the predicted relevance for noisy images (middle) and representative noisy images (right) by descending order of relevance, with relevance value reported below. Test accuracy without and with additional data using class prototypes (5) is shown next to class names.

Fig. 3 .

 3 Fig. 3. Noisy data statistics for Low-Shot ImageNet(top) and Low-Shot Places365(bottom). (a) Number of additional images collected from YFCC-100M per class c for all novel classes. (b) Number of classes per group, when groups are created according to |X c N | in logarithmic scale. (c) Accuracy improvement ∆ Acc (difference of accuracy between our method with noisy examples and the baseline without noisy examples) for prototype classifier, for same groups as in (b).

Figure 3 (Fig. 4 .

 34 Fig. 4. Impact of λ on the validation set of the extended Low-shot ImageNet benchmark with YFCC-100M for noisy examples using class prototypes (5).

(Fig. 5 .

 5 Fig. 5. Examples of clean images on Low-Shot Places365 Benchmark (left) for 1-shot classification, cumulative histogram of the predicted relevance for noisy images (middle), and representative noisy images (right), each having its position in the descending ranked list according to relevance value reported below. Test accuracy without and with additional data using prototypes (5) is shown next to class names.

Table 1 .

 1 where x is the class prototype created with the features of clean examples. 3. Linear learns a linear binary classifier where the positive instances are the k labeled examples, and the negative examples are chosen randomly from other classes. Comparison with baselines using noisy examples on the Low-shot ImageNet benchmark. We report top-5 accuracy on novel classes with classification by class prototypes (5).

	Method	k=1	2	5	10	20
		Few Clean Examples
	Class proto. [9]	45.3±0.65	57.1±0.37	69.3±0.32	74.8±0.20	77.8±0.24
		Few Clean & Many Noisy Examples
	Similarity	49.8±0.29	56.3±0.27	64.2±0.32	68.4±0.14	71.2±0.12
	β-weighting, β = 1	56.1±0.06	56.4±0.08	57.1±0.05	57.7±0.08	58.7±0.06
	β-weighting, β *	55.6±0.24	58.3±0.14	63.4±0.25	67.5±0.34	71.0±0.22
	Linear	59.8±0.00	59.3±0.00	58.4±0.00	58.6±0.00	59.4±0.00
	Label Propagation	62.6±0.35	67.0±0.41	74.6±0.30	76.3±0.23	77.7±0.18
	MLP	63.6±0.41	68.8±0.42	73.7±0.25	75.6±0.21	77.6±0.21
	Ours	67.8±0.10	70.9±0.30	73.9±0.17	76.1±0.12	78.2±0.14

Table 2 .

 2 Comparison to the state of the art on the Low-shot ImageNet benchmark. We report top-5 accuracy on novel classes. We use class prototypes[START_REF] Douze | Low-shot learning with large-scale diffusion[END_REF], cosine classifier learning[START_REF] Finn | Model-agnostic meta-learning for fast adaptation of deep networks[END_REF] and deep network fine-tuning for classification with our GCN-based data addition method. † denotes numbers taken from the corresponding papers. All other experiments are re-implemented by us.

	Method		Top-5 accuracy on novel classes
		k=1	2	5	10	20
		ResNet-10 -Few Clean Examples
	Proto.-Nets [33] †	39.3	54.4	66.3	71.2	73.9
	Logistic reg. w/ H [41] † 40.7	50.8	62.0	69.3	76.5
	PMN w/ H [41] †	45.8	57.8	69.0	74.3	77.4
	Class proto. [9]	45.3±0.65 57.1±0.37 69.3±0.32 74.8±0.20 77.8±0.24
	Class proto. w/ Att. [9] 45.8±0.74 57.4±0.38 69.6±0.27 75.0±0.29 78.2±0.23
	ResNet-10 -Few Clean & Many Noisy Examples
	Ours -class proto. (5) 67.8±0.10 70.9±0.30 73.7±0.20 76.1±0.16 78.2±0.14
	Ours -cosine (6)	73.2±0.14 75.3±0.25 75.6±0.24 78.5±0.32 80.7±0.26
	Ours -fine-tune	74.1±0.19 76.2±0.28 77.7±0.23 80.6±0.31 82.6±0.24
		ResNet-50 -Few Clean Examples
	Proto.-Nets [33] †	49.6	64.0	74.4	78.1	80.0
	PMN w/ H [41] †	54.7	66.8	77.4	81.4	83.8
	ResNet-50 -Few Clean & Many Unlabeled Examples
	Diffusion [5] †	63.6±0.61 69.5±0.60 75.2±0.40 78.5±0.34 80.8±0.18
	Diffusion -logistic [5] † 64.0±0.70 71.1±0.82 79.7±0.38 83.9±0.10 86.3±0.17
	ResNet-50 -Few Clean & Many Noisy Examples
	Ours -class proto. (5) 69.7±0.44 73.7±0.56 77.0±0.20 79.9±0.30 81.9±0.29
	Ours -cosine (6)	78.0±0.38 80.2±0.33 80.9±0.17 83.7±0.19 85.7±0.11
	Ours -fine-tune	80.2±0.33 82.6±0.14 83.3±0.26 85.9±0.22 88.3±0.21

Table 3 .

 3 Comparison with baselines using noisy examples on the Low-shot Places365 benchmark. We report top-5 accuracy on novel classes.

For any set X ⊂ X , we denote by X c its subset of examples labeled in class c ∈ C.

For instance, after training on a different task or a set of classes other than C. Learning of the feature extractor used in our experiments is described in Appendix.

We denote the 2-normalized counterpart of vector x by x. Similarly, if y = f (x), we denote ŷ by f (x).

Acknowledgements. This work is funded by MSMT LL1901 ERC-CZ grant and OP VVV funded project CZ.02.1.01/0.0/0.0/16 019/0000765 "Research Center for Informatics".