Dr Suzanne Lesage
email: suzanne.lesage@upmc.fr

Marion Houot

Graziella Mangone

Christelle Tesson

Hélène Bertrand

Sylvie Forlani

Mathieu Anheim

Christine Brefel-Courbon

Emmanuel Broussolle

Stéphane Thobois

Philippe Damier

Franck Durif

Emmanuel Roze

François Tison

David Grabli

Fabienne Ory-Magne

Bertrand Degos

François Viallet

Florence Cormier-Dequaire

Anne-Marie Ouvrard-Hernandez

Marie Vidailhet

Ebba Lohmann

Andrew Singleton

Jean-Christophe Corvol

Alexis Brice

Genetic and phenotypic basis of autosomal dominant Parkinson's disease in a large multi-center cohort

Keywords: Parkinson's disease, LRRK2, G2019S, SNCA, VPS35, Autosomal dominant inheritance, Genotype-Phenotype correlations AAO, age at onset, AD, autosomal dominant, LRRK2, leucine-rich repeat kinase 2, MMSE, mini mental state examination, PD, Parkinson's disease, VPS35, vacuolar protein sorting 35

LRRK2, SNCA and VPS35 are unequivocally associated with autosomal dominant Parkinson's disease (PD). We evaluated the prevalence of LRRK2, SNCA and VPS35 mutations and associated clinical features in a large French multi-center cohort of PD patients. Demographic and clinical data were collected for 1805 index cases (592 with autosomal dominant inheritance and 1213 isolated cases) since 1990. All probands were screened with TaqMan assays for LRRK2 Gly2019Ser. In the absence of this mutation, the coding sequences of the three genes were analyzed by Sanger sequencing and/or next-generation sequencing. The data for the three genes were analyzed according to age at onset, family history, ethnic origin and clinical features. We identified 160 index cases (8.9%) with known pathogenic variants: 138 with pathogenic LRRK2 variants (7.6%), including 136 with the Gly2019Ser mutation, 19 with SNCA point mutations or genomic rearrangements (1.1%), and three with the VPS35 Asp620Asn mutation (0.16%). Mutation frequencies were higher in familial than isolated cases, consistent with autosomal dominant inheritance (12.0% vs. 7.3%; OR 1.7, 95% CI [1.2-2.4], p = 0.001). PD patients with LRRK2 variants were more likely to have higher rates of late-onset PD (> 50 years; OR 1.5, 95% CI [1.0-2.1], p = 0.03), whereas those with SNCA mutations tended to have earlier age at onset disease (≤ 50 years, p = 0.06). The clinical features of LRRK2 carriers and those without any pathogenic variants in known PD-associated genes were similar. The likelihood of detecting disease-causing mutations was higher in cases compatible with autosomal dominant inheritance.

Contribution to the field

Two decades after the identification of SNCA as the first causal gene of Parkinson's disease (PD) and the realization that genetic factors play a substantial role in PD pathogenesis, our knowledge of the genetic architecture underlying this disease has increased considerably, with numerous genes and genetic loci now known to cause familial forms of PD or to influence the risk of PD. This study of a large cohort of genotyped and extensively phenotyped patients, including more than 190 mutation carriers, provides information about the frequency of the most three relevant genes known to cause autosomal dominant PD, by age at onset, family history, ethnicity, and associated phenotype. These findings are critical for guiding genetic testing in clinical practice and identification of patients for gene-targeting therapeutic trials.

INTRODUCTION

Heterozygous sequence variants of LRRK2 or VPS35, and mutations or genomic rearrangements in SNCA cause monogenic Parkinson's disease (PD) with autosomal dominant (AD) inheritance [reviewed in [START_REF] Lunati | The genetic landscape of Parkinson's disease[END_REF]]. Only seven of the hundred or so variants of LRRK2 reported to date (Asn1437His, Arg1441Gly/Cys/His, Tyr1699Cys, Gly2019Ser, and Ile2020Thr), which appear to be clustered in functionally important regions highly conserved throughout evolution, have been demonstrated to be pathogenic on the basis of co-segregation with the disease and absence or rarity in specific control populations [START_REF] Ross | Association of LRRK2 exonic variants with susceptibility to Parkinson's disease: a casecontrol study[END_REF]. The most common of these mutations, LRRK2 Gly2019Ser, has a reported frequency of 0% to above 40%, depending on the population considered [START_REF] Trinh | Genotypephenotype relations for the Parkinson's disease genes SNCA, LRRK2, VPS35: MDSGene systematic review[END_REF], mostly due to a common founder effect [START_REF] Lesage | Parkinson's disease-related LRRK2 G2019S mutation results from independent mutational events in humans[END_REF]. SNCA mutations are the second most common cause of autosomal dominant inherited PD; genomic duplications have been detected in ~1-2% of families with AD PD. Other SNCA mutations, such as whole-locus triplications and a few missense mutations (Ala53Thr/Glu/Val, Glu46Lys, Ala30Pro, Gly51Asp), are extremely rare [reviewed in [START_REF] Lunati | The genetic landscape of Parkinson's disease[END_REF]]. Finally, VPS35 was the first PD-causing gene to be identified by next-generation sequencing (NGS) in large multi-incident families [START_REF] Vilariño-Güell | VPS35 mutations in Parkinson disease[END_REF][START_REF] Zimprich | A mutation in VPS35, encoding a subunit of the retromer complex, causes late-onset Parkinson disease[END_REF].

Subsequent studies in multiple ethnic groups, including a large multi-center study, indicated that Asp620Asn was the only pathogenic variant, with a relative frequency ranging from 0.1% to 1% in familial PD, depending on population background [START_REF] Sharma | A multi-centre clinico-genetic analysis of the VPS35 gene in Parkinson disease indicates reduced penetrance for disease-associated variants[END_REF]. As the phenotype of AD PD closely resembles that of idiopathic PD, we assumed that rare variants of genes causing AD PD might also contribute to the etiology of isolated PD in the French population.

In this study, we aimed at determining the relative frequencies of known mutations of three genes, LRRK2, SNCA, and VPS35, in a large cohort of familial and isolated cases. The high prevalence of the LRRK2 Gly2019Ser mutation provided us with a unique opportunity to compare in details clinical characteristics between carriers of this mutation and patients with no known mutations of PDassociated genes.

MATERIAL AND METHODS

Patients

In total, 673 PD patients from 592 families and 1213 isolated cases without known consanguinity were recruited from 1990 onwards, through the French Parkinson Disease Genetics Network (the PDG group, Supplementary Material). All participants underwent a detailed medical and family history, and a family tree were drawn. Familial cases compatible with AD inheritance, and referred to here as AD PD cases, were defined as AD cases with at least one other affected relative in a Informed consent was obtained from all participants, and genetic studies were approved by local ethics committees.

Most index cases were of European ancestry (n = 1530; 84.8%), mostly French (n = 1202; 78.6%); the others were North African (n = 221; 12.2%) or of other origins, including Asian, Sub-Saharan African or "mixed" origins (n = 42; 2.3%); or of unknown origin (n = 12; 0.7%). This study included 226 index cases from families with PD compatible with AD inheritance reported elsewhere [START_REF] Lesage | Molecular analyses of the LRRK2 gene in European and North African autosomal dominant Parkinson's disease[END_REF][START_REF] Lesage | Identification of VPS35 mutations replicated in French families with Parkinson disease[END_REF].

Methods

Genomic DNA was obtained from peripheral blood lymphocyte or saliva samples (Oragene TM DNA Self-Collection Kit, DNA Genotek), by standard protocols. Patients with variants of the GBA risk factor (n = 153), or with variants of genes for which the causal role in AD PD was uncertain, such as GIGYF2 (n = 6), EIF4G1 (n = 2), and c9ORF72 (n = 4), were not included in this study. In addition, we excluded 25 ( All index cases were genotyped in duplicate for LRRK2 Gly2019ser, by the TaqMan allelic discrimination Assay-By-Design method, in accordance with the manufacturer's instructions, with 8 ng of DNA mixed with the TaqMan Genotyping Master Mix (Thermo Fisher Scientific Inc.) and custom-produced TaqMan SNP genotyping assays [C_63498123_10 (rs34637584), Thermo Fisher Scientific Inc.] on an Applied Biosystems PRISM 7000 sequence detection system (Thermo Fisher Scientific Inc.) or LightCycler ® 480 machine (Roche, Life Technologies SAS). All patients found not to carry LRRK2 Gly2019ser were then screened for pathogenic variants of the coding sequences of LRRK2, SNCA, and VPS35, by Sanger sequencing (n = 855), targeted sequencing of a customized next-generation sequencing (NGS) gene panel containing the 22 most prevalent PDassociated genes (n = 404) (Supplementary Table 1), or available whole-exome sequencing (n = 410), as previously described [START_REF] Bouhouche | Mutation analysis of consanguineous Moroccan patients with Parkinson's disease combining microarray and gene panel[END_REF][START_REF] Lesage | Loss of VPS13C function in autosomal recessive parkinsonism causes mitochondrial dysfunction and increases PINK1/Parkin-dependent mitophagy[END_REF]. We considered known pathogenic mutations of the three genes.

Sanger sequencing was used to confirm variants and co-segregation analyses were performed, where possible. SNCA rearrangements were detected by semi-quantitative multiplex PCR [START_REF] Ibanez | Alpha-synuclein gene rearrangements in dominantly inherited parkinsonism: frequency, phenotype, and mechanisms[END_REF] or by the SALSA multiplex ligation-dependent probe amplification method (MLPA, MRC Holland, Amsterdam, the Netherlands; http://www.mlpa.com), according to the manufacturer's instructions.

Statistical analysis

Demographic and clinical characteristics are expressed as means and standard deviations for continuous variables and as counts and percentages for qualitative variables, separately for each group (i.e. with and without mutation). These characteristics were compared between patients with the LRRK2 Gly2019ser mutation (LRRK2+) and those with no mutations of genes known to be associated with PD (genetically undefined PD), in Welch's t-tests for continuous variables and Fisher's exact tests for qualitative variables.

We used generalized linear models (GLMs) to compare clinical features between patients with the LRRK2 Gly2019ser mutation and those with genetically undefined PD, with adjustment for sex, age at onset (AAO), and disease duration. Disease duration was not included in models of clinical features at onset. We used GLMs with identity links and normal distributions for continuous clinical features, and GLMs with logit links and Bernoulli distributions for binary clinical features. Fisher type II tests were performed to test each effect, and effect size was estimated with Cohen's f2. We corrected for multiple testing by the Benjamini-Hochberg method. Residual normality and heteroskedasticity were checked visually. Influencers and outliers were checked by calculating hat values and Cook distances.

Only patients with no missing data for the covariables included in the models, such as AAO, sex, and disease duration, were retained for analysis. Statistical analyses were performed with R 3.6.1.

RESULTS

Demographic and clinical data

Table 1 shows the baseline demographic and clinical characteristics of the 1805 PD index cases included in the study. Men (n = 1106, 61.3%) and early AAO cases (mean 47.4 [SD 12.5]; range 9-86 years; Table 1) were overrepresented, particularly among isolated cases (mean 46.1 [SD 12.5]; range: 9-79 years) as opposed to familial cases (mean 50.0 [SD 12.1]; range 10-86 years).

Summary of the mutations identified

We identified seven different known mutations of LRRK2, SNCA, and VPS35 in 160 of the 1805 PD index cases (8.9%, 95% confidence interval (CI): [7.6-10.3]). With the exception of rare cases with the homozygous LRRK2 Gly2019Ser mutation (n = 6), all the known mutations and gene multiplications identified were heterozygous. We found that 136 of the 138 LRRK2 mutation carriers (7.5% of index cases, 95% CI [6.4-8.9]) carried the Gly2019Ser mutation; 13 of these patients had already been reported elsewhere [START_REF] Lesage | Molecular analyses of the LRRK2 gene in European and North African autosomal dominant Parkinson's disease[END_REF]. Two index cases carried the rare LRRK2 Arg1441His mutation [START_REF] Lesage | Molecular analyses of the LRRK2 gene in European and North African autosomal dominant Parkinson's disease[END_REF]. SNCA mutations were found in 19 index cases (1.1%, 95% CI [0.63-1.6]), including 11 families already reported [START_REF] Ibanez | Alpha-synuclein gene rearrangements in dominantly inherited parkinsonism: frequency, phenotype, and mechanisms[END_REF][START_REF] Lesage | G51D α-synuclein mutation causes a novel parkinsonian-pyramidal syndrome[END_REF][START_REF] Book | A Meta-analysis of α-synuclein multiplication in familial parkinsonism[END_REF]: one with whole-gene triplications, 14 with duplications, one with the Gly51Asp mutation, and three with the Ala53Thr mutation. The three PD index cases with the only VPS35 mutation identified, Asp620Asn (0.17%) were described in a previous report [START_REF] Lesage | Identification of VPS35 mutations replicated in French families with Parkinson disease[END_REF]. Overall, mutations were more frequently identified in familial (71/592, 12.0%; 95% CI [9.5-14.9]) than in isolated cases (89/1213, 7.3%; 95% CI [5.9-9.0], Fisher's exact test: OR 1.7, 95% CI [1.2-2.4], p = 0.001), particularly for SNCA (2.4%, 14/592 vs. 0.41%, 5/1213, Fisher's exact test: OR 5.9, 95% CI [2.1-16.3], p = 0.0003; Table 2). An analysis of PD cases according to age at onset (≤ 50 years vs. > 50 years) showed that LRRK2 mutations were more frequent among late-onset PD cases (61/636, 9.6%; 95% CI [7.4-12.2] vs. 71/1063, 6.7%; 95% CI [5.3-8.4], Fisher's exact test: OR 1.5, 95% CI [1.0-2.1], p = 0.03; Table 2). By contrast, SNCA mutation carriers tended to have an earlier AAO (16/1063, 1.5% vs. 3/636, 0.5%, p = 0.06).

Clinical characteristics of mutation carriers and comparison of LRRK2 Gly2019Ser mutation carriers (LRRK2+) with individuals with no known PD mutations (genetically undefined PD)

Co-segregation analyses identified 193 PD patients as mutation carriers: 151 with LRRK2 2). By contrast, AAO was highest in patients with SNCA duplications (mean AAO: 45.3 years [SD 6.3], range: 36-56 years), consistent with typical PD and a good response to levodopa (100%), with more than 50% of these patients reporting levodopa-induced motor complications. Non-motor symptoms, including depression/psychosis and dysautonomia, were present in about one third of the reported cases, but cognitive decline was less frequent (18.8%, 3/16). Detailed clinical characteristics for PD patients with SNCA multiplications are provided in Supplementary Table 3.

VPS35

Five of the eight PD patients carrying VPS35 Asp620Asn mutations have been described before [START_REF] Lesage | Identification of VPS35 mutations replicated in French families with Parkinson disease[END_REF].

The three newly genotyped patients were relatives of patient 838-006 (Supplementary Table 4).

Briefly, patients carrying VPS35 mutations had features similar to those with idiopathic PD, with a mean AAO of ~57 years (range: 38-71 years, Table 3): all patients presented the classical triad, with akinesia as the predominant symptom at onset (100%), but a much lower frequency of tremor as an initial symptom (37.5%), a good response to levodopa (100%), with less than 37% of those treated developing dyskinesias and motor fluctuations, and a low rate of dysautonomia (2/8, 25%), with no cognitive or neuropsychiatric symptoms or atypical signs.

LRRK2

We compared the clinical features of the LRRK2 G2019S+ PD patient (LRRK2+) group with those of PD patients with no mutations of known PD-associated genes, excluding subjects with missing data for the covariables included in the models, such as sex, AAO and disease duration: 135/151 LRRK2+ and 1552/1693 PD patients without mutations were included in the final analysis (Table 4).

The proportion of men was greater in the genetically undefined PD patient group than in the LRRK2+ group (61.1% vs. 51.9%, p = 0.04). The mean AAO of the LRRK2 Gly2019Ser carriers was four years higher than that of non-carriers (p < 0.001). The Gly2019Ser carriers were more likely to be of North-African ancestry (p < 0.001) and to report a family history of PD (p < 0.02). They had a higher UPDRS III score during the "OFF" state and a higher Hoehn and Yahr score during the "ON" state than non-carriers, but these results were no longer significant after adjustment for AAO and disease duration. The frequencies of signs at onset and at examination, the degree of response to treatment, motor complications and non-motor signs, including cognitive impairment and autonomic dysfunction were similar in both groups.

Clinical comparisons between heterozygous (n = 144) and homozygous (n = 7) Gly2019Ser mutation carriers revealed no significant differences in sex (men: 52.1% vs. 42.9%, OR 1.

DISCUSSION

This is one of the largest national multi-center studies to investigate the frequency of variants of the three major genes unequivocally linked to AD PD -LRRK2, SNCA, and VPS35and their associated phenotypes in a large cohort of >1800 French and North African index PD cases. We report an overall mutation frequency of 8.9% across both populations, the LRRK2 G2019S mutation being the most frequently identified variant (7.5%), particularly in familial rather than isolated cases.

However, the frequency of mutations differed considerably between populations. We confirm here that the LRRK2 Gly2019Ser mutation is the principal genetic cause of PD in our cases of North-African ancestry, reaching an overall frequency of 45% (100/221) and 62% (34/55) in familial cases. By contrast, this mutation was present at a much lower rate of 2.4% (36/1530) in our native French PD cases. Our findings are consistent with those of previous multi-center studies [START_REF] Healy | Phenotype, genotype, and worldwide genetic penetrance of LRRK2-associated Parkinson's disease: a case-control study[END_REF]. SNCA duplications were the second most common type of mutations, identified in 14 index cases (0.78%).

SNCA duplication carriers were mostly of European ancestry, particularly French (93%), tended to be predominantly females, probably due to random or recruitment bias, and had a higher frequency of a family history of PD. We also identified three unrelated PD patients carrying the SNCA Ala53Thr mutation. Although generally rare, this mutation appears to be particularly common in the Italian and Greek populations, due to a founder effect [START_REF] Papadimitriou | motor and nonmotor features of carriers of the p.a53t alpha-synuclein mutation: a longitudinal study[END_REF][START_REF] Tambasco | A53T in a parkinsonian family: a clinical update of the SNCA phenotypes[END_REF]. Only four individuals without Greek or Italian ancestry have been reported to carry this mutation, in haplotypes different from those reported in Greek and Italian families [START_REF] Michell | A case of late onset sporadic Parkinson's disease with an A53T mutation in alpha-synuclein[END_REF][START_REF] Ki | The Ala53Thr mutation in the alpha-synuclein gene in a Korean family with Parkinson disease[END_REF][START_REF] Puschmann | A Swedish family with de novo alpha-synuclein A53T mutation: evidence for early cortical dysfunction[END_REF][START_REF] Xiong | The heterozygous A53T mutation in the alpha-synuclein gene in a Chinese Han patient with Parkinson disease: case report and literature review[END_REF]. Additional haplotype analysis would determine the ancestral origin of our three French mutation carriers. Other rare known mutations were also identified in our study: VPS35 Asp620Asn and LRRK2 Arg1441His in three and two AD PD families, respectively. At least 25 LRRK2 Arg1441His carriers, including those described here, have been reported to date [(3); www.mdsgene.org)]. Most were Caucasian, and all but one case reported a family history of the disease. This pathogenic variant was not found in more than 6,000 healthy controls tested [START_REF] Lubbe | Additional rare variant analysis in Parkinson's disease cases with and without known pathogenic mutations: evidence for oligogenic inheritance[END_REF] and is very rare in the Genome Aggregation Database (GnomAD) (1/31,298 alleles); it is therefore very likely to be pathogenic. Consistent with this conclusion, the Arg1441His variant affects the same amino-acid residue as two other recurrent PD-causing mutations (Arg1441Cys and Arg1441Gly).

Finally, previous haplotype analyses did not support the hypothesis of a common founder for the Arg1441His variant, instead suggesting that there might be a mutational hotspot. Following initial reports of the existence of several VPS35 variants [START_REF] Vilariño-Güell | VPS35 mutations in Parkinson disease[END_REF][START_REF] Zimprich | A mutation in VPS35, encoding a subunit of the retromer complex, causes late-onset Parkinson disease[END_REF], pathogenicity has been confirmed only for the Asp620Asn variant. Consistent with our findings, this recurrent mutation has been identified predominantly in families of Caucasian descent affected by AD PD. A meta-analysis of 21,824 PD patients from 15 case-control studies performed worldwide from 2011 to 2016 identified an overall mutation frequency of 0.12% (0.29% in familial cases and 0.023% in isolated cases) [reviewed in [START_REF] Gambardella | vacuolar protein sorting genes in parkinson's disease: a re-appraisal of mutations detection rate and neurobiology of disease[END_REF]], and an absence of this mutation from healthy controls and the GnomAD public database.

The clinical features of our LRRK2 mutation carriers, whether heterozygous or homozygous, were indistinguishable from those of patients with no mutations in known PD-associated genes. These features overlapped those of typical, idiopathic PD. In our study, patients with the G2019S mutation had a mean AAO of ~52 years, a high proportion of patients with late AAO (> 50 years), a good response to levodopa, a predominance of tremor as a first symptom of PD, about a quarter had cognitive impairment, about 10% had dysautonomia, but no other atypical signs after a mean disease duration of approximately 10 years. Although the clinical features of the LRRK2 Gly2019Ser carriers compared with patients with idiopathic PD in literature are conflicting [meta-analysis in [START_REF] Shu | Clinical heterogeneity among LRRK2 variants in Parkinson's disease: a meta-analysis[END_REF]], even for the same ethnic PD population [i.e. of North-African origin, [START_REF] Ishihara | Screening for Lrrk2 G2019S and clinical comparison of Tunisian and North American Caucasian Parkinson's disease families[END_REF][START_REF] Lesage | Is the common LRRK2 G2019S mutation related to dyskinesias in North African Parkinson disease?[END_REF][START_REF] Nishioka | A comparative study of LRRK2, PINK1 and genetically undefined familial Parkinson's disease[END_REF][START_REF] Trinh | Comparative study of Parkinson's disease and leucine-rich repeat kinase 2 p.G2019S parkinsonism[END_REF][START_REF] Nabli | Motor phenotype of LRRK2associated Parkinson's disease: a Tunisian longitudinal study[END_REF][START_REF] Ben Romdhan | LRRK2 G2019S Parkinson's disease with more benign phenotype than idiopathic[END_REF]], our data are consistent with those of 724 LRRK2 mutation carriers listed in the MDSGene database. Like LRRK2 mutation carriers, VPS35 Asp620Asn carriers had a phenotype very similar overall to that of idiopathic PD: absence of atypical signs, excellent levodopa response, normal cognition and absence of neuropsychiatric features. However, the mean AAO of our patients appeared to be later (57 years) than that reported in a recent meta-analysis (51 years) [START_REF] Deutschländer | VPS35-Related Parkinson Disease[END_REF], due to the presence of multiple affected relatives with a late onset of disease within the same family (see Supplementary Table 4). Lastly, SNCA mutation carriers had motor features similar to those of idiopathic PD, but an overall earlier AAO, a shorter disease course, a higher frequency of motor complications, a higher frequency of nonmotor signs and symptoms (cognitive decline in 17%, autonomic dysfunction in 39%, and psychotic symptoms and depression in 32%). Atypical signs have also been observed in rare PD patients carrying the SNCA Gly51Asp mutation [START_REF] Lesage | G51D α-synuclein mutation causes a novel parkinsonian-pyramidal syndrome[END_REF]. In this study, we also identified a rare known variant of SNCA, His50Gln, in the homozygous state. This variant has been described as a causal variant associated with late-onset PD, dementia and dystonia [START_REF] Appel-Cresswell | Alphasynuclein p.H50Q, a novel pathogenic mutation for Parkinson's disease[END_REF][START_REF] Proukakis | A novel αsynuclein missense mutation in Parkinson disease[END_REF], but a revaluation in larger datasets of PD patients and controls, including the GnomAD database (23/282,808 alleles), provided no evidence of pathogenicity for this variant [START_REF] Blauwendraat | Insufficient evidence for pathogenicity of SNCA His50Gln (H50Q) in Parkinson's disease[END_REF]. However, interestingly, the 42 year-old female patient carrying the His50Gln variant in our cohort had clinical features similar to those observed in carriers of other types of SNCA mutation carriers. She presented an early AAO (32 years), an excellent response to levodopa, motor complications, akinetic-rigid parkinsonism and dystonia, an absence of cognitive decline and neuropsychiatric symptoms, but the presence of autonomic dysfunction and atypical neurological signs, such as postural instability, REM sleep behavior disorder (RBD) and impulse control disorders (See Supplementary Table 2). However, in this study, the SNCA His50Gln was found using our customized gene panel and in absence of whole exome/genome sequencing to detect other possible pathogenic mutations, its pathogenicity remains inconclusive.

The principal strength of this national multi-center study is the large group of well-phenotyped and genotyped patients and family members recruited at the 16 different PDG centers, and the use of a standardized protocol, ensuring comparable and consistent clinical data reporting and diagnoses at each center. This enabled us to refine the estimated prevalence of mutations in genes causing AD PD in France. We show that our population, although mixed, has a relatively high frequency of SNCA, LRRK2 and VPS35 mutations. However, the clinical data were cross-sectional, most patients were European or North African, and our populations were biased towards EO cases.

In most instances, the phenotypes of cases due to AD PD mutations are indistinguishable from those of cases without mutations, demonstrating the need for genetic analysis for their identification. Genespecific disease-modifying therapies are currently being developed and tested. More generalized genetic testing is therefore required in PD patients, to identify those most likely to benefit from personalized care [START_REF] Tolosa | LRRK2 in Parkinson disease: challenges of clinical trials[END_REF].

  different generation, identified by an examination of secondary cases (n = 146) or on the basis of family history (n = 446). PD was diagnosed according to the clinical diagnostic criteria of the UK Parkinson Disease Society Brain Bank (PDSBB) (8). Comprehensive standardized interviews and neurological examinations were performed by movement disorder experts. Motor and non-motor symptoms were assessed by evaluating Unified Parkinson Disease Rating Scale (UPDRS) scores, Hoehn and Yahr staging, autonomic dysfunction, sleep, cognitive [Mini Mental State Examination (MMSE)], neuropsychological and behavioral scores. Early onset was defined as the onset of symptoms before the age of 51 years.

  23 with bi-allelic PRKN mutations and two with bi-allelic PINK1 mutations) of the 1089 PD index cases who have been screened for autosomal recessive (AR)-PD associated genes [814 by gene panel/exome sequencing (see below) and 275 by direct sequencing of the two most frequent AR PD genes, Parkin and PINK1].

The

  LRRK2 Gly2019Ser mutation was more common in PD index cases of North-African origin (100/221, 45.2%; 95% CI [38.6-52.1]) than in Europeans (36/1530, 2.4%; 95% CI [1.7-3.2]) (Fisher's exact test: odds ratio (OR) = 34.3, 95% CI [22.5-52.4], p <0.0001; Table 2). By contrast, SNCA point mutations and locus triplications/duplications were found mostly in PD index cases of European ancestry, accounting for 1.2% (18/1530) of such patients, whereas only one case of North-African ancestry (0.45%, 1/221) carried an SNCA duplication. VPS35 mutations were identified exclusively in patients of European origin, accounting for 0.19% (3/1530) of these individuals.

  Gly2019Ser and five with Arg1441His mutations, 29 with SNCA (see below) and eight with VPS35 Asp620Asn mutations. SNCA The clinical characteristics of the 29 PD patients carrying either SNCA rearrangements (triplications (n = 2) and duplications (n = 21)) or missense mutations [Ala53Thr (n = 3) and Gly51Asp (n = 3)] are shown in Table 3. All but three of the families concerned originated from France. The remaining three families, originating from Italy, Turkey and Morocco, all had SNCA duplications. Within this cohort, SNCA duplications were the most frequent mutation identified (14/19, 73.7%), followed by the Ala53Thr mutation (3/19, 18.8%). Disease onset occurred earliest in patients with the Ala53Thr mutation (mean 34.7 [SD 7.6], range 26-40 years). Patients carrying the Ala53Thr mutation had an extra-pyramidal parkinsonian syndrome, but with heterogeneity between patients with the same mutation: Patient 1172-001, with both SNCA Ala53Thr and a heterozygous PRKN Thr240Met variant, had atypical PD, with a poor response to levodopa, early motor fluctuations and cerebellar signs. He rapidly developed impulse control disorders. This patient currently displays no cognitive decline. He had a bilateral subthalamic deep brain stimulation (STN-DBS). He received clozapine treatment for delusions with a beneficial effect. Patient 1219-001 presented a parkinsonian syndrome that responded well to levodopa, but developed severe dysarthria. She underwent unilateral internal globus pallidus (GPi)-DBS. This patient presented no major cognitive and behavioral signs other than an alteration of executive functions. A third PD patient, 196-016 presented early-onset (26 years) typical PD that responded well to levodopa. Detailed clinical data are provided in Supplementary Table 2. Both SNCA locus triplications and Gly51Asp mutation were associated with early-onset atypical parkinsonism (mean AAO: 42.0 years [SD 8.5], range: 36-48 years and mean AAO: 42.0 [SD 15.7], range: 31-60 years, respectively). The patients with SNCA triplications were characterized by severe cognitive impairment in one of two carriers, dysautonomia, a poor response to levodopa in both patients. The three Gly51Asp mutation carriers had shorter disease duration (mean: 3.7 years [SD 3.1], range: 1-7 years), a mild-to-moderate response to levodopa, frequent psychiatric symptoms but no dementia or autonomic dysfunction (Supplementary Table

  4, CI [0.31-6,7], p = 0.71), AAO (mean 51.4 [SD 12.1] vs. mean 53.7 [SD 11.2] years, p = 0.62), disease duration (mean 8.7 [SD 6.7] vs. 9.1 [SD 4.5] years, p = 0.87) or clinical presentation, but the number of homozygous carriers was small. Unlike LRRK2 Gly2019Ser carriers, all patients with the Arg1441His mutation were French and all reported a family history of PD. They had a shorter disease duration (mean: 5.2 years [SD 5.1], range: 2-14 years vs. mean 9.0 years [SD 8.0], range: 0.5-63 years), were more likely to develop an akineticrigid motor phenotype (80% vs. 53%), had a slightly better response to levodopa (100% vs. 90%), and an absence of cognitive and neuropsychiatric symptoms, but a similar mean age at onset (52.6 years [SD 9.6], range: 39-64 years vs. 51.4 years [SD 12.1], range: 29-86 years).

Table 1 . Baseline demographics for our study population All index cases AD PD index cases

 1

	Isolated cases

Table 2 . Overall frequencies of LRRK2, SNCA and VPS35 mutations according to index case ethnicity, family history of Parkinson's disease and age at onset

 2

		Europeans	North-	North-Africans	AD PD	Isolated cases AD PD vs.	EO-PD	LO-PD	LO-PD vs.
			Africans	vs. Europeans			Isolated cases			EO-PD
				OR, 95% CI, p			OR, 95% CI, p			OR, 95% CI,
										p
		n = 1530	n = 221		n = 592	n = 1213		n = 1063	n = 636
	All LRRK2, n	(2.5%, [1.8-	100 (45.2%,	OR = 32.5,	54 (9.1%, [6.9-	84 (6.9%, [5.6-	OR = 1.3, [0.9-	71 (6.7%,	61 (9.6%,	OR = 1.5,
	(%, 95% CI)	3.4])	[38.6-52.0])	[21.4-49.2], p <	11.7])	8.5])	1.9], p = 0.11	[5.3-8.4])	[7.4-12.2])	[1.0-2.1], p =
				0.0001						0.03
	Gly2019Ser	(2.4%)	100 (45.2%)		52 (8.8%)	84 (6.9%)		71 (6.7%)	59 (9.3%)
	n (%)
	Arg1441His	(0.13%)	0 (0%)		2 (0.34%)	0 (0%)		0 (0%)	2 (0.31%)
	n (%)
	All SNCA, n	(1.2%, [0.7-	1 (0.45%,	OR = 0.38,	14 (2.4%, [1.3-	5 (0.41%, [0.1-	OR = 5.9, [2.1-	16 (1.5%,	3 (0.47%,	OR = 0.31,
	(%, 95% CI)	1.9])	[0.01-2.5])	[0.05-2.9], p =	3.9])	1.0])	16.3], p =	[0.9-2.4])	[0.1-1.4])	[0.09-1.1], p =
				0.50			0.0003			0.06
	Triplications	1 (0.065%)	0 (0%)		1 (0.17%)	0 (0%)		1 (0.094%)	0 (0%)
	n (%)
	Duplications	13 (0.85%)	1 (0.5%)		11 (1.9%)	3 (0.25%)		11 (1.0%)	3 (0.47%)
	n (%)
	Ala53Thr n	(0.20%)	0 (0%)		1 (0.17%)	2 (0.16%)		3 (0.28%)	0 (0%)
	(%)
	Gly51Asp n	1 (0.065%)	0 (0%)		1 (0.17%)	0 (0%)		1 (0.094%)	0 (0%)
	(%)

Table 3 . Summary of the clinical data for patients carrying SNCA, VPS35 and LRRK2 Arg1441His mutations identified in this study SNCA rearrangements SNCA missense mutations VPS35 LRRK2 Triplications a Duplications a Ala53Thr Gly51Asp b Asp620Asn c Arg1441His d

 3

	No. of carriers
	(index cases)

a Ibanez et al. (13) and Book et al. (15) b Lesage et al. (14) c Lesage et al. (10) d Lesage et al. (9)

Table 4 . Comparison of demographic and clinical characteristics of patients with Parkinson's disease by LRRK2 mutation status (LRRK2 carriers versus patients without mutations in known Parkinson's disease-associated genes) LRRK2 G2019S+ and patients without mutations; unadjusted comparisons LRRK2 G2019S + and patients without mutations; adjusted comparisons LRRK2+ n = 135 (8%)

 4 Data are given as the mean ± standard deviation for continuous variables and as counts (percentages) for qualitative variables.Welch's t-test was used to compare the groups for continuous variables and Fisher's exact test was used for binary variables. Coefficients for continuous clinical features and odds ratios (ORs) for binary clinical features, confidence intervals (CIs) and P-values were calculated from GLMs with mutation status, sex, age, and disease duration, for all variables except for onset variables, for which only mutation status, sex and age at onset were added. Linear models were used for continuous variables; GLMs with logit link and Bernouilli distributions were used for binary variables # Levodopa responsiveness was defined as a >30% improvement in subjective perceived motor symptoms *Dysautonomia included at least one of the following three signs: orthostatic hypotension, erectile dysfunction and/or urinary problems AD, autosomal dominant; MMSE, CI, Confidence Intervals; Mini Mental State Examination; OR, Odds Ratio; PD, Parkinson's disease; UPDRS III, the motor subsection of the Unified Parkinson's Disease Rating Scale ¥ P corrected for multiple testing by the Benjamini-Hochberg procedure

	Genetically	p-value	Coefficient or	p-	p-value
	undefined PD		OR (CIs)	value	adjusted ¥
	n = 1552 (92%)		(reference:
			non-mutation
			carriers)

* p < 0.05

[START_REF] Lunati | The genetic landscape of Parkinson's disease[END_REF] 21[START_REF] Lesage | G51D α-synuclein mutation causes a novel parkinsonian-pyramidal syndrome[END_REF]

(3) 3 (1) 8 (3) 5(2)No. of index cases with family history of PD

ACKNOWLEDGMENTS

We thank the patients and their families. We thank the DNA and Cell Bank of the ICM for sample preparation.

FUNDING

This work was supported by the Fondation de France, France-Parkinson Association, la Fédération pour la Recherche sur le Cerveau (FRC) and the French program "Investissements d'avenir" (ANR-10-IAIHU-06). This work was also supported in part by the Intramural Research Program of the National Institute on Aging, National Institutes of Health, Department of Health and Human Services; project number ZO1 AG000949.

DATA AVAILABILITY STATEMENT

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

ETHICS STATEMENT

The studies involving human participants were reviewed and approved by INSERM, CCPPRB du Groupe Hospitalier Pitié-Salpêtrière, Paris, France. Written informed consent to participate in this study was provided by the participant' legal guardian/next of kin.

AUTHOR CONTRIBUTIONS

SL conceived, designed and organized the study, wrote the first draft, reviewed, and critically revised the manuscript. JCC and AB conceived the project, reviewed, and critically revised the manuscript.

MH contributed to the statistical analysis and critically revised the manuscript. GM, CT, HB, SF, MA, CB-C, EB, ST, PD, FD, ER, FT, DG, FO-M, BD, FV, FC-D, A-MO-H, MV, EL, and AS contributed to the execution of the research project and critically revised the manuscript. All authors contributed to the article and approved the submitted version.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fneur. 2020.00682/full#supplementary-material VPS35 Asp620Asn, n (%, 95% CI)


PhD Suzanne Lesage
email: suzanne.lesage@upmc.fr

MSc Ariane Lunati

MSc Marion Houot

Ben Sawssan

PhD Romdhan

PhD Fabienne Clot

PhD Christelle Tesson

MD Graziella Mangone

MSc Benjamin Le Toullec

MSc Thomas Courtin

MSc Kathy Larcher

MD Mustapha Benmahdjoub

MD Mohammed Arezki

MD Ahmed Bouhouche

MD Mathieu Anheim

MD Emmanuel Roze

MD François Viallet

MD François Tison

MD Emmanuel Broussolle

MD Murat Emre

MD Hasmet Hanagasi

MD Basar Bilgic

MD Meriem Tazir

Mouna Ben Djebara

MD Riadh Gouider

MD Christine Tranchant

MD Marie Vidailhet

MD Eric Le Guern

PhD Olga Corti

MD Chokri Mhiri

MD Ebba Lohmann

PhD Andrew Singleton

MD Jean-Christophe Corvol

MD Alexis Brice
email: alexis.brice@icm-institute.org

Prof Alexis Brice

Characterization of recessive Parkinson's disease in a large multicenter study

Keywords: Parkinson's disease, PRKN, PINK1, autosomal recessive inheritance, genotypephenotype correlations AAO, age-at-onset, AR, autosomal recessive, EO, early-onset, GLM, generalised linear model, PD, Parkinson's disease, LRRK2, leucine-rich repeat kinase 2, PINK1, PTEN-induced putative kinase

Studies of the phenotype and population distribution of rare genetic forms of parkinsonism are required, now that gene-targeting approaches for Parkinson's disease have reached the clinical trial stage. We evaluated the frequencies of PRKN, PINK1, and DJ-1 mutations in a cohort of 1587 cases. Mutations were found in 14.1% of patients: 27.6% were familial and 8% were isolated. PRKN was the gene most frequently mutated in Caucasians whereas PINK1 mutations predominated in Arab-Berber individuals. Patients with PRKN mutations had an earlier age at onset, and less asymmetry, levodopa-induced motor complications, dysautonomia, and dementia than those without mutations.

Our understanding of the genetic basis of Parkinson's disease (PD) has been improved with the identification of several disease-causing genes. [START_REF] Lunati | The genetic landscape of Parkinson's disease[END_REF] Trials targeting these genes are underway and the development of cohorts ready for precision clinical trials that target genetic forms of PD are now required. [START_REF] Strafella | Application of Precision Medicine in Neurodegenerative Diseases[END_REF][START_REF] Vollstedt | 's Disease Study Group. Using global team science to identify genetic parkinson's disease worldwide[END_REF] PRKN, PINK1, and DJ-1 mutations are the most frequent cause of early-onset (EO) autosomal recessive (AR) typical PD. We investigated the frequency and nature of pathogenic variants of these three genes in a cohort of 1587 PD probands, comparing clinical characteristics between patients with PRKN mutations (PRKN-PD), and those without pathogenic variants of known PD genes (PD-NM).

Patients and Methods

Patient selection

Patients were enrolled between 1990 and 2018, through the French PDG network and North African and Turkish collaborations. PD was diagnosed according to the clinical diagnostic criteria of the UK Parkinson Disease Society Brain Bank (PDSBB). [START_REF] Hughes | Accuracy of clinical diagnosis of idiopathic Parkinson's disease: a clinico-pathological study of 100 cases[END_REF] Cases with mutations responsible for recessive atypical parkinsonism or those carrying the common LRRK2 Gly2019Ser variant were not included. We selected a cohort of 1587 probands from 497 AR PD families (at least two affected siblings and isolated cases born to consanguineous parents), and 1090 isolated cases, all screened for PRKN.

Standardized neurological examinations were performed by movement disorder experts, and 28 variables were used to obtain comparable data.

This article is protected by copyright. All rights reserved.

Most probands were Caucasian (n=1324, 83.4%; 927 French, 134 Turkish), Arab-Berber (n=213, 13.4%) or of other ethnicities (n=50, 3.2%). We included 1587 PD probands and 52 mutation-carrying relatives in the genotype/phenotype correlation analysis. Informed consent and approval from institutional review boards were obtained for sample collection.

Procedures

Probands were screened by denaturing high-performance liquid chromatography (dHPLC) and/or direct sequencing, [START_REF] Lesage | Deletion of the parkin and PACRG gene promoter in early-onset parkinsonism[END_REF][START_REF] Ibáñez | Mutational analysis of the PINK1 gene in earlyonset parkinsonism in Europe and North Africa[END_REF][START_REF] Leutenegger | Juvenile-onset Parkinsonism as a result of the first mutation in the adenosine triphosphate orientation domain of PINK1[END_REF][START_REF] Lohmann | Genetic bases and phenotypes of autosomal recessive Parkinson disease in a Turkish population[END_REF] next-generation sequencing (NGS) with a targeted gene panel or whole-exome sequencing [START_REF] Bouhouche | Mutation analysis of consanguineous Moroccan patients with Parkinson's disease combining microarray and gene panel[END_REF][START_REF] Lesage | Loss of VPS13C function in autosomalrecessive Parkinsonism causes mitochondrial dysfunction and increases PINK1/Parkin-dependent mitophagy[END_REF] .

PRKN was screened in 1587 probands, PINK1 and DJ-1 in 1223. Sanger sequencing was performed to confirm variants and cosegregation analyses, where possible. Exon rearrangements were detected by semi-quantitative PCR for PRKN [START_REF] Lesage | Deletion of the parkin and PACRG gene promoter in early-onset parkinsonism[END_REF] or with the MRC Holland Salsa MLPA P051/P052 Parkinson kits. Patients with an AAO ≤40 years and lymphoblastoid cells available for RT-PCR analysis (n=15) [START_REF] Lesage | Deletion of the parkin and PACRG gene promoter in early-onset parkinsonism[END_REF] or unaffected relatives for cosegregation analysis (n=30) were investigated for possible PRKN rearrangements undetectable by MLPA.

Statistical analysis

The PD-NM and PRKN-PD groups were compared with Welch's t-tests for continuous variables and Fisher's exact tests for categorical variables.

We used generalized linear models (GLMs) to compare clinical features between PRKN-PD with bi-allelic or double heterozygous mutations and PD-NM, adjusting for sex, age-at-onset This article is protected by copyright. All rights reserved.

(AAO), disease duration and dopaminergic medication. We used GLMs with identity links and normal distributions for continuous clinical features, and GLMs with logit links and Bernoulli distributions for binary clinical features. Interactions between AAO and disease duration were also included. Disease duration and dopaminergic medication were not included in models for clinical features at onset. Effects were assessed in Fisher type II tests, and effect size was estimated with Cohen's f2. Benjamini-Hochberg correction for multiple testing was performed. GLMs were generated for the 15/28 clinical features for at least 20% of the patients in each group with available data.

Results

Demographic and clinical data

In our cohort, men (n=951, 60%) and EO cases (mean AAO 40.2 [SD 12.1] years) were overrepresented, particularly among isolated cases (p<0.0001) (Table 1).

Distribution and nature of recessive PD-associated gene mutations

Bi-allelic or double heterozygous mutations of known AR PD-causing genes were present in 224 of the 1587 probands (14.1%): 27.6% (137/497) of familial and 8% (87/1090) of isolated cases. The most frequently mutated genes were PRKN (199/1587, 12.5%), then PINK1 (23/1223, 1.9%), and DJ-1 (2/1223, 0.16%). We identified 56 patients with single heterozygous variants in the three genes in whom AAO was significantly later than cases with bi-allelic or double heterozygous mutations (36.9 [SD 10.6] vs. 31. Most of the 23 probands with pathogenic PINK1 variants were Arab-Berbers (n=12, 52.2%); 8 carried the homozygous Gln456* mutation. Most variants were homozygous (n=18, 78.3%).

We identified 21 different pathogenic variants, including three not in databases (Fig 1B ).

Distribution of PRKN and PINK1 mutation carriers by age-at-onset, pattern of presentation and ethnicity

The proportion of probands with PRKN mutations decreased with increasing AAO: 42. This article is protected by copyright. All rights reserved.

Comparison of PRKN mutation carriers (PRKN-PD) with non-mutation carriers (PD-NM): genotype-phenotype correlations

We compared the clinical features between the 228/241 PRKN-PD and 1181/1307 PD-NM subjects without missing data (Table 2; Supplementary Table 1). The proportion of men (padj=0.016) and AAO (padj<0.0001) were greater in the PD-NM than the PRKN-PD group. Dopaminergic treatment was similar between groups (padj=0.69), but levodopa responsiveness was higher in PRKN-PD than PD-NM (padj=0.045).

After adjustment for covariables, PRKN-PD had a higher initial frequency of tremor (padj=0.0076), but lower frequencies of akinesia (padj=0.0003), micrographia (padj=0.010), and asymmetry (padj=0.0005) than PD-NM patients (Table 2). Dystonia at onset and cardinal symptom (bradykinesia, rest tremor and rigidity) frequencies were similar in both groups.

Motor severity after adjustment for disease duration was lower in PRKN-PD than in PD-NM (UPDRS score, padj=0.014), and PRKN-PD patients developed fewer levodopa-induced motor complications (dyskinesia: padj=0.0005; motor fluctuations: padj<0.0001). Non-motor symptoms, including dysautonomia (padj<0.0001) and dementia (padj=0.014), were less frequent in PRKN-PD patients.

Mutational and phenotypic characteristics of PINK1 and DJ-1 mutation carriers

The PINK1-associated phenotype of the 33 carriers resembled that of PRKN, but with a slightly later AAO (mean 34. frequency of tremor (16.7% vs. 68.5%; p=0.0005) and a higher rate of dystonia (90% vs. 18.2%; p<0.0001) at onset (Supplementary Table 2). Non-motor symptoms, such as dysautonomia (72.2% vs. 19.6%; p<0.0001) and dementia (25% vs. 3.8%; p=0.0007) were more frequent in patients with PINK1 variants, who were also more likely to display levodopa-induced dyskinesia (93.3% vs. 54.1%; p=0.0024) and motor fluctuations (66.7% vs.

46.2%; p=0.06).

The two pathogenic DJ-1 variant carriers (one Caucasian, one North African) each carried a previously unknown homozygous Glu94* and a compound heterozygous variant affecting the same highly conserved amino acid (Thr154Ile/Thr154Ala). They developed PD, with the four cardinal signs, at the ages of 29 and 28. Dystonia at onset, dyskinesia and orthostatic hypotension were noted in one patient, without cognitive signs.

Discussion

We established the spectrum and relative frequencies of mutations in a large cohort of AR PD cases, elucidating the genotype-clinical phenotype relationship. Homozygous/compound or double heterozygous pathogenic variants of PRKN, PINK1, and DJ-1 account for 14.1% of our PD patients, with PRKN the most frequently mutated. This study included a large number of genotyped and extensively phenotyped patients (276 mutation carriers) compared with a group of cases not carrying mutations of known AR PD-causing genes. However, the clinical data were cross-sectional, the numbers of patients with mutations of genes other than PRKN were small, and our populations were biased towards EO cases. In addition, 24% (48/199) of This article is protected by copyright. All rights reserved.

our PD patients with two PRKN mutations lacked information on their mutational phasing.

However, given that we found no mutations in cis in a co-segregation analysis of 59 index cases, we are confident that the vast majority of patients with two mutations carry them in trans. Nevertheless, our findings may have major implications for patient selection for genetic testing based on AAO, pattern of disease presentation, and ethnicity. Indeed, the frequency of pathogenic variants of AR PD-associated genes i) decreased with increasing AAO, to zero for an AAO beyond 60 years, ii) in cases with a positive family history or consanguinity was more than triple that in isolated cases, and iii) was much higher for PRKN than PINK1 in Caucasians, but similar for these two genes in Arab-Berbers. However, pathogenic PRKN variants were more frequent in our EO PD cases (<50 years, 190/1273, 14.9%) than in four other cohorts (10.1% in a Taiwanese cohort [START_REF] Lin | A clinical and genetic study of early-onset and familial parkinsonism in Taiwan: An integrated approach combining gene dosage analysis and next-generation sequencing[END_REF] and 2.8% in a Norwegian cohort [START_REF] Gustavsson | Genetic identification in early onset Parkinsonism among Norwegian Patients[END_REF] , both with EO defined as <45 years; 5.9% in a UK series [START_REF] Kilarski | Systematic review and UK-based study of PARK2 (parkin), PINK1, PARK7 (DJ-1) and LRRK2 in early-onset Parkinson's disease[END_REF] and 2.8% in a larger multicenter sample [START_REF] Marder | Predictors of parkin pathogenic variants in early-onset Parkinson disease: the consortium on risk for early-onset Parkinson disease study[END_REF] , both with EO <50 years). A meta-analysis of more than 5800 PD patients, found PRKN variants in 8.6% of PD cases with an AAO <50. [START_REF] Marder | Predictors of parkin pathogenic variants in early-onset Parkinson disease: the consortium on risk for early-onset Parkinson disease study[END_REF] We provide more precise data for PRKN and PINK1 genes, according to AAO. GLM analyses revealed that AAO was lower, disease progression slower and the response to levodopa stronger in patients with PRKN mutations than in those without pathogenic variants, as previously reported. [START_REF] Lin | A clinical and genetic study of early-onset and familial parkinsonism in Taiwan: An integrated approach combining gene dosage analysis and next-generation sequencing[END_REF][START_REF] Marder | Predictors of parkin pathogenic variants in early-onset Parkinson disease: the consortium on risk for early-onset Parkinson disease study[END_REF][START_REF] Lohmann | How much phenotypic variation can be attributed to parkin genotype?[END_REF][START_REF] Khan | Parkin disease: a phenotypic study of a large case series[END_REF] However, these mutations were not associated with higher rates of dystonia at onset, a trait more strongly associated with EO (see Supplementary Table 1) than with genetic status. Patients with PRKN mutations also had a distinctive non-motor symptom profile, with lower frequencies of dementia and dysautonomia, consistent with previous reports. [START_REF] Alcalay | Cognitive and motor function in long-duration PARKIN-associated Parkinson disease[END_REF] After adjustment for disease duration and dopaminergic medication, these This article is protected by copyright. All rights reserved.

patients had fewer treatment-induced complications, such as dyskinesia and motor fluctuations, than PD-NM patients. This very pure and slowly progressive phenotype makes patients with PRKN variants, very good candidates for deep-brain stimulation (DBS). [START_REF] Pal | Genetic and clinical predictors of deep brain stimulation in young-onset Parkinson's disease[END_REF][START_REF] Rizzone | Genetic background and outcome of deep brain stimulation in Parkinson's disease[END_REF] The causal gene(s) remained unidentified for a number of families with AR PD (~72.5%), suggesting that pathogenic variants of known genes may have been missed or the involvement of unknown genes. These findings will help to guide routine genetic testing and to establish cohorts of patients for clinical trials targeting the gene defects or their physio-pathological consequences.

This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved.

Coefficients for continuous clinical features and odds ratios (ORs) for binary clinical features and standard error (SE), Cohen's f2 and p-values were calculated from GLMs with mutation status, sex, age-at-onset, disease duration, L-DOPA group and age-at-onset vs. disease duration for all 15 variables except for onset variables for which only mutation status, sex and age-at-onset were added. Linear models were used for continuous variables; GLMs with logit links and Bernouilli distributions were used for binary variables # Levodopa responsiveness was defined as a >30% improvement in subjective perceived motor symptoms ¥ p corrected for multiple testing by the Benjamini-Hochberg procedure *p < 0.05

  3 [SD 11.1], p=0.0009); they were removed from genotype/phenotype correlation analyses. This article is protected by copyright. All rights reserved. The 199 cases carried either homozygous (n=92), or compound heterozygous (n=59), confirmed by segregation analysis of all available unaffected relatives or double heterozygous (n=48) PRKN mutations for whom phasing was still unknown. These carriers displayed 77 different variants, including 19 absent from public databases (www.mdsgene.org) 11 (Fig 1A). RT-PCR identified a single French family with PRKN Ex3del/Ex3dup compound heterozygous rearrangements not detectable by MLPA but elucidated by co-segregation analysis.

  2% for an AAO ≤20 years, 29% (21 to 30 years), 13% (31 to 40 years), and 4.4% (41 to 60 years) (Fig 2A). This decrease was more marked in isolated than familial cases (Fig 2B). PINK1 mutations were less frequent than PRKN mutations (1.9% vs. 12.5%) but more evenly distributed among familial cases for AAOs up to 60 years. Neither PRKN nor PINK1 variants were found in patients with onset after 60 (Fig 2A-D). PRKN mutations were more frequent in Caucasians (179/1324, 13.5%) than in Arab-Berbers (16/213, 7.5%) (Fig 2E). Conversely, PINK1 mutations were more common in Arab-Berbers (12/188, 6.4%) than Caucasians (9/1005, 0.9%) (Fig 2E).

  6 [SD 12.2] years vs. 31.3 [SD 10.9] years; p=0.20), a lower This article is protected by copyright. All rights reserved.

FIGURE 1 :

 1 FIGURE 1: Schematic representation of the (A) PRKN and (B) PINK1 genes and respective

FIGURE 2 :

 2 FIGURE 2: Distribution of PRKN and PINK1 mutation carriers by age-at-onset, pattern of

TABLE 1 .

 1 Demographic data for our study population

		Whole PD	n	AR PD, including	n	Isolated	n	AR PD vs.
		group		isolated cases		cases		isolated case
				with				p value
				consanguinity
	% men (n)	60.0 (951)	1587	57.1 (284)	497	61.2 (667)	1090	0.14
	Ethnic background		1587		497		1090	<0.0001*
	% Caucasian (n)	83.4 (1324)		75.5 (375)		87.1 (949)
	% Arab-Berbers (n)	13.4 (213)		22.1 (110)		9.4 (103)
	% others/mixed (n)	3.2 (50)		2.4 (12)		3.5 (38)
	% Consanguinity (n)	12.9 (202)	1564	40.9 (202)	494	0	1070	NA
	Age-at-onset (SD), y	40.2 (12.1)	1543	43.8 (15.0)	485	38.6 (10.1)	1058	<0.0001*
	Range	2-81		3-81		2-74
	Age at examination (SD), y	49.7 (13.2)	1575	54.2 (14.5)	495	47.7 (12.0)	1080	<0.0001*
	Range	9-87		16-87		9-79
	Disease duration (SD), y	9.8 (8.6)	1530	10.9 (9.6)	482	9.2 (8.0)	1048	0.0005*
	Range	0-63		0-63		0-48
	Frequencies were compared in Fisher's exact tests for qualitative traits and means were compared in t-tests for continuous
	variables.
	*p < 0.05
	NA, Not appropriate.

TABLE 2 .

 2 Demographic and clinical characteristics of patients with Parkinson's disease with pathogenic PRKN variants (PRKN-PD) and of patients without pathogenic variants (PD-NM) PD-NM and PRKN-PD unadjusted comparisons PD-NM and PRKN-PD adjusted comparisons Data are expressed as mean (standard deviation) for continuous variables, and as counts (percentages) for categorical variables. We used t-tests to compare the two groups for continuous variables and Fisher's exact tests for binary variables.

	Characteristics	PD-NM	PRKN-PD	p value	p value	Coefficient or	Cohen's	p value	p value
		n=1181	n=228		adjusted ¥	odds ratio	f2		adjusted ¥
						(OR) (SE)
	Baseline
	Sex (% male)	727/1181	118/228	0.0063*	0.016*
		(61.6%)	(51.8%)
	Age at	50.5 (13.2)	45.4 (12.9)	<0.0001*	<0.0001*
	examination
	(SD), y
	Disease duration	8.8 (7.8)	14.1 (10.4)	<0.0001*	<0.0001*
	(SD), y
	Age-at-onset	41.6 (12.0)	31.2 (10.7)	<0.0001*	<0.0001*
	(SD), y
	L-DOPA-treated 791/1181	148/228	0.54	0.69
		(67%)	(64.9%)
	Levodopa	659/738	142/149	0.022*	0.045*	1.9 (0.82)	0.007	0.11	0.15
	responsiveness #	(89.3%)	(95.3%)
	Motor symptoms and signs
	Dystonia at onset 164/990	37/201	0.54	0.69	0.84 (0.18)	0.001	0.42	0.46
		(16.6%)	(18.4%)
	Akinesia at onset 590/946	99/206	0.0002*	0.0010*	0.51 (0.09)	0.015	0.0001*	0.0003*
		(61.3%)	(48.1%)
	Tremor at onset	570/960	142/205	0.0091*	0.021*	1.7 (0.29)	0.007	0.0030*	0.0076*
		(59.4%)	(69.3%)
	Micrographia at	308/927	42/204	0.0003*	0.0013*	0.58 (0.11)	0.007	0.0047*	0.010*
	onset	(33.2%)	(20.6%)
	Asymmetry	1001/1035	181/198	0.0016*	0.0049*	0.26 (0.09)	0.010	0.0002*	0.0005*
		(96.7%)	(91.4%)
	Bradykinesia	1033/1069	201/208	1.0000	1.0000	1.4 (0.63)	0.002	0.43	0.46
		(96.6%)	(96.6%)
	Rigidity	1008/1066	194/204	0.87	0.90	1.1 (0.42)	<0.001	0.77	0.77
		(94.6%)	(95.1%)
	Tremor	796/1056	168/205	0.031*	0.057	1.5 (0.32)	0.002	0.048*	0.072
		(75.4%)	(82%)
	UPDRS-Part III	19.6 (13.8)	15.9 (11.9)	0.0017*	0.0049*	-3.3 (1.2)	0.008	0.0081*	0.014*
	ON (SD)
	Hoehn&Yahr ON	2 (0.91)	2.00 (0.93)	0.61	0.74	-0.14 (0.09)	0.003	0.13	0.16
	(SD)
	Dyskinesia	457/667	97/178	0.0007*	0.0025*	0.44 (0.09)	0.020	0.0001*	0.0005*
		(68.5%)	(54.5%)
	Motor	485/663	82/177	<0.0001*	<0.0001*	0.32 (0.07)	0.041	<0.0001*	<0.0001*
	fluctuations	(73.2%)	(46.3%)
	Non-motor symptoms and signs
	Dysautonomia	254/470	36/192	<0.0001*	<0.0001*	0.19 (0.05)	0.095	< 0.0001*	<0.0001*
		(54.0%)	(18.8%)
	Dementia	67/667	6/153 (3.9%) 0.017*	0.037*	0.34 (0.16)	0.009	0.0087*	0.014*
		(10.0%)

This article is protected by copyright. All rights reserved.

This article is protected by copyright. All rights reserved.his article is protected by copyright. All rights reserved.

Acknowledgement

We thank the patients and their families. We thank the DNA and Cell Bank of the ICM Brain Institute for sample preparation. This work was supported by the Fondation de France, France-Parkinson Association, la Fédération pour la Recherche sur le Cerveau (FRC) and the French program "Investissements d'avenir" (ANR-10-IAIHU-06). This work was supported in part by the Intramural Research Program of the National Institute on Aging, National Institutes of Health, Department of Health and Human Services; project number ZO1 AG000949.

Supplementary Materials

This file includes: Supplementary Table 1 Supplementary Table 2 Supplementary Table 3 Author Contributions SL, J-CC, and AB contributed to the conception and design of the study. All authors contributed to the acquisition and analysis of data. SL, AL, MH, CT, ER, J-CC, AB contributed to the drafting of the text and figure preparation.

Potential Conflicts of Interest

This article is protected by copyright. All rights reserved.

None to report.


Fellow,IEEE Michel Barlaud
email: michel.barlaud@i3s.unice.fr

Frédéric Guyard
email: frederic.guyard@orange.com

Learning sparse deep neural networks using efficient structured projections on convex constraints for green AI

Deep neural networks (DNN) have been applied recently to different domains and perform better than classical state-of-the-art methods. However the high level of performances of DNNs is most often obtained with networks containing millions of parameters and for which training requires substantial computational power. To deal with this computational issue proximal regularization methods have been proposed in the literature but they are time consuming. In this paper, we propose instead a constrained approach. We provide the general framework for this new projection gradient method. Our algorithm iterates a gradient step and a projection on convex constraints. We studied algorithms for different constraints: the classical 1 unstructured constraint and structured constraints such as the 2,1 constraint (Group LASSO). We propose a new 1,1 structured constraint for which we provide a new projection algorithm. Finally, we used the recent "Lottery optimizer" replacing the threshold by our 1,1 projection. We demonstrate the effectiveness of this method with three popular datasets (MNIST, Fashion MNIST and CIFAR). Experiments with these datasets show that our projection method using this new 1,1 structured constraint provides the best decrease in memory and computational power.

I. MOTIVATION

Deep neural networks have been applied recently to different domains and have shown a dramatic improvement in accuracy of image recognition [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], speech recognition [START_REF] Nassif | Speech recognition using deep neural networks: A systematic review[END_REF] or natural language processing [START_REF] Otter | A survey of the usages of deep learning in natural language processing[END_REF]. These studies relied on deep networks with millions or even billions of parameters. For instance, the original training of ResNet-50 [START_REF] He | Deep residual learning for image recognition[END_REF] (image classification) contains 25.6M parameters and required 29 hours of processing using 8 GPUs. Storing the model requires 98MB. The cost in memory of the inference on a single 224x224 image is about 103MB and 4 GFLOPs are needed [START_REF] Canziani | An analysis of deep neural network models for practical applications[END_REF]. The recent development of DNNs, hardware accelerators like GPUs and the availability of deep learning frameworks for smartphones [START_REF] Ignatov | AI benchmark: Running deep neural networks on android smartphones[END_REF] suggest seamless transfer of DNN models trained on servers onto mobile devices. However, it turns out that the memory [START_REF] Rallapalli | Are very deep neural networks feasible on mobile devices[END_REF] and energy consumption [START_REF] Han | Ese: Efficient speech recognition engine with sparse lstm on fpga[END_REF] are still the main bottlenecks for running DNNs on such devices. Thus the computational cost has an impact on the carbon footprint. It has been argued that this trend is environmentally unfriendly [START_REF] Strubell | Energy and policy considerations for deep learning in nlp[END_REF]. Some authors of [START_REF] Schwartz | Green ai[END_REF] advocate a practical solution by providing an efficient evaluation criterion. In this paper, we propose a new splitting projection-gradient method with an efficient structured constraint to cope with these computational and memory issues. In the formulation of our method, a constraint defines a convex set and the regularization is replaced by projection onto this convex set. The benefits of this formulation are twofold. Firstly, the constraint has a direct geometric interpretation whereas the impact of parameter values in traditional regularization methods are more difficult to understand. Secondly, the convergence of this new method is formally proved. The paper is organized as follows. We first present related works in Section II, then in Section III, we develop the theoretical background of our constrained projection method. In Section IV, we compare experimentally the methods. The tests involve several datasets with different neural network architectures.

II. RELATED WORKS

Weights sparsification

It is well known [START_REF] Denil | Predicting parameters in deep learning[END_REF] that DNN models are largely overparametrized and that in practice, relatively few network weights are actually necessary to accurately learn data features. Based on this result, numerous methods have been proposed in order to remove network weights (weight sparsification) either on pre-trained models or during the training phase. A basic idea to sparsify the weights of the neural network is to use the Least Absolute Shrinkage and Selection Operator (LASSO) formulation [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]. The 1 penalty added to the classification cost can be interpreted as a convexification of the 0 penalty. In [START_REF] Han | Learning both weights and connections for efficient neural network[END_REF], weights with the smallest amplitude in pretrained networks are removed. Model sensitivity to weights can also be used [START_REF] Tartaglione | Learning sparse neural networks via sensitivity-driven regularization[END_REF], [START_REF] Gomez | Learning sparse networks using targeted dropout[END_REF]. where weights with a weak influence on the network output are pruned. Constraint optimization is used in order to learn sparse networks with 0 , 1 or 2 constraints on the weights [START_REF] Carreira-Perpiñán | Learning-compression algorithms for neural net pruning[END_REF].

Accepted for publication ICPR 2020 Milan

These methods generally produce networks with random sparse connectivity, i.e. high-dimensional but sparse weight matrices. They only partially reduce the computational demand since they result in networks with sparse weight matrices, requiring the availability of sparse matrix multiplication to effectively take advantage of the sparsity. Decreasing both memory and computational requirements can however be achieved by suppressing neurons instead of weights. This approach is frequently referred to as structured sparsification or neuron level sparsification. The two main approaches for structured sparsity are based on group regularization and lowrank factorization. Many regularizing techniques have been proposed to allow structured sparsification. Pruning methods are sparsifying pretrained networks. Filters in CNN are pruned based on the 1 norm of their kernel weights [START_REF] Li | Pruning filters for efficient convnets[END_REF]. Some authors perform channel pruning using LASSO regression and least squared reconstruction [START_REF] He | Channel pruning for accelerating very deep neural networks[END_REF]. Neurons are pruned based on the average percentage of zeros (APoZ) after the ReLU activation [START_REF] Hu | Network trimming: A datadriven neuron pruning approach towards efficient deep architectures[END_REF].

Learning structured sparse DNNs using regularization methods

In contrast to the case of weight sparsification, neuron level sparsification introduces a new challenge forcing adoption of other types of regularization. The most common approaches are based on group LASSO 2,1 [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF] or on sparse group LASSO 2,1 + 1 [START_REF] Friedman | Regularization path for generalized linear models via coordinate descent[END_REF] regularization.

Numerous other methods include regularization during the training of the DNN. It is customary in DNN learning to train networks with Stochastic Gradient Descent (SGD) with momentum, even in the case where non-smooth penalization is used [START_REF] Scardapane | Group sparse regularization for deep neural networks[END_REF]. Group LASSO regularization in a number of studies in [START_REF] Scardapane | Group sparse regularization for deep neural networks[END_REF], [START_REF] Wen | Learning structured sparsity in deep neural networks[END_REF]. Group LASSO and filter decorrelation regularization are used in order to discard CNN filters [START_REF] Friedman | A note on the group lasso and a sparse group lasso[END_REF]. Group LASSO and group variance regularization have been used in [START_REF] Torfi | Attentionbased guided structured sparsity of deep neural networks[END_REF].

To deal with non-smooth 1 regularization, subgradient descent is used in [START_REF] Liu | Learning efficient convolutional networks through network slimming[END_REF]. Here, structured sparsification is performed without group regularization. The idea is to scale the neurons output with a given factor λ i and apply 1 regularization to push the various factors λ i towards 0.

Fully connected layers can be represented by their weights matrix (i.e. 2d tensor) whereas convolutional layers correspond to 4d tensors. One of the popular compression methods for DNN is nuclear regularization (Nuclear norm penalty) [START_REF] Cheng | A survey of model compression and acceleration for deep neural networks[END_REF]. Nuclear norm penalty was successfully used in matrix low rank approximation [START_REF] Cheng | A survey of model compression and acceleration for deep neural networks[END_REF], matrix completion [START_REF] Zhang | Matrix completion by truncated nuclear norm regularization[END_REF], matrix factorization [START_REF] Cavazza | Dropout as a low-rank regularizer for matrix factorization[END_REF] and DNN dropout modeling [START_REF] Srivastava | Dropout: a simple way to prevent neural networks from overfitting[END_REF].

Learning structured sparse DNNs using proximal regularization methods

A different approach is however based on optimization under convex constraint where proximal methods are the most natural tools. We recall the proximal operator of a function f (x) [START_REF] Moreau | Proximité et dualité dans un espace hilbertien[END_REF]:

prox τ f (x) := arg min x f (x) + x -x 2 2τ , (1)
Let W be the weight matrix of a neural network, L(W ) be a gradient Lipschitz loss and R(w) be a convex penalty. We define the penalty criterion by min W L(W ) + λR(W ). This criterion can be minimized using a classical forward-backward method belonging to the class of splitting methods [START_REF] Lions | Splitting algorithms for the sum of two nonlinear operators[END_REF], [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF], [START_REF] Mosci | Solving structured sparsity regularization with proximal methods[END_REF], [START_REF] Sra | Optimization for Machine Learning[END_REF]. Using SGD with penalization is limited and time consuming due to the tuning of the corresponding penalization hyper-parameters [START_REF] Hastie | The entire regularization path for the support vector machine[END_REF], [START_REF] Mairal | Complexity analysis of the lasso regularization path[END_REF].

Proximal gradient descent with group LASSO constraints was used in [START_REF] Zhou | Less is more: Towards compact cnns[END_REF] and in [START_REF] Alvarez | Learning the number of neurons in deep networks[END_REF]. In [START_REF] Huang | Data-driven sparse structure selection for deep neural networks[END_REF], the output neurons are scaled using a factor λ i and an accelerated proximal gradient is used with a 1 constraint to push as many coefficients λ i towards 0 without significantly decreasing the performance. In [START_REF] Yoon | Combined group and exclusive sparsity for deep neural networks[END_REF], neuron sparsification is performed with proximal gradient descent with group LASSO constraint and an additional 1,2 constraint (Exclusive Sparsity) enforces neurons to fit disjoint sets of features. Similarly, in [START_REF] Zhang | Learning to share: Simultaneous parameter tying and sparsification in deep learning[END_REF] proximal gradient descent with group OWL constraint (grOWL [START_REF] Oswal | Representational similarity learning with application to brain networks[END_REF]) is used to simultaneously sparsify neurons and enforce parameter sharing. Proximal gradient descent is also used in [START_REF] Zhang | Simultaneous sparsity and parameter tying for deep learning using ordered weighted 1 regularization[END_REF] where Ordered Weighted 1 regularization (OWL [START_REF] Figueiredo | Ordered weighted l1 regularized regression with strongly correlated covariates: Theoretical aspects[END_REF]) allowing simultaneous sparsify weights and optimized weight sharing. In [START_REF] Lin | Toward compact convnets via structure-sparsity regularized filter pruning[END_REF], filters in CNN layers are pruned by solving an optimization problem using a dedicated optimizer with either group LASSO or 2,0 regularization.

Goal of the work

Classical Learning structured sparse DNNs are based on proximal regularization methods. In this paper, we propose an alternative constrained approach that takes advantage of available efficient projections 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], [START_REF] Perez | A filtered bucketclustering method for projection onto the simplex and the 1 -ball[END_REF] and on the 2,1 ball [START_REF] Liu | Multi-task feature learning via efficient l2, 1-norm minimization[END_REF], [START_REF] Barlaud | Robust supervised classification and feature selection using a primal-dual method[END_REF]. We further propose a new 1,1 projection.

III. LEARNING SPARSE DNN

A Projection gradient algorithm for constrained learning

In this work, we propose a constrained approach in which the constraint is directly related to the number of zero-weights. Moreover it takes advantage of an available efficient projection on the 1 -ball [START_REF] Condat | Fast projection onto the simplex and the l1 ball[END_REF], [START_REF] Perez | A filtered bucketclustering method for projection onto the simplex and the 1 -ball[END_REF]. Let L(W ) be a gradient Lipschitz loss, R(w) be a convex constraint, and C its convex set. Lets us define the following criterion min

W L(W ) s.t. R(W ) η (2)
where the scalar η 0 is the constraint parameter. We use a splitting gradient-projection method to minimize this criterion based on the following forward-backward scheme to generate a sequence of iterates [START_REF] Barlaud | Classification and regression using an outer approximation projection-gradient method[END_REF]:

V n := W n -γ∇L(W n ), (3)
W n+1 := proj(V n ) + ε n , (4)
where proj denotes the projection on the convex constraint. We can therefore apply the algorithm to any constraint for which an exact or approximate projection can be computed. We derive the following algorithm Algorithm 1 Splitting gradient-projection algorithm where ∇L(W ) is provided by the net and proj(ηV ) is the projection on the constraint Input: X, Y, W 0 , N, γ, η

for n = 1, . . . , N do V ← W -γ • ∇L(W ) W ← proj(ηV ) end for Output: W

Optimizer with structured constraints

In the case of the constraint R(w) = W 1 , efficient algorithms have been proposed [START_REF] Condat | Fast projection onto the simplex and the l1 ball[END_REF], [START_REF] Perez | A filtered bucketclustering method for projection onto the simplex and the 1 -ball[END_REF]. Unfortunately this 1 constraint does not induce a sparse structure.

W 1 := i,j |w i,j |.
Classical optimizer with 2,1 norm constraint (Group LASSO): The Group LASSO was first introduced in [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF]. The main idea of Group LASSO is to enforce model parameters for different classes to share features. Group sparsity reduces complexity by eliminating entire features. Group LASSO consists in using the 2,1 norm for the constraint on W . The row-wise 2,1 norm of a n × d matrix W (whose columns are denoted w i , i = 1, . . . , d) is defined as follows:

W 2,1 := d i=1 w i .
We use the approach proposed in [START_REF] Barlaud | Robust supervised classification and feature selection using a primal-dual method[END_REF] to compute the projection W on the 2,1 -ball of radius η of a n × d matrix V (whose columns are denoted

v i , i = 1, • • • , d) : compute t i which is the projection of the vector ( v i i ) d
i=1 on the 1 ball of radius η in R n . Each column of the projection is then obtained according to

w i = t i v i max{t i , v i } , i = 1, . . . , d.
This last operation is denoted as

W i := proj 2 (V i , t i ) in Algorithm 2.
This algorithm requires the projection of the vector

( v i i ) d i=1
on the 1 ball of R n of radius η whose complexity is only O(d×log(d)). We can note that another approach was proposed in [START_REF] Liu | Multi-task feature learning via efficient l2, 1-norm minimization[END_REF]. The main drawback of their method is to compute the roots of an equation using bisection, which is quite slow.

Algorithm 2 Projection on the 2,1 norm-proj 1 (V, η) is the projection on the 1 -ball of radius η Input: V, η t := proj 1 ((

v i i ) d i=1 , η) for i = 1, . . . , d do w i := proj 2 (v i , t i ) end for Output: W
A new optimizer with an adaptive weighted 1,1 norm constraint: Unfortunately, algorithm 2 does not provide efficient sparsity. Thus we propose the algorithm 3.

Algorithm 3 Projection on the 1,1 norm-proj 1 (V, η) is the projection on the 1 -ball of radius η Input: V, η t := proj 1 (( v i 1 ) d i=1 , η) for i = 1, . . . , d do w i := proj 1 (v i , t i ) end for Output: W
This "projection" can be seen in the following way: Given a matrix V = (v i,j ), 1 ≤ i ≤ d, 1 ≤ j ≤ n, the algorithm computes t as the solution of:

min d i=1 |ti|≤η d i=1   n j=1 |v i,j | -t i   2 (5)
and then for all i = 1, . . . , d, the solution w of:

min n j=1 |wi,j |≤ ti n j=1 (v i,j -w i,j ) 2 . ( 6
)
Clearly both t and w are bounded by η in their respective 1 norm.

Let us consider, for ε > 0, the problem

(P ε ) min i ti≤η j |wi,j |-ti≤0 d i=1   n j=1 |v i,j | -t i   2 +ε d i=1 n j=1 (v i,j -w i,j ) 2
It turns out that this is a (strongly) convex problem, with solution (t ε , w ε ). It is easy to show that (t ε , w ε ) → ( t, w) as ε → 0. Indeed , since ( t, w) is admissible i ti ≤ η and j | wi,j | ≤ ti , and then, using the strong convexity of the objective with respect to t i ,

d i=1 (t ε i -ti ) 2 + d i=1   n j=1 |v i,j | -t ε i   2 + ε d i=1 n j=1 (v i,j -w ε i,j ) 2 ≤ d i=1   n j=1 |v i,j | -ti   2 + ε d i=1 n j=1 (v i,j -wi,j ) 2 .
Using first that t is a minimizer of the first term in the righthand side, we find:

d i=1 (t ε i -ti ) 2 +ε d i=1 n j=1 (v i,j -w ε i,j ) 2 ≤ ε d i=1 n j=1 (v i,j -wi,j ) 2
Sending ε → 0, we obtain that t ε → t. Then, dividing the expression by ε, and considering ε → 0 again along a subsequence for which w ε converges to some limit w, we obtain:

d i=1 n j=1 (v i,j -w i,j ) 2 ≤ d i=1 n j=1 (v i,j -wi,j ) 2 .
Now, since in the limit, t ε → t, we have n j=1 |w i,j | ≤ ti for all i = 1, . . . , d, so that w is admissible in the minimization problem [START_REF] Ignatov | AI benchmark: Running deep neural networks on android smartphones[END_REF] whose solution is w. Hence w = w, and in fact we get that lim ε→0 w ε = w. Observe also that this shows also t εt = o( √ ε). Similarly, one can show that w εw = o(ε1/4 ). Actually a more precise analysis shows that t εt = O(ε 2/3 ) and w ε -w = O(ε 1/3 ). A further remark is that one can show the problem (P ε ) to be equivalent to (P + ε ) defined with

min i ti≤η j |wi,j |-ti≤0 d i=1   n j=1 |v i,j | -t i +   2 +ε d i=1 n j=1 (v i,j -w i,j ) 2 .
whose value is now convex with respect to V . Indeed, if i,j |v i,j | ≤ η, it is easy to see that the value of both problems is zero, with w = v (while the value of t is no longer unique in this case in (P ε )). On the other hand, when i,j |v i,j | > η, then at least for one i one must have j |v i,j | > t i so that, if for some other i , j |v i ,j | < t i , it is clear that decreasing slightly t i and increasing t i of the same amount, one can reduce the objective. Hence, in that case, the unique solution of (P + ε ) is given by the solution of (P ε ) and the value is the same. This shows that in fact, the values of both problems are equal, and hence, also, convex with respect to V (since the objective in the (P + ε ) is globally convex in (V, t, w)).

Lottery optimizer

Following the work of Frankle and Carbin [START_REF] Frankle | The lottery ticket hypothesis: Finding sparse, trainable neural networks[END_REF], [START_REF] Zhou | Deconstructing lottery tickets: Zeros, signs, and the supermask[END_REF] proposed a simple algorithm to find sparse sub-networks within larger networks that are trainable from scratch. Their approach to finding these sparse networks is as follows: after training a network, set all weights smaller than some threshold to zero, rewind the rest of the weights to their initial configuration, and then retrain the network from this starting configuration but with the zero weights frozen (not trained). We replaced the thresholding by our 1,1 projection and devised the following algorithm: Algorithm 4 Projection on the 1,1 norm-proj 1 (V, η) is the projection on the 1 -ball of radius η, ∇L(W, M 0 ) is the masked gradient with binary mask M 0 , and f is the ADAM optimizer, γ is the learning rate

Input: W * , γ, η for n = 1, . . . , N (epochs) do V ← f (W, γ, ∇L(W )) end for t := proj 1 (( v i 1 ) d i=1 , η) for i = 1, . . . , d do w i := proj 1 (v i , t i ) end for Output: W, M 0 Input: W * for n = 1, . . . , N (epoch) do W ← f (W, γ, ∇L(W, M 0 )) end for Output: W IV. EXPERIMENTAL RESULTS
We used the pytorch framework to implement our sparse learning method using a constrained approach. We chose the Adam optimizer [START_REF] Kingma | a method for stochastic optimization[END_REF], a standard optimizer in PyTorch as baseline comparison to our optimizer with 1 and 2,1 constraints. We denoted as "PGL1", the algorithm with 1 constraint, "PGL21", the algorithm with 2,1 constraint and "PGL11" the algorithm with 1,1 constraint. We used the entropy (bit/weight) of the weights distributions to compute estimations of the model storage memory cost. To this end, the weights can for instance be coded using JPEG2000 1 an image coding system that uses state-of-the-art compression techniques based on wavelet theory. The classical computational cost evaluates FLOPs (floating point operations) as a measure. When using FLOPs, additions (accumulates) and multiplications are counted separately. However, a lot of hardware can compute multiply-add operations in a single instruction. We therefore use MACCs (multiply-accumulate operations) as computational cost for which one multiplication and one addition are counted as a single instruction. We provide the results in normalized bytes and MACCs: we divided the number of bytes or MACCs by the number of bytes or MACCs obtained with Adam (i.e. without constraint). For all experiments we used Algorithm 4. Computation was performed on a Cocolink Klimax 210 HPC with 10 GPUs (Nvidia Quadro P6000, P100 and GeForce GTX 1080).

Results on MNIST with a convolutionnal Network

We selected the popular MNIST dataset [START_REF] Lecun | The mnist database of handwritten digits[END_REF] containing 28 × 28 grey-scale images of handwritten digits of 10 classes (from 0 to 9). This dataset consists of a training set of 60,000 instances and a test set of 10,000 instances. We consider a neural network with two convolutional layers and two linear layers denoted as Net4. The size of its weight matrices are (1 × 10 × 5 × 5), (10 × 20 × 5 × 5), (320 × 50) and (50 × 10) respectively. Thus the total number of elements of these weight matrices are 250, 5000, 16000 and 500 respectively. To apply the 1,1 constraint to the tensor, we unfolded the tensor in a matrix form. The first layer, which interacts directly with the input image and the last one interacting directly with the output are most sensitive to sparsity and thus we did not apply sparsity constraint.

The sizes of matrix weights was very unbalanced. Thus our strategy was to sparsify the 2 most numerous layers, i.e. the 2 nd convolutional and the 1 st linear layer with the same optimizer.

We studied the layer-wise influence of the constraint parameter η on the accuracy and the weight sparsity, defined as the percentage of weights set to zero. For comparison purposes, the weights using Adam for the two linear layers are depicted in figure 2 (top shows unstructured sparsity, bottom shows structured sparsity using PGL11).

Figure 1 represents the weight distributions using a Parzen kernel method. The top part shows that weights distributions follow a Gaussian shape for Adagrad and the Adam optimizer. The bottom part shows that weights distributions follow a Laplacian shape (thus lower entropy) for PGL1 and PGL11. Results on MNIST with a Linear fully connected Network

We used a linear fully connected network (LFC4) with an input layer of d neurons, 4 hidden layers followed by a RELU activation function and a latent layer of dimension k. Comparison with public results on MNIST using Le Net 300/100

Le Net 300/100 is a popular Linear Fully connected Network. Table III shows that our method outperforms the state-of-the-art [START_REF] Tartaglione | Learning sparse neural networks via sensitivity-driven regularization[END_REF] in terms of bytes accuracy compromise. Note that, to the best of our knowledge no results have been published in terms of FLOP reduction on this basis.

Results on Fashion MNIST using a Linear Fully connected Network. We observe a similar behavior to that of the previous experiment with the MNIST dataset. Table (IV) shows a large decrease in the global memory by a factor 9. On the other hand, we observed a decrease in the calculation cost by a factor of 9 for the 1,1 constraint and a very small decrease for the 1 constraint.

Fashion-MNIST

Results on CIFAR10

The CIFAR-10 data set is composed of 60,000 32x32 color images, 6,000 images per class, for a classification in 10 classes. The training set is made up of 50,000 images, while the remaining 10,000 are used for the testing set. We use Simplenet 3 , the highly optimized architecture [START_REF] Hasanpour | Lets keep it simple, using simple architectures to outperform deeper and more complex architectures[END_REF]. This network is composed of 13 blocks B i = (convolutional2D/Batch N ormalization/ReLU ) for i = 1, .., 13 with sequences M axP ool2d/Dropout after the blocks B 4 , B 7 , B 9 , B 10 , B 12 and B 13 followed by a classifier layer. Results are reported in Figures 5 andin the Table V.

Table (V) shows a large global decrease in memory by a factor of 10. On the other hand the decrease in the calculation cost was about 30% for the 1,1 constraint and almost null for

V. DISCUSSION

To the best of our knowledge, entropy of the weights (bit/weight) as a measure of memory has never been reported in the DNN literature. Thus comparison of memory with that of previous reports is not straightforward. A more in-depth study will be performed as well as the JPEG2000 compression of the model for storage and reported in a forthcoming paper. Energy consumption is directly related to the number of instructions (Flops or MACCs) [START_REF] Grochowski | Energy per instruction trends in intel ® microprocessors[END_REF]. Thus we report MACCs rather than Flops. The experimental results with MNIST and Fashion MNIST using Lenet 300/100 showed an improvement in memory by a factor 12.9 and 8 and by a factor 3 and 2 for computational power. We obtain on CIFAR10 a good trade-off between a 0.98% drop in accuracy and a substantial improvement in memory by a factor of 8.5 in comparison with the Adam optimizer. Note that, in this paper, the different projections are not layerwise optimized. The same constraint was applied to each layer of the network. There is a priori no reason to believe that the same constraint value is adapted to the sparsification of all the layers of a network.

VI. CONCLUSION

To deal with the computational issue associated to DNNs, a lot of studies into proximal regularization methods which are time consuming have been published. In this paper, we propose an alternative constrained approach. We provide a general framework with a new projection gradient method. We designed algorithms for the classical 1 constraint and the new 1,1 constraint. Our experiments show the benefit of the Lottery optimizer that uses only one projection for deep neural network sparsification. Experiments using three popular datasets (MNIST, FASHION MNIST and CIFAR) show that our new projection method on the 1,1 constraint provides better structured sparsity resulting in a substantial decrease in the cost of memory and of computation. Furthermore, we are currently applying our method to other large Neural Networks.

Fig. 1 :Fig. 2 :

 12 Fig. 1: MNIST, Net4, Top: Distribution of convolutional layer Conv2 with Adam and Adagrad optimizers, Bottom : Distribution of the convolutional layer Conv2 with PGL1 and PGL11 optimizers

Figure 3 ,Fig. 4 :

 34 Figure 3, figure 4 and tables I and II 2 show that the main

  [START_REF] Xiao | Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms[END_REF] is a dataset from the publication by Zalando of article images consisting of a training set of 60,000 examples and a test set of 10,000 examples. Each example is a 28x28 grayscale image, associated with a label from 10 classes. Fashion-MNIST is to serve as a direct drop-in replacement for the original MNIST dataset for benchmarking machine learning algorithms. Fashion-MNIST and MNIST share the same image size and structure of training and testing splits.

Fig. 5 :

 5 Fig. 5: CIFAR 10 SimpleNet, MACCs as a function of η, PGL1 (red) and PGL11 (green)

TABLE I :

 I MNIST Net 4 total memory, MACCs, and accuracy

	Methods	Memory (kBytes) (k-MACCs) MACCs	Accuracy (%)
	Adam	33.64	480	99.
	PGL1 η = 80	10.9	477	99.01
	PGL11 η = 40	3.7	336	98.03
	PGL11 η = 25	2.1	122	97.4

TABLE II :

 II MNIST wth LFC4, total memory, MACCs, and accuracy

	Methods	Memory (kBytes) (k-MACCs) MACCs	Accuracy (%)
	ADAM	3989	2379	98.3
	PGL1(η = 200)	438	1960	98.29
	PGL11(η = 200)	72	150	97.7
	PGL11(η = 400)	215	480	98.07
	PGL21(η = 50)	1810	1408	98.05

TABLE III :

 III MNIST Le Net 300/100 total memory, MACCs, and accuracy

	Methods	Memory (kBytes) (k-MACCs) MACCs	Accuracy (%)
	ADAM	477	266.2	98.21
	PGL1(η = 200)	61	189	98.03
	PGL11(η = 200)	32	62	96.4
	PGL11(η = 400)	83	150	97.8
	PGL21 (η = 50)	164	257	98.1
	Tartaglione [15]	33.7	-	96.6

TABLE IV :

 IV Fashion MNIST LFC4 : Memory, MACCs, and accuracy

	Methods	Memory (kBytes) (k-MACCs) MACCs	Accuracy (%)
	ADAM	3989	2379	89.9
	PGL1(η = 400)	567	2114	89.2
	PGL11(η = 400)	131	267	87.5

TABLE V :

 V CIFAR10 total memory, and accuracy using Simplenet

	Methods	MACCs (M-MACCs) (M-Bytes) Memory	Accuracy %
	Adam	631.51	9.44	93.8
	PGL1 (η = 13000)	626.08	1.45	91.12
	PGL11(η = 14000)	441	0.86	91

https://jpeg.org/jpeg2000/index.html

The projections PGL1, PGL11 and PGL21 being different, the impact of the projection parameter η is different on each of them. When comparing the projections (tables II and III), we provide parameter η such that the models PGL1, PGL11 and PGL21 have a similar accuracy.

constraint (Figure5). 3 https://github.com/Coderx7/SimpleNet_Pytorch

Acknowledgement The authors would like to thank Antonin Chambolle for his contribution to the theoretical background of the 1,1 "projection".