Marta F Riesco

Félix Francisca

Matias Domitilia

Joaquim Sandra

Suquet Marc

Cabrita Elsa

Elsa Cabrita
email: ecabrita@ualg.pt

First study in cryopreserved Crassostrea angulata sperm

Keywords: Crassostrea angulata, Sperm, Cryopreservation, Motility, Viability, Fertilization Crassostrea angulata, sperm, cryopreservation, motility, viability, fertilization

Sperm cryopreservation is a widely employed technique that promotes alternative techniques to contribute to broodstock management or restoration programs for species of commercial interest, endangered species or species with an interesting genotype. The preservation of genetic material from improved stocks or from the original population is extremely important for the oyster aquaculture industry to prevent the potential impacts of epidemic diseases and natural disasters. The Portuguese oyster, Crassostrea angulata, was the most important species commercialized by the shellfish industry. However, inadequate management of this industry and pathology occurrences resulted in a significant decrease in natural populations. For this reason, in this work a successful sperm cryopreservation protocol for this important species has been developed for the first time. Different internal cryoprotectants (DMSO, ethylene glycol, polyethylene glycol and methanol) at several concentrations (5, 10, 20%), containers (straws vs cryovials) and freezing rates (slow and fast rates) were tested. Cryoprotectant toxicity tests corroborated that this assay did not take into account the following steps of cryopreservation protocol as sperm agglutination. A fast freezing rate of cells diluted in10% DMSO and the use of straws as containers were the best cryopreservation conditions for Portuguese oyster sperm. Finally, fertilization assays confirmed the efficiency of the cryopreservation protocol in oyster sperm. These results demonstrated that different susceptibilities have been detected concerning sperm cryopreservation depending on oyster species or genetic material composition.

Highlights

► Toxicity tests do not take into account the following steps of cryopreservation protocol as sperm agglutination. ► 10% DMSO, fast freezing and the use of straws as container represent the best cryopreservation conditions for Portuguese oyster sperm. ► Fertilization ability is the definitive tool to evaluate the cryopreservation protocol efficiency in oyster sperm. ► Sperm cryopreservation protocol was successfully established in C. angulata for the first time.

Introduction

Most bivalve production still relies on the collection of wild seed [START_REF] Suquet | Survival, Growth and Reproduction of Cryopreserved Larvae from a Marine Invertebrate, the Pacific Oyster (Crassostrea gigas)[END_REF]. Seed collection can have a negative environmental impact or be a limiting factor in the development of the industry due to the unpredictable seed supply [START_REF] Corporeau | Proteomic identification of quality factors for oocytes in the Pacific oyster Crassostrea gigas[END_REF]. Seed production in hatcheries constitutes a complement to the wild seed supply for the shellfish industry, and can offer new opportunities for genetic improvement, including the creation of special lines (e.g. tetraploids, disease resistant). Moreover, a number of bivalve species worldwide, including oysters, have been affected by mass mortality events associated with herpesviruses (European Food Safety Authority, [START_REF] Efsa | Oyster mortality. EFSA Panel on Animal Health and Welfare (AHAW)[END_REF], resulting in significant losses. In this respect, the development of selection programs to obtain improved selected genotypes combined with mid-and long-term cryopreservation in cryobanks of these genotypes could be a successful tool for restoration programs in some endangered species or populations.

In this regard, sperm cryopreservation as a tool for preserving genetic material has already been established in several fish and oyster species. These cryobanks allow the availability of gametes all year round for fertilization outside the normal reproductive seasons [START_REF] Hu | High-throughput cryopreservation of spermatozoa of blue catfish (Ictalurus furcatus): establishment of an approach for commercial-scale processing[END_REF]Yang et al., 2012). In terms of sperm cryopreservation, oysters are the most widely studied mollusc species, in particular the Pacific oyster, Crassostrea gigas [START_REF] Hassan | Sperm cryopreservation in oysters: A review of its current status and potentials for future application in aquaculture[END_REF]. There are other species of particular interest in which cryopreservation could play a particular role in the conservation of native populations and in improving reproductive strategies for production. In the Portuguese oyster, C. angulata, sperm has not been cryopreserved until now. Some decades ago in Europe, C.angulata was the major species commercialized by the shellfish industry [START_REF] Boudry | Differentiation between populations of the Portuguese oyster, Crassostrea angulata (Lamark) and the Pacific oyster, Crassostrea gigas (Thunberg), revealed by mtDNA RFLP analysis[END_REF]. However, inadequate management of this industry, together with diminished water quality, exposed the species to several pathologies that almost lead to its extinction [START_REF] Comps | Une virose de l'huitre Portugaise (Crassostrea angulata Lmk)[END_REF].

Nowadays, pure wild populations of C. angulata are rare to find. Cryopreservation technology could promote alternative techniques to contribute to the management of restoration programs in this important species. Although protocols have been developed for a closely related species, C. gigas, different susceptibilities have been detected concerning sperm cryopreservation depending on oyster species or genetic material composition [START_REF] Dong | The need for standardization in cryopreservation: a case study with oysters[END_REF]. Therefore, the direct application of one protocol to a wider number of species may cause a decrease in sperm cryopreservation success due to variability in the cryopreservation process (sperm collection, selection of extenders, concentration and type of cryoprotectants (CPAS), packaging of sperm and freezing and thawing rates) within and among species [START_REF] Tiersch | Introduction[END_REF].

For this reason, the aim of the present work was to establish a sperm cryopreservation protocol for C. angulata sperm for the first time. Previous studies in different oyster species were taken into account, and some preliminary assays, such as caffeine addition [START_REF] Suquet | Marathon vs sprint racers: an adaptation of sperm characteristics to the reproductive strategy of Pacific oyster, turbot and seabass[END_REF] were carried out for the improvement of postthaw sperm motility.

Material and methods

Oyster maintenance

Oyster broodstock (C. angulata) with a commercial weight > 70g was acquired from the Ria de Alvor and Mira river bivalve farms during the natural spawning season and maintained at the Experimental Shellfish Aquaculture Station (IPMA, I.P.), Portugal. During this period, oysters were fed daily with a mixture of two microalgae (Chaetoceros calcitrans and Skeletonema costatum: 50/50 cells/µL) and kept in open systems. Temperature was maintained at 20°C to 22°C.

Sperm collection

The oysters were opened and a small sample of gonad tissue was examined microscopically to determine sex. Sperm was collected by the dry method, extracting the sperm directly from the gonad using a micropipette. The gonadal area was previously wiped to remove any contamination. Depending on the experiment, sperm was pre-diluted in an extender media and maintained at 4ºC until used. Sperm from at least 3 males was used in all the experiments.

Effect of the extender solution

In order to test two extender solutions for C. angulata oyster sperm, samples were collected as previously described and immediately pre-diluted 1:10 (v/v) in artificial sea water (Asw) [START_REF] Gwo | Semen cryopreservation of small abalone (Haliotis diversicolor supertexa)[END_REF] or in Asw supplemented with 10 mM caffeine (pH=8.1; osmolarity: 1150 mOsm/Kg). The diluted sperm was filtered using two different sieves: 100 µm to retain larger impurities such as gonad clumps, and 20 µm for smaller debris.

The samples were stored at 4ºC for 3 days. Motility was analyzed daily until the end of the experiment. Prior to this, a set of experiments was performed to test the effect of sperm temperature during activation (Figure 1 Supplementary Material). Thus, sperm motility was analyzed further by waiting 30 minutes at RT (Room Temperature) before loading the sperm into the Makler chamber and activating it with Asw or Asw+caffeine.

Total motility was determined using ISAS software (ISAS, Proiser R+D, S.L., Spain). Using a 10 µL sample, sperm movement and concentration were determined using CASA software. For all the analyses, motility was assessed in a Makler chamber using a phase-contrast microscope (Nikon 200, Japan) with a 10 x negative contrast objective, and a digital camera (Basler A312f C-mount, Germany) set for 25 fps. The CASA software settings were adapted to oyster sperm. The parameter settings were: 1 < Particles Area (in microns 2 ) < 90; connectivity: 6. Only males showing total motility rate values equal to or higher than 40% were used. Sperm concentration was measured using the same software and values ranged from 1 to 2 x 10 9 spermatozoa/ml in all analyzed samples. The experiments were performed using 4 individual males.

Cryoprotectant toxicity assays

The aim of the toxicity test was to determine the best cryoprotectants (CPAs) to use and their ideal concentration. For this, four different CPAs were tested, polyethylene glycol (PEG), dimethyl sulfoxide (DMSO), ethylene glycol (EG) and methanol (MetOH), at three concentrations (5, 10 and 20%, v/v final concentration) and in two dilution mediums, Asw and Asw supplemented with 10 mM caffeine. Sperm motility and viability were the analyzed parameters chosen in the toxicity study.

Pre-diluted sperm was diluted 1:1 (v/v) under each condition and exposed to the CPAs effect for 10 minutes (equilibration time) at 4°C. After this, 10 µL of diluted sperm were used to determine motility as previously described, and 15 µL were used for the viability test. The percentage of motile cells and cell concentration were recorded.

Spermatozoa viability was determined mixing 15 µL of diluted sperm, 0.5 µL SYBR Green (final concentration 100 nM) and 1.5 µL propidium iodide (final concentration 12 µM) and observed in a fluorescence microscope (Nikon E200, Japan) equipped with a triple excitation filter block DAPI-FITC-Texas Red (excitation filter wavelengths: 395-410 nm (bandpass, 403 CWL), 490-505 nm (bandpass, 498 CWL), and 560-580 nm (bandpass, 570 CWL). The percentage of live cells (cells with membrane integrity-green cells) was recorded. At least 100 cells per slide were counted, and two slides per pool and per treatment were observed. Both motility parameters and cell viability were measured in fresh sperm as the control. The lowest toxicity conditions were chosen for the next steps (n=3 individual males).

Cryopreservation assays

Comparison of two different freezing rates

In this experiment, two different freezing rates were tested. Sperm was diluted in DMSO and PEG at two concentrations (10 and 20%), and diluted in Asw or Asw+ 10 mM caffeine. The conditions for the slower freezing were: 0.3 °C/min from 0 to -30 °C, after which the samples were plunged directly into liquid nitrogen, and for the faster freezing, 6 °C/min from 0 to -70 °C also followed by plunging the samples directly into liquid nitrogen. The freezing rates were adapted from those previously employed by other authors in C. gigas [START_REF] Ieropoli | Effects of extender composition, cooling rate and freezing on the fertilisation viability of spermatozoa of the Pacific oyster (Crassostrea gigas)[END_REF][START_REF] Suquet | Survival, Growth and Reproduction of Cryopreserved Larvae from a Marine Invertebrate, the Pacific Oyster (Crassostrea gigas)[END_REF]. A portable programed biofreezer (Asymptote Grant EF600, UK) was used for both freezing rates.

All samples were cryopreserved in 0.5 mL French straws (IMV, France).

For thawing, the straws were removed from the liquid nitrogen and placed in a water bath at 37 °C for 10 s and the samples were immediately used to evaluate sperm quality, through motility and viability tests, as described before. Samples presenting agglutination were discarded after thawing in order to preserve sperm viability.

Agglutination was verified by analyzing photos of cryopreserved samples (Canon G12 digital camera) and classified according to the method described by [START_REF] Dong | Control of sperm concentration is necessary for standardization of sperm cryopreservation in aquatic species: Evidence from sperm agglutination in oysters[END_REF].

This method was based on an agglutination scale with five different levels of sperm agglutination: (i) homogeneous suspension, level 0; (ii) few clumps discernable, level 1;

(iii) many clumps evident, level 2; (iv) aggregation of clumps, level 3; (v) formation of elongated clumping ("noodles"), level 4; (vi) formation of well-developed noodles, level 5. After the photographic evaluation, viability photos were acquired from the same samples in order to determine whether agglutination correlated with cell survival.

All the experiments were repeated using at least three different males and the same samples were analyzed in fresh as the control.

Establishment of sperm packaging: straws vs cryovials

The type of container was also tested using 0.5 mL French straws and 1mLcryovials. To cryopreserve the samples, the best conditions obtained previously were chosen. Samples were diluted in Asw containing 10% DMSO and the faster freezing rate was used. The comparison between straws and cryovials was based on the motility and viability results.

Samples cryopreserved in straws and cryovials were thawed at 37°C during 10 and 90 s, respectively. These different times were established according to the sample volume, sperm concentration, and container type.

Fertilization capacity

Fertilization capacity was tested in order to evaluate the designed protocol. For fertilization procedures both gametes were extracted using two different methods.

Male gametes were obtained by the dry method, as described before, and female gametes by the wet method. In the wet method, the oocytes were released by the gonad, using a scalpel, performing small horizontal and vertical cuts. Artificial seawater was used to wash the gonad and to collect the oocytes into a 1 L beaker. For each fertilization, gametes from at least four different females were collected and pooled in equal amounts. The oocytes were then filtered using two sieves: 100 µm to retain larges debris and 20 µm to retain the oocytes, and left to incubate in 1L Asw for 30 min to induce final maturation. Oocyte concentration was determined counting 5 different 1mL subsamples from each pool (n=5) in a counting chamber (Sedgewick rafter, Portugal). Calculations for the fertilization procedures were made in order to have 200 000 eggs/L for each incubator.

For the sperm, approximately 40 straws per pool (number of males per pool=3 and number of pools=5) were thawed using the conditions previously established (10% DMSO in Asw, and fast freezing method) and an aliquot was removed from the cryopreserved samples to determine concentration after thawing, using the CASA system. The same procedure was followed for the fresh samples. Four different spermatozoa/egg ratios (1 000, 2 500, 5 000 and 10 000 spz/egg) were tested for the cryopreserved material. For the control (pre-diluted fresh sperm in Asw), a ratio of 1000 spz/egg was used according to [START_REF] Song | Setting of a procedure for experimental fertilization of Pacific oyster (Crassostrea gigas) oocytes[END_REF] for C.gigas. For each fertilization, oocytes were maintained in 500 mL Asw and the correct proportion of sperm was added and mixed. After 20-30 min, 500 mL of fresh seawater was added to the 1L beaker. Fertilizations were done using 5 replicates under controlled environmental conditions and kept at 21-22 °C for 24 hours. Five subsamples of eggs were removed after homogenization and the number of fertilized and non-fertilized eggs counted. The percentage of fertilized eggs was calculated in respect to the total number of incubated eggs.

Statistical analysis

Data collected were treated and analyzed using the software SPSS Statistics v.22 (IBM, USA). To analyze the effects of caffeine on sperm motility, a general linear model with the Bonferroni correction was used (p< 0.05). One-way ANOVA (p<0.05) was performed followed by Student-Newman-Keuls (S-N-K) as a post hoc test for the motility, viability and fertilization tests after Portuguese oyster sperm cryopreservation and in toxicity studies. Results are expressed as a mean of percentages in all analyses of 3-5 individual males.

Results

Effect of the extender composition

Two methods were tested in this assay in order to check the long term effect of sperm dilution prior to cryopreservation. Preliminary tests were done to check the necessity of sperm acclimation to RT prior to being used when previously stored at 4ºC. It was shown that C. angulata sperm can be stored successfully at 4ºC for 3 days without losing motility and that acclimation at room temperature prior to the motility analysis enhances (p<0.05) the percentage of motile cells (Figure 1 supplementary data). Incubation with caffeine did not stimulate an increase in the percentage of motile cells (Figure 1 supplementary data, p< 0.05).

Toxicity test

In general, exposure to CPAs decreased sperm motility. The effect of caffeine supplementation depended on the CPA used. It significantly increased motility percentages in: 10 and 20% PEG, 5% DMSO and 5% MetOH (p< 0.05). The other conditions tested showed no significant differences between the samples exposed to CPAs and the respective fresh control (Figure 1). The CPA with the least toxic effect in terms of motility was the 10% PEG diluted in Asw supplemented with caffeine (Figure 1A) and 20% MetOH was the most toxic when diluted in both mediums (Figure 1C) (p< 0.05).

With regard to sperm viability, caffeine supplementation in the dilution medium did not have any drastic effects on cell viability. Also, as expected, 5% CPA concentration showed higher viability values for all the CPAs used in both dilution mediums (Figure 2), and the most harmful conditions were 20%MetOH in terms of cell survival rate (Figure 2D) (p< 0.05). In view of these results, MetOH was discarded in the next step of the cryopreservation trials.

Linking the results for motility and viability, the toxicity test defined 10% PEG as the least toxic combination for C. angulata spermatozoa (viable sperm > 85%), as no significant differences were observed in either parameter (motility and viability) when compared with fresh sperm.

Cryopreservation assays

3.3.1 DMSO using a faster freezing method reported the highest viability rates.

Two CPAs were chosen according to the previous results and to literature references for Crassostrea genus. Therefore, PEG was selected as the least toxic CPA in terms of sperm viability and motility and DMSO was selected in terms of producing a reduction in sperm agglutination after post-thawing, a process described by several authors in oyster sperm cryopreservation and confirmed in this study (Ierepoli et al., 2004[START_REF] Hassan | Sperm cryopreservation in oysters: A review of its current status and potentials for future application in aquaculture[END_REF].

At the slower freezing rate, post-thaw sperm motility was very low (<5%) in all treatments and significantly different from the control (p<0.05) (Figure 3A). No significant differences were observed between CPAs, their concentrations or extender mediums. The highest viability rates (>40%) were obtained using 10 and 20% DMSO and 10% PEG, regardless of the extender used (with and without caffeine) (Figure 3B). The same motility results were observed at the faster freezing rate. Sperm motility after cryopreservation was very low in comparison with the fresh control (>40%) (Figure 4A). These motility results were independent from the conditions tested (CPAs, their concentrations and dilution medium) and no significant differences were observed between the cryopreserved samples. According to the viability results, higher viability rates were obtained using 10 and 20% DMSO (with and without caffeine as additive) (Figure 4B). The viability rates were, in all cited cases, higher than 70% and no significant differences were observed in comparison with the fresh control (Figure 4B).

The results of the agglutination test coincided with previously described results. The use of PEG increased the agglutination level in sperm when compared with DMSO (Figure 2, supplementary material) and the respective images taken of the viability test confirmed that most of the cells cryopreserved with PEG were all non-viable (stained with PI, red cells) (Figure 2, supplementary material), contrary to the results obtained using DMSO (SYBR green positive cells).

In summary, PEG proved to be inadequate for oyster sperm cryopreservation, since it produced the worst results in terms of agglutination and sperm viability in both freezing rates and at the two concentrations used (Figures 3B and4B). However, regardless of the results obtained in the toxicity assays, 10% DMSO diluted in Asw (Figure 4B) was the best CPA to use with a faster freezing rate.

The type of container was not critical for sperm viability and motility after cryopreservation.

There were no significant differences in terms of sperm post-thaw motility and viability between samples stored in 0.5 mL French straws and 1mL cryovials. Motility values were lower in cryopreserved sperm compared with fresh (Figure 5A). However, no significant differences were found in terms of cell viability (Figure 5B). Taking into account that no significant differences were found among these containers, we chose the straws as they were the best and most economical way to store and transport large quantities of these cells.

Fertilization ability is the definitive assay to ensure sperm quality after cryopreservation.

Taking into account the low motility rates, a fertilization assay was necessary to demonstrate C. angulata sperm functionality. The fertilization trials performed with four different spermatozoa/egg ratios revealed significant differences between the different conditions. The fresh sperm, as expected, showed the highest percentage (>90%) of fertilized eggs (Figure 6). As shown in Figure 6, higher fertilization rates (58-72%) were obtained, ranging from 2.500 to 10 000 spz/egg, the highest ratio being the one that showed better results (>70%), even if not significantly different from the rest.

Discussion

C.angulata sperm showed motility for 3 days without decreasing significantly.

These observations are in accordance with previous studies carried out in oyster sperm motility [START_REF] Rose | Viability of American oyster, Crassostrea virginica, spermatozoa exposed to stress[END_REF] which demonstrated that sperm movement usually lasts longer in marine invertebrates than in marine fish species. Sperm motility lasted 4 to 5 hours in Eastern oyster (C. virginica) [START_REF] Faure | Paramètres physiologiques de l'émission et de l'activation des gamètes mâles de deux mollusques bivalves: la coquille Saint Jacques Pecten maximus (L.) et l'huîtrecreuse Crassostrea gigas (Thunberg)[END_REF] and >24 h in Pacific oyster (C. gigas) [START_REF] Boulais | Involvement of mitochondrial activity and OXPHOS in ATP synthesis during the motility phase of spermatozoa in the Pacific Oyster, Crassostrea gigas[END_REF]. C. angulata sperm proved to be very resistant and to have long duration movement, since motility did not decrease until the third day (Figure 1, supplementary material). Long-term sperm motility duration can be attributed to the reproductive strategy of the different species. According to published literature, many parameters can influence sperm motility, such as temperature and extender media composition including additives such as caffeine [START_REF] Suquet | Marathon vs sprint racers: an adaptation of sperm characteristics to the reproductive strategy of Pacific oyster, turbot and seabass[END_REF]. For this reason, we evaluated the influence of several parameters to obtain the best possible sperm conditions before sperm cryopreservation and to determine standard conditions for sperm motility analysis. In our results, sperm motility significantly increased after refrigerated sperm acclimation to room temperature for 30 minutes (Figure 1, supplementary material). This motility temperature influence could be explained by an increase in the metabolism of these cells at higher temperatures, triggering the activation of non-motile cells. Like in humans and other species, when spermatozoa are incubated at lower temperatures than the natural fertilization conditions, they adopt a resting state, which allows sperm to preserve their energy [START_REF] Gallup | On the origin of descended scrotal testicles: the activation hypothesis[END_REF]. This mechanism is known as "the activation hypothesis". This hypothesis possibly explains the diminished motility of refrigerated spermatozoa compared to those acclimated to RT.

The addition of caffeine to the dilution medium has been widely studied in fish species such as Pacu (Piaractus mesopotamicus), where sperm motility in fresh samples increased with a linear correlation with the increase in caffeine concentration [START_REF] Carvalho | Effect of Caffeine Added to the Activating Solution on Sperm Motility of Fresh and Thawed Semen of Pacu, Piaractus mesopotamicus, and Curimba, Prochilodus lineatus[END_REF]. In Pacific oyster, Suquet and colleagues (2012) reported that the sperm maturation process could be stimulated through the addition of 10 mM caffeine to sea water, increasing the beat frequency of flagella. This reaction can also be obtained with the addition of theophylline [START_REF] Robles | Sperm cryopreservation of sexreversed rainbow trout (Oncorhynchus mykiss): parameters that affect its ability for freezing[END_REF]. Caffeine and theophylline are methylxanthines that act as phosphodiesterase inhibitors and can increase the intracellular cAMP (a derived form of ATP), which stimulates protein phosphorylation and triggers sperm movement [START_REF] Suquet | Marathon vs sprint racers: an adaptation of sperm characteristics to the reproductive strategy of Pacific oyster, turbot and seabass[END_REF]. As a consequence of the action of methylxanthine, we can assume that its addition to the activation solution should be similar to the addition of AMPc. In our study, the effect of caffeine supplementation was beneficial during the toxicity assays. However, when we carried out different cryopreservation assays (with two different freezing rates) using PEG and DMSO with and without 10mM caffeine, we found that the addition of caffeine to PEG caused sample agglutination (Figure 2, supplementary material). Sperm agglutination is a phenomenon that appears in sperm solutions after thawing and has previously been described in other oyster species such as Pacific and Eastern oyster [START_REF] Dong | Control of sperm concentration is necessary for standardization of sperm cryopreservation in aquatic species: Evidence from sperm agglutination in oysters[END_REF]. Moreover, the freezing rate also played an important role in this phenomenon, since agglutination was higher in samples cryopreserved with a slower freezing rate and the use of faster freezing was avoided. These results are similar to those in previous studies by [START_REF] Dong | Control of sperm concentration is necessary for standardization of sperm cryopreservation in aquatic species: Evidence from sperm agglutination in oysters[END_REF], demonstrating that CPAs at low concentrations, cooling at slow rates and high sperm concentration led to the formation of higher levels of agglutination. Agglutination drastically affected C.angulata sperm viability and helped to explain the fact that toxicity studies do not take into account the possible interactions of CPAs with the following step of the cryopreservation process. For all these reasons, toxicity assays are a previous step that should be corroborated with a cryopreservation trial before discarding or choosing any condition. Thus, in the present study, although 10% PEG was the CPA with the least toxic effect on the cells (Figure 1 and 2), 10-20% DMSO proved to have a better CPA effect. DMSO is one of the most suitable CPAs due to its ability to penetrate cells rapidly, thus protecting cells from freezing and thawing injury [START_REF] Hassan | Sperm cryopreservation in oysters: A review of its current status and potentials for future application in aquaculture[END_REF].

Taking into account that motility rates are lower in cryopreserved samples compared to fresh samples, a fertility assay was included to demonstrate sperm functionality after the selected cryopreservation protocol (10% DMSO, the faster freezing rate and 0.5mL straws as a container). Correlations between motility and fertilization capacity have been found in different oyster species [START_REF] Suquet | Changes in motility, ATP content, morphology and fertilization capacity during the movement phase of tetraploid Pacific oyster (Crassostrea gigas) sperm[END_REF]. cryopreservation, this phenomenon being avoided by using some CPAs and discarding some conditions (slow freezing). In our study, fertilization assays were the definitive tool to demonstrate the functionality of the cryopreservation protocol. In the future, other quality assays should be performed in this species to check for cryopreservation effects at different levels with the aim of guaranteeing the total safety of the technique.

Conclusions

• Toxicity tests do not take into account the following steps of the cryopreservation protocol as agglutination phenomenon.

• 10% DMSO, the faster freezing rate and the use of straws as containers are the best cryopreservation conditions for Portuguese oyster sperm.

• Fertilization ability is the definitive tool for evaluating the efficiency of a cryopreservation protocol efficiency in oyster sperm.

• A sperm cryopreservation protocol was successfully established in C.angulata for the first time.

Acknowledgments

The authors are grateful to Marcelo Livramento (CCMAR) and Ana Margarete Ramos (IPMA) for their technical assistance during oyster sampling and oyster conditioning. This work was supported by project CRIOBIV-31-03-05-FEP-59. Marta 8. DMSO in Asw, using the faster freezing method, was tested and compared with the pre-diluted fresh control (Asw). Motility (A) and viability (B) were expressed as a mean of percentages ± SE in 4 individual males. One-way ANOVA (p<0.05) was

Figure 1 .

 1 Figure 1. C. Angulat motility results after toxicity tests. Four different

Figure 2 .

 2 Figure 2. C. Angulata viability results after toxicity tests. Four different

Figure 3 .

 3 Figure 3. C. Angulata motility and viability results after slower freezing. Two

Figure 4 .

 4 Figure 4. C. Angulata motility and viability results after faster freezing. Two

Figure 5 .

 5 Figure 5. C.angulata motility and viability results comparing two containers,

However, some studies report that low motility cryopreserved sperm is able to fertilize, as is the case of the Eastern oyster, C. virginica, described by Yang et al., (2012). Similar to this previous study, in our results, post-thaw motility did not show correlation with fertility. Oyster sperm can swim continuously for hours upon suspension in seawater (considerably longer than for sperm of most fishes), providing an advantage in fertilization ability. As a consequence, post-thaw motility is not always effective at predicting fertilizing ability and this fact was previously reported by other authors in oyster sperm [START_REF] Paniagua-Chávez | Assessment of gamete quality for the eastern oyster (Crassostrea virginica) by use of fluorescent dyes[END_REF]. Although motility and structural integrity are important sperm quality assessment indicators, theoretically, fertilization trials should be the most unequivocal method for determining sperm quality. According to studies performed in C. gigas by Adams et al., (2004), the fertilization ability of fresh sperm was 30-to 100-fold higher than that of sperm cryopreserved with DMSO. According to [START_REF] Hassan | Sperm cryopreservation in oysters: A review of its current status and potentials for future application in aquaculture[END_REF] in several other oyster species, 1 000 to 10 000 post-thaw spermatozoa per egg is enough to produce similar fertilization rates as those obtained using fresh sperm, if sperm functionality was not affected by cryopreservation. Based on these results, we selected the sperm per egg ratios in the present study obtaining fertilization percentages higher than 70% when using 10000 spermatozoa/egg. The results obtained in this work can be comparable to others using a similar protocol developed for C. virginica (10% DMSO, 20 minutes equilibration time, fast freezing rate, use of straws, thawing at 40 °C for 8 s), where the authors achieved fertilization rates of 77% (Yang et al., 2012).

To summarize, in C. angulata sperm cryopreservation, CPA toxicity tests are important to discard some CPAs and concentrations, but are not definitive. We observed that agglutination is also a significant problem for C. Angulata sperm performed followed by Student-Newman-Keuls (SNK) as a post hoc test. Different letters show significant differences between conditions (p <0.05). Figure 6. C angulata fertilization results comparing fresh and cryopreserved sperm using different fertilization rates. Percentage of fertilized eggs using fresh and cryopreserved sperm. Four different ratios spermatozoa/egg (number of males per pool=3, number of pools=5); mean ± SE) were used for the thawed sperm. Oneway ANOVA (p<0.05) was performed followed by Student-Newman-Keuls (SNK) as a post hoc test. Different letters show significant differences between conditions (p <0.05).

Supplementary material


Romain Couillet
email: romain.couillet@centralesupelec.fr

Abla Kammoun
email: abla.kammoun@kaust.edu.sa

Random Matrix Improved Subspace Clustering

This article introduces a spectral method for statistical subspace clustering. The method is built upon standard kernel spectral clustering techniques, however carefully tuned by theoretical understanding arising from random matrix findings. We show in particular that our method provides high clustering performance while standard kernel choices provably fail. An application to user grouping based on vector channel observations in the context of massive MIMO wireless communication networks is provided.

I. INTRODUCTION

Clustering, be it for graphs (community detection) or multivariate data, is an important leg of machine learning by which one aims at grouping together alike elements in a completely unsupervised manner (that is based on no training examples). With the advent of the big data paradigm, clustering of large dimensional datasets becomes a key ingredient of machine learning. Because the underlying problem (that of devising an optimal grouping) is inherently discrete and often computationally expensive, most clustering strategies are based on relaxed optimization schemes. This is the case of spectral kernel clustering [START_REF] Kammoun | A tutorial on spectral clustering[END_REF] and notably of the popular Ng-Weiss-Jordan algorithm [START_REF] Ng | On spectral clustering: Analysis and an algorithm[END_REF], which will be our starting point here.

Precisely analyzing the performance of spectral kernel algorithms is quite challenging for two essential reasons: (i) studying generically the eigenvectors of a data-driven adjacency (or similarity) matrix is intractable, and (ii) even if made possible, the non-linear kernel function induces an involved correlation structure within the kernel matrix. To tackle this difficulty, [START_REF] Von Luxburg | Consistency of spectral clustering[END_REF] proved that kernel spectral clustering methods are consistent when the number of observations tends to infinity while the data dimensions remain fixed. This is a desirable result but that makes non desirable assumptions from a big data perspective, where the observation dimensions may be larger than their number. Recently, in [START_REF] Couillet | Kernel spectral clustering of large dimensional data[END_REF], a novel approach was considered, whereby the size and number of samples were assumed simultaneously large and the data to be clustered arising from a Gaussian mixture. In this case, it is shown that, thanks to a concentration of measure effect, the kernel function can be expanded in a Taylor series, resulting in the kernel matrix being asymptotically equivalent to a well-understood random matrix model. The results of [START_REF] Couillet | Kernel spectral clustering of large dimensional data[END_REF] notably evidence the inner mechanisms under play in spectral clustering and allow for many improvements. Besides, simulations on real datasets (MNIST database) suggest an extremely close fit in performance when compared to inputs extracted from a Gaussian mixture with same (empirical) means and covariances as the dataset.

A particularly striking finding of [START_REF] Couillet | Kernel spectral clustering of large dimensional data[END_REF] is that, for data vectors with vanishing differences in (empirical or statistical) means across classes, a very specific kernel choice allows for a dramatic (asymptotic) performance improvement -corresponding in fact to a phase transition towards a faster convergence regime. The purpose of this article is to study this setting in depth. Precisely, we shall consider the kernel spectral clustering of a Gaussian zero mean mixture and shall focus on clustering the data upon their "normalized shape" (i.e., irrespective of their amplitude). This setting places us naturally in a subspace clustering context. Subspace clustering is an important branch of clustering, whereby the objective is to identify groups of data living in similar subspaces [START_REF] Parsons | Subspace clustering for high dimensional data: a review[END_REF].

We will place ourselves in a regime where classical spectral clustering kernel choices provably fail, while our proposed approach achieves arbitrarily good performance. Our theoretical results are then appended into a novel algorithm for zero-mean subspace clustering when multiple independent copies of each sample is available. This algorithm finds concrete applications in user grouping for modern massive MIMO wireless communications [START_REF] Haghighatshoar | Channel vector subspace estimation from low-dimensional projections[END_REF], [START_REF] Hoydis | Massive mimo in the ul/dl of cellular networks: How many antennas do we need?[END_REF]. Practically speaking, the performances of the proposed massive MIMO method dramatically improves over alternatives [START_REF] Haghighatshoar | Channel vector subspace estimation from low-dimensional projections[END_REF], [START_REF] Adhikary | Joint spatial division and multiplexing: Opportunistic beamforming and user grouping[END_REF] where an extremely large number of channel observations from each user is assumed, while we obtain good performances already for very few channel observations.

II. MAIN RESULTS

Let x 1 , . . . , x n ∈ R p be n independent vectors such that, for n 1 , . . . , n k with i n i = n, x n1+...+nj-1+1 , . . . , x n1+...+nj ∼ N (0, p -1 C j )

where C 1 , . . . , C k ∈ R p×p are nonnegative definite covariance matrices satisfying 1 p tr C a = 1 for each a = 1, . . . , k. The latter constraint is inconsequential for the remainder and is merely imposed for convenience. The fact that the x i 's labels are sorted per class is merely for convenience and is no more restrictive. We shall say that x i ∈ C a when x i ∼ N (0, p -1 C a ).

The objective is to devise an appropriate spectral clustering method to group the x i 's within their respective classes C 1 , . . . , C k . We recall that spectral clustering [START_REF] Kammoun | A tutorial on spectral clustering[END_REF] consists first in building a matrix K ∈ R n×n with K ij = κ(x i , x j ) containing some affinity metric between vectors x i and x j . Here we shall consider a radial kernel κ(x i , x j ) = f ( x i -x j 2 ). It can then be shown by intuitive arguments that, when clustering is "not too difficult", K tends to have large isolated eigenvalues, the eigenvectors of which look like noisy step functions (when the x i are sorted per class as above). Each plateau of the step functions is mapped to one class and thus clustering can be performed by smartly exploiting these eigenvectors. Calling u 1 , . . . , u the dominant eigenvectors of K, the natural method to do this smart exploitation is to perform standard low-dimensional clustering (such as k-means or expectation maximization) on the n vectors ([u 1 ] i , . . . , [u ] i ) ∈ R for i = 1, . . . , n. The top graph in Figure 3 at the end of this document provides an = 2-dimensional representation of n = 400 vectors

([u 1 ] i , [u 2 ] i ) ∈ R 2 with colors corresponding to ground truth classes.
Our precise objective is to select an appropriate function f such that the aforementioned clustering approach provides non-trivial performances in difficult scenarios. To this end, we shall place ourselves in a regime where p, n → ∞ and shall impose growth rates of n, p and C 1 , . . . , C k in such a way that the probability of misclustering remains of order O(1).

Assumption 1 (Growth Rate):

As p → ∞, n/p → c 0 ∈ (0, ∞) and, for each a ∈ {1, . . . , k}, n a /n → c a ∈ (0, 1). Be- sides, for each a, b ∈ {1, . . . , k}, denoting C • = k i=1 ni n C i and C • a = C a -C • , 1 √ p tr C • a C • b converges in [0, ∞).
We shall further define

T = lim p→∞ c a c b p tr C • a C • b ω = √ 2 lim p→∞ 1 p tr(C • ) 2 .
The fundamental assumption here is that tr

C • a C • b = O( √ p). It is shown in [4] that, if tr C • a C • b = o(p)
, then it is in general impossible to recover the classes using the kernel approach, while non-trivial clustering probability can be achieved when tr

C • a C • b = O(p).
However, [START_REF] Couillet | Kernel spectral clustering of large dimensional data[END_REF]Remark 12] points to an exception to this statement for a very specific choice of the kernel. Under this choice, it is proved that asymptotically perfect clustering is achieved when tr

C • a C • b = O(p).
One of our contributions here is to show that, for this very kernel option, tr C • a C • b can be made as small as O( √ p) with nontrivial class recovery. We make this discussion more precise below by introducing the kernel matrix of interest.

Assumption 2 (Kernel Matrix):

For x 1 , . . . , x n ∈ R p , denoting x = x x , let K = f ( xi -xj 2 ) n i,j=1
where f : R + → R + is a three-times differentiable function satisfying f (2) = 0 and f (2) = 0.

Examples of functions f satisfying the conditions of Assumption 2 are f poly (t) ≡ a(t -2) 2 + b for some a, b > 0 or, to anticipate evident problems of robustness to outliers using polynomials, f exp (t) ≡ exp(-a(t -2) 2 ) for some a > 0.

Letting D = diag(K1 n ) with 1 n ∈ R n the vector of ones, we define the (normalized centered) Laplacian matrix of K as

L = nD -1 2 KD -1 2 -n D 1 2 1 n 1 T n D 1 2
1 T n D1 n which is the central object of interest (rather than K itself) here, reminiscent of the Ng-Weiss-Jordan Laplacian [START_REF] Ng | On spectral clustering: Analysis and an algorithm[END_REF].

Note that D 1 2 1 n is the eigenvector of D -1 2 KD -1 2 corresponding to its leading eigenvalue 1 (a well-known property of Laplacian matrices). Thus, the matrix L is (up to the scalar n) the projection of the matrix D -1 2 KD -1 2 on the subspace orthogonal to D 1 2 1 n . This projection, also exploited in [START_REF] Couillet | Kernel spectral clustering of large dimensional data[END_REF], allows for a simplified spectral study of nD -1 2 KD -1 2 which is composed of (i) a dominant unit rank eigenspace carried by D 1 2 1 n with associated eigenvalue n (thus diverging) and (ii) an n -1 orthogonal space with associated eigenvalues of order O(1). As shown in [START_REF] Couillet | Kernel spectral clustering of large dimensional data[END_REF], the subspace (i) does not contain any valuable information and thus the dominant eigenvector of nD -1 2 KD -1 2 asymptotically contain no clustering information. We shall therefore restrict ourselves to the study of L as defined above.

The central hypothesis of Assumption 2 is the choice of f (2) = 0 without which kernel spectral clustering is asymptotically infeasible. Equipped with these intuitions, we are in position to introduce our main result which provides a tractable random matrix equivalent for L. This matrix is much simpler to study than L itself and will allow for a clear understanding of the eigenvectors content.

Theorem 1 (Random Matrix Equivalent): Under Assumption 1, as n, p → ∞, almost surely,

L = f (0) -f (2) f (2) P + 2f (2) f (2) 1 p tr C • a C • b 1 na 1 T n b p k a,b=1 + 2f (2) f (2) P ΦP + O • (p -1 2 )
where

P = I n -1 n 1 n 1 T n and Φ ∈ R n×n is defined by Φ ij = δ i =j (x T i x j ) 2 - 1 p 2 tr C a C b with x i ∈ C a , x j ∈ C b ,
and • denotes the operator norm.

In particular, we can evaluate, for i

= j, x i ∈ C a , x j ∈ C b , E[Φ ij ] = 0 Var[Φ ij ] = 2 p 4 (tr C a C b ) 2 + 6 p 4 tr(C a C b ) 2 = O(p -2 ).
As a corollary of Theorem 1, it is easily shown that

L ≡ √ p f (2) 2f (2) L - f (0) -f (2) f (2) P = 1 p tr C • a C • b 1 na 1 T n b p k a,b=1 + P ( √ pΦ)P + o • (1)
is a matrix having asymptotically the same dominant eigenvectors as those of L (with eigenvalues defined up to a mapping).

As such, the eigenvectors of interest for clustering are those of L. In the rest of the article, we shall then exclusively focus on L rather than L.

Note first that L takes the form of the sum of: (i) a random matrix with zero mean and entries of variance O(p -1 ) and (ii) a maximum rank-k (in fact even k -1) matrix with eigenvectors made of linear combinations of the canonical class vectors j 1 , . . . , j k with j i = (0, . . . , 0, 1 T ni , 0, . . . , 0) ∈ R n , modulated by the class informations 1 p tr C • a C • b . This is a spiked random matrix model of the information-plus-noise type [START_REF] Benaych-Georges | The singular values and vectors of low rank perturbations of large rectangular random matrices[END_REF]. For such models, there exists a phase transition phenomenon by which, if the eigenvalues of matrix (ii) are large enough, L will contain isolated large amplitude eigenvalues with associated eigenvectors much aligned to those of (ii) itself. If instead no such large eigenvalue exists, no isolated eigenvalue is found in the spectrum of L and no information is exploitable for clustering. This intuitive discussion is made clear in the following result.

Theorem 2 (Eigenvalue Localization): Let Assumption 1 hold. Then, as p → ∞, the eigenvalue distribution µ n ≡ 1 n n i=1 δ λi(L) (with λ i (X) the eigenvalues of X) almost surely converges (in the weak limit of probability measures) to the probability measure µ with

µ(dt) = 1 2πc 0 ω 2 (4c 0 ω 2 -t 2 ) + dt with support S = [-2 √ c 0 ω, 2 √ c 0 ω].
Besides, for all large n almost surely, there are at most k -1 eigenvalues of L found at a macroscopic distance of S. These are defined as follows.

Let ν 1 ≥ . . . ≥ ν k be the eigenvalues of T . Then, for each i ∈ {1, . . . , k}, if √ c 0 |ν i | > ω, for all large n almost surely, L has an isolated λ i satisfying

λ i a.s. -→ ρ i ≡ c 0 ν i + ω 2 ν i
(since T has a zero eigenvalue, the inequality √ c 0 |ν i | > ω is not met at least once). Theorem 2 unveils a phase transition effect by which, if |ν i |/ω or c 0 are large enough, then isolated eigenvalues are found in the limiting spectrum of L. Figure 1 depicts the histogram of the eigenvalues of L versus the asymptotic semicircle law µ, where one can observe isolated eigenvalues on the left side of the main bulk. When this occurs, the eigenvector u i associated with the isolated eigenvalue λ i of L will correlate the eigenvector with eigenvalue ν i of

{ 1 p 2 tr C • a C • b • 1 na 1 T n b } k a,b=1
, this correlation being all the stronger that |ν i | is large. As such, for sufficiently large |ν i |, the eigenvectors u i will tend to behave like (noisy) step vectors. We make this intuition more rigorous in what follows.

Assume now that λ i is an isolated eigenvalue of L as per Theorem 2 with unit multiplicity. From the statistical interchangeability of vectors x i within each class C a , we may write the eigenvector u i associated with λ i as

u i = k a=1 α a i j a √ n a + σ a i w a i for j a = [0 T n1 , . . . , 0 T na-1 , 1 T na , 0 T na+1 , . . . , 0 T n k ] T ∈ R n ,
w a i a vector of unit norm, supported on the indices of class C a and orthogonal to j a , and α a i ∈ R, σ a i > 0 scalars to be determined. Similarly, for two isolated eigenvalues λ i , λ i of L, it shall be interesting to study the correlation

σ a i,i ≡ u a i -α a i j a √ n a T u a i -α a i j a √ n a
with u a i = diag(j a )u i ∈ R n the restriction of u i to its indexes in class C a . In particular, (σ a i ) 2 = σ a ii . These quantities characterize completely the mutual behavior of the isolated eigenvectors of L used for spectral clustering and thus allow for anticipating the performance of kernel spectral clustering. In the next theorem, we provide the asymptotic values of those parameters.

Theorem 3 (Expression of Eigenvectors): For each (ν i , v i ) eigenpair of T with ν i of unit multiplicity satisfying √ c 0 |ν i | > ω and for each a ∈ {1, . . . , k}, let α a i = 1 na u T i j a with (λ i , u i ) the eigenpair of L mapped to ν i as per Theorem 1. Then, as p → ∞, under the conditions of Assumption 1, for each a, b ∈ {1, . . . , k},

α a i α b i a.s. -→ 1 - 1 c 0 ω 2 ν 2 i v i v T i ab .
Besides, for (λ i , u i ), (λ i , u i ) two such eigenpairs of L, letting σ a ii = 1 na (u a i -α a i j a ) T (u a i -α a i j a ), we have

σ a ii a.s. -→ δ ii c a c 0 ω 2 ν 2 i .
Figure 2 depicts the leading two eigenvectors under the setting of Figure 1 and the theoretical values for α j i and σ j i . Remark 1 (Relation to Subspace Clustering): Subspace clustering consists in grouping vectors x i in classes defined through the distance between the covariance matrices E[x i x T i ] (or between their dominant subspaces). In particular, for k = 2 classes, such a metric may be the Frobenius norm tr(C 1 -C 2 ) 2 between C 1 and C 2 . It appears that this is exactly what is implemented by the proposed method. For k ≥ 3, the method instead considers a metric based on the eigenvalues of

{tr(C • a -C • b ) 2 } k a,b=1
which is one among multiple possibilities of such distance definitions.

III. APPLICATION TO WIRELESS COMMUNICATIONS

The setting under study is of broad interest in massive MIMO wireless communications where, to avoid the bottleneck problem known as pilot contamination [START_REF] Hoydis | Massive mimo in the ul/dl of cellular networks: How many antennas do we need?[END_REF], it is necessary to smartly schedule data transmissions to users having channels belonging to as far subspaces as possible. That is, given a massive MIMO transmitter equipped with a large number p antennas serving n mobile users with respective channels

= 3, c 1 = c 2 = 1/4, c 3 = 1/2, C i ∝ Ip + (p/8) -5 4 W i W T i , W i ∈ R p×(p/8) with i.i.d. N (0, 1) entries, f = fexp. 1 √ n 1 (α 1 1 -σ 1 1 ) 1 √ n 2 (α 2 1 + σ 2 1 ) 1 √ n 2 α 2 1 0
200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 Fig. 2. Leading two eigenvectors of L (or equivalently of L) versus deterministic approximations of α a i ± σ a i as per Theorem 3. Setting identical to that of Figure 1. h 1 , . . . , h n ∈ R p , it is a first objective to the transmitter to group users into subsets of alike channel subspaces. 1 Here the so-called subspaces relate to the covariances of the zero mean random channels h i (randomness arising from mobility and the null mean for the absence of line of sight components). This problem is traditionally handled by assuming a method to extract good estimators of each covariance E[h i h T i ], see e.g., [START_REF] Haghighatshoar | Channel vector subspace estimation from low-dimensional projections[END_REF], [START_REF] Adhikary | Joint spatial division and multiplexing: Opportunistic beamforming and user grouping[END_REF]. For large p (traditionally ∼ 400 in massive MIMO) and n, this means accessing a tremendous amount of independent channels which is unrealistic.

Instead, we propose here to use a single channel observation to meet the same results, which we then refine to several independent observations. The underlying reason for such a gain is that, while [START_REF] Haghighatshoar | Channel vector subspace estimation from low-dimensional projections[END_REF], [START_REF] Adhikary | Joint spatial division and multiplexing: Opportunistic beamforming and user grouping[END_REF] believe in the need to retrieve the extremely large matrices E[h i h T i ], we claim that only estimating tr C • a C • b for groups a, b of users is enough, and this is easily obtained through kernel spectral clustering.

In the case of a single channel acquisition, the idea here consists in building a kernel matrix K, and its associated (modified) Laplacian L, with K ij = f ( hihj

2 ) for hi = hi hi and f with f (2) = 0 and f (2) = 0, and then to perform subspace clustering as previously introduced upon L. When T independent copies of the channels h for each user i are obtained, we suggest to proceed as follows: (i) build a T n × T n matrix K, and subsequently L, for the ordered channels [h were arising from T n rather than n users, then (ii) to exploit the fact that the vectors h (1) i , . . . , h (T ) i are mapped to a single user (and therefore a single point for the clustering problem at hand), average the dominant nT -dimensional eigenvectors u i of L across the T indexes corresponding to channels of the same users, resulting into the n-dimensional vectors

ūi ≡ 1 T (I T ⊗ 1 T T )u i (1)
with 1 T ∈ R T the all-ones vector, and finally (iii) proceed to clustering (from k-means or expectation-maximization procedures) over the dominant vectors ūi . The main effect of the folding operation ( 1) is to reduce the variance of the rows of matrix [u 1 , . . . , u n ] by a factor T .

To simulate the performance of the method, we model the h (j) i ∈ R 2p as two-dimensional real vectors of the stacked real and imaginary parts of Gaussian circularly symmetric zero mean channels with covariance matrices C a ∈ R 2p×2p for users i in class C a . Letting Γ a ∈ C p×p be the complex representation of C a , we consider the popular solid angular model

[Γ a ] ij = 1 ∆ a + -∆ a - θ a +∆ a + θ a +∆ a - exp -2πı d λ sin(t)(j -i) dt
for a linear antenna array with inter-antenna distance d and transmission wavelength λ taken here such that d = λ. The array beam focuses uniformly across the angles [θ a +∆ a -, θ a + ∆ a + ]. We shall take here ∆ a -= -∆ a + = π/20 for each a, while θ 1 = -π/30, θ 2 = 0, and θ 3 = π/30. The performances associated with the setting of Figure 3 are provided in Figure 4, where a comparison between the classical Gaussian kernel f (t) = exp(-t 2 ) and our proposed improved kernel f (t) = exp(-(t -2) 2 ) is provided. The last step of clustering is performed either via a k-means or an expectation-maximization (EM) method (assuming Gaussian eigenvector fluctuations). As is shown, under the considered setting, irrespective of T , the Gaussian kernel does not manage to provide any efficient clustering. In comparison, the kernel choice with f (2) = 0 achieves 100% clustering probability already for 8 independent channel acquisitions. As is standard in clustering, k-means operates better than EM at low correct clustering rates but is then overtaken by EM for higher rates.

IV. CONCLUDING REMARKS

In this article, we have exploited the recent theoretical analysis [START_REF] Couillet | Kernel spectral clustering of large dimensional data[END_REF] on the asymptotic performance of kernel spectral clustering to produce an improved method for subspace clustering of large dimensional datasets. Possibly the most striking outcome of the study is its demonstrating that smartly chosen kernel choices allow for data clustering where traditional kernels provably fail. This unfolds from a change in regime for the specific problem at hand when using appropriate kernels.

Generically speaking, this article may be considered as a first step towards enabling random matrix-based improvements in large dimensional machine learning problems. It is envisioned that, based again on the investigation method from [START_REF] Couillet | Kernel spectral clustering of large dimensional data[END_REF] naturally leading to spiked equivalent models of random kernel matrices, questions such as deep theoretical understanding of semi-supervised learning and support vector machine methods for large dimensional datasets could be similarly addressed, thereby possibly leading to improved versions of these so popular methods.

Fig. 1 .

 1 Fig. 1. Eigenvalues of L and semi-circle law, for p = 1000, n = 2000, k= 3, c 1 = c 2 = 1/4, c 3 = 1/2, C i ∝ Ip + (p/8) -5 4 W i W T i , W i ∈ R p×(p/8) with i.i.d. N (0, 1) entries, f = fexp.

Figure 3

 3 Figure 3 illustrates the gain of the per-user folding strategy (1) on a single random scenario (see figure caption for details). It is seen that the large spread of data points prior to folding (top figure) is significantly reduced after T = 10 copies folding (bottom figure). In both graphs are shown in blue the theoretical 1σ and 2σ standard deviations obtained from Theorem 3 and the T -fold reduction in variance from (1).The performances associated with the setting of Figure3are provided in Figure4, where a comparison between the classical Gaussian kernel f (t) = exp(-t 2 ) and our proposed improved kernel f (t) = exp(-(t -2) 2 ) is provided. The last step of clustering is performed either via a k-means or an expectation-maximization (EM) method (assuming Gaussian

2 Fig. 3 .

 23 Fig. 3. Leading two eigenvectors before (left figure) and after (right figure) T -averaging. Setting: p = 400, n = 40, T = 10, k = 3, c 1 = c 3 = 1/4, c 2 = 1/2, Γa's built from angular spread model with angles -π/30 ± π/20, 0 ± π/20, and π/30 ± π/20. Kernel function f (t) = exp(-(t -2) 2 ).

Fig. 4 .

 4 Fig.[START_REF] Couillet | Kernel spectral clustering of large dimensional data[END_REF]. Comparison of the correct clustering probability between the optimal kernel f (x) (here f (x) = exp(-(x -2) 2 )) and the Gaussian kernel exp(-x 2 ), for different number of observations T per source, using the kmeans or EM algorithms. Setting identical to Figure3.

For simplicity, we assume here that channels are modeled as real vectors. When complex, real and imaginary parts can be stacked into a

2p-dimensional vector.

V. ACKNOWLEDGEMENTS

Couillet's work is supported by the ANR RMT4GRAPH (ANR-14-CE28-0006) and the HUAWEI RMTin5G projects.


John C Semple
email: jcsemple@uwaterloo.ca

Florian Jabbour

TYPE SPECIMENS OF INULA (PITYOPSIS) GRAMINIFOLIA (ASTERACEAE: ASTEREAE)

Recent proposed changes to the nomenclature of the goldenaster genus Pityopsis disagreed significantly on the application of the name Inula graminifolia, the basionym of Pityopsis graminifolia. In order to clarify the problem, the three known collections of Inula graminifolia in the Herbarium of the Muséum national d'Histoire naturelle, Paris (P) collected by André Michaux were examined. The holotype collection has small heads with many stipitate glands on the distal portion of the phyllaries and obvious long hairs on the peduncles. This is also the case for the isotype collection. A third collection by Michaux is mixed with a shoot with larger heads and shoots with smaller heads and all with phyllaries with few or no stipitate glands. Thus, the third specimen is not part of the same taxon as the holotype and isotype.

The Pityopsis graminifolia (Michx.) Nutt. complex includes goldenasters native to the southeastern USA, southern Mexico, Guatemala, Belize, western Honduras, and the Bahamas. The complex includes either a single species with multiple varieties [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF][START_REF] Semple | Pityopsis Nuttall[END_REF] or the complex might be divided into multiple species with more narrowly defined morphological and ecological limits. [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF] presented a revision of the genus Pityopsis Nutt. recognizing seven species. Four species were placed in P. sect. Pityopsis and all are narrowly to very narrowly distributed species. Pityopsis pinifolia (Elliott) Nutt. is a rare species found along the eastern edge Piedmont along the "fall line" counties of North Carolina, South Carolina, and Georgia, (and Mississippi, Nesom 2019). Pityopsis falcata (Pursh) Nutt. is native to the Atlantic Coastal plain from Cape Cod, Massachusetts to southern New Jersey. Pityopsis flexuosa (Nash) Small is native to a few counties around Tallahassee, Florida. Pityopsis ruthii (Small) Small is confined to a small area in southeastern Tennessee along short sections of the Hiwassee and Ocoee Rivers [START_REF] Moore | Current knowledge, threats, and future efforts to sustain populations of Pityopsis ruthii (Asteraceae), an endangered southern Appalachian species[END_REF]. Three additional species were included in P. sect. Graminifoliae (Small) Semple. Pityopsis oligantha (Chapm. ex Torr. & A. Gray) Small has big glandular heads and occurs in the Florida Pandle and adjacent counties in Alabama and Georgia. Pityopsis aspera (A. Gray) ex Small has glandular peduncles and includes two varieties: var. aspera is confined to a small area in southern Georgia and adjacent northern Florida; var. adenolepis (Fernald) Semple & F.D. Bowers occurs from northeastern Virginia to northern Florida and west to southern Mississippi. Lastly, there is the more widely distributed species P. graminifolia in the broad sense, occurring from Delaware to southern Ohio and south to southern Florida, and west through the southern states to eastern Texas and disjunct in Latin America as noted above; most forms of the species have long narrow lower stem and rosette leaves and all have peduncles with long white cobwebby hairs. Five varieties were recognized within P. graminifolia by [START_REF] Semple | Pityopsis Nuttall[END_REF][START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF]followed in Semple 2006). Diploid var. graminifolia was defined on the basis of having smaller heads with inner phyllaries that are obviously stipitate glandular (Fig. 1); such plants occur from southern North Carolina to northern Florida and west to Mississippi on the outer coastal plain. Diploid var. tenuifolia (Torr.) Semple & F.D. Bowers has small heads with inner phyllaries that are not strongly glandular; Chrysopsis microcephala Small and combinations on that name were treated as synonyms; var. tenuifolia in this sense occurs from North Carolina to southern Florida and west to east and south Texas and extending north into Arkansas and extreme eastern Oklahoma. Diploid var. aequilifola F.D. Bowers & Semple has smaller heads and stiff densely-arranged ovate-lanceolate stem leaves and occurs in Peninsula Florida. Tetraploid and rarely hexaploid var. latifolia (Fernald) Semple and F.D. Bowers has mid-sized heads with phyllaries that are not glandular (or with a few obscure glands); the stem leaves are narrowly lanceolate; the range occupies the entire area described for the species. Hexaploid var. tracyi (Small) Semple has large heads without obvious glands and is confined to the Florida Panhandle. Ploidy level data was presented and summarized in [START_REF] Semple | Cytogeography of Pityopsis Nutt., the grass-leaved goldenasters (Compositae: Astereae)[END_REF] with additional counts reported by [START_REF] Semple | Chromosome number determinations in fam. Compositae, Tribe Astereae. VII. Mostly eastern North American and some Eurasian taxa[END_REF][START_REF] Semple | Chromosome numbers in Fam. Compositae, Tribe Astereae. VIII. Eastern North American taxa. II[END_REF]Semple et al. , 2019 in press) in press).

Recently, [START_REF] Bridges | Combinations, rank changes, and nomenclatural and taxonomic comments in the vascular flora of the southeastern United States. III[END_REF], focusing on the Peninsula Florida taxa proposed breaking up P. graminifolia in the broad sense into five species. Pityopsis graminifolia was noted to have small heads with glandular phyllaries which follows the application of the name by [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF]. Pityopsis aequilifolia (F.D. Bowers & Semple) E.L. Bridges & Orzell occurs "exclusively on xeric sands, in both Florida scrub and fire-prone sandhills within central Florida" [START_REF] Bridges | Combinations, rank changes, and nomenclatural and taxonomic comments in the vascular flora of the southeastern United States. III[END_REF]. Pityopsis microcephala Small was noted to have very narrow leaves and that further study supporting it as a separate species distinct from P. graminifolia was needed across the range of the two taxa. Pityopsis tracyi (Small) Small was noted to have large involucres 12-14 mm high and to have robust lower stem and rosette leaves and was confined to "somewhat poorly to poorly drained pine savannas and seasonally wet grasslands with a historical high fire return interval" in Peninsula Florida. Pityopsis latifolia (Fernald) Bridges & Orzell was considered to have involucres 8-12 mm high and to have few soft narrow stem leaves and be restricted to southern Florida. [START_REF] Bridges | Combinations, rank changes, and nomenclatural and taxonomic comments in the vascular flora of the southeastern United States. III[END_REF] noted their uncertainty about how to deal with plants included in P. graminifolia var. latifolia by [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF] outside Peninsula Florida and taxonomic problems with applying other names listed in synonymy in [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF] to possibly separate races, e.g., names based on and Erigeron nervosus Willd., synonym Pityopsis nervosa (Willd.) Dress and Chrysopsis correllii Fernald.

Even more recently [START_REF] Nesom | Taxonomic synopsis of Pityopsis (Asteraceae)[END_REF] proposed an alternative approach to the broadly delimited Pityopsis graminifolia with very different results from [START_REF] Bridges | Combinations, rank changes, and nomenclatural and taxonomic comments in the vascular flora of the southeastern United States. III[END_REF] in how some names were applied. [START_REF] Nesom | Taxonomic synopsis of Pityopsis (Asteraceae)[END_REF] recognized 12 species of Pityopsis with the delimitations of P. aspera, P. falcata, P. flexuosa, P. oligantha, P. pinifolia and P. ruthii being in agreement with [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF] and [START_REF] Semple | Pityopsis Nuttall[END_REF]. The dot distribution map for P. pinifolia shows many more collections and one from Alabama, which were not known by [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF] because the location was uncertain and was the neotype for the species selected by [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF]. [START_REF] Nesom | Taxonomic synopsis of Pityopsis (Asteraceae)[END_REF] split S. graminifolia sensu lato into 6 separate species that do not correspond to varieties in [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF] and [START_REF] Semple | Pityopsis Nuttall[END_REF]. Most significantly, the name P. graminifolia is applied to what was P. graminifolia var. latifolia in [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF] and [START_REF] Semple | Pityopsis Nuttall[END_REF] and P. tracyi was expanded to include diploid and tetraploid plants from Peninsula Florida as well as the large headed hexaploids included in P. graminifolia var. tracyi by [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF] and [START_REF] Semple | Pityopsis Nuttall[END_REF]. [START_REF] Nesom | Taxonomic synopsis of Pityopsis (Asteraceae)[END_REF] delimited S. aequilifolia the same way that [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF] and [START_REF] Semple | Pityopsis Nuttall[END_REF] delimited S. graminifolia var. aequilifolia but at species rank. [START_REF] Nesom | Taxonomic synopsis of Pityopsis (Asteraceae)[END_REF] applied the name P. microcephala to small headed plants with densely stipitate glandular phyllaries and mapped a range that is similar to that of P. graminifolia var. graminifolia in [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF]. Plants that were included in var. microcephala in [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF] and [START_REF] Semple | Pityopsis Nuttall[END_REF] were treated as P. tenuifolia (Torr.) Nesom occurring west of the Mississippi River, and into other taxa east of the Mississippi. [START_REF] Nesom | Taxonomic synopsis of Pityopsis (Asteraceae)[END_REF], working from online digital images, considered that P00742815 (Fig. 2) was the holotype of Inula graminifolia and that P00742818 (Fig. 3) included one isotype shoot of P. latifolia and two shoots of P. microcephala. These conclusions needed further investigation and the results are presented below.

In 1984, the first author visited the Muséum national d'Histoire naturelle in Paris and examined a Michaux collection (Fig. 1) included in the P-MICH historical collection (Thiers, continuously updated). This was a short visit and only specimens on a list sent earlier by mail were provided for examination. The holotype of Inula graminifolia was one of the specimens included on the list. Stipitate glands were observed on the small heads of the single shoot on the specimen (Fig. 4), which bore a TYPE label at the time and an annotation by William T. Gillis dated 1 juin 1976 and noting that the specimen was the type of Inula graminifolia Michx. The first author annotated the specimen as the holotype of Inula graminifolia and took several black and white photographs during the 1984 visit. This was the only Michaux collection of Inula graminifolia examined in 1984 and is the only one of the three Michaux collections examined in preparing this publication that includes a handwritten label indicating the classification of the specimen at the time. The second author compared the writing with known hand script of André Michaux and concluded that the label was likely written by Michaux himself. Thus, this collection is the only real choice for holotype of Inula graminifolia, the basionym of Pityopsis graminifolia. Only plants with small heads and stipitate glands on the phyllaries can be included in P. graminifolia var. graminifolia, as was done by [START_REF] Semple | A revision of the goldenaster genus Pityopsis Nutt. (Compositae: Astereae)[END_REF] and [START_REF] Semple | Pityopsis Nuttall[END_REF], or in a narrowly defined P. graminifolia as done by [START_REF] Bridges | Combinations, rank changes, and nomenclatural and taxonomic comments in the vascular flora of the southeastern United States. III[END_REF]. Involucres on the holotype averaged about 5.6 mm high.

The other two Michaux collections (P00742818 and P00742815) were not seen by the first author in 1984 and were examined at his request by the second author in 2019. Both collections have a printed ISOTYPE label on the sheets. The three shoots on P00742818 have small heads (involucres averaging 6.1 mm tall) and phyllaries with glands (Figs. 2 and5). Thus, P00742818 is similar to the holotype and is treated here as an isotype of Inula graminifolia. The two shoots on P00742815 differ in head size (taller left shoot with involucres averaging 9.1 mm high; right shoot with involucres averaging 8.8 mm); both shoots have phyllaries with few or no glands (Figs. 3 and6). Therefore, P00742815 is not the same taxon as the holotype of Inula graminifolia and should not be considered as an isotype. Shoot "a-b" and shoot "d" are all part of the Michaux collection, acquired at one point by the E. Drake herbarium. The material was included in P in a second step.

A much larger study is needed to resolve questions on whether splitting up Pityopsis graminifolia into more narrowly defined separate species and how many species should be recognized. If additional research determines that P. latifolia should be limited to plants in Florida, then Pityopsis nervosa (Willd.) Dress [Baileya 19: 166. 1975.] based on Erigeron nervosus Willd. [Sp. Pl., 3: 1953. 1803.] is the oldest species epithet to apply to the much more widely distributed species level taxon occurring in much of the southeastern USA, southern Mexico, Guatemala, and Belize. The Bahamian populations could belong to P. latifolia or P. nervosa depending upon study results. Further work is needed to determine whether it is appropriate to split these small headed non-glandular P. microcephala into an eastern taxon under that name and a western P. tenuifolia. A multivariate study of the kind being done on multiple groups of Solidago recently might provide a solution to these problems (e.g., [START_REF] Semple | A multivariate morphometric analysis of Solidago subsect. Thyrsiflorae (Asteraceae: Astereae)[END_REF][START_REF] Semple | A multivariate morphometric analysis of Solidago subsect. Nemorales (Asteraceae: Astereae)[END_REF]. Cutting edge DNA based research may also provide solutions, if polygenomic sampling is utilized on sufficient numbers of specimens (see [START_REF] Wadl | Development of microsatellite loci for the endangered species Pityopsis ruthii (Asteraceae)[END_REF]. Single gene or internal transcribed spacer sequencing will probably not be sufficient [START_REF] Brouillet | Systematics, Evolution and Biogeography of the Compositae[END_REF]). The first author has split Solidago into many more species in recent years (Semple 2019) than were recognized in 2006 and would likely conclude that splitting a broadly defined P. graminifolia into 5-6 or more species now is an appropriate solution at this time pending results of new studies. Selecting the correct names to apply will necessitate doing more than looking at online digital images of type collections because critical indument details are not clearly visible even on the higher resolution images now available.

Counting the numbers of hairs and glands on phyllaries will be tedious but may be necessary after assembling a large sample of Pityopsis herbarium collections from the entire range of the species and including all races. Using a dissecting scope at 40-70. magnification is still a necessity in modern systematic botany.

Figures 4 .

 4 Figures 4. Phyllaries details of the holotype of Inula graminifolia. Scale bar = 1 mm in A and B

ACKNOWLEDGEMENTS

An early portion of this work was supported by a Natural Sciences and Engineering Research Council of Canada Operating and Discovery Grants to JCS. We thank Didier Geffard-Kuriyama for his assistance in taking images at the Atelier Iconographie Scientifique, UMS 2700 2AD CNRS-MNHN.