Christophe Arnoult

Guillaume Brachet

Diana Cadena Castaneda

Nicolas Azzopardi

Christophe Passot

Celine Desvignes

Gilles Paintaud

Nathalie Heuzé-Vourc'h

Herve Watier

Valérie Gouilleux-Gruart

Nathalie Heuzé-Vourc'h § ¶

Hervé Watier

Crucial Role for Immune Complexes but Not FcRn in Immunization against Anti-TNF-α Antibodies after a Single Injection in Mice

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Anti-TNF- antibodies (Abs) are widely used, and some of them are even blockbuster biopharmaceuticals. Anti-TNF- Abs, like adalimumab or infliximab are indicated in the case of severe rheumatoid arthritis, in first-line treatment or in mild cases when disease-modifying antirheumatic drugs or methotrexate alone have failed to control disease progression [START_REF]Remicade -Summary of product characteristics[END_REF][START_REF][END_REF].

Yet, immunization issues leading to inter-individual variability tarnish their success, frequently forcing clinicians to switch to another treatment line.

Immunogenicity of infliximab and adalimumab is a major concern, since at least 14% and up to a third of the patients develop Abs also called Anti-Drug Abs (ADA), directed against these anti-TNF- after only a few weeks of treatment [START_REF] Hanauer | Maintenance infliximab for Crohn's disease: the ACCENT I randomised trial[END_REF][START_REF] Schaeverbeke | Immunogenicity of biologic agents in rheumatoid arthritis patients: lessons for clinical practice[END_REF]. These ADAs can lead to adverse effects during infusion or to a decrease in biologic concentration, which was associated with lower treatment efficacy (4).

Many factors have been suspected to be involved in the immunization against therapeutic antibodies. Even though some were refuted, like the degree of Ab humanization [START_REF] Van Meer | Immunogenicity of mAbs in non-human primates during nonclinical safety assessment[END_REF], or discussed, like the IgG allotypes [START_REF] Bartelds | Surprising negative association between IgG1 allotype disparity and anti-adalimumab formation: a cohort study[END_REF][START_REF] Magdelaine-Beuzelin | IgG1 heavy chain-coding gene polymorphism (G1m allotypes) and development of antibodies-to-infliximab[END_REF][START_REF] Montes | Rheumatoid arthritis response to treatment across IgG1 allotype -anti-TNF incompatibility: a case-only study[END_REF], several other factors were showed to be involved, like dosing and administration schedule, under-exposure to the anti-TNF- drug [START_REF] Baert | Influence of immunogenicity on the long-term efficacy of infliximab in Crohn's disease[END_REF], coadministration of other drugs [START_REF] Krieckaert | Methotrexate reduces immunogenicity in adalimumab treated rheumatoid arthritis patients in a dose dependent manner[END_REF] and finally formation of aggregates [START_REF] Ahmadi | Small amounts of sub-visible aggregates enhance the immunogenic potential of monoclonal antibody therapeutics[END_REF]. Complexes can be formed between several molecules of the therapeutic Ab (aggregates) or with its target antigen as in classical immune complexes (ICs) to constitute target-related ICs (TRICs) [START_REF] Chaigne | Monoclonal antibodies in excess: A simple way to avoid immunogenicity in patients?[END_REF]. It has been shown that monoclonal Abs (mAbs) are immunogenic in vivo when they form TRICs [START_REF] Jonker | The internal image-like anti-idiotypic response to a CD3-specific monoclonal antibody in primates is dependent on the T cell-binding properties of the injected antibody[END_REF]. This phenomenon can be understood in the light of the recent literature concerning IC presentation by immune cells via the neonatal Fc receptor (FcRn) and with the work of Foss et al. who recently demonstrated that TRICs are more efficiently transcytosed than monomeric mAb via FcRn [START_REF] Foss | Enhanced FcRn-dependent transepithelial delivery of IgG by Fc-engineering and polymerization[END_REF]. These mechanisms bring into play the well described functions of FcRn; extension of IgG half-life, IgG biodistribution and involvement in humoral and anti-tumoral immune response through IC presentation by immune cells [START_REF] Baker | The Role of FcRn in Antigen Presentation[END_REF][START_REF] Baker | Neonatal Fc receptor expression in dendritic cells mediates protective immunity against colorectal cancer[END_REF].

The present work focuses on the ability of TRICs to trigger a primary immune response (i.e. after a single injection) against anti-TNF- antibodies in the presence or absence of FcRn. In order to answer this question, anti-TNF- antibodies were administered to wild type and FcRn knock out mice and the formation of ADA was measured. The first one, infliximab specifically recognizes hTNF- whilst the other one, adalimumab cross-reacts with mTNF-, allowing to evaluate the role of TRICs and FcRn in ADA formation.

Materials and methods

Antibodies

The following mAbs were used: infliximab and golimumab (anti-TNF- IgG1, MSD), adalimumab (anti-TNF- IgG1, Abbvie) and rituximab (anti-CD20 IgG1, Roche).

Administration of anti-TNF- mAb in wild type (WT) and fcgrt-/-(FcRn KO) mice

Six-to eight-week-old C57BL/6 WT mice were obtained from Janvier® (Saint-Berthevin, France). The B6.129X1-Fcgrttm1 Dcr/DcrJ (fcgrt-/-) were generated by the Jackson Laboratory (Bar Harbor, Maine, USA). A targeting vector was designed to replace 1588 nucleotide fragments (encoding promoter sequence 5' of the transcriptional start site, exon 1, intron 2, and most of exon 2) with a PGK-Neor cassette. The vector was electroporated into 129X1/SvJ-derived ESV/J-1182 embryonic stem (ES) cells. Correctly targeted ES cells were injected into recipient C57BL/6J blastocysts. The resulting chimeric animals were crossed to C57BL/6J mice. The mice were then backcrossed to C57BL/6J for 11 generations. All experiments were performed in accordance with national animal care guidelines (EC directive 86/609/CEE, French decree n°87-848), and were approved by the Val-de-Loire Animal Experiments committee of ethics (approval n° 2012-04-09). Body weight was monitored throughout the study, as an indicator of health status. After one week of adaptation period in a same room, WT and FcRn KO mice were anesthetized with isoflurane 2.5% and intravenously injected by retro-orbital route, with a single dose of 4mg/kg adalimumab or infliximab. Blood samples were collected at 2 and 6h, and on days 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14 and 21 after adalimumab or infliximab injection. Blood samples were centrifuged at 1500×g for 15 minutes, and sera were stored at -20°C until measurement of infliximab, adalimumab or ADA levels.

Injection of hTNF- /anti-TNF- mAb immune complexes in WT and FcRn KO mice.

ICs were generated in vitro by mixing anti-TNF- Ab and hTNF- (Peprotech, Neuilly-sur-Seine, France), respectively in a 1:2 molar ratio, and left for 1h at room temperature. The final concentrations in PBS were 500µg/mL for anti-TNF- Ab and 116µg/mL for hTNF-. Two hundred µL were intravenously injected by retro-orbital route. Blood samples were collected before mAb injection and on days 5, 7, 10 and 21 for infliximab and adalimumab, or on day 3, 10 and 20 for golimumab.

Pharmacokinetic (PK) analysis

The serum concentrations of infliximab and adalimumab were measured using validated ELISA techniques [START_REF] Desvignes | Development and validation of an enzyme-linked immunosorbent assay to measure adalimumab concentration[END_REF][START_REF] Ternant | An enzyme-linked immunosorbent assay for therapeutic drug monitoring of infliximab[END_REF]. Briefly, recombinant human TNF- was coated on the solid phase and recognized by infliximab or adalimumab, and the therapeutic antibody was detected by an anti-human IgG Fc-specific antibody conjugated to horseradish peroxidase.

ADA and TNF- ELISA

Ninety six flat-bottomed plates were coated with one µg per well of adalimumab, infliximab or golimumab in 0.1M NaCO 3 /NaHCO 3 buffer (pH9.6) in case of ADA ELISA or with 200ng per well of human or murine TNF-(Peprotech, Neuilly-sur-Seine, France) in TNF-ELISA.

The plates were incubated 12h at 4°C. Washing steps were performed with PBS, 0.1% tween 20, plus 5% casein in blocking step or in mAb dilutions. Serum dilutions (1:100 unless otherwise specified) were incubated at room temperature for 1h. Specific binding was detected by using either peroxidase-labeled donkey anti-mouse IgG (H+L) (Jackson laboratory, Bar Harbor, Maine, USA) for 1h at room temperature in ADA ELISA, peroxidaselabeled goat anti-human IgG (H+L), peroxidase-labeled anti-IgG antibody (anti-Fc, Jackson laboratory, Bar Harbor, Maine, USA) or with a peroxidase-labeled anti-IgM antibody (anti-Fc, Jackson laboratory, Bar Harbor, Maine, USA). The reaction was developed by adding 100μL of tetramethylbenzidine (Sigma Aldrich, Saint Louis, Missouri, USA) for 20min and stopped with 50µl H 2 SO 4 0.5M. Absorbance was measured at 450nm with a Mithras LB 940 Multimode Microplate Reader (Berthold Technologies, Thoiry, France).

Flow cytometry analysis.

Mice spleens were analyzed 2 or 6 days after Ab injection. Single-cell suspensions were isolated with Spleen Dissociation Kits (Miltenyi Biotec, Paris, France). Total leukocyte cell counts were performed and 1×10 6 cells were analyzed by flow cytometry. FcR were first saturated by cell incubation with an anti-FcRII/FcRIII mAb (2.4G2, BD Pharmingen, San Diego, CA) for 20min. Then, cells were incubated with fluorochrome-conjugated specific Abs, in PBS containing 2% SVF and 0.1% sodium azide at 4°C for 60min. Cells were analyzed using a FACS Gallios™ equipped with Kaluza software (Beckman Coulter, Villepinte, France). Anti-mouse CD3-FITC, Ly-6c PE-CF594, CD11c PerCP-Cy5.5, CD45R/B220 PE-Cy7, CD11b APC-Cy7 and corresponding control isotypes were purchased from BD Pharmingen (San Diego, CA).

Statistical analysis

Flow cytometry and ELISA data are expressed as mean and standard deviation of independent experiments. The Mann-Whitney test was used to determine significant differences unless otherwise specified. Statistical analysis was performed using GraphPad Prism 5. The alpha risk was set at 5%. The level of significance is indicated on the figures as * for p≤0.05, ** for p≤0.01 and *** for p≤0.001.

Results

Adalimumab and infliximab have different elimination profiles in WT mice

To measure the relation between anti-TNF- mAb concentrations and the formation of ADA, adalimumab and infliximab concentrations were monitored during 21 days in the serum of WT and KO mice after a single 4 mg/kg mAb injection (Figure 1). As expected, serum concentrations of both mAbs were undetectable after 7 days in FcRn KO mice. An acceleration of mAb elimination, with a nonlinear decrease, was observed especially for adalimumab in WT mice, suggesting the formation of ADA.

ADA follow-up formation reveals immunization against adalimumab but not infliximab in both WT and FcRn KO mice

ADA formation against adalimumab or infliximab was monitored in both WT and FcRn KO mice after a single mAb injection. The results, presented in figure 2, show that ADA are detected earlier in FcRn KO than in WT mice after adalimumab injection (day 4 and day 6 respectively). Even though ADAs occurred earlier in FcRn KO mice, the titers obtained in WT mice seem to be higher (Figure 2C). The two main mechanistic explanations for this phenomenon are a booster role of FcRn in the humoral response, leading to increased Ab production in WT mice, and the protection of these IgGs from lysosomal degradation by FcRn. ADA detection with an anti-mouse-IgG-specific or an anti-mouse-IgM-specific antibody revealed that the response is mainly an IgG response with very low IgM levels. By contrast, infliximab did not induce any ADA response up to 21 days after injection in both WT and FcRn KO mice (data not shown).

TRICs are involved in ADA development

Our hypothesis was that infliximab did not induce ADA due to the lack of TRIC formation with the endogenous mTNF-, whilst adalimumab cross-reacts with mTNF-and therefore generates TRICs following injection. First, the species-specificity of anti-TNF- mAbs was assessed. As shown in figure 3, adalimumab binds both human and murine TNF- while infliximab only binds human TNF-. Rituximab, an anti-CD20 mAb, was used as negative control. Then, infliximab-containing TRICs were constituted in vitro by mixing human TNF- and infliximab prior to injection to WT and FcRn KO mice. No ADA could be detected by ELISA in WT or FcRn KO mice injected with infliximab alone. On the contrary, high amounts of infliximab ADA were measured in WT as well as in FcRn KO mice injected with in vitro-formed infliximab-containing TRICs (figure 4). No anti-TNF- Abs were detected in mice injected with recombinant human TNF- alone. Altogether these data showed that artificially formed TRICs are able to elicit immune response to infliximab. In FcRn KO mice, ADA formation has different kinetics depending on the injected mAb with anti-adalimumab Abs appearing more rapidly than anti-infliximab Abs, whilst ADA decreased more quickly in FcRn KO mice than in WT mice for both mAbs. We injected a 10 fold higher concentration of adalimumab (40mg/kg), and showed that immunization occurred later in these mice, suggesting that dilution of the immune complexes in a tide of uncomplexed antibodies delays ADA formation.

Degree of humanization is not responsible for adalimumab immunogenicity

The lack of immunogenicity of infliximab alone could be due to the fact that its variable domains are murine while adalimumab has human variable domains. To address this question, a new set of experiments was performed with golimumab, a fully-human therapeutic anti-TNF- mAb, which is specific to human TNF-, as shown in figure 5A. Golimumab was injected once, either alone or after in vitro IC generation by incubation of golimumab with human TNF-. Golimumab alone did not elicit ADA response in WT mice, in contrast to preformed golimumab-TRICs (Figure 5B). These results were confirmed by repeating the experiment and performing ELISA with an anti-IgG HRPO-labeled murine-Fc-specific antibody (data not shown).

TRICs induced a cellular immune response in both WT and FcRn KO mice

Spleens of WT and FcRn KO animals injected with adalimumab were harvested on day 2 or day 6 in order to examine the cellular population. Spleens from naive animals were used as control ("day 0" in figure 6). As showed in figure 6A, the size of the spleen increased 2 days after adalimumab injection with a largely more pronounced effect in FcRn KO mice. This phenomenon was described in all mice by total splenocyte count (figure 6A), and a return to a normal count was observed on day 6. Flow cytometry analysis revealed that the increase in cell count on day 2 after adalimumab injection was observed for B cells (B220+), T lymphocytes (CD3+), dendritic cells (Ly6c+/CD11b+) and neutrophils (CD11c+/CD11b+) with a 1.43 to 1.97 ratio (day 2/day 0), compiling all FcRn KO and WT mice data (figure 6B). As expected, ratios were higher in FcRn KO (1.73 to 2.24) as compared to WT mice (1.40 to 1.57) for all cell types. This increase in cell count was more pronounced for B cell ratio (2.24 in FcRn KO versus 1.45 in WT mice), and dendritic cells (1.97 in FcRn KO versus 1.40 in WT mice), followed by a milder increase in neutrophils (1.84 in FcRn KO versus 1.43 in WT mice) and T cells (1.73 in FcRn KO versus 1.57 in WT mice).

Discussion

The immunogenicity of therapeutic mAbs is a complex phenomenon that can occur during treatment with therapeutic mAbs leading to the formation of ADA-related ICs (ARICs) and a profound alteration in mAb pharmacokinetics in most patients [START_REF] Baert | Influence of immunogenicity on the long-term efficacy of infliximab in Crohn's disease[END_REF][START_REF] Bendtzen | Individualized monitoring of drug bioavailability and immunogenicity in rheumatoid arthritis patients treated with the tumor necrosis factor α inhibitor infliximab[END_REF][START_REF] Ducourau | Antibodies toward infliximab are associated with low infliximab concentration at treatment initiation and poor infliximab maintenance in rheumatic diseases[END_REF]. These ARICs are defined as complexes formed by the binding of ADA to mAbs, particularly to the mAb idiotypes, since paratopes remain immunogenic whatever the degree of humanization of mAbs [START_REF] Van Meer | Immunogenicity of mAbs in non-human primates during nonclinical safety assessment[END_REF]. This phenomenon has been evidenced in patients treated with infliximab or adalimumab and measurement of ADA is often a warning sign of upcoming treatment failure after an initial response [START_REF] Baert | Influence of immunogenicity on the long-term efficacy of infliximab in Crohn's disease[END_REF][START_REF] Bartelds | Development of antidrug antibodies against adalimumab and association with disease activity and treatment failure during long-term follow-up[END_REF][START_REF] Krintel | The frequency of anti-infliximab antibodies in patients with rheumatoid arthritis treated in routine care and the associations with adverse drug reactions and treatment failure[END_REF]. In order to prevent immunization, it is necessary to understand the mechanisms leading to the formation of ADA. Our study shows that TRICs but not FcRn are essential to trigger an immune response (i.e. after a single injection) against anti-TNF- antibodies. Moreover, although the crucial role played by FcRn in the development of humoral immune responses has been demonstrated by several studies [START_REF] Liu | The neonatal FcR-mediated presentation of immune-complexed antigen is associated with endosomal and phagosomal pH and antigen stability in macrophages and dendritic cells[END_REF][START_REF] Baker | Neonatal Fc receptor for IgG (FcRn) regulates cross-presentation of IgG immune complexes by CD8-CD11b+ dendritic cells[END_REF], our data fit better with a booster role for FcRn [START_REF] Qiao | Dependence of antibody-mediated presentation of antigen on FcRn[END_REF]. Most of these studies being carried out using ovalbumin because the T-epitopes are well characterized and can be followed-up, the specific case of anti-TNF- antibodies is different. In our study, the antigens are antibodies which form TRICs that elicit an immune response after the first injection (or not in the case of infliximab). This point could modify the kinetics of the immune response. In our animal model, we first confirmed that the presence of ADA modifies the pharmacokinetics of therapeutic antibodies. Adalimumab displayed a faster log-linear elimination than infliximab but, above all, an accelerated terminal elimination, characteristic of the development of ADA [START_REF] Chen | A mathematical model of the effect of immunogenicity on therapeutic protein pharmacokinetics[END_REF][START_REF] Ng | The genetic variants underlying breast cancer treatment-induced chronic and late toxicities: a systematic review[END_REF]. This acceleration was observed in all mice treated with adalimumab, independently of the FcRn status even if elimination was clearly faster in FcRn KO mice [START_REF] Chen | The effect of the neonatal Fc receptor on human IgG biodistribution in mice[END_REF]. Adalimumab binds mouse TNF-asdescribed in the European Medicine Agency document (29) and confirmed byELISA. We hypothesized that with adalimumab treatment TRICs would be formed and could in turn cause immunization. To address this question, we incubated infliximab and TNF- to form TRICs, which were able to induce anti-infliximab Abs after injection in both WT and FcRn KO mice.

The antibodies used differ in several ways, and one could wonder whether the results could be biased due to confounding factors. Adalimumab is a fully human therapeutic mAb. In order to discard the possibility that immunization could be a result of the presence of human portions in the protein sequence, we performed similar experiments with golimumab, another fully human anti-TNF-mAb very specific for human TNF-The absence of anti-golimumab Abs supports the idea that mAb humanization is not responsible in the development of ADA in our model. However, as observed with infliximab, golimumab-containing TRICs injected to mice were able to elicit an anti-golimumab response. Apart from the degree of humanization, allotypes have been suspected to participate in the immunization process in chronic inflammatory diseases, even though the results were negative [START_REF] Bartelds | Surprising negative association between IgG1 allotype disparity and anti-adalimumab formation: a cohort study[END_REF]. All the antibodies used here shared the same allotype (G1m17; 1), which discards this hypothesis. Finally, the intrinsic characteristics of the TRICs themselves, such as their size [START_REF] Qiao | Dependence of antibody-mediated presentation of antigen on FcRn[END_REF], are involved in their trafficking and condition the type of immune response elicited [START_REF] Vidarsson | FcRn: an IgG receptor on phagocytes with a novel role in phagocytosis[END_REF]. But since FcRn -/-mice also mount an immune response against these ICs discards this possibility. Moreover, infliximab incubated with hTNF- also elicits an immune response. Altogether, these data confirmed that TRICs are necessary to elicit a primary immune response in the case of anti-TNF- antibodies.

FcRn is involved in the recycling and transcytosis of IgGs and monovalent immune complexes [START_REF] Baker | Crosspresentation of IgG-containing immune complexes[END_REF]. But surprisingly, ADAs were detected in both WT and KO FcRn mice suggesting that FcRn is not necessary for immunization against anti-TNF- Abs. Even more, anti-adalimumab Abs appeared more rapidly in FcRn KO mice than in WT animals. Several studies performed in FcRn KO models show that mAb pharmacokinetics is clearly impaired and that IgG biodistribution is differently affected depending of the organ tested [START_REF] Chen | The effect of the neonatal Fc receptor on human IgG biodistribution in mice[END_REF][START_REF] Guilleminault | Fate of inhaled monoclonal antibodies after the deposition of aerosolized particles in the respiratory system[END_REF]. We hypothesized that, in absence of FcRn, adalimumab would not be recycled and that, its biodistribution outside the vascular compartment being altered, it would be fully available to form ICs with its target, especially in places such as mesenchymal lymph nodes, where ADA concentrations remain sufficient to form TRICs due to resident plasma cells as described by Baker et al. [START_REF] Baker | Neonatal Fc receptor expression in dendritic cells mediates protective immunity against colorectal cancer[END_REF]. TRICs would then be able to rapidly induce an immune response, characterized by IgG and low IgM ADAs, probably via FcR binding and MHC class II presentation [START_REF] Baker | Crosspresentation of IgG-containing immune complexes[END_REF][START_REF] Amigorena | Type II and III receptors for immunoglobulin G (IgG) control the presentation of different T cell epitopes from single IgG-complexed antigens[END_REF][START_REF] Simitsek | Modulation of antigen processing by bound antibodies can boost or suppress class II major histocompatibility complex presentation of different T cell determinants[END_REF]. Another non-exclusive hypothesis is that the concentrations of free adalimumab are lower in FcRn -/-mice, and that their tolerogenic effect over TRICs is therefore shorter. Although dependence of Ab-mediated presentation of Ag on FcRn has been described in the case of secondary humoral response [START_REF] Baker | The Role of FcRn in Antigen Presentation[END_REF][START_REF] Qiao | Dependence of antibody-mediated presentation of antigen on FcRn[END_REF], its absence could probably be overcome by other molecules for IC routing and MHC class II presentation as suggested in our model. The ADA response has a cellular component as observed by the increase in dendritic/monocyte counts as well as B or T-cell counts in the spleen of immunized animals. This cellular immune response is necessary to lead to the humoral response and is proportional to the amount of ADA generated. Chen et al. have shown that in the liver or the spleen, FcRn may exert its transcytosis function to transport IgGs out of the tissue [START_REF] Chen | The effect of the neonatal Fc receptor on human IgG biodistribution in mice[END_REF]. In the absence of FcRn, mAbs/TRICs might be retained in the spleen, thus participating in the increased immune response. Simultaneously, the absence of FcRn leading to accelerated clearance of adalimumab could also explain the faster clearance of ADA in FcRn KO mice as compared to WT. The involvement of this mechanism in ADA kinetic differences is sustained by the kinetics observed with preformed TRICs in our model. In this case, anti-infliximab Ab formation occurs without delay. The question remains as to which FcRs participate in this immune response. The cellular counts performed on the harvested spleen show an increase in multiple cell populations 2 days after the injection of preformed TRICs among which dendritic cells (CD11c+/CD11b+), neutrophils and monocytes (LY6c+/CD11b+). These cells express the activating murine FcRI (monocytes, dendritic cells), FcRIII and FcRIV (dendritic cells, neutrophils and monocytes) that could all play a role in this process.

However, because of the numerous differences between humans and mice in terms of FcR, [START_REF] Bruhns | Mouse and human FcR effector functions[END_REF] caution must be used for extrapolation to the clinical situation.

In humans, ADAs have been detected in a large number of patients treated with anti-TNF-mAbs. Under-exposure to mAbs has been clearly demonstrated in patients as a risk factor of immunization [START_REF] Bendtzen | Individualized monitoring of drug bioavailability and immunogenicity in rheumatoid arthritis patients treated with the tumor necrosis factor α inhibitor infliximab[END_REF][START_REF] Ducourau | Antibodies toward infliximab are associated with low infliximab concentration at treatment initiation and poor infliximab maintenance in rheumatic diseases[END_REF]. This approach does not take into account the evaluation of specific TRICs, which are known to skew the dosage of ADAs in the serum [START_REF] Van Schouwenburg | A novel method for the detection of antibodies to adalimumab in the presence of drug reveals "hidden" immunogenicity in rheumatoid arthritis patients[END_REF]. The nature of the target may also participate to the formation of ICs. In that context, TNF- is a good candidate since it is composed of a trimer and exists as soluble Ag that is more prone to form multimeric ICs. The three antibodies used here target TNF- in different ways.

Adalimumab and infliximab stabilize TNF- as a trimer, while golimumab binding does not prevent monomer exchange with unbound trimers [START_REF] Van Schie | Therapeutic TNF Inhibitors can Differentially Stabilize Trimeric TNF by Inhibiting Monomer Exchange[END_REF], which could explain the rather important variability observed in the anti-golimumab responses. This could explain the higher levels of ADA found with anti-TNF- than with biologics directed against other molecules, along with the inflammatory profile of the pathologies involving high titers of TNF-.

It is important to bear in mind that this study was performed using FcRn KO mice, meaning that the bioavailability of the TRICs is altered compared to WT mice, and that full FcRn deficiency has never been evidenced so far in humans.

It has been documented that low concentrations of anti-TNF- antibodies is associated with increased ADA formation in patients [START_REF] Bartelds | Development of antidrug antibodies against adalimumab and association with disease activity and treatment failure during long-term follow-up[END_REF], and hypothesized that maintaining high concentrations of monomeric antibody could prevent the immunization process [START_REF] Chaigne | Monoclonal antibodies in excess: A simple way to avoid immunogenicity in patients?[END_REF]. In the case of anti-TNF- therapy and primary immune response in mice, we show that this is true.

Foonotes: 1 1-

 1 This work was supported by a public grant overseen by the French National Research 2 Agency (ANR) as part of the "Investissements d'Avenir" program (reference: ANR-10-3 LABX -53-01) and by European funds (FEDER Grant Agreement: Presage 4940-37478; 4 OutExFon). Measurement of infliximab serum concentrations were carried out within the 5 CePiBAc platform. CePiBAc is cofinanced by the European Union. Europe is committed to 6 the region Centre with the European Regional Development Fund.

7 2 -

 2 Abbreviations used in this article: FcRn: neonatal IgG Fc receptor; mAb: monoclonal 8 antibody; ADA: Anti-drug Abs; ICs: immune complexes; TRICs: target-related ICs; WT 9 mice: wild type mice; FcRn KO mice: fcgrt-/-mice.

10 11 Figure legends Figure 1 :

 1011legends1 Figure legends

16 17Figure 2 :

 162 Figure 2: Detection of ADA against adalimumab and infliximab in WT (black lines) and

21 A. 26 and 1 :

 21261 Detection of IgM ADA against adalimumab in WT (n=7) and FcRn KO mice (n=8). ADA 22 formation was evaluated by blood sampling over three weeks and measured by ELISA in the 23 serum (serum dilution 1:100). B. and C. Detection of IgG ADA against adalimumab in WT 24 (n=7) and FcRn KO mice (n=8). ADA formation was evaluated by blood sampling over three 25 weeks and measured by ELISA in the serum, at two different serum dilutions (1:100 for B 1000 for C). D. Detection of IgG ADA against adalimumab at 4mg/kg in WT (n=7) and 27 FcRn KO mice (n=8) or at 40mg/kg in WT mice (n=7). ADA formation was evaluated by 28 blood sampling over three weeks and measured by ELISA in the serum (diluted at 1:100). P-29 values are represented as follows: * for p < 0.05, ** for p < 0.01 and *** for p<0.001.

30 31Figure 3 :

 303 Figure 3: Analysis of anti-TNF- mAb specificity by ELISA. Adalimumab and infliximab

Figure 4 :

 4 Figure 4: In vitro-formed infliximab containing TRICs leads to infliximab ADA

Figure 5 :

 5 Figure 5: A) Analysis of golimumab specificity for human TNF- by ELISA. B) Injection of

Acknowledgements

The authors would like to thank the animal facility of Université François Rabelais for their kind help.

Disclosure

The authors declare that the research was conducted in the absence of any financial relationships that could be construed as a potential conflict of interest.

WT mice with golimumab alone or with in vitro-formed TRICs with golimumab and human


Virgile Robles

Nikolai Kosmatov

Virgile Prevosto

Louis Rilling
email: louis.rilling@irisa.fr

Pascale Le Gall
email: pascale.legall@centralesupelec.fr

MetAcsl: Specification and Verification of High-Level Properties

Modular deductive verification is a powerful technique capable to show that each function in a program satisfies its contract. However, function contracts do not provide a global view on which high-level (e.g. security-related) properties of a whole software module are actually established, which makes it very difficult to express such properties. To address this issue, this paper proposes a new specification mechanism, called meta-properties, able to express a rich set of high-level properties. A meta-property can be seen as an enhanced global invariant specified for all or a subset of functions, and capable to express predicates on values of variables, as well as memory related conditions (such as separation) and read or write access constraints. We also propose an automatic transformation technique translating meta-properties into usual contracts and assertions, that can be proved by deductive verification tools in a usual way. This technique has been implemented as a Frama-C plugin called MetAcsl and successfully applied to specify and prove several safety-and security-related meta-properties in two illustrative case studies.

Introduction

Modular deductive verification is a well-known technique for formally proving that a program respects some user-defined properties. It consists in providing for each function of the program a contract, which basically contains a precondition describing what the function expects from its callers, and a postcondition indicating what it guarantees when it successfully returns. Logical formulas, known as verification conditions or proof obligations (POs), can then be generated and given to automated theorem provers. If all POs are validated, the body of the function fulfills its contract. Many deductive verification frameworks exist for various programming and formal specification languages. We focus here on Frama-C [START_REF] Kirchner | Frama-C: A software analysis perspective[END_REF] and its deductive verification plugin Wp, which allows proving a C program correct with respect to a formal specification expressed in ACSL [START_REF] Kirchner | Frama-C: A software analysis perspective[END_REF].

However, encoding high-level properties spanning across the entire program in a set of Pre/Post-based contracts is not always immediate. In the end, such high-level properties get split among many different clauses in several contracts, without an explicit link between them. Therefore, even if each individual clause is formally proved, it might be very difficult for a verification engineer, a code reviewer or a certification authority to convince themselves that the provided contracts indeed ensure the expected high-level properties. Moreover, a software product frequently evolves during its lifetime, leading to numerous modifications in the code and specifications. Maintaining a high-level (e.g. security-related) property is extremely complex without a suitable mechanism to formally specify and automatically verify it after each update.

The purpose of the present work is to propose such a specification mechanism for high-level properties, which we call meta-properties, and to allow their automatic verification on C code in Frama-C thanks to a new plugin called MetAcsl. Motivation This work was motivated by several previous projects. During the verification of a hypervisor, we observed the need for a mechanism of specification and automatic verification of high-level properties, in particular, for global properties related to isolation and memory separation. Isolation properties are known as key properties in many verification projects, in particular, for hypervisors and micro-kernels.

A similar need for specific high-level properties recently arose from a case study on a confidentiality-oriented page management system submitted by an industrial partner. In this example, each page and each user (process) are given a confidentiality level, and we wish to specify and verify that in particular:

-(P read ) a user cannot read data from a page with a confidentiality level higher than its own; -(P write ) a user cannot write data to a page with a confidentialtiy level lower than its own. This case study will be used as a running example in this paper. As a second case study (also verified, but not detailed in this paper), we consider a simple smart house manager with several interesting properties such as: "a door can only be unlocked after a proper authentication or in case of alarm" or "whenever the alarm is ringing, all doors must be unlocked". Again, these examples involve properties that are hard to express with function contracts since they apply to the entire program rather than a specific function. 1Contributions The contributions of this paper2 include:

a new form of high-level properties, which we call meta-properties, and an extension of the ACSL language able to express them (Sect. 2), a set of code transformations to translate meta-properties into native ACSL annotations that can be proved via the usual methods (Sect. 3), a Frama-C plugin MetAcsl able to parse C code annotated with metaproperties and to perform the aforementioned code transformations (Sect. 4), a case study: a confidentiality-oriented page system, where important security guarantees were expressed using meta-properties and automatically verified thanks to the code transformation with MetAcsl (Sect. 4).

Specification of Meta-properties

A meta-property is a property meant to express high-level requirements. As such, it is not attached to any particular function but instead to a set of functions. It is thus defined in the global scope and can only refer to global objects.

To define a meta-property, the user must provide (i) the set of functions it will be applied to, (ii) a property (expressed in ACSL) and (iii) the context, i.e. a characterization of the situations in which they want the property to hold in each of these functions (everywhere in the function, only at the beginning and the end, upon writing in a variable, etc.). Furthermore, depending on the context, the property can refer to some special variables which we call meta-variables. Figure 1 features a few examples of meta-properties further explained below.

Let F denote the set of functions defined in the current program, and P the set of native ACSL properties. Formally, we can define a meta-property as a triple (c, F, P ), where c is a context (see Sect. 2.2), F ⊆ F and P ∈ P. Intuitively, we can interpret this triple as "∀f ∈ F , P holds for f in the context c". For the meta-property to be well-formed, P must be a property over a subset of G ∪ M(c), where G is the set of variables available in the global scope of the program and M(c) is the set of meta-variables provided by the context c.

The actual MetAcsl syntax for defining a meta-property (c, F, P ) is meta [specification of F] c, P; An example is given by property M 1 (cf. lines 10-12 in Figure 1), where F = F, c = strong_invariant and P is the predicate stating that the status of any page should be either FREE or ALLOCATED.

Target Functions and Quantification

Meta-properties are applied to a given target set of functions F defined as F = F + \F -by providing explicit lists of considered and excluded functions F + , F -⊆ F. If not provided, F + and F -are respectively equal to F and ∅ by default, i.e. the meta-property should hold for all functions of the program. F -is useful when the user wants to target every function except a few, since they do not have to explicitly provide every resulting target function.

The MetAcsl syntax for the specification of F uses the built-in ACSL construction \forall, possibly followed by \subset with or without logic negation ! (to express f ∈ F + and f / ∈ F -). It can be observed in property M 2 (lines 13-16), where F + = F and F -= {page_encrypt} excludes only one function.

Notion of Context

The context c of a meta-property defines the states in which property P must hold, and may introduce meta-variables that can be used in the definition of P .

Beginning/Ending Context (Weak Invariant) A weak invariant indicates that P must hold at the beginning and at the end of each target function f ∈ F .

Everywhere Context (Strong invariant) A strong invariant is similar to a weak invariant, except that it ensures that P holds at every point3 of each target function. For example, property M 1 specifies that at every point of the program, the status of any page must be either FREE or ALLOCATED.

Writing Context This ensures that P holds upon any modification of the memory (both stack and heap). It provides a meta-variable \written that refers to the variable (and, more generally, the memory location) being written to.

A simple usage of this context can be to forbid any direct modification of some global variable, as in property M 2 . This property states that for any function that is not page_encrypt, the left-hand side of any assignment must be separated from (that is, disjoint with) the global variable metadata[page].level for any page with the ALLOCATED status. In other words, only the page_encrypt function is allowed to modify the confidentiality level of an allocated page.

An important benefit of this setting is a non-transitive restriction of modifications that cannot be specified using the ACSL clause assigns, since the latter is transitive over function calls and necessarily permits to modify a variable when at least one callee has the right to modify it. Here, since we only focus on direct modifications, a call to page_encrypt (setting to public the level of the page it has encrypted) from another function does not violate meta-property M 2 .

Furthermore, modification can be forbidden under some condition (i.e. that the page is allocated), while assigns has no such mechanism readily available.

Reading Context Similar to the writing context, this ensures that the property holds whenever some memory location is read, and provides a meta-variable \read referring to the read location. It is used in property M 3 (lines 17-20), which expresses the guarantee P read of the case study (see Sec. 1) by imposing a separation of a read location and the contents of allocated confidential pages when the user does not have sufficient access rights. As another example, an isolation of a page can be specified as separation of all reads and writes from it. These few simple contexts, combined with the native features of ACSL, turn out to be powerful enough to express quite interesting properties, including memory isolation and all properties used in our two motivating case studies.

Verification of Meta-properties

Figure 2 shows an (incorrect) toy implementation of two functions of Figure 1 that we will use to illustrate the verification of meta-properties M 1 -M 3 .

The key idea of the verification is the translation of meta-properties into native ACSL annotations, which are then verified using existing Frama-C analyzers. To that end, the property P of a meta-property (c, F, P ) must be inserted as an assertion in relevant locations (as specified by context c) in each target function f ∈ F , and the meta-variables (if any) must be instantiated. We define a specific translation for each context. For weak invariants, property P is simply added as both a precondition and a postcondition in the contract of f . This is also done for the strong invariant, for which P is additionally inserted after each instruction potentially modifying the values of the free variables in P4 For example, Figure 3a shows the translation of M 1 on page_alloc. Our property (defined on lines 11-12 in Figure 1, denoted P M1 here) is inserted after the modification of a status field (line 6) since the property involves these objects, but not after the modification of a level field (line 8).

For Writing (resp. Reading) contexts, P is inserted before any instruction potentially making a write (resp. read) access to the memory, with the exception of function calls. In addition, each meta-variable is replaced by its actual value. For example, in the translation of M 2 on page_alloc (Figure 3b), the property is inserted before the two modifications of fp, and \written is replaced respectively by fp->status and fp->level. In this case M 2 does not hold. While its first instantiation (lines 4-6) is easily proved, it is not the case for the second one (lines 8-10). Indeed, there exists a page (the one being modified) that has a status set to ALLOCATED because of the previous instruction (line 7) and for which the \separated clause is obviously false. Hence, the assertion fails, meaning that the whole meta-property M 2 cannot be proved. The fix consists in swapping lines 6 and 7 in Figure 2. After that, all assertions generated from M 2 are proved.

A similar transformation for M 3 on page_read shows that the proof fails since the implementation allows an agent to read from any page without any check. Adding proper guards allows the meta-property to be proved. Conversely, if a meta-property is broken by an erroneous code update, a proof failure after automatically re-running MetAcsl helps to easily detect it.

Results on Case Studies and Conclusion

Experiments The support of meta-properties and the proposed methodology for their verification were fully implemented in OCaml as a Frama-C plugin called MetAcsl. We realized a simple implementation of the two case studies mentioned in Sect. 1) and were able to fully specify and automatically verify all aforementioned properties (in particular P read and P write ) using MetAcsl. The transformation step is performed in less than a second while the automatic proof takes generally less than a minute.

Conclusion

We proposed a new specification mechanism for high-level properties in Frama-C, as well as an automatic transformation-based technique to verify these properties by a usual deductive verification approach. The main idea of this technique is similar to some previous efforts e.g. [START_REF] Pavlova | Enforcing High-Level Security Properties for Applets[END_REF] . Meta-properties provide a useful extension to function contracts offering the possibility to express a variety of high-level safety-and security-related properties. They also provide a verification engineer with an explicit global view of high-level properties being really proved, avoiding the risk to miss some part of an implicit property which is not formally linked to relevant parts of several function contracts, thus facilitating code review and certification. Another benefit of the new mechanism is the possibility to easily re-execute a proof after a code update. Initial experiments confirm the interest of the proposed solution.

Future Work We plan to establish a formal soundness proof for our transformation technique, thereby allowing MetAcsl to be reliably used for critical code verification. Other future work directions include further experiments to evaluate the proposed approach on real-life software and for more complex properties.

1Fig. 1 :

 1 Fig. 1: Partial meta-specification of a confidentiality case study

1 2 /Fig. 2 :

 22 Fig. 2: Incorrect code w.r.t. M 2 and M 3

1 2 structFig. 3 :

 23 Fig.3: Examples of code transformations for functions of Figure2

These examples are publicly available at https://huit.re/metacas.

A longer version is available at https://arxiv.org/abs/1811.10509

More precisely, every sequence point as defined by the C standard.

The AST is normalized so that every memory modification happens through an assignment. Then we conservatively determine if the object being assigned is one of the free variables of P : in presence of pointers, we assume the worst case.

Acknowledgment This work was partially supported by the project VESSEDIA, which has received funding from the EU Horizon 2020 research and innovation programme under grant agreement No 731453. The work of the first author was partially funded by a Ph.D. grant of the French Ministry of Defense. Many thanks to the anonymous referees for their helpful comments.


Michel Barlaud

Antonin Chambolle

Jean-Baptiste Caillau

Robust supervised classification and feature selection using a primal-dual method

This paper deals with supervised classification and feature selection in high dimensional space. A classical approach is to project data on a low dimensional space and classify by minimizing an appropriate quadratic cost. A strict control on sparsity is moreover obtained by adding an 1 constraint, here on the matrix of weights used for projecting the data. It is well known that using a quadratic cost is not robust to outliers. We cope with this problem by using an 1 norm both for the constraint and for the loss function. Another second issue is that we optimize simultaneously the projection matrix and the centers used for classification. In this paper, we provide a novel tailored constrained primal-dual method to compute jointly selected features and classifiers. Extending our primal-dual method to other criteria is easy provided that efficient projection (on the dual ball for the loss data term) and prox (for the regularization term) algorithms are available. We illustrate such an extension in the case of a Frobenius norm for the loss term. We provide a convergence proof of our primal-dual method, and demonstrate its effectiveness on three datasets (one synthetic, two from biological data) on which we compare 1 and 2 costs.

Introduction

In this paper we consider methods where feature selection is embedded into a classification process, see [START_REF] Furey | Support vector machine classification and validation of cancer tissue samples using microarray expression data[END_REF][START_REF] Guyon | Gene selection for cancer classification using support vector machines[END_REF]. However, classification in high dimension space suffers from the curse of dimensionality [START_REF] Aggarwal | On k-anonymity and the curse of dimensionality[END_REF][START_REF] Radovanovic | Hubs in space: Popular nearest neighbors in high-dimensional data[END_REF]. In order to overcome this issue, the main idea of Linear Discriminant Analysis (LDA) (de la [START_REF] De La Torre | Discriminative cluster analysis[END_REF][START_REF] Ding | Adaptive dimension reduction using discriminant analysis and k-means clustering[END_REF] is to project data into a lower dimensional space. In parallel, sparse learning based methods have received a great attention in the last decade because of their high performance. The basic idea is to use a sparse regularizer which forces coefficients to be zero. To achieve feature selection, the Least Absolute Shrinkage and Selection Operator (LASSO) formulation [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF][START_REF] Hastie | The entire regularization path for the support vector machine[END_REF][START_REF] Ng | Feature selection, l 1 vs. l 2 regularization, and rotational invariance[END_REF][START_REF] Friedman | Regularization path for generalized linear models via coordinate descent[END_REF][START_REF] Hastie | Statistcal learning with sparsity: The lasso and generalizations[END_REF] adds an 1 penalty term to the classification cost, which can be interpreted as convexifying an 0 penalty [START_REF] Donoho | Optimally sparse representation in general (nonorthogonal) dictionaries via 1 minimization[END_REF][START_REF] Donoho | Compressed sensing[END_REF][START_REF] Candès | Enhancing sparsity by reweighted l1 minimization[END_REF]. However, an issue is that using the Frobenius norm Y µ -XW F (that is the 2 norm of the vectorized matrix) for the data term is not robust to outliers. (In the previous expression, W is the projection matrix, µ the matrix of centers, and Y the binary matrix mapping each line to its class; see Section 1.) In order to overcome this issue, Nie et al proposed a feature selection based on 2,1 norm minimization (Nie et al., 2010) on both loss function and the regularization. Note that 2,1 norm regularization has strong connections with group LASSO methods [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF][START_REF] Zou | Sparse principal component analysis[END_REF][START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF][START_REF] Li | Network-constrained regularization and variable selection for analysis of genomic data[END_REF][START_REF] Jacob | Group lasso with overlap and graph lasso[END_REF][START_REF] Liu | A robust and efficient doubly regularized metric learning approach[END_REF][START_REF] Simon | A sparse-group lasso[END_REF][START_REF] Hastie | Statistcal learning with sparsity: The lasso and generalizations[END_REF][START_REF] Li | Feature selection: A data perspective[END_REF]. In this paper, we propose a more drastic approach that uses an 1 norm both on the regularization term and on the loss function Y µ -XW 1 . In this case, the criterion is convex but not gradient Lipschitz. The basic idea is to use a splitting method [START_REF] Lions | Splitting algorithms for the sum of two nonlinear operators[END_REF] together with proximal methods. Proximal methods were introduced in [START_REF] Moreau | Proximité et dualité dans un espace hilbertien[END_REF] and have been intensively used in signal processing; see, e.g., [START_REF] Combettes | Signal recovery by proximal forward-backward splitting[END_REF][START_REF] Chaux | Nested iterative algorithms for convex constrained image recovery problems[END_REF][START_REF] Mosci | Solving structured sparsity regularization with proximal methods[END_REF][START_REF] Liu | Moreau-yosida regularization for grouped tree structure learning[END_REF][START_REF] Combettes | Proximal splitting methods in signal processing[END_REF][START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF][START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF][START_REF] Sra | Scalable nonconvex inexact proximal splitting[END_REF][START_REF] O'connor | Primal-dual decomposition by operator splitting and applications to image deblurring[END_REF][START_REF] Chambolle | On the ergodic convergence rates of a first-order primal-dual algorithm[END_REF][START_REF] Flammarion | Robust discriminative clustering with sparse regularizers[END_REF]. The first issue is the computation of the proximal operator involving the affine transform Y µ -XW in the criterion. We tackle this point by dualizing the norm computation and use a primal-dual method. When one uses an 1 penalization to ensure sparsity, the computational time due to the treatment of the corresponding hyper-parameter has a worst case in O(3 d ), see [START_REF] Mairal | Complexity analysis of the lasso regularization path[END_REF]. We propose instead a constrained approach that takes advantage of an available ef-ficient projection on the 1 ball [START_REF] Condat | Fast projection onto the simplex and the l1 ball[END_REF][START_REF] Duchi | Efficient projections onto the l 1-ball for learning in high dimensions[END_REF].

The paper is organized as follows. We first present our setting that combines dimension reduction, classification and feature selection. Then we propose a Lagrangian primaldual scheme in Section 2. An efficient alternative is to replace the 1 penalization by a hard constraint: we provide in Section 3 an updated primal-dual scheme for this constrained formulation of the classification problem. Section 5 is devoted to analyze the convergence of this approach. In the last section, we give some experimental comparisons between 1 and Frobenius loss function. The tests involve three different bases: the first one is a synthetic dataset; the second base is a proteomics dataset on patients with ovarian or prostate cancer; the third and last database comes from single cell analysis.

Problem Statement

Projection of the data on a low dimensional space

Let X be the data m × d matrix made of m line samples x 1 , . . . , x m belonging to the d-dimensional space of features. Let Y ∈ {0, 1} m×k be the label matrix where k 2 is the number of clusters. Each line of Y has exactly one nonzero element equal to one, y ij = 1 indicating that the sample x i belongs to the j-th cluster. Projecting the data in lower dimension is crucial to be able to separate them accurately. Let W ∈ R d×k be the projection matrix, k d. (Note that the dimension of the projection space is equal to the number of clusters.) One of the goals is to compute the projection matrix W .

Robust classification using 1 centers

In order to build a robust classifier for the projected data, XW , we compute an 1 -center for each cluster. This is more robust to outliers than the outcome of the classical 2 modelling [START_REF] Witten | A framework for feature selection in clustering[END_REF]. It is standard to define the 1 -medoid µ j of the j-th cluster as a vector that minimizes the average dissimilarity inside the class in the projected space: µ j := (XW )(i * , :) where

i * = arg min i s.t. yij =1 l (XW )(i, :) -(XW )(l, :) 1 .
By analogy, we compute the matrix of centers, µ ∈ R k×k , by minimizing the following 1 norm:

min µ Y µ -XW 1 .

Primal-dual scheme, Lagrangian formulation

We propose to minimize 1 loss cost with an 1 penalty term (a Lagrangian parameter λ is introduced) so as to promote sparsity and induce feature selection. So, given the matrix of labels, Y , and the matrix of data, X, we consider the following convex supervised classification problem where both µ and W are unknowns and I k the identity matrix:

min (W,µ) Y µ -XW 1 + λ W 1 + ρ 2 I k -µ 2 F . (1)
Note that an 2 -regularization term has been added in order to avoid the trivial solution µ = 0, W = 0. (An additional hyperparameter ρ is used.) We dualize the computation of the first 1 norm, and rely on a min-max / primal-dual approach to recast the problem as

min (W,µ) max Z ∞ 1 Z, Y µ-XW +λ W 1 + ρ 2 I k -µ 2 F (2)
where Z is an m × k matrix and Z ∞ = max i,j |z ij |. We consider the following scheme, which consists in a proximal descent with respect to the primal variables (W, µ) followed by a proximal ascent with respect to the dual variable Z (see Section 5):

W n+1 := arg min W 1 2τ W -W n 2 F -X T Z n , W + λ W 1 µ n+1 := 1 1 + τ µ ρ (µ n + ρτ µ I k -τ µ Y T Z n ) Z n+1 := proj B∞ (Z + σ(Y µ -X(2W n+1 -W n )))
where B ∞ is the closed unit in the space of R m×k matrices endowed with the ∞ norm. The update on W can be computed using a suitable soft thresholding operator:

W n+1 i,j = W n i,j -ρ • τ µ I k + τ (X Z n ) i,j -λθ i,j , θ i,j 1 and θ i,j = sign(W i,j ) if W i,j = 0 (3)
Writing explicitly the projection operator on B ∞ , we eventually derive Algorithm 1. The convergence condition discussed in Section 5 imposes that

σ τ µ 1 + τ µ (ρ/4) Y 2 + τ X 2 < 1. (4)
The norms involved in the previous expression are operator norms, that is, e.g.,

X = sup W F ≤1 XW F = sup v 2≤1 X(:)v 2 . (5)
Algorithm 1 Primal-dual algorithm-soft(V, λ) is the standard soft thresholding , and µ is the 1 center of clusters. 1: Input: X, Y, Z, N, σ, τ, λ, ρ, µ 0 , W 0 , Z 0 2: W := W 0 3: µ := µ 0 4: Z := Z 0 5: for n = 1, . . . , N do 6:

W old := W 7:

µ old := µ 8:

W := W + τ (X T Z) 9:
W := soft(W, λ) 10:

µ := 1 1+τµρ (µ + ρτ µ I k -τ µ (Y T Z))
11:

Z := max(-1, min(1, Z + σ(Y (2µ -µ old ) - X(2W -W old )))) 12: end for 13: Output: W, µ
The main drawback of this penalty minimization is in the cost associated with the computation of the Lagrange multiplier λ using homotopy algorithms [START_REF] Friedman | Regularization path for generalized linear models via coordinate descent[END_REF][START_REF] Hastie | The entire regularization path for the support vector machine[END_REF]. The worst complexity case is O(3 d ) [START_REF] Mairal | Complexity analysis of the lasso regularization path[END_REF], which is usually intractable on highdimensional data sets such as genomics data sets. To overcome this computational issue, we introduce instead a constrained formulation of the problem.

Primal-dual scheme, constrained formulation

Constrained primal-dual approach

We consider the convex constrained supervised classification problem,

min (W,µ) Y µ-XW 1 + ρ 2 I k -µ 2 F s.t. W 1 η, (6)
that we dualize as before:

min (W,µ) max Z ∞ 1 Z, Y µ-XW + ρ 2 I k -µ 2 F s.t. W 1 η. (7)
We adapt the update of W of Algorithm 1 by using a projected gradient step instead of thresholding, and devise Algorithm 2.

Algorithm 2 Primal-dual algorithm, constrained caseproj(V, η) is the projection on the 1 ball of radius η (see [START_REF] Condat | Fast projection onto the simplex and the l1 ball[END_REF]).

1: Input: X, Y, N, σ, τ, η, ρ, µ 0 , W 0 , Z 0 2: W := W 0 3: µ := µ 0 4: Z := Z 0 5: for n = 1, . . . , N do 6:

W old := W 7:

µ old := µ 8:

W := W + τ • (X T Z) 9: W := proj(W, η) 10: µ := 1 1+τµ•ρ (µ old + ρ • τ µ I k -τ µ • (Y T Z))
11:

Z := max(-1, min(1, Z + σ • (Y (2µ -µ old ) - X(2W -W old ))
)) 12: end for 13: Output: W, µ

Constrained primal-dual approach with over-relaxation

An over-relaxed variant of the previous algorithm is presented below (Algorithm 3). In this case, condition (4) should be replaced with (see section 5.2)

σ τ µ 1 + τµρ 4 1-2γ 1-γ Y 2 + τ X 2 < 1, (8)
at least if γ < 1/2. For γ ≥ 1/2, the condition obtained for ρ = 0 is better and simply reads

σ τ µ Y 2 + τ X 2 < 1. (9)
Algorithm 3 Primal-dual algorithm, constrained case with over-relaxation. 1: Input: X, Y, N, σ, τ, η, ρ, µ 0 , W 0 , Z 0 , γ ∈ (-1, 1) 2: W := W 0 3: µ := µ 0 4: Z := Z 0 5: for n = 1, . . . , N do 6:

W old := W 7:

µ old := µ 8:

Z old := Z 9:

W := W + τ • (X T Z) 10: W := proj(W, η) 11: µ := 1 1+τµρ (µ + ρτ µ I k -τ µ (Y T Z))
12:

Z := max(-1, min(1, Z + σ(Y (2µ -µ old ) - X(2W -W old ))))
13:

W := W + γ(W -W old )
14:

µ := µ + γ(µ -µ old )
15:

Z := Z + γ(Z -Z old ) 16: end for 17: Output: W, µ

Extension to other criteria: Frobenius loss

Our method can be extended straightforwardly to other criteria provided that (i) we can compute the projection on the dual ball for the loss data term, (ii) we can compute the prox for the regularization term. Such examples include combinations of 1 and 2 norms, as seen in group LASSO [START_REF] Jacob | Group lasso with overlap and graph lasso[END_REF]. In this paper, we study an algorithm for the Frobenius norm. Note that our approach based on a dual computation of the norm allows us to use the norm itself, instead of the squared Frobenius norm (for with other approaches, taking care of the smoothness of the loss term, would be available; see, e.g., [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF]).

Frobenius loss Minimization

We consider the following update of (6):

min (W,µ) Y µ -XW F + ρ 2 I k -µ 2 F s.t. W 1 η, (10
) and dualize according to

min (W,µ) max Z F 1 Z, Y µ-XW + ρ 2 I k -µ 2 F s.t. W 1 η. ( 11
)
Obvious modifications of the previous scheme lead to Algorithm:

Algorithm 4 Primal-dual algorithm for Frobenius loss minimization: constrained case. 1: Input: X, Y, N, σ, τ, η, ρ, µ 0 , W 0 , Z 0 2: W := W 0 3: µ := µ 0 4: Z := Z 0 5: for n = 1, . . . , N do 6:

W old := W 7:

µ old := µ 8:

W := W + τ • (X T Z) 9:
W := proj(W, η) 10:

µ := 1 1+τµ•ρ (µ old + ρ • τ µ I k -τ µ • (Y T Z))
11:

Z := Z + σ • (Y (2µ -µ old ) -X(2W -W old ))))
12:

Z := Z/ max{1, Z F } 13: end for 14: Output: W, µ

Frobenius loss minimization with over-relaxation

Obvious modifications of the previous scheme with overrelaxation lead to Algorithm 5. (Note that the algorithm requires computation of Z F at each iteration.)

Algorithm 5 Primal-dual algorithm for Frobenius loss minimization with over-relaxation.

1: Input: X, Y, N, σ, τ, η, ρ, µ 0 , W 0 , Z 0 , γ ∈ (-1, 1) 2: W := W 0 3: µ := µ 0 4: Z := Z 0 5: for n = 1, . . . , N do 6:

W old := W 7:

µ old := µ 8:

Z old := Z 9:

W := W + τ (X T Z) 10: W := proj(W, η) 11: µ := 1 1+τµρ (µ + ρ • τ µ I k -τ µ (Y T Z))
12:

Z := Z + σ • (Y (2µ -µ old ) -X(2W -W old ))))
13:

Z := Z/ max{1, Z F } 14:

W := W + γ(W -W old ) 15: µ := µ + γ(µ -µ old )
16:

Z := Z + γ(Z -Z old ) 17: end for 18: Output: W, µ
A convergence analysis is drawn in Section 5. We provide in Section 6 a comparison between Algorithms 3 and 5.

Convergence Analysis

Convergence of primal-dual algorithms

The proof of convergence of the algorithms relies on Theorems 1 and 2 in [START_REF] Chambolle | On the ergodic convergence rates of a first-order primal-dual algorithm[END_REF] which we slightly adapt for our setting. The algorithms we present here correspond to Alg. 1 and 2 in that reference, adapted to the particular case of problem (1) and its saddle-point formulation (2). In addition, here, the primal part of the objective is "partially strongly convex" (ρ-strongly convex with respect to the variable µ, thanks to the term (ρ/2) µ -I 2 . (We could exploit this to gain "partial acceleration" [START_REF] Valkonen | Acceleration of the PDHGM on partially strongly convex functions[END_REF], however at the expense of a much more complex method and no clear gain for the variable W , while translated in the Euclidean setting, [START_REF] Chambolle | On the ergodic convergence rates of a first-order primal-dual algorithm[END_REF] remains simple and easy to improve.) For our setting we consider a general objective of the form:

min x,x max y f (x) + g(x ) + Kx + K x , y -h * (y) (12)
for f, g, h convex functions whose "prox" (see below) are easy to compute and K, K linear operators, and we assume moreover f is ρ-strongly convex for some ρ > 0. We will show how this last property can be exploited to "boost" the convergence, allowing for larger steps than usually suggested by other authors.

When computing the "prox" x at point x of a ρ-strongly convex function x → f (x), with parameter τ , that is, the minimizer

x = prox τ f (x) := arg min x f (x) + x -x 2 2τ , (13)
one has for all test point x:

f (x) + x -x 2 2τ ≥ f (x) + x -x 2 2τ + x -x 2 2τ + ρ 2 x -x 2 .
However, combined with non-strongly convex iterates, the slight improvement given by the factor ρ is hard to exploit (whereas for simple gradient descent type iterates one obviously can derive linear convergence to the optimum), see for instance [START_REF] Valkonen | Acceleration of the PDHGM on partially strongly convex functions[END_REF] for a possible strategy.

We exploit here this improvement in a different way. We combine the parallelogram identity

x -x 2 + x -x 2 = 1 2 x -x 2 + 2 x -x+x 2 2
with the previous inequality to obtain:

f (x) + (1 + τ ρ 2 ) x -x 2 2τ ≥ f (x) + (1 + τ ρ 4 ) x -x 2 2τ + (1 + τ ρ 2 ) x -x 2 2τ . (14)
The first type of algorithm we consider is Algorithm 1, which corresponds to Alg. 1 in [START_REF] Chambolle | On the ergodic convergence rates of a first-order primal-dual algorithm[END_REF] (see also [START_REF] Pock | An algorithm for minimizing the mumford-shah functional[END_REF][START_REF] Esser | A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science[END_REF][START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF]). It consists in tackling problem (12) by alternating a proximal descent step in x, x followed by an ascent step in y:

x n+1 = prox τ f (x n -τ K * y n ), x n+1 = prox τ g (x n -τ K * y n ), y n+1 = prox σh * (y n + σ(K(2x n+1 -x n ) + K (2x n+1 -x n ))). (15)
We then introduce the "ergodic" averages

X N = 1 N N n=1 x n , X N = 1 N N n=1 x n , Y N = 1 N N n=1 y n .
Theorem 1 in [START_REF] Chambolle | On the ergodic convergence rates of a first-order primal-dual algorithm[END_REF], shows with an elementary proof the estimate, for any test point (x, x , y):

L(X N , X N , y) -L(x, x , Y N ) ≤ 1 2N   x x y   -   x 0 x 0 y 0   2 M τ,τ ,σ (16)
where L is the Lagrangian function in ( 12) and provided the matrix M τ,τ ,σ , given by

M τ,τ ,σ =   I τ 0 -K * 0 I τ -K * -K -K I σ   (17)
is positive-definite. Before exploiting the estimate ( 16), let us express the conditions on τ, τ , σ which ensure that this is true. We need that for any (ξ, ξ , η) = 0,

1 τ ξ 2 + 1 τ ξ 2 + 1 σ η 2 > 2 Kξ, η + 2 K ξ , η
and obviously, this is the same as requiring that for any a, a , b positive numbers,

a 2 τ + a 2 τ + b 2 σ > 2( K a + 2 K a )b.
The worst b in this inequality is b = σ( K a + 2 K a ), then one checks easily that the worse a, a are of the form ā K τ , ā K τ respectively, so that one should have for all ā = 0:

ā2 K 2 τ + K 2 τ > σā 2 K 2 τ + K 2 τ 2 ,
yielding the condition

σ(τ K 2 + τ K 2 ) < 1.
We notice in addition that under such a condition, one also has

M τ,τ ,σ ≤ 2   I τ I τ I σ  
which allows to simplify a bit the expression in the righthand side of (16) (at the expense of a factor 2 in front of the estimate).

We have not made use of the strong convexity up to now, and in particular, of (14). A quick look at the proof of Theorem 1 in [START_REF] Chambolle | On the ergodic convergence rates of a first-order primal-dual algorithm[END_REF] shows that it will improve slightly the latter condition, allowing to replace τ with the smaller effective step τ /(1 + τ ρ/4), yielding the new condition

σ τ 1 + τ ρ 4 K 2 + τ K 2 < 1. ( 18
)
This ensures now that ( 16) holds with M τ,τ ,σ replaced with

M τ,τ ,σ,ρ =   1 τ + ρ 2 I 0 -K * 0 I τ -K * -K -K I σ   ≤   2 τ + 3ρ 4 I 0 0 0 2 τ I 0 0 0 2 σ I   (19)
where the last inequality follows from (18). Applied to problem (2), which is ρ-convex in µ, we find that (18) becomes the condition

σ τ µ 1 + τ µ ρ 4 Y 2 + τ X 2 < 1. (20)
When (20) holds, then the ergodic iterates (here we denote W n , etc, the value of W computed at the end of iteration n):

W N = 1 N N n=1 W n , μN = 1 N N n=1 µ n , ZN = 1 N N n=1 Z n .
(21) satisfy for all W, µ, Z:

L( W N , μN , Z) -L(W, µ, ZN ) ≤ 1 N 3ρ 8 µ -µ 0 2 + µ -µ 0 2 τ µ + W -W 0 2 τ + Z -Z 0 2 σ . (22)
Here the Lagrangian L is:

L(W, µ, Z) = Z, Y µ -XW + λ W 1 + ρ 2 I -µ 2 .
We denote E(W, µ) = sup Z L(W, µ, Z) the primal energy (which appears in ( 1)) and remark that in (2), Z is bounded (|Z i,j | ≤ 1 for all i, j) so that Z -Z 0 2 ≤ 4mk in ( 22). Hence, taking the supremum on Z and choosing for (W, µ) a primal solution (minimizer of E) (W * , µ * ), we deduce:

E( W N , μN ) -E(W * , µ * ) ≤ 1 N 4mk σ + 3ρ 8 + 1 τ µ µ * -µ 0 2 + W * -W 0 2 τ . ( 23
)
In general, if one can compute reasonable estimates ∆ µ , ∆ W for these quantities, one should take:

τ = ∆ W 2 √ mk X , τ µ =    1 2 √ mk Y ∆µ -ρ 4 if 8 √ mk Y ρ∆µ > 1 τ µ >> 1 else, σ = 1 τµ 1+τµ ρ 4 Y 2 + τ X 2 .
to obtain (considering here only the case ρ small, that is when

ρ∆ µ ≤ 8 √ mk Y ): E( W N , μN )-E(W * , µ * ) ≤ √ mk(5∆ µ Y + 4∆ W X ) N ,
There is no clear way how to estimate a priori the norm W * -W 0 in the Lagrangian approach.

Remark 1 Note that for the

1 constrained problem (6) ∆ W is bounded. Since W 1 ≤ η: W * -W 0 ≤ W * -W 0 1 ≤ 2η, we use the estimate ∆ W ≤ 2η.
Using the initial value µ 0 = I k , ∆ µ is also easily shown to be bounded (as W is). Empirically, we found that we can use the estimate ∆ µ α I k F = α √ k where α ∈ [0, 1] is a parameter to be tuned. Thus ρ being small we have 8 √ m Y ρα > 1. Moreover, using X = 1 (X can be normalized), we obtain the following reasonable parameter choice:

τ = ∆ W 2 √ mk , τ µ = α 2 √ m Y -(1/4)αρ σ = 1 τµ 1+τµ ρ 4 Y 2 + τ . ( 24
)
In the case of Problem (11) (Sec. 4), Z is also bounded but then, one has simply Z * -Z 0 2 ≤ 4, hence ( 23) must be replaced with

E( W N , µ N ) -E(W * , µ * ) ≤ 1 N 4 σ + 3ρ 8 + 1 τ µ µ * -µ 0 2 + W * -W 0 2 τ . (25)
(Obviously, now, the energy E is the primal energy in (10).)

The same analysis as before remains valid, but now with mk replaced with 1.

Convergence with over-relaxation

For the over-relaxed variant (Algorithm 3), the adaption is a little bit more complicated, and one does not benefit much from taking into account the partial strong convexity. One approach is to rewrite the improved descent rule ( 14) as follows:

f (x) ≤ f (x) + 1 + τ ρ/2 2τ x -x 2 -x -x 2 -x -x 2 + ρ 8 x -x 2 = f (x) + 1 2 τ x -x, x -x + ρ 8 x -x 2 (26)
where τ = τ /(1 + τ ρ/2) is an effective time-step.

As a result, we observe that the first (primal) update in (15) yields the same rule as an explicit-implicit primal update of a nonsmooth+smooth functions with effective step τ and Lipschitz constant ρ/4, cf Eq. ( 9) in [START_REF] Chambolle | On the ergodic convergence rates of a first-order primal-dual algorithm[END_REF]. Hence, the analysis of these authors (see Sec. 4.1 in the above reference) can be reproduced almost identically and will yield for the over-relaxed algorithm (3) similar convergence rates, cf. ( 16)-( 23), now, with the factor 1/N replaced with 1/((1 + γ)N ). It requires that the matrix

M =   ( 1 τ -ρ/4 1-γ )I 0 -K * 0 I τ -K * -K -K I σ   (27)
be positive definite. Observe however that the estimates hold for the ergodic averages (cf ( 21)) of the variables obtained at the end of Step 12 of Algorithm 3 and Step 13 of Algorithm 5, rather than for the over-relaxed variables (which could not even be feasible). We derive that for this method, condition (20) should be replaced with

σ τ µ 1 + τµρ 4 1-2γ 1-γ Y 2 + τ X 2 < 1, (28)
at least if γ < 1/2.
As seen, for γ ≥ 1/2, the condition obtained for ρ = 0 is better (hence, the partial strong convexity does not seem to yield any reasonable improvement for this algorithm). It simply reads

σ τ µ Y 2 + τ X 2 < 1, (29)
and one gets the estimate from [START_REF] Chambolle | On the ergodic convergence rates of a first-order primal-dual algorithm[END_REF] (Eq. ( 24), further simplified thanks to (28)):

E( W N , μN ) -E(W * , µ * ) ≤ 1 (1 + γ)N 4mk σ + µ * -µ 0 2 τ µ + W * -W 0 2 τ .
(30)

Numerical experiments 6.1. Experimental settings

In this section, we compare the constrained primal-dual 1 approach with the one based on the Frobenius norm. Our primal-dual method can be applied to any classification problem with feature selection on high dimensional dataset stemming from computational biology, image recognition, social networks analysis, customer relationship management, etc., We provide an experimental evaluation in computational biology on simulated and real single-cell sequencing dataset.

There are two advantages of such biological dataset. First, many public data are now available for testing reproductibility; besides, these dataset suffer from outliers ("dropouts") with different levels of noise depending on sequencing experiments. Single-cell is a new technology which has been elected "method of the year" in 2013 by Nature Methods [START_REF] Evanko | Method of the year 2013: Methods to sequence the dna and rna of single cells are poised to transform many areas of biology and medicine[END_REF]. We provide also evaluation on proteomics mass-spectrometric dataset. A test query x (a dimension d row vector) is classified according to the following rule: it belongs to the unique class j * such that

j * ∈ arg min j=1,...,k µ j -xW 1 . ( 31
)
Feature selection is based on the sparsity inducing 1 constraint. The projection on the 1 ball P roj(V, η) aims at sparsifying the W matrix so that the gene j will be selected if |W (j, :) > ε. The set of non-zero column coefficients is interpreted as the signature of the corresponding cluster. We use the Condat method [START_REF] Condat | Fast projection onto the simplex and the l1 ball[END_REF] to compute the projection on the 1 ball.

We report the classical accuracy versus η using cross validation (4 folds). We also define selectivity as the number of required genes to obtain a desired accuracy. Processing times are obtained on a laptop computer using an i7 processor (3.1 Ghz). In our experiments, we normalize the features according to X = 1, and we set µ 0 = I k and ρ = 0.0001. We choose η in connection with the desired number of genes.

As ∆ W and η are bounded, we can set a maximum value for τ , that we denote τ 0 . Then we tune τ µ and compute σ using equation ( 24). Based on results in Fig. 2, we set N = 40 for synthetic dataset and Tabula Muris dataset, while Ovarian data set requires N = 60 iterations. We report the 1 and Frobenius error (log-log plot) in the training set (normalized by the value of the first iterate) both for standard algorithm Fig. 4 and over-relaxed one in Fig. 5.

Synthetic dataset

The simulated dataset is a realistic simulation of single cell sequencing experiments (The dataset is provided in supplementary material). This dataset is composed of 600 samples 15,000 genes and k = 4 clusters. 6.3. Dataset: ovarian [START_REF] Guyon | Feature extraction, foundations and applications[END_REF] The data available on UCI data base were obtained from two sources: the National Cancer Institute (NCI) and the Eastern Virginia Medical School (EVMS). All the data consist of mass-spectra obtained with the SELDI technique. The samples include patients with cancer (ovarian or prostate cancer), and healthy or control patients. The dataset is composed of 216 samples and 15000 features. 6.5. Accuracy, signature and scalability

Our numerical experiments demonstrate that our 1 norm minimization method is more robust to outliers (Fig. 6) and to dropouts on real large single cell datasets (Fig. 7). Fig. 3 shows that minimization of the 1 norm is robust to parameter τ 0 . We report in Fig. 8 the sorted absolute value of the matrix W (divided by the maximum of each column). As expected by biologists, Fig. 8 shows that the sparsity of the signature of W columns is well adapted to each cluster. The accuracy for each cluster is given Fig. 7, illustrating the reliability of the signature. Fig. 4 shows that over relaxation improves convergence both for the 1 and Frobenius norm minimization. The complexity of our primal-dual algorithm is O(d × m) for primal iterates and O(m × k) for dual iterates. One must then add the cost of projection on the 1 ball; this cost is expected to be O(d× k) in average, in order that the our constrained Primal-dual method is scalable.

As pointed in Section ( 4) the squared Frobenius loss is smooth and we can use Fista algorithm [START_REF] Beck | A fast iterative shrinkagethresholding algorithm for linear inverse problems[END_REF][START_REF] Chambolle | On the convergence of the iterates of "fista[END_REF] combined with centroid classifier [START_REF] Witten | A framework for feature selection in clustering[END_REF] to minimize a squared Frobenius criterion. To do so, one performs a full run of Fista to retrieve W , then a single centroid computation to estimate µ. Table 1 shows that the primal-dual method outperforms Fista. (Note that the complexity of gradient based methods such as Fista is O(d 2 ).)

Conclusion

We have provided a new primal-dual method for supervised classification based on an 1 norm both for the data loss and the constraint (feature selection). Our algorithm computes jointly variables transform W and centers µ that are used to devise a classifier, and we establish convergence results. We demonstrate the effectiveness of our method on three datasets (one synthetic, two from biological data), and provide a comparison between 1 and 2 costs. Both accuracy, selectivity and computational time are improved with the purely 1 approach. Extending the method to other criteria is easy on condition that efficient projection (on the dual ball for the loss data term) and prox (for the regularization term) algorithms are available. We have treated an example using the Frobenius norm of the loss. Current research extends our approach to other loss or regularization criteria such as sparse group LASSO. A code is available in supplementary material.

8. Appendix: minimization of the square Frobenius loss using Fista

Let us consider the classical feature selection criterion with the squared Frobenius loss:

min (W,µ) Y -XW 2 F s.t. W 1 η. ( 32
)
We compute an estimator of W using Fista algorithm.

Algorithm 6 Feature selection algorithm with FISTA.

Input: X, Y, µ 0 = I, W 0 , N, γ, η W ← W 0 t ← 1 for n = 0, . . . , N do V ← W -γX T (XW -Y µ) W new ← P 1 η (V ) t new ← (n + 5)/4 λ ← 1 + (t -1)/t new W ← (1 -λ)W + λW new t ← t new end for Output: W

Figure 1 .

 1 Figure 1. Synthetic dataset. The evolution of the number of selected genes versus the constraint η is a smooth monotonous function. The bound η for the 1 constraint is thus easily tuned.

Figure 2 .

 2 Figure 2. Synthetic dataset. This figure shows that roughly 40 iterations are optimal in order to avoid over-fitting.

6. 4 .

 4 Lung Tabula Muris single cell scRNA-seq dataset[START_REF] Schaum | Single-cell transcriptomics of 20 mouse organs creates a tabula muris[END_REF] Tabula Muris is a compendium of single cell transcriptome data from the model organism Mouse musculus, containing nearly 100,000 cells from 20 organs and tissues. The data allow comparison between gene expression in cell types. (The authors thank Pr Pascal Barbry team (IPMC Laboratory) for the group's biological expertise.) Lung Tabua Muris sub-dataset is a subset of Lung organ composed of 5,400 cells, 10,516 genes and k=14 clusters.

Figure 3 .

 3 Figure3. Synthetic dataset. The accuracy versus number of genes shows that a minimum number of genes is required to get the best possible classification followed by a large plateau. We get similar accuracy and selectivity for different values of parameter τ0 and for over-relaxation. 1 norm improves selectivity over Frobenius norm minimization.

Figure 4 .

 4 Figure 4. Synthetic dataset. Convergence rate is faster for 1 minimization.

Figure 8 .

 8 Figure 8. Tabula Muris: the signature is sparse and adapted to the clustering.

Figure 5 .

 5 Figure 5. Synthetic dataset. Over-relaxation provides O(1/N) convergence rate for both 1 and Frobenius minimization.

Figure 6 .

 6 Figure 6. Ovarian dataset: using 1 norm minimization outperforms Frobenius minimization both for accuracy and feature selectivity.

Figure 9 .

 9 Figure 9. Tabula Muris Confusion Matrix: vertically Ground truth, Horizontally: Predicted True positive in %.

Figure 7 .

 7 Figure 7. Tabula Muris dataset: accuracy: 1 norm minimization outperforms Frobenius norm minimization by 5 %. Moreover 1 minimization improves feature selectivity.

Table 1 .

 1 Time (seconds) according to methods: primal-dual is slightly faster for 1 norm minimization on large dataset since Frobenius norm minimization requires computation of Frobenius norm of O(m × k) matrix at each iteration. Primal-dual outperforms Fista primal algorithm.

	Dataset	1	1 OR Fro	Fista
	Synthetic 0.7	0.43	0.7	7.6
	Ovarian 0.26	0.21	0.26 6.83
	Tumlung	3.3	2.5	3.5	7.