Gaël Letarte
email: gael.letarte.1@ulaval.ca

Emilie Morvant
email: emilie.morvant@univ-st-etienne.fr

Pascal Germain
email: pascal.germain@inria.fr

Pseudo-Bayesian Learning with Kernel Fourier Transform as Prior

.

INTRODUCTION

Kernel methods [START_REF] Shawe | Kernel Methods for Pattern Analysis[END_REF], such as support vector machines [START_REF] Boser | A training algorithm for optimal margin classifiers[END_REF][START_REF] Vapnik | Statistical learning theory[END_REF], map data in a high dimension space in which a linear predictor can solve the learning problem at hand. The mapping space is not directly computed and the linear predictor is represented implicitly thanks to a kernel function. This is the powerful kernel trick: the kernel function computes the scalar product between two data points in this high dimension space. However, kernel methods notoriously suffer from two drawbacks. On the first hand, computing all the scalar products for all the learning samples is costly: O(n 2 ) for many kernel-based methods, where n is the number of training data point. On the other hand, one has to select a kernel function adapted to the learning problem for the algorithm to succeed.

The first of these drawbacks has motivated the development of approximation methods making kernel methods more scalable, such as Nyström approximation [START_REF] Christopher | Using the Nyström method to speed up kernel machines[END_REF][START_REF] Drineas | On the nyström method for approximating a gram matrix for improved kernel-based learning[END_REF]) that constructs a low-rank approximation of the Gram matrix1 and is data dependent, or random Fourier features (RFF) [START_REF] Rahimi | Random features for large-scale kernel machines[END_REF] that approximates the kernel with random features based on the Fourier transform and is not data dependent (a comparison between the two approaches have been conducted by [START_REF] Yang | Nyström method vs random fourier features: A theoretical and empirical comparison[END_REF]. In this paper, we revisit the latter technique.

We start from the observation that a predictor based on kernel Fourier features can be interpreted as a weighted combination of those features according to a data independent distribution defined by the Fourier transform. We introduce an original viewpoint, where this distribution is interpreted as a prior distribution over a space of weak hypotheses-each hypothesis being a simple trigonometric function obtained by the Fourier decomposition. This suggests that one can improve the approximation by adapting this distribution in regards to data points: we aim at learning a posterior distribution. By this means, our study proposes strategies to learn a representation to the data. While this representation is not as flexible and powerful than the ones that can be learned by deep neural networks [START_REF] Goodfellow | Deep Learning[END_REF], we think that it is worthwhile to study this strategy to eventually solve the second drawback of kernel methods that currently heavily rely on the kernel choice. This in mind, while the majority of work related to random Fourier fea-tures focus on the study and improvement of the kernel approximation, we propose here a reinterpretation in the light of the PAC-Bayesian theory [START_REF] Mcallester | Some PAC-Bayesian theorems[END_REF][START_REF] Catoni | PAC-Bayesian supervised classification: the thermodynamics of statistical learning[END_REF]. We derive generalization bounds that can be straightforwardly optimized by learning a pseudo-posterior thanks to a closed-form expression.

The rest of the paper is organized as follows. Section 2 recalls the RFF setting. Section 3 expresses the Fourier transform as a prior leading (i) to a first PAC-Bayesian analysis and a landmarks-based algorithm in Section 4, (ii) to another PAC-Bayesian analysis in Section 5 allowing to justify the kernel alignment method of [START_REF] Sinha | Learning kernels with random features[END_REF] and to propose a greedy kernel learning method. Then Section 6 provides experiments to show the usefulness of our work.

RANDOM FOURIER FEATURES

Problem setting. Consider a classification problem where we want to learn a predictor f : R d → Y , from a d-dimensional space to a discrete output space (e.g.,

Y = {0, 1, . . . , |Y |-1}). The learning algorithm is given a training set S = {(x i , y i )} n i=1 ∼ D n of n i.i.d. samples, where D denotes the data generating distribution over R d × Y . We consider a positive- semidefinite (PSD) kernel k : R d × R d → [-1, 1]. Ker- nel machines learn predictors of the form f (x) = n i=1 α i k(x i , x) , (1)
by optimizing the values of vector α ∈ R n .

Fourier features. When n is large, running a kernel machine algorithm (like SVM or kernel ridge regression) is expensive in memory and running time.

To circumvent this problem, [START_REF] Rahimi | Random features for large-scale kernel machines[END_REF] introduced the random Fourier features as a way to approximate the value of a shift-invariant kernel, i.e., relying on the value of δ = x-x ∈ R d , which we write

k(δ) = k(x -x ) = k(x, x )
interchangeably. Let the distribution p(ω) be the Fourier transform of the shift-invariant kernel k,

p(ω) = 1 (2π) d R d k(δ) e -i ω•δ d δ .
(2)

Now, by writing k as the inverse of the Fourier transform p, and using trigonometric identities, we obtain:

k(x -x ) = R d p(ω)e i ω•(x-x ) d ω = E ω∼p e i ω•(x-x ) = E ω∼p cos ω • (x-x ) +i sin ω • (x-x ) = E ω∼p cos ω • (x -x ) .
(3) [START_REF] Rahimi | Random features for large-scale kernel machines[END_REF] suggest expressing the above cos ω • (x-x ) as a product of two features. One way to achieve this is to map every input example into

z ω (x) = cos(ω • x), sin(ω • x) . (4)
The random variable z ω (x) • z ω (x ), with ω drawn from p, is an unbiased estimate of k(x -x ). Indeed, we recover from Equation (3) and Equation ( 4):

E ω∼p z ω (x) • z ω (x ) = E ω∼p cos(ω • x) cos(ω • x ) + sin(ω • x) sin(ω • x ) = E ω∼p cos ω • (x -x ) .
To reduce the variance in the estimation of k(x-x ), the idea is to sample D points i.i.d. from p:

ω 1 , ω 2 , . . . , ω D . Then, each training sample x i ∈ R d is mapped to a new feature vector in R 2D : φ(x i ) = 1 √ D cos(ω 1 • x i ) , . . . , cos(ω D • x i ) , (5) sin(ω 1 • x i ) , . . . , sin(ω D • x i ) . Thus, we have k(x -x ) ≈ φ(x) • φ(x ) when D is "large enough"
. This provides a decomposition of the PSD kernel k that differs from the classical one (as discussed in Bach, 2017). By learning a linear predictor on the transformed training set S → {(φ(x i ), y i )} n i=1 through an algorithm like a linear SVM, we recover a predictor equivalent to the one learned by a kernelized algorithm. That is, we learn a weight vector w = (w 1 , . . . , w 2D ) ∈ R 2D and we predict the label of a sample x ∈ R d by computing, in place of Equation (1),

f (x) = 2D j=1 w j φ j (x) . (6)

THE FOURIER TRANSFORM AS A PRIOR DISTRIBUTION

As described in the previous section, the random Fourier features trick has been introduced to reduce the running time of kernel learning algorithms. Consequently, most of the subsequent work study and/or improve the properties of the kernel approximation (e.g., [START_REF] Yu | Orthogonal random features[END_REF][START_REF] Rudi | Generalization properties of learning with random features[END_REF][START_REF] Francis | On the equivalence between kernel quadrature rules and random feature expansions[END_REF][START_REF] Choromanski | The geometry of random features[END_REF] with some notable exceptions, as the kernel learning algorithms of [START_REF] Yang | A la carte -learning fast kernels[END_REF], [START_REF] Sinha | Learning kernels with random features[END_REF][START_REF] Sinha | Learning kernels with random features[END_REF][START_REF] Junier | Bayesian nonparametric kernel-learning[END_REF], that we discuss and relate to our approach in Section 5.

We aim at reinterpreting the Fourier transform-i.e., the distribution p of Equation (2)-as a prior distribution over the feature space. It can be seen as an alternative representation of the prior knowledge that is encoded in the choice of a specific kernel function, that we denote k p from now on. In accordance with Equation (3), each feature obtained from a vector ω ∈ R d can be seen as a hypothesis

h ω (δ) := cos(ω • δ) .
Henceforth, the kernel is interpreted as a predictor performing a p-weighed aggregation of weak hypotheses. This alternative interpretation of distribution p as a prior over hypotheses naturally suggests to learn a posterior distribution over the same hypotheses. That is, we seek a distribution q giving rise to a new kernel

k q (δ) := E ω∼q h ω (δ) .
In order to assess the quality of the kernel k q , we define a loss function based on the consideration that its output should be high when two samples share the same label, and low otherwise. Hence, we evaluate the kernel on two samples (x, y) ∼ D and (x , y ) ∼ D through the linear loss

k q (δ), λ := 1 -λ k q (δ) 2 , (7)
where δ = x-x denotes a pairwise distance and λ denotes the pairwise similarity measure:

λ = λ(y, y ) := 1 if y = y , -1 otherwise.
Furthermore, we define the kernel alignment generalization loss L ∆ (k q ) on a "pairwise" probability distribution ∆, defined over R d ×[-1, 1] as

L ∆ (k q ) := E (δ,λ)∼∆ k q (δ), λ . (8)
Note that any data generating distribution D over input-output spaces R d × Y automatically gives rise to a "pairwise" distribution ∆ D . By a slight abuse of notation, we write L D (k q ) the corresponding generalization loss, and the associated kernel alignment empirical loss is defined as

L S (k q ) := 1 n(n -1) n i,j=1,i =j k q (δ ij ), λ ij , (9)
where for a pair of examples {(x i , y i ), (x j , y j )} ∈ S 2 we have δ ij := (x i -x j ) and λ ij := λ(y i , y j ).

Starting from this reinterpretation of the Fourier transform, we provide in the rest of the paper two PAC-Bayesian analyses. The first one (Section 4) is obtained by combining n PAC-Bayesian bounds: instead of considering all the possible pairs of data points, we fix one point and we study the generalization ability for all the pairs involving it. The second analysis (Section 5) is based on the fact that the loss can be expressed as a second-order U-statistics.

PAC-BAYESIAN ANALYSIS AND LANDMARKS

Due to the linearity of the loss function , we can rewrite the loss of k q as the q-average loss of every hypothesis. Indeed, Equation ( 8) becomes

L D (k q ) = E (δ,λ)∼∆ D E ω∼q h ω (δ), λ = E ω∼q E (δ,λ)∼∆ D (h ω (δ), λ) = E ω∼q L D (h ω ) .
The above q-expectation of losses L D (h ω ) turns out to be the quantity bounded by most PAC-Bayesian generalization theorems (sometimes referred as the Gibbs risk in the literature), excepted that such results usually apply to the loss over samples instead of distances. Hence, we use PAC-Bayesian bounds to obtain generalization guarantees on L D (k q ) from its empirical estimate of Equation ( 9), that we can rewrite as

L S (k q ) = 1 n 2 -n n i,j=1;i =j E ω∼q h ω (δ), λ ij = E ω∼q L S (h ω ) .
However the classical PAC-Bayesian theorems cannot be applied directly to bound L D (k q ), as the empirical loss L S (k q ) would require to be computed from i.i.d. observations of ∆ D . Instead, the empirical loss involves dependent samples, as it is computed from n 2 -n pairs formed by n elements from D.

First Order KL-Bound

A straightforward approach to apply classical PAC-Bayesian results is to bound separately the loss associated with each training sample. That is, for each (x i , y i ) ∈ S, we define

L i D (h ω ) := E (x,y)∼D h ω (x i -x), λ(y i , y) , (10)
and

L i S (h ω ) := 1 n-1 n j=1,j =i h ω (x i -x j ), λ(y i , y j ) .
Thus, the next theorem gives a generalization guarantee on L i D (k q ) relying namely on the empirical estimate L i S (k q ) and the Kullback-Leibler divergence KL(q p) = E ω∼q ln q(ω) p(ω) between the prior p and the learned posterior q. Note that the statement of Theorem 1 is obtained straightforwardly from Alquier et al. (2016, Theorem 4.1 and Lemma 1), but can be recovered easily from [START_REF] Lever | Tighter PAC-Bayes bounds through distribution-dependent priors[END_REF]. Theorem 1. For t > 0, i ∈ {1, . . . , n}, and a prior distribution p over R d , with probability 1-ε over the choice of S ∼ D n , we have for all q on R d :

L i D (k q ) ≤ L i S (k q ) + 1 t KL(q p) + t 2 2(n-1) + ln 1 ε .
By the union bound, and using the fact that L D (k q ) = E (xi,yi)∼D L i D (k q ), we prove the following corollary in the supplementary material.

Corollary 2. For t > 0 and a prior distribution p over R d , with probability 1-ε over the choice of S ∼ D n , we have for all q on R d :

L D (k q ) ≤ L S (k q )+ 2 t KL(q p) + t 2 2(n-1) + ln n+1 ε .
Pseudo-Posterior for KL-bounds. Since the above result is valid for any distribution q, one can compute the bound for any learned posterior distribution. Note that the bound promotes the minimization of a trade-off-parameterized by a constant t-between the empirical loss L S (k q ) and the KLdivergence between the prior p and the posterior q:

L S (k q ) + 2 t KL(q p) .
It is well-known that for fixed t, p and S, the minimum bound value is obtained with the pseudo-Bayesian posterior q * , such that for ω ∈ R d ,

q * (ω) = 1 Z p(ω) exp -τ L S (h ω ) , (11)
where τ := 1 2 t and Z is a normalization constant.2 Note also Corollary 2's bound converges to the generalization loss L D (k q ) at rate O ln n n for the param-

eter choice t = √ n ln n.
Due to the continuity of the feature space, the pseudoposterior of Equation ( 11) is hard to compute. To estimate it, one may make use of Monte Carlo (e.g., [START_REF] Dalalyan | Sparse regression learning by aggregation and langevin monte-carlo[END_REF] or variational Bayes methods (e.g., [START_REF] Alquier | On the properties of variational approximations of gibbs posteriors[END_REF]. In this work, we explore a simpler method: we work solely from a discrete probability space.

Landmarks-Based Learning

We now propose to leverage on the fact that Theorem 1 bounds the kernel function for the distances to a single data point, instead of learning a kernel globally for every data point as in Corollary 2. We thus aim at learning a collection of kernels (which we can also interpret as similarity functions) for a subset of the training points. We call landmarks these training points. The aim of this approach is to learn a new representation of the input space, mapping the datapoints into compact feature vectors, from which we can learn a simple predictor.

Concretely, along with the learning sample S of n examples i.i.d. from D, we consider a landmarks sample L = {(x l , y l )} n L l=1 of n L points i.i.d. from D, and a prior Fourier transform distribution p. For each landmark (x l , y l ) ∈ L, let sample D points from p, denoted Ω L = {ω l m } D m=1 ∼ p D . Then, consider a uniform distribution P on the discrete hypothesis set Ω L , such that

P (ω l m ) = 1 D and h l m (δ) := cos(ω l m • δ). We aim at learning a set of kernels { k Q l } n L l=1
, where each k Q l is obtained from a distinct x l ∈ L with a fixed parameter β > 0, by computing the pseudo-posterior distribution Q l given by

Q l m = 1 Z l exp -β √ n L l S (h l m ) , (12)
for m=1, . . . , D ; Z l being the normalization constant. Note that Equation ( 12) gives the minimum of Theorem 1 with t = β √ n. That is, β = 1 corresponds to the regime where the bound converges. Moreover, similarly to Corollary 2, generalization guarantees are obtained simultaneously for the n L computed distributions thanks to the union bound and Theorem 1. Thus, with probability 1-ε, for all {Q l } n L l=1 :

L l D ( k Q l ) ≤ L S ( k Q l )+ 1 t KL(Q l P )+ t 2 2(n-1) + ln n L ε ,
where KL(

Q l P ) = ln D + D j=1 Q l j ln Q l j .
Once all pseudo-posterior are computed thanks to Equation ( 12), our landmarks-based approach is to map samples x ∈ R d to n L similarity features:

ψ(x) := k Q 1 (x 1 -x), . . . , k Q n L (x n L -x) , (13)
and to learn a linear predictor on the transformed training set. Note that, this mapping is not a kernel map anymore and is somehow similar to the mapping proposed by Balcan et al. (2008b,a); [START_REF] Zantedeschi | Fast and provably effective multi-view classification with landmark-based svm[END_REF] for a similarity function that is more general than a kernel but fixed for each landmark.

LEARNING KERNEL (REVISITED)

In this section, we present PAC-Bayesian theorems that directly bound the kernel alignment generalization loss L D (k q ) on a "pairwise" probability distribution ∆ D -as defined by Equation ( 8)-even if the empirical loss L D (k q ) is computed on dependent samples. These bounds suggest a kernel alignment (or kernel learning) strategy similar to the one of [START_REF] Sinha | Learning kernels with random features[END_REF]. We stress that our guarantees hold solely for the kernel alignment loss, but not for the predictor trained with this kernel. Hence, our proposed algorithm learns a kernel independently of the prediction method to be used downstream. This is in contrast with the one-step frameworks of [START_REF] Yang | A la carte -learning fast kernels[END_REF] and Oliva et al. ( 2016), which learn a mixture of random kernel features in a fully Bayesian way; they rely on a data-generating model, whereas our approach assumes only that the observations are i.i.d.

Second Order KL-bound

The following result is based on the fact that

L S (h ω ) := 1 n 2 -n n i =j (h ω (δ ij ), λ ij
) is an unbiased second-order estimator of E (δ,λ)∼∆ D (h ω (δ), λ), allowing us to build on the PAC-Bayesian analysis for U-statistics of Lever et al. (2013, Theorem 7). Indeed, the next theorem gives a generalization guarantee on the kernel alignment loss L D (k q ). Theorem 3 [START_REF] Lever | Tighter PAC-Bayes bounds through distribution-dependent priors[END_REF]. For t > 0 and a prior distribution p over R d , with probability 1-ε over the choice of S ∼ D n , we have for all q on R d :

L D (k q ) ≤ L S (k q ) + 1 t KL(q p) + t 2 2n + ln 1 .
Except for some constant terms, the above Theorem 3 is similar to Corollary 2. Indeed, both are minimized by the same pseudo-posterior q * (Equation 11, with τ := t for Theorem 3). Interestingly, we get rid of the ln(n + 1) term of Corollary 2, making in Theorem 3's bound to converge at rate O( 1 √ n ) when t = √ n.

Second Order Bounds for f -Divergences

In the following, we build on a recent result of [START_REF] Alquier | Simpler PAC-Bayesian bounds for hostile data[END_REF] to express a new family of PAC-Bayesian bounds for our dependent samples, where the KL term is replaced by other f -divergences.

Given a convex function f such that f (1)=0, a f -divergence is given by D f (q p) := E ω∼p f q(ω) p(ω) . The following theorem applies to f -divergences such that f (x) = x µ -1. Theorem 4. For µ > 1 and a prior distribution p over R d , with probability 1-ε over the choice of S ∼ D n , we have for all q on R d :

L D (k q ) ≤ L S (k q ) +      1 2 √ n µ-1 D µ (q p) + 1 1 µ 1 ε 1-1 µ if 1 < µ ≤ 2, 1 4n 1-1 µ D µ (q p) + 1 1 µ 1 ε 1-1 µ if µ > 2,
where

D µ (q p) := E ω∼p q(ω) p(ω) µ -1 . Proof. Let M µ := E ω∼p E S ∼D n L D (h ω ) -L S (h ω ) µ .
We start from Alquier and Guedj (2018, Theorem 1):

L D (k q ) ≤ L S (k q )+ M µ ε 1-1 µ D µ (q p)+1 1 µ . (14) Let us show M µ ≤ 1 2 √ n µ for 1 < µ ≤ 2 : M µ = E ω∼p E S ∼D n L D (h ω ) -L S (h ω ) 2 µ 2 ≤ E ω∼p E S ∼D n L D (h ω ) -L S (h ω ) 2 µ 2 (15) = E ω∼p Var S ∼D n (L S (h ω )) µ 2 ≤ E ω∼p 1 4n µ 2 = 1 4n µ 2 . ( 16
)
Line ( 15) is obtained by Jensen's inequality (since 0 < µ 2 ≤ 1), and the inequality of Line ( 16) is proven by Lemma 6 of the supplementary material. Note that the latter is based on the Efron-Stein inequality and Boucheron et al. (2013, Corollary 3.2).

The first case of Theorem 4 statement (1 < µ ≤ 2) is obtained by inserting Line (16) in Equation ( 14). The second case (µ > 2) is obtained by upper-bounding

M µ by M 2 = 1 4n , as |L D (h ω ) -L S (h ω )| ≤ 1 .
As a particular case, with µ = 2, we obtain from Theorem 4 a bound that relies on the chi-square divergence χ 2 (q p) = E ω∼p q(ω) p(ω)

2 -1 .

Corollary 5. Given a prior distribution p over R d , with probability 1-ε over the choice of S ∼ D n , we have for all q on R d :

L D (k q ) ≤ L S (k q ) + χ 2 (q p) + 1 4 n ε .
It is noteworthy that the above result looks alike other PAC-Bayesian bounds based on the chi-square divergence in the i.i.d. setting, as the one of Honorio and Jaakkola (2014, Lemma 7), Bégin et al. (2016, Corollary 10) or Alquier and Guedj (2018, Corollary 1). Interestingly, the latter has been introduced to handle unbounded (possibly heavy-tailed) losses, and one could also extend our Corollary 5 to this setting.

PAC-Bayesian Interpretation of Kernel Alignment Optimization

Sinha and Duchi (2016) propose a kernel learning algorithm that weights random kernel features. To do so, their algorithm solves a kernel alignment problem.

As explained below, this method is coherent with the PAC-Bayesian theory exposed by our current work.

Kernel alignment algorithm. Let us consider a

Fourier transform distribution p, from which N points are sampled, denoted Ω = {ω m } N m=1 ∼ p N . Then, consider a uniform distribution P on the discrete hypothesis set Ω, such that P (ω m ) = 1 N and h m (δ) := cos(ω m • δ). Given a dataset S, and constant parameters µ > 1, ρ > 0, the optimization algorithm proposed by Sinha and Duchi solves the following problem. maximize

Q∈R N + n i=1 n j=1 λ ij N m=1 Q m h m (δ ij ) , (17)
such that

N m=1 Q m =1 and D µ (Q P ) ≤ ρ . ( 18
)
The iterative procedure proposed by Sinha and Duchi finds an -suboptimal solution to the above problem in O(N log( 1)) steps. The solution provides a learned kernel k

Q (δ) := 1 N N m=1 Q m h m (δ).
Sinha and Duchi propose to use the above alignment method to reduce the number of features needed compared to the classical RFF procedure (as described in Section 2). Albeit this method is a kernel learning one, empirical experiments show that with a large number of random features, the classical RFF procedure achieves as good prediction accuracy. However, one can draw (with replacement) D < N features from Ω according to Q. For a relatively small D, learning a linear predictor on the random feature vector (such as the one presented by Equation 5) obtained from Q achieves better results than the classical RFF method on the same number D of random features.

PAC-Bayesian interpretation. The optimization problem of Equations (17-18) deals with the same trade-off as the one promoted by Theorem 4. Indeed, maximizing Equation ( 17) amounts to minimizing L S (k q ), and the constraint of Equation ( 18) controls the f -divergence D µ (Q P ), which is the same complexity measure involved in Theorem 4. Furthermore, the empirical experiments performed by [START_REF] Sinha | Learning kernels with random features[END_REF] focus on the χ 2 -divergence (case µ=2), which corresponds to tackling the trade-off expressed by Corollary 5.

Greedy Kernel Learning

The method proposed by [START_REF] Sinha | Learning kernels with random features[END_REF] can easily be adapted to minimize the bound of Theorem 3 instead of the bound of Theorem 4. We describe this kernel learning procedure below.

Q m = 1 Z exp -β √ n L S (h m ) . (19)
Then, we sample with replacement D < N features from Ω according to the pseudo-posterior Q. The sampled features are used to map every x ∈ R d of the training set into a new vector φ(x) ∈ R 2D according to Equation ( 5). The latter transformed dataset is then given as input to a linear learning procedure.

In summary, this learning method is strongly inspired by the one described in Section 5.3, but the posterior computation phase is faster, as we benefit from a closed-form expression (Equation 19). Once L S (h m ) is computed for all h m , 3 we can vary the parameter β and get a new posterior in O(N ) steps.

EXPERIMENTS

All experiments use a Gaussian (a.k.a. RBF) kernel of variance σ 2 : k σ (x, x ) = exp -1 2σ 2 xx 2 , for which the Fourier transform is given by

p σ (ω) = σ 2 2π d 2 e -1 2 σ 2 ω 2 = N (ω; 0, 1 σ 2 I) . ( 20
)
Apart from the toy experiment of Figure 1, the experiments on real data are conducted by splitting the available data into a training set, a validation set and a test set. The kernel parameter σ is chosen among {10 -7 , 10 -6 , . . . , 10 2 } by running an RBF SVM on the training set and keeping the parameter having the best accuracy score on the validation set. That is, this σ defines the prior distribution given by Equation ( 20) for all our pseudo-Bayesian methods. Unless otherwise specified, all the other parameters are selected using the validation set. More details about the experimental procedure are given in the supplementary material.

Landmarks-Based Learning

Toy experiment. To get some insight from the landmarks-based procedure of Section 4.2, we generate a 2D dataset S toy , illustrated by Figure 1.

We randomly select five training points L={x 1 , x 2 , x 3 , x 4 , x 5 } ⊂ S toy , and compare two procedures, described below. RBF-Landmarks: Learn a linear SVM on the empirical kernel map given by the five RBF kernels centered on L. That is, each x ∈ S toy is mapped such that x → kσ(x1, x), kσ(x2, x), kσ(x3, x), kσ(x4, x), kσ(x5, x) .

3 We show in supplementary material (section A.2) that each LS(hm) can be computed in O(n) steps. PB-Landmarks: Generate 20 random samples according to the Fourier transform of Equation ( 20). For every landmark of L, learn a similarity measure thanks to Equation ( 12) (with β = 1), minimizing the PAC-Bayesian bound. We thus obtain five posterior distributions

Q 1 , Q 2 , Q 3 , Q 4 , Q 5
, and learn a linear SVM on the mapped training set obtained by Equation ( 13).

Hence, the RBF-Landmarks method corresponds to the prior, from which we learn a posterior by landmarks by the PB-Landmarks procedure. Right-most plots of Figure 1 show that the PB-Landmarks setting successfully finds a representation from which the linear SVM can predict well.

Experiments on real data. We conduct similar experiments as the above one on seven real binary classification datasets. Figure 2 studies the behavior of the approaches according to the number of selected landmarks. We select a percentage of the train- ing points as landmarks (from 1% to 25%), and we compare the classification error of a linear SVM on the mapping obtained by the original RBF functions (as in the RBF-Landmarks method above), with the mapping obtained by learning the landmarks posterior distributions (PB-Landmark method). We also compare the case where the landmarks are selected at random among the training data (curves postfixed "-R"), to another scenario where we use the centroids obtained with a k-Means clustering as landmarks (curves postfixed "-C"). Note that, the latter case is not rigorously backed by our PAC-Bayesian theorems, since the choice of landmarks is now dependent of the whole observed training set. The results show that the classification error of both cases are similar, but the clustering strategy leads to a more stable behavior, probably since the landmarks are more representative of the original space. Moreover, the pseudo-Bayesian method improves the results on almost all datasets.

Table 1 compares the error rate of an SVM (trained along with the full Gram matrix and a properly selected σ on the validation set) with four landmarksbased approaches: (RBF) the landmarks are RBF kernel of parameter σ; (PB) the PB-Landmarks approach where the number of features per landmarks D and the β parameter are selected using the validation set;

(PB β=1 ) the PB-Landmarks approach where β=1 is fixed and D is selected by validation; and (PB D=64 ) the PB-Landmarks approach where D=64 is fixed and β is selected by validation. For all landmarks-based approaches, we select the landmarks by clustering, and use 10% of the training set size as the number of landmarks; we want to study the methods in the regime where it provides relatively compact representations. We observe that learning the posterior improves the RBF-Landmarks (except on "mnist56") and that the validation of both β and D parameters are not mandatory to obtain satisfactory results. The SVM RBF is better than all landmarks-based approaches on 4 datasets out of 7, but requires a far less compact representation of the data as it uses the full Gram matrix.

Greedy Kernel Learning

Figure 3 presents a study of the kernel learning method detailed in Section 5.4, inspired from the one of [START_REF] Sinha | Learning kernels with random features[END_REF]. We first generate N =20000 random features according to p σ as given by Equation ( 4), and we learn a posterior using two strategies: (OKRFF) the original optimized kernel of Sinha and Duchi given by Equations (17)(18), where ρ is selected on the validation set; and (PBRFF) the pseudo-posterior given by Equation ( 19) where β is selected on the validation set. For both obtained posteriors, we subsample an increasing number of features D ∈ [1, 5000] to create the mapping given by Equation ( 5), on which we learn a linear SVM. We also compare to (RFF) the standard random Fourier features as described in Section 2, with D randomly selected features according to the prior p σ .

We see that our PBRFF approach behaves similarly as OKRFF, with a slight advantage for the latter. However, we recall that computing the posterior of former method is faster. Both kernel learning methods have better accuracy than the classical RFF algorithm for a small number of random features, and similar ones for a large number of random features.

CONCLUSION & PERSPECTIVES

We elaborated an original viewpoint of the random Fourier features, proposed by [START_REF] Rahimi | Random features for large-scale kernel machines[END_REF] to approximate a kernel. By looking at the Fourier transform as a prior distribution over trigonometric functions, we present two kinds of generalization theorems that bound a kernel alignment loss. Based on classical first-order PAC-Bayesian results, we derived a landmarks-based strategy that learns a compact representation of the data. Then, we proposed two second-order generalization bounds. The first one is based on the U-statistic theorem of [START_REF] Lever | Tighter PAC-Bayes bounds through distribution-dependent priors[END_REF]. The second one is a new PAC-Bayesian theorem for f -divergences (replacing the usual KLdivergence term). We show that the latter bound provides a theoretical justification to the kernel alignment method of [START_REF] Sinha | Learning kernels with random features[END_REF], and we also empirically evaluate a similar but simpler algorithm where the alignment distribution is obtained by the PAC-Bayesian pseudo-posterior closed-form expression.

Our current guarantees hold solely for the kernel alignment loss, and not for the predictor trained with this kernel. An important research direction is to extend the guarantees to the final predictor, which could in turn be the bedrock of a new one-step learning procedure (in the vein of [START_REF] Yang | A la carte -learning fast kernels[END_REF][START_REF] Junier | Bayesian nonparametric kernel-learning[END_REF]. Other research directions include the study of the RKHS associated with the learned kernel, and the extension of our study to wavelet transforms [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]. Furthermore, considering the Fourier transform of a kernel as a (pseudo-)Bayesian prior might lead to other original contributions. Among them, we foresee new perspectives on representation and metric learning, namely for unsupervised learning.

A Supplementary Material

A.1 Mathematical Results

Corollary 2. For t > 0 and a prior distribution p over R d , with probability 1-ε over the choice of S ∼ D n , we have for all q on R d :

L D (k q ) ≤ L S (k q ) + 2 t KL(q p) + t 2 2(n -1) + ln n + 1 ε .
Proof. We want to bound

L D (k q ) = E (x,y)∼D E (x ,y )∼D E ω∼q h ω (x -x ), λ(y, y ) = E (x ,y )∼D L D (k q ) ,
where L D (k q ) is the alignment loss of the kernel k q centered on (x , y ) ∼ D (see Equation ( 10)).

Let t > 0 and p a distribution on R d . By applying the PAC-Bayesian theorem, with ε 0 ∈ (0, 1), we have

Pr S∼D n ∀q on R d : L D (k q ) ≤ 1 n n i=1 L i D (k q ) + 1 t KL(q p) + t 2 2n + ln 1 ε 0 ≥ 1 -0 .
Moreover, we have that for each i ∈ {1, . . . , n}, with a ε i ∈ (0, 1), we have

Pr S∼D n ∀q on R d : L i D (k q ) ≤ L i S (k q ) + 1 t KL(q p) + t 2 2(n -1) + ln 1 ε i ≥ 1 -i .
By combining above probabilistic results with ε

0 = ε 1 = • • • = ε n = ε n+1
, we obtain that, with probability at least 1 -ε ,

L D (k q ) = E (x ,y )∼D L D (k q ) ≤ 1 n n i=1 L i S (k q ) + 1 t KL(q p) + t 2 2(n -1) + ln n + 1 ε + 1 t KL(q p) + t 2 2n + ln n + 1 ε = L S (k q ) + 1 t KL(q p) + t 2 2(n -1) + ln n + 1 ε + 1 t KL(q p) + t 2 2n + ln n + 1 ε ≤ L S (k q ) + 2 t KL(q p) + t 2 2(n -1) + ln n + 1 ε .
Lemma 6. For any data-generating distribution D:

Var S ∼D n (L S (h ω )) ≤ 1 4n . Proof. Given S = {(x i , y i )} n i=1 ∼ D n , we denote F ω S ) := F ω (x 1 , y 1 ), . . . , (x n , y n ) := L S (h ω ) = 1 n(n -1) n i =j h ω (x i -x j ), λ(y i , y j ) .
The function F ω above has the bounded differences property. That is, for each i ∈ {1, . . . , n} : sup

S ,x * ∈R d ,y * ∈Y F ω (x 1 , y 1 ), . . . , (x n , y n ) -F ω (x 1 , y 1 ), . . . , (x i-1 , y i-1 ), (x * , y * ), (x i+1 , y i+1 ), . . . , (x n , y n ) ≤ 1 n , Proposition 8 (Multi-class classification). When S = (x i , y i ) n i=1 ∈ (R d × {1, . . . , L}) n , we have L S (h ω ) = n 2(n-1) - 1 2n(n-1)   2 L y=1 (c 2 y + s 2 y ) -

  Proceedings of the 22 nd International Conference on Artificial Intelligence and Statistics (AISTATS) 2019, Naha, Okinawa, Japan. PMLR: Volume 89. Copyright 2019 by the author(s).

  Given a Fourier transform prior distribution p, let sampleN points Ω = {ω m } N m=1 ∼ p N . Let P (ω m ) = 1 N and h m (δ) := cos(ω m • δ).Given a dataset S, and constant parameters β > 0, compute the following pseudo-posterior for m = 1, . . . , N :

Figure 1 :Figure 2 :

 12 Figure 1: First row shows selected RBF-Landmarks kernel outputs, while second row shows the corresponding learned similarity measures on random Fourier features (PB-Landmarks). The rightmost column displays the classification learned by a linear SVM over the mapped dataset.

Figure 3 :

 3 Figure 3: Train and test error of the kernel learning approaches according to the number of random features D.

λλ

  That is, in the multi-class classification case with L classes (y ∈ {1, . . . , L}) n ), one can compute the empirical alignment lossL S (h ω ) in O(n) steps. ij h ω (x i -x j ) = ij cos(ω • (x i -x j )) -n y i = y j ] -1) cos(ω • (x i -x j )) -n i = y j ] cos(ω • (x i -x j )) -• (x i -x j )) -nLet's denote S y := {x i |(x i , y) ∈ S}. We have i = y j ] cos(ω • (x i -x j )

Figure 4 :

 4 Figure 4: Repetition of Figure 1's experiment, with another toy dataset. First row shows selected RBF-Landmarks kernel outputs, while second row shows the corresponding learned similarity measures on random Fourier features (PB-Landmarks). The rightmost column displays the classification learned by a linear SVM over the mapped dataset.

Figure 5 :

 5 Figure5: Train and test error of the kernel learning approaches according to the number of random features D on the remaining 4 datasets (not reported by Figure3).

Figure 6 :

 6 Figure6: Behavior of the landmarks-based approach according to the percentage of training points selected as landmarks on the remaining 6 datasets (not reported by Figure2).

Table 1 :

 1 Test error of the landmarks-based approach.

			Landmarks-based
	Dataset	SVM RBF	PB PB β=1 PB D=64
	ads	3.05 10.98 4.88	5.12	5.00
	adult	19.70 19.60 17.99 17.99	17.99
	breast	4.90 6.99	3.50	3.50	2.80
	farm	11.58 17.47 15.73 14.19	15.73
	mnist17 0.34 0.74	0.42	0.32	0.32
	mnist49 1.16 2.26 1.80	2.09	2.50
	mnist56 0.55 0.97	1.06	1.55	1.03

The Gram matrix is the n × n matrix constituted by all the kernel values computed on the learning samples.

This trade-off is the same one involved in some other PAC-Bayesian bounds for i.i.d. data (e.g.,[START_REF] Catoni | PAC-Bayesian supervised classification: the thermodynamics of statistical learning[END_REF]. As discussed in[START_REF] Zhang | Information-theoretic upper and lower bounds for statistical estimation[END_REF];[START_REF] Grünwald | The safe Bayesian -learning the learning rate via the mixability gap[END_REF];[START_REF] Germain | PAC-Bayesian theory meets Bayesian inference[END_REF], there is a similarity between the minimization of such PAC-Bayes bounds and the Bayes update rule.

Acknowledgments. P. Germain wants to thank Francis Bach for insightful preliminary discussions. This work was supported in part by the French Project APRIORI ANR-18-CE23-0015 and in part by NSERC. This research was enabled in part by support provided by Compute Canada (www.computecanada.ca).

Thus, we apply the Efron-Stein inequality (following Boucheron et al., 2013, Corollary 3.2) to obtain

A.2 Kernel Alignment Loss Computation

The kernel learning algorithms presented in Section 5 require to compute the empirical kernel alignment loss for each hypothesis h ω , given by

A naive implementation of Equation ( 21) would need O(n 2 ) steps. Propositions 7 and 8 below show how to rewrite Equation ( 21) in a form that needs O(n) steps. Proposition 7 is dedicated to the binary classification, and is equivalent to the computation method proposed by [START_REF] Sinha | Learning kernels with random features[END_REF]. By Proposition 8, we extend the result to the multi-classification case.

Proposition 7 (Binary classification). When S = (x i , y i )

Proof. Using the cosine trigonometric identity

A farm https://archive.ics.uci.edu/ml/datasets/Farm+Ads mnist http://yann.lecun.com/exdb/mnist/ As [START_REF] Sinha | Learning kernels with random features[END_REF], binary classification tasks are compiled with the following digits pairs: 1 vs. 7, 4 vs. 9, and 5 vs. 6.

We split the datasets into training and testing sets with a 75/25 ratio except for adult which has a training/test split already computed. We then use 20% of the training set for validation. Table 2 presents an overview. We use the following parameter values range for selection on the validation set: 5 gives extra results for the greedy kernel learning experiment (Subsection 6.2).


Filipe Rocha

Vera Homem

Javier Castro-Jimenez

Nuno Ratola

Javier Castro-Jiménez

Marine vegetation analysis for the determination of volatile methylsiloxanes in coastal areas

Keywords:

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Siloxanes are a vast group of chemicals consisting of a backbone of alternating silicon (Si) and oxygen (O) atoms with organic side chains. Polymeric forms of siloxanes are usually designated as silicones and depending on the functional groups selected to the side chains and the size of the polymers, their physicochemical properties may vary considerably, allowing an extensive range of uses and applications [START_REF] Rücker | Environmental chemistry of organosiloxanes[END_REF]. Due to their hydrophobicity, these compounds are part of formulations of adhesives, sealants and coatings in water repellent paints [START_REF] Arespacochaga | Understanding the effects of the origin, occurrence, monitoring, control, fate and removal of siloxanes on the energetic valorization of sewage biogasa review[END_REF], and in paper, leather and textile industries [START_REF] Graiver | A review of the fate and effects of silicones in the environment[END_REF] for the enhancement of material properties. Siloxanes are also used in the production of rubbers, resins and other matrices with low thermal and electric conductivity [START_REF] Rücker | Environmental chemistry of organosiloxanes[END_REF], and in antifoaming and wetting agents, reducing the surface tension of liquids. They also take part in agrochemical and pharmaceutical formulations, in household and industrial cleaning products [START_REF] Genualdi | Global distribution of linear and cyclic volatile methyl siloxanes in air[END_REF], and even as additives in the food processing industry [START_REF] Andriot | Silicones in industrial applications[END_REF]. One other major application of siloxanes-derived products is in cosmetics and personal care products (PCPs), such as skin and hair care products, soaps, perfumes and fragrances, antiperspirants, sunscreens, balms or make-up [START_REF] Wang | Low molecular weight cyclic volatile methylsiloxanes in cosmetic products sold in Canada: implication for dermal exposure[END_REF]Capela et al., 2016aCapela et al., , 2016b)). Under the generic designation of dimethicone and cyclomethicone, linear VMSs (lVMSs), polydimethylsiloxanes (PDMSs) and cyclic VMSs (cVMSs) are commonly used as softeners and moisturizers, facilitating the application of creams and lotions (Capela et al., 2016a), with the cyclic congeners D5 and D6 being the predominant VMSs used [START_REF] Wang | Review of recent advances in research on the toxicity, detection, occurrence and fate of cyclic volatile methyl siloxanes in the environment[END_REF][START_REF] Dudzina | Concentrations of cyclic volatile methylsiloxanes in European cosmetics and personal care products: prerequisite for human and environmental exposure assessment[END_REF]Capela et al., 2016a). Consequently, PCPs are considered a major source of human exposure to VMSs, as well as one of the most important source of their presence in the environment (Capela et al., 2016a). It is estimated that 90% of the VMSs present in PCPs formulations volatilize during use (spreading via atmospheric transport), while most of the remainder are washed-off and discharged to wastewater treatment plants (WWTPs) [START_REF] Allen | Industrial organic materials, their environmental entry and predicted fate[END_REF][START_REF] Van Egmond | Monitoring and modelling of siloxanes in a sewage treatment plant in the UK[END_REF][START_REF] Mackay | Bioconcentration and aquatic toxicity of superhydrophobic chemicals: a modelling case study of cyclic volatile methyl siloxanes[END_REF]Capela et al., 2016a).

Taking into account the large (and increasing) amount of siloxanes being produced worldwide, the concern about these compounds is growing, especially regarding VMSs. Considered innocuous for a long time, VMSs were recently classified as emerging pollutants. A report from the Norwegian Pollution Control Authority in 2007 placed D5 on a priority list of substances of which the emissions should be considerably reduced or halted [START_REF] Schlabach | Siloxanes in the Environment of the Inner Oslofjord -TA-2269[END_REF]. Environment Canada and Health Canada published in 2008 reports assessing the risk of cVMS to the environment, stating that D4 and D5 were identified as persistent, bioaccumulative, toxic and prone to ecological harm, while D6 was not considered likewise (EC/HC, 2008a, 2008b, 2008c). The European Commission classified D4 as an endocrine disruptor, possibly interfering with the human endocrine function and impairing human fertility (EC no. 1272(EC no. /2008(EC no. , 2008)), and in 2009 a risk assessment report by the UK Environment Agency considered D4 and D6 as a threat to the environment, while D5 was considered not hazardous (Brooke et al., 2009a[START_REF] Brooke | Environmental Risk Assessment Report: Decamethylcyclopentasiloxane[END_REF][START_REF] Brooke | Environmental Risk Assessment Report: Dodecamethylcyclohexasiloxane[END_REF]. Also the United States Environmental Protection Agency (USEPA) included D4 in a list of chemicals to be reviewed for further assessment under the Toxic Substance Control Act [START_REF] Sanchís | Unexpected occurrence of volatile dimethylsiloxanes in Antarctic soils, vegetation, phytoplankton, and krill[END_REF]. These risk assessment reports are mostly based on model predictions, and the lack of experimental data results in great uncertainties and controversies. However, studies developed in mice suggest that a continuous inhalation exposure to D4 is related to the occurrence of uterine endometrial adenocarcinomas in female rats, as well as increased liver weight (hepatic hyperplasia) and inducing of drug metabolizing liver enzymes [START_REF] Zhang | Induction of rat hepatic drug metabolizing enzymes by dimethylcyclosiloxanes[END_REF]. D4 was also correlated with hormonal changes on oestrogen levels, resulting in impairment of fertility and reproductive complications [START_REF] Mckim | Potential estrogenic and antiestrogenic activity of the cyclic siloxane octamethylcyclotetrasiloxane (D4) and the linear siloxane hexamethyldisiloxane (HMDS) in immature rats using the uterotrophic assay[END_REF][START_REF] He | Octamethylcyclotetrasiloxane exhibits estrogenic activity in mice via ER alpha[END_REF][START_REF] Quinn | In vitro and in vivo evaluation of the estrogenic, androgenic, and progestagenic potential of two cyclic siloxanes[END_REF][START_REF] Meeks | An inhalation reproductive toxicity study of octamethylcyclotetrasiloxane (D4) in female rats using multiple and single day exposure regiments[END_REF]. On the other hand, D5 although not affecting fertility in mice, showed possible carcinogenic effects related to high chronic exposure to liver and lungs in animal studies (EC/HC, 2008b). In the specific case of marine environments, siloxanes were reported as generally non-toxic to organisms at their low levels of solubility in the aqueous media [START_REF] Arespacochaga | Understanding the effects of the origin, occurrence, monitoring, control, fate and removal of siloxanes on the energetic valorization of sewage biogasa review[END_REF], but the knowledge on this subject is still scarce.

The presence of VMSs has been reported in numerous environmental compartments like sea water, sediment and biota samples in European Nordic Countries [START_REF] Kaj | Siloxanes in the Nordic environment[END_REF], river sediment in the United Kingdom [START_REF] Sparham | Determination of decamethylcyclopentasiloxane in river and estuarine sediments in the UK[END_REF], the pelagic food webs of two Norwegian lakes [START_REF] Borgå | Consistency in trophic magnification factors of cyclic methyl siloxanes in pelagic freshwater food webs leading to brown trout[END_REF], in wastewater, surface water and sediments in Catalonia [START_REF] Sanchís | Occurrence of linear and cyclic volatile methylsiloxanes in wastewater, surface water and sediments from Catalonia[END_REF], in seawater, marine sediment and fish samples in Northeast China [START_REF] Hong | Distribution, source, fate and bioaccumulation of methyl siloxanes in marine environment[END_REF], in the marine food web of Dalian Bay, Northern China [START_REF] Jia | Trophic transfer of methyl siloxanes in the marine food web from coastal area of northern China[END_REF], and in pine needles, soil and air in Portugal [START_REF] Ratola | Using air, soil and vegetation to assess the environmental behaviour of siloxanes[END_REF]. Some studies suggest that VMSs are able to undergo long-range atmospheric transport, reaching areas far away from their emission sources: cVMSs were detected in Arctic air [START_REF] Genualdi | Global distribution of linear and cyclic volatile methyl siloxanes in air[END_REF][START_REF] Krogseth | Occurrence and seasonality of cyclic volatile methyl siloxanes in Arctic air[END_REF], sediment and biota [START_REF] Warner | Volatile siloxanes in the European arctic: assessment of sources and spatial distribution[END_REF], as well as in Antarctic soils, vegetation, phytoplankton and krill [START_REF] Sanchís | Unexpected occurrence of volatile dimethylsiloxanes in Antarctic soils, vegetation, phytoplankton, and krill[END_REF]. Considering that some of the emission routes of these compounds are through WWTPs discharges and the common use of siloxanes in PCPs formulations, marine and coastal environments are potential hotspots for the presence of VMSs, especially considering the high anthropogenic pressure and the increased use of protective PCPs in the summer months with the rise of the temperature and the exponential growth of tourism. [START_REF] Ratola | Using air, soil and vegetation to assess the environmental behaviour of siloxanes[END_REF] found much higher levels of VMSs in the air of one beach from the south of Portugal in summer than in winter. Monitoring the distribution levels of VMSs in such areas is therefore very important, and to date there is not much information available on the distribution, fate and trends of siloxanes in marine environments in Southern European latitudes. Considering marine vegetation, to the authors' best knowledge, only one study on VMSs in seaweed is available [START_REF] Jia | Trophic transfer of methyl siloxanes in the marine food web from coastal area of northern China[END_REF]. Due to the tendency to accumulate pollutants in their tissues and surface area, marine vegetation has been used in the biomonitoring of pollutants in aquatic environments [START_REF] Roberts | Contamination of marine biogenic habitats and effects upon associated epifauna[END_REF]. Moreover, these matrices occupy low trophic levels in food chains, making them a potential vehicle for the accumulation of pollutants into higher trophic levels [START_REF] Gutow | Experimental evaluation of seaweeds as a vector for microplastics into marine food web[END_REF].

This work aimed to determine the presence, levels and possible trends of three linear (L3-L5) and four cyclic (D3-D6) VMSs in marine vegetation, using a QuEChERS extraction followed by gas chromatography-mass spectrometry (GC/MS) analysis. Different species (algae and seaweed) were collected from several sites in the Atlantic (North and centre regions of Portugal), and the Mediterranean (coast of the Region of Murcia in southeast Spain and surroundings of Marseille city in the south of France).

Materials and methods

Sampling strategy

In this study, different marine vegetation was sampled in three different geographical areas (Fig. 1): three species of macroalgae (Fucus vesiculosus, Porphyra sp. and Ulva lactuca), collected in the Atlantic coast of the north and centre of Portugal, and a species of sea grass (Posidonia oceanica) endemic from the Mediterranean Sea, collected in the Region of Murcia, southeast of Spain, and in the surroundings of Marseille, south of France. Samples were collected using disposable nitrile gloves, stored in sealed polyethylene bags and kept refrigerated and protected from light until proper storage in the lab at -20 °C until analysis.

Macroalgae samples from 7 different locations in the seashore of the North Region of Portugal, in Porto Metropolitan Area, and 2 other further South, close to the city of Aveiro, were detached from the rocks during low tide conditions in the months of May, June, July, August, September and December 2016, to assess possible seasonal variations. Locations were selected based on their different surroundings and anthropogenic pressure in its vicinities and most of all consist of beach areas, with different affluence levels during the bathing season in Portugal. The most common macroalgae species were collected according to their availability, hence it was not possible to guarantee the collection of the samples from the same species in the same spots in every sampling campaign. Nevertheless, three different species of seaweed were found frequently enough to ensure the coverage of 9 different spots throughout the 6 sampling months. Fucus vesiculosus was collected in the beaches of Barra and Biarritz (the latter in a salt water lagoon close to the shore) near Aveiro (an urban area with about 120,000 inhabitants), as well as in 3 other beaches in the North coast of Portugal in the Porto Metropolitan Area (the second biggest urban area in the country, with almost two million inhabitants): Paraíso, Conchinha and Afurada (the latter in the estuary of the Douro river, very close to the mouth). Another two species were collected in the North coast of Portugal: Porphyra sp. in the beaches of Paraíso, Aterro, Matosinhos, Canide Norte and Miramar, and Ulva lactuca in the beach of Conchinha.

Posidonia oceanica, a seagrass species with ribbon-like leaves, endemic to the Mediterranean Sea, was also part of this study, covering regions with different socio-geographic and climatic patterns. Samples were collected in 7 sites in the Region of Murcia (Spain) in two sampling campaigns: summer (July 2015) and winter (February 2016), collecting the samples in the shore. The Region of Murcia coastline is a very popular summer destination, which increases considerably the urban pressure in those months. Four sites (La Manga North Mediterranean, La Manga North Mar Menor, La Manga South Mediterranean and La Manga South Mar Menor) were located in La Manga, a 21 km long and 100 m wide land strip that separates the Mediterranean Sea from the Mar Menor lagoon and has an intense urban pressure in the summer. Cala Cortina is an urban beach with high occupancy levels in the city of Cartagena, very close to a very important industrial complex that includes the Port and a petroleum refinery and a power plant. La Azohía is a smaller town with strong afflux of tourists in the summer, while Cala Blanca, a remote beach in an area with no urban development, can be considered a background site. Samples from the surroundings of Marseille were taken from 5 different spots a few meters from the coastline with the help of trained divers, in a single winter campaign in 2016. Marseille is a highly populated area (about 2 million people in the whole conurbation) with a strong social, economic and industrial activity, and with several beaches in the city and in the surrounding areas. Sites were in this case also selected to cover different backgrounds. Frioul (a small archipelago just in front of Marseille) and La Vesse (about 20 km away from Marseille, in the Côte Bleue) are considered low-density populated spots but with high tourist affluence and, in the case of Frioul, intense maritime traffic. Plateau de Chèvres and Cassis are located in the Calanques National Park, but in the vicinity of the WWTP outlet from Marseille and Cassis conurbations, respectively. Marseille Rade Sud is located in the south harbour of the city, close to the Huveaune river outlet.

Chemicals and materials

Individual standards of octamethyltrisiloxane (L3), decamethyltetrasiloxane (L4), dodecamethylpentasiloxane (L5), hexamethylcyclotrisiloxane (D3), octamethylcyclotetrasiloxane (D4), decamethylcyclopentasiloxane (D5) and dodecamethylcyclohexasiloxane (D6) with purity levels higher than 97%, were purchased from Sigma-Aldrich (St. Louis, MO, USA), as well as tetrakis (trimethylsilyloxy)silane (M4Q), used as the internal standard. The standards and all solutions were prepared in analytical grade n-hexane, obtained from VWR (Fontenay-sous-Bois, France), and protected from light in amber glass vials. All preparations were kept in the dark at -20 °C until use. Sodium acetate (CH 3 COONa), acquired from Merck (Darmstadt, Germany), anhydrous magnesium sulphate (MgSO 4 ) from Panreac AppliChem (Barcelona, Spain), primary and secondary amine (PSA)-bonded silica and octadecyl-silica (C18), from Supelco (Bellefonte, PA, USA), were used for the QuEChERS preparation. MgSO 4 was baked overnight at 450 °C before use. Nitrogen (99.995%) for sample evaporation and helium (99.9999%) used as carrier gas in the GC/MS system were supplied by Air Liquide (Maia, Portugal).

Sample extraction and quantification

The development of the extraction procedure was initially based on the protocol validated by [START_REF] Ramos | Solvent-saving approaches for the extraction of siloxanes from pine needles, soils and passive air samples[END_REF] for the extraction and quantification of VMSs in pine needles and soils, and recently optimized and validated for the analysis of 7 VMSs in marine vegetation samples (work submitted for publication).

In brief, 2.5 g of finely cut sample was weighted directly into a 50 ml polypropylene conical bottom centrifuge tube, spiked with internal standard (M4Q) at 15 ng•g -1 , and extracted with 10 ml of n-hexane in a 720 W JP Selecta ultrasonic bath (Barcelona, Spain) for 15 min after 3 min of vortex. Then, a first QuEChERS, containing 6 g of anhydrous MgSO 4 and 1.5 g of CH 3 COONa, was added and the mixture was vortexed for 3 min and centrifuged for 10 min at 2670g. The supernatant was transferred to the second QuEChERS, consisting of 900 mg of MgSO 4 , 300 mg of PSA-bonded silica and 150 mg of C18, used to remove undesired compounds in a dispersive solid-phase clean-up step. The mixture was vortexed and centrifuged as before, with the resulting supernatant being collected and its volume reduced under a nitrogen stream until a final extract of 150 μl. Samples were extracted in duplicate and stored at -20 °C prior to analysis. Instrumental analysis was performed using a Varian Ion Trap GC-MS system (Walnut Creek, CA, USA), equipped with a 4000-GC gas chromatograph, a 240-MS ion trap mass spectrometer, a CP-1177 split/splitless injector adapted with a Merlin microseal system (a microvalve alternative to the conventional silicone rubber septa) and an CP-8410 auto-sampler. To prevent bleeding from common chromatographic columns and minimize background contaminations of siloxanes, compound separation was obtained in a low-bleed Agilent DB-5 ms ultra-inert column (30 m × 0.25 mm I.D., 0.25 μm film thickness) at a constant flow of helium (1.0 ml•min -1 ). The temperature programme was the following: 35 °C hold for 5 min, raised at 10 °C•min -1 to 95 °C, then 5 °C•min -1 to 140 °C and finally 35 °C•min -1 to 300 °C (hold for 5.43 min), in a total runtime of 30 min. Injection volume was 1 μl and the temperatures of manifold, ion trap, transfer line and injector were 50, 200, 250 and 200 °C, respectively. The mass spectrometer was operated in the electron ionization (EI) mode (70 eV) and for quantitative analysis of target compounds, time-scheduled selected ion storage (SIS) mode was applied. The filament emission current was 50 μA.

Quality assurance and quality control

The linearity responses were good, ranging from 1 to 750 μg•l -1 for each VMSs studied, with R 2 values higher than 0.996 and low LODs (between 11.7 pg•g -1 dw (D5) and 62.9 pg•g -1 dw (L3)), suitable for the detection of these contaminants at trace levels. Global mean of recoveries of 93 ± 7% were obtained, and good intraday and interday precision values were achieved (RSD values lower than 10 and 15%, respectively, for most target compounds). LODs, limits of quantification (LOQs) and recovery values for each of the target compounds are presented in Supplementary material (Table S1).

Due to the ubiquitous presence of siloxanes and their reported use in PCPs and in some laboratory equipment, special precautions were taken to minimize possible external contaminations throughout the different steps of the experiment. Analysts avoided the use of lotions, perfumes, hand creams and other PCPs, and nitrile gloves were used and constantly changed during sample manipulation. All glass material was rinsed with acetone and the non-calibrated pieces baked at 400 °C for at least 4 h before use. Instrumental analysis was also enhanced to eradicate possible contamination sources of siloxanes: together with the use of the Merlin microseal system in the injection port, the inlet liner was free from silanized glass wool and frequently changed. A low bleed ultra-inert column was used and at the end of each chromatographic run, a temperature of 300 °C was kept for about 5.5 min as a clean-up step to remove impurities possibly remaining in the column. To avoid possible memory effects and guarantee the column rinsing, chromatographic blanks of n-hexane were frequently run. Procedural blanks (extractions performed with no sample) were also performed and analysed with every extraction batch (Table S2 in Supplementary material) and, whenever needed, the results were blank corrected. Injections of pure solvent were also performed from time to time to assess possible contamination of the solvent itself, but this never happened.

The concentrations of the target compounds in each sample were presented in a dry weight (dw) basis. The water content of each sample was determined according to a procedure proposed by [START_REF] Littler | Ecological field methods: macroalgae[END_REF] for macroalgae: approximately 2.5 g of sample was left to dry at 105 °C until constant weight was achieved. A portion of each collected species was reserved for the dry weight determination and tests were performed in duplicate. The results are presented in Supplementary Material (Table S3).

Statistics

Quantification of the target compounds was performed using the VARIAN, INC. Mass Spectrometry Workstation software, version 6.9.3. Univariate statistic tests for a 95% confidence interval were applied to the results in order to clarify their interpretation and discussion, according to [START_REF] Miller | Statistics for analytical chemistry[END_REF].

Results and discussion

The concentrations found in samples of marine vegetation collected in the Atlantic coast of Portugal and in the Mediterranean coasts of the Region of Murcia (Spain) and Marseille (South of France) will be presented and discussed separately, to cover different points of view in the whole study.

VMSs in macroalgae from the Atlantic Ocean

As mentioned previously, 6 sampling campaigns in 6 different months (May, June, July, August, September and December) were conducted in several locations in the surroundings of the city of Aveiro and of Porto Metropolitan Area (Portugal). Several species of macroalgae were collected in the selected sampling sites, upon availability. Fucus vesiculosus and Porphyra sp. were collected in the majority of the sampling campaigns in 5 locations, while Ulva lactuca was only frequent in one of the selected sites. The presence of VMSs was confirmed in 51 out of the 55 samples analysed, and at least one of the target compounds was found at concentrations higher than the LOQ in 48 of them. The levels and mean values of VMSs are presented separately for each species and per month, in Figs. 2, 3 and4. Complete concentrations for each VMS, beach and macroalgae species are presented in Supplementary Material (Tables S4 to S6).

Overall, 99% of the VMSs quantified in all the samples were cVMSs, with a predominance of D5 (44%) and D6 (38%), followed by D4 (12%) and D3 (5%). Considering lVMS, L3 was only detected below the LOQ in Ulva lactuca from Conchinha in July, while L4 and L5 were found in concentrations ranging from below LOD to 1.9 ± 0.9 ng•g -1 dw and 0.6 ± 0.2 ng•g -1 dw , respectively, both in samples of Porphyra sp. from the beach of Miramar in June and August. On the other hand, cVMSs were detected in a higher frequency and amount, with concentrations ranging between below the LOQ for all cVMS and a maximum of 2.7 ± 0.8 ng•g -1 dw for D3 (Fucus vesiculosus, Conchinha in August), 5.0 ± 2.7 ng•g -1 dw for D4 (Porphyra sp., Canide Norte in August), 27.0 ± 20.4 ng•g -1 dw for D5 (Fucus vesiculosus, Conchinha in August) and 33.3 ± 19.6 ng•g -1 dw for D6 (Porphyra sp., Canide Norte in August). It is important to note that in some cases the levels have high standard deviations, which reflects the complexity of the selected matrixes (due to natural variability of biological materials, such as marine vegetation), and the challenges associated with the analyses of these volatile compounds at trace levels. Similar standard deviations were reported in the only work found so far in the literature regarding siloxanes in seaweed [START_REF] Jia | Trophic transfer of methyl siloxanes in the marine food web from coastal area of northern China[END_REF], in which the trophic transfer of cVMSs was assessed in the Dalian Bay, Northern China. The siloxanes found in this study at higher concentrations in sea lettuce (Ulva pertusa) were also D4, D5 and D6, with 6.3 ± 2.3, 5.8 ± 3.4 and 10.5 ± 11.7 ng•g -1 ww (wet weight), respectively. Those values were lower when compared to the same compounds concentration in other biota samples in the same study, with an estimated higher trophic level (for instance, mean D4, D5 and D6 concentrations detected in 5 different fish species were 14.0 ± 8.9, 31.7 ± 29.6 and 19.1 ± 2.2 ng•g -1 ww , respectively).

The site with the highest concentrations of VMSs throughout this study was Canide Norte, followed by Miramar. Both are highly-visited beaches, located in the suburban areas of Vila Nova de Gaia, across the Douro River from Porto. If in Canide Norte VMSs were only detected during the official bathing season (July, August and September), in concentrations up to 50.4 ± 20.8 ng•g -1 dw , in Miramar a background concentration is more constant throughout the 6 sampling months, ranging between 9.0 ± 4.6 and 22.9 ± 1.4 ng•g -1 dw , in December and August, respectively. In this beach there is a small stream (Ribeira do Espírito Santo) which flows through small urbanized areas until it enters the ocean in this location. This may be a potential contribution to the background concentrations of VMSs observed. The third location with higher incidence of VMSs was Conchinha, a small beach with great occupancy rate in summer, despite being located near the industrial complex of the Matosinhos refinery and in the vicinities of the Matosinhos WWTP. The highest concentration of the sum of VMSs was 29.8 ± 20.4 ng•g -1 dw in a Fucus vesiculosus sample in August. Other target locations near the same infrastructures, such as the beaches of Paraíso, Aterro and even Matosinhos (the latter closer to Leixões, one of the major seaports in the country), registered mean total VMSs values in samples of Porphyra sp. of 6.8, 3.6 and 6.1 ng•g -1 dw , respectively. The lowest values of VMSs were found in Afurada, an estuarine area (Douro River) with no sunbathing activities where Fucus vesiculosus was sampled and the sum of the target compounds in each month ranged between not detected (December) and 1.2 ± 0.5 ng•g -1 dw . Further south, the two sites near Aveiro (with Fucus vesiculosus) revealed that Biarritz, also not in the open sea (salt water lagoon), presented a concentration of VMSs (global mean value of the sum of VMSs of 2.0 ng•g -1 dw ) lower than that found for Barra, 3.9 ng•g -1 dw , a very popular beach in the warmer months.

Looking into the results under a temporal perspective, seasonal tendencies are in most cases unclear, with the sum of the concentrations of VMSs per month suggesting a background level of contamination, with differences among months not statistically significant most of the times. Nevertheless, some exceptions must be pointed out. For instance, in Canide Norte, VMSs were obtained in Porphyra sp. in much higher concentrations in July, August and September (mostly D6), with the highest total VMSs value in August (50.4 ± 20.8 ng•g -1 dw ) statistically different (p b 0.05) from all other months. The same species from the beach of Matosinhos also showed a significantly higher concentration of the sum of VMSs in August (16.4 ± 1.0 ng•g -1 dw ), followed by September (9.5 ± 1.6 ng•g -1 dw ), while the other months presented much lower values. In Conchinha, Fucus vesiculosus in registered a total amount of VMSs of 29.8 ± 20.4 ng•g -1 dw in August (91% of which D5), significantly different from the other months. However, a sample of Ulva lactuca, another species collected in Conchinha from June to September, revealed a different seasonal pattern, as well as different levels of total VMSs in each month. July had the highest value (9.9 ± 2.3 ng•g -1 dw ), followed by August (8.1 ± 4.4 ng•g -1 dw , D5 representing 89%). Comparing the results between these two species collected in the same site, the total VMSs values vary considerably, possibly a consequence of their significant differences in shape, superficial area, composition, life cycle and even position on the shore [START_REF] Skene | Key differences in photosynthetic characteristics of nine species of intertidal macroalgae are related to their position on the shore[END_REF][START_REF] Parmar | Bioindicators: the natural indicator of environmental pollution[END_REF]. Interestingly, D5 is the compound in higher proportion in both, an indication of consistent sources of VMSs. Finally, in Paraíso, samples of Porphyra sp. collected between July and September showed that July was the month with the highest values of VMSs (17.0 ± 3.8 ng•g -1 dw ), significantly different from August and December, but not from September (7.8 ± 2.1 ng•g -1 dw ). However, Fucus versiculosus collected from June to December in the same location, revealed much lower concentrations of VMSs during summer, peaking in December instead. Paraíso is a beach with very limited sunbathing which receives the discharge of a nearby WWTP through a local stream. This can be an indication that VMSs have background values in this site, being the seasonal variations not evident. In fact, even the total VMSs levels are some of the lowest found in the study, which concurs with the previous explanation. Overall, these different levels in different species from the same sampling site suggest that inter-species comparison of the uptake of VMSs in macroalgae must be taken with caution. Besides the structural and biological differences between species, it is impossible to determine in which conditions and for how long each organism has been exposed to the different environmental elements in their natural habitat.

VMSs in Posidonia oceanica from the Mediterranean Sea

Concentrations of VMSs in this marine plant collected in those two regions are presented in Figs. 5 and6. Complete concentrations for each VMS and sampling site are presented in Supplementary Material (Tables S7 to S9). Similarly to the results obtained in the macroalgae samples in Portugal, VMSs were found in most of the samples analysed (17 out of 19), with a clear prevalence of cVMSs (96%) over lVMSs.

Mediterranean coastline of the Region of Murcia, Spain

In the samples from the Region of Murcia, the total concentration of VMSs ranged between not detected (Cala Blanca, in summer) and 292.0 ± 41.5 ng•g -1 dw (Cala Cortina, also in summer), with D5 and D6 being the predominant VMSs, representing 57 and 38%, respectively, of the total concentrations. D3 and D4 are almost absent in the samples collected in summer, being usually more frequent in winter samples. This suggests their volatilization from the water phase, due to higher temperatures during the warmer months (consistent with their higher vapour pressure), making these compounds less available for uptake by Posidonia oceanica seagrass.

Comparing the results between summer and winter seasons, the total VMSs concentrations were higher in the summer in 4 sites (La Azohía, La Manga North in both Mediterranean and Mar Menor sides, and Cala Cortina), and in the winter in the remaining ones. The remotest area considered in this sampling was Cala Blanca, a pristine beach with very limited access to population. VMSs were not detected in the summer, and in the winter, only very low levels of L4, L5, D5 and D6, with a total concentration of 3.6 ± 0.7 ng•g -1 dw ), consistent with its remote location. Having no direct sources of siloxanes identified, the VMSs found may be more subject to the common incidence of VMSs in the atmosphere, which was reported as higher in winter by several authors, due to lower transformation phenomena in this season, such as the reaction of VMSs with OH radicals in the air [START_REF] Mclachlan | Concentrations and fate of decamethylcyclopentasiloxane (D5) in the atmosphere[END_REF]. In La Azohía, the total concentration of VMSs in summer (15.1 ± 1.5 ng•g -1 dw ) is about 4 times higher to that of the winter season (4.3 ± 0.7 ng•g -1 dw ), with the prevalence of D5 (63%), followed by D6 (22%). Despite being a small urbanized area, La Azohía faces in summer the aforementioned increase of its population due to the afflux of tourists to its beaches. These results suggest the increasing use of PCPs such as sunscreens, deodorants and other body lotions in these areas. Similar trends were observed in three other beaches: in both samples from La Manga North (in both beaches from Mediterranean Sea and Mar Menor) and in Cala Cortina. In La Manga North, similar values of VMSs were registered in both sides of the land strip, with a total maximum for the sample collected in the Mediterranean side in summer (8.0 ± 2.6 ng•g -1 dw ), followed by the same season in the Mar Menor side (6.4 ± 0.9 ng•g -1 dw ), while the values of VMSs found in winter in both locations were 5 to 4 times lower.. However, in La Manga South, both beach sites in Mediterranean Sea and Mar Menor showed slight higher total concentrations of VMSs in the samples collected in Winter (6.7 ± 1.0 and 26.8 ± 3.1 ng•g -1 dw , respectively), with the prevalence of D5 (74%) followed by D6 (18%). This result was not consistent with the seasonal pattern verified for La Manga North, suggesting that other sources of VMSs in these locations were present. It is also interesting to note that in La Manga South, total VMSs were about 4 times higher in both seasons in the Mar Menor side when compared to the Mediterranean side. Mar Menor is an enclosed salt water lagoon with limited water exchanges and high levels of pollution reported, mostly due to the impacts of mining and agricultural activities, as well as a major urban development in its surroundings during the last decades [START_REF] Conesa | The Mar Menor lagoon (SE Spain): a singular natural ecosystem threatened by human activities[END_REF][START_REF] Moreno-González | Seasonal distribution of pharmaceuticals in marine water and sediment from a mediterranean coastal lagoon (SE Spain)[END_REF][START_REF] Carratalá | Occurrence and seasonal distribution of polycyclic aromatic hydrocarbons and legacy and current-use pesticides in air from a Mediterranean coastal lagoon (Mar Menor, SE Spain)[END_REF]. Also, the south part of La Manga suffers more strongly with the urban pressure, and although not as intensively as in summer, even in winter the nice weather conditions attract a considerable number of tourists, adding to the local population, which is more concentrated in this area. Moreover, recent reports highlight the occurrence of plastic debris in the Mar Menor area, and their role as transport vectors of organic pollutants, including PCPs, to this enclosed environment [START_REF] León | Potential transfer of organic pollutants from littoral plastics debris to the marine environment[END_REF]. Although the authors did not measure VMSs specifically, plastic debris could be a currently unexplored source for this family of compounds. The sample collected in the summer in Cala Cortina showed the highest levels of total VMSs (292.0 ± 41.5 ng•g -1 dw ), mostly D5 (53%) and D6 (46%). Being a highlyfrequented urban beach close to a heavy industrial complex, Cala Cortina is prone to suffer the effects of the use of PCPs by the population during the bathing season, but also the potential use of silicone-based polymers commonly applied as coatings in ship hulls, producing a surface which discourages the settling and accumulation of marine fouling, mostly in cruise and fast container ships [START_REF] Bohlander | Review of Options for in-Water Cleaning of Ships[END_REF]. However, the total concentrations of VMSs in winter are significantly (p b 0.05) lower (3.6 ± 0.7 ng•g -1 dw ), suggesting the importance of the afflux of sunbathers in the summer and the enhanced use of sunscreens and other PCPs.

Overall, statistically significant differences between summer and winter (p b 0.05) were only found in Cala Cortina (values in summer much higher than in winter). The absence of a clear seasonal or geographical pattern considering the other locations in both seasons can also be attributed to the characteristics of the marine vegetation, which can have different responses in terms of development depending on the exposure. Also, the extremely high temperatures observed in the Region of Murcia in the summer may induce phenomena of revolatilization of VMSs into the atmosphere and of enhanced reaction with OH radicals that have a stronger presence in the warmer months [START_REF] Vaughan | Seasonal observations of OH and HO in the remote tropical marine boundary layer[END_REF], counteracting the effect of more local sources of VMSs via the enhanced urban pressure and use of PCPs. Posidonia oceanica is considered an indicator of seawater quality [START_REF] Gobert | Assessment of the ecological status of Mediterranean French coastal waters as required by the Water Framework Directive using the Posidonia oceanica Rapid Easy Index: PREI[END_REF], being absent if the exposure to pollution is high. Consequently, the ubiquity of VMSs and their behaviour, transport and distribution in the environment is a challenging task to take on.

Mediterranean coastline of Marseille, South France

The sum of VMSs varied between not detected (in Cassis) and 457.5 ± 25.8 ng•g -1 dw (in Frioul) in the Posidonia oceanica samples collected in winter, and the prevalent individual congeners were D6 (65%), followed by D5 (30%). Once again, D5 and D6 are found at higher concentrations, consistent with the results in the other Mediterranean region and with the trends found in previous studies [START_REF] Bletsou | Mass loading and fate of linear and cyclic siloxanes in a wastewater treatment plant in Greece[END_REF][START_REF] Xu | Critical review and interpretation of environmental data for volatile methylsiloxanes: partition properties[END_REF][START_REF] Sanchís | Unexpected occurrence of volatile dimethylsiloxanes in Antarctic soils, vegetation, phytoplankton, and krill[END_REF]Capela et al., 2016a).

VMSs were not detected (Cassis) or exhibited the lowest concentrations (Plateau de Chèvres, only D5 at 5.0 ± 0.6 ng•g -1 dw ) in samples collected in the Calanques National Park, in spite of their vicinity to the WWTPs. The other three locations are spots facing the city of Marseille directly, thus bearing higher and more diverse anthropogenic pressures in its surroundings, which is reflected on the total amount of VMSs found. In La Vesse, a total concentration of VMSs of 93.1 ± 84.4 ng•g -1 dw was measured, mostly D6 (65%) and D5 (29%), and samples from Marseille Rade Sud (South harbour of the city of Marseille), also showed high values (173.1 ± 195.4 ng•g -1 dw ), with D6 (72%), D5 (23%) and L5 (5%) being the only compounds detected and showing a potential riverine/WWTP influence. Interestingly, the samples collected close to the Frioul archipelago exhibited the highest VMSs concentrations. This area is known for its intense maritime traffic of commercial and leisure vessels and important tourist pressures. D6 and D5 were again the predominant compounds (290.8 ± 2.2 and 146.3 ± 25.7 ng•g -1 dw , respectively), followed by the lVMSs L5 (17.1 ± 1.4 ng•g -1 dw ) and L4 (3.3 ± 1.0 ng•g -1 dw ). In addition, this area has been recently found as a microplastic accumulation spot in the bay of Marseille [START_REF] Schmidt | Plastic organic additives in surface seawater and zooplankton from the Gulf of Lion (NW Mediterranean Sea). Poster Communication[END_REF], consistent with our hypothesis of plastic debris as potential sources of VMSs.

The observed VMSs relative distribution in some samples, like la Vesse or Marseille Rade Sud may also point towards a potential influence of WWTPs [START_REF] Wang | Fate of anthropogenic cyclic volatilemethylsiloxanes in a wastewater treatment plant[END_REF]. However, more sampling sites and longer and more frequent observation periods are needed to really ascertain the potential influence of the WWTPs on the VMSs occurrence on Posidonia samples in the study area due to the high variability on the time and intensity of the discharges. It is interesting to note that in the three locations with the highest VMSs concentrations, D6 is predominant over D5, with L5 also being quantified in all of the three determinations. Since samples were collected in winter, these levels are unlikely to be related to the direct inputs by beach attendants, so probably a combination of the intense urban and industrial pressures, the presence of nearby WWTPs and plastic debris could have influenced the observed pattern.

The mean concentrations found in Marseille are higher than in Murcia and in addition only two sites in Murcia (Cala Blanca and Cala Cortina) showed the same predominance of D6 in winter as the samples in Marseille. This may reflect different patterns of VMSs usage and emissions. However, there is a high variability, and factors such as the different sampling strategies could have contributed to the observed differences.

The clear predominance of cVMSs over lVMSs verified in all samples considered in this study is consistent with the results on the determination of siloxanes in biota samples in marine environments [START_REF] Kaj | Siloxanes in the Nordic environment[END_REF][START_REF] Warner | Volatile siloxanes in the European arctic: assessment of sources and spatial distribution[END_REF][START_REF] Hong | Distribution, source, fate and bioaccumulation of methyl siloxanes in marine environment[END_REF][START_REF] Jia | Trophic transfer of methyl siloxanes in the marine food web from coastal area of northern China[END_REF], as well as in terrestrial vegetation (pine needles) in Portugal [START_REF] Ratola | Using air, soil and vegetation to assess the environmental behaviour of siloxanes[END_REF]. Besides, these results are also in line with the prevalence of cyclomethicones use in wash-off PCPs, mostly D5 and D6, and the resulting higher mass loadings of these compounds to WWTPs when compared to lVMSs [START_REF] Van Egmond | Monitoring and modelling of siloxanes in a sewage treatment plant in the UK[END_REF][START_REF] Xu | Occurrence and fate of volatile siloxanes in a municipal wastewater treatment plant of Beijing, China[END_REF]Capela et al., 2016a), suggesting that these facilities are one of the possible route of these pollutants into coastal environments. Another source could probably be the release of siloxanes due to its presence in sun care and other PCPs used by the population attending the beaches during the bathing season. In addition, plastic fragments, in particular those coming from PCPs containers could constitute a not-yet investigated source of VMSs in marine waters and vegetation. Nevertheless, marine vegetation is not directly impacted, in the sense that receives the input from the amount of VMSs present in seawater or air. Since VMSs are not strongly water-soluble, the variations in the levels found or the absence of significant seasonal trends in some cases highlight the difficulty in quantifying VMSs. But their detection in almost all samples is an evidence of their ubiquitous presence. Thus the need to pursue the research in coastal areas, involving more matrices, with a wider temporal coverage, and eventually including the analysis of other siloxanes. For instance, higher molecular lVMSs and the lowest one, L2, (the latter excluded from the current study due to an overlap with the solvent peak in the GC/MS analysis). The influence of other factors such as pH, temperature, seawater dissolved oxygen or nutrient availability (among others) may also be relevant in future efforts.

Conclusions

An analytical methodology based on QuEChERS extraction followed by GC/MS was successfully employed for the assessment of 3 linear (L3-L5) and 4 cyclic (D3-D6) VMSs in different marine vegetation. Sampling campaigns were conducted in the central and North Atlantic coastline of Portugal with three species of seaweed, Fucus vesiculosus, Porphyra sp. and Ulva lactuca, collected in 9 different sites, as well as in the Mediterranean coastlines of the Spanish Region of Murcia (7 sites in summer and winter) and South of France, in the surroundings of Marseille (5 locations only in winter), using Posidonia oceanica with samples. From a total of 74 samples analysed, a clear prevalence of cVMSs over lVMSs was seen, mostly D5 and D6. In general, higher concentrations of siloxanes were reported in locations with higher urban and industrial pressures, and WWTPs as well as the use of PCPs by the population attending beach areas during the bathing seasons appear to be two of the main sources of VMSs in the coastal environments investigated. However, our results also indicate that other sources may exist, such as the plastic debris introduced in the marine environment, in particular those coming from PCPs containers. Seasonal patterns were also investigated, with results suggesting a tendency for higher concentrations of VMSs (mostly the cyclic ones) in some of the beach areas with high attendances during summer. However, there were also some results pointing otherwise, hence more studies in coastal areas should be conducted in the future for a better disclosure on this topic. Marine vegetation was a hard matrix to work with for the analysis of such ubiquitous compounds at trace levels, with difficulties and limitations being accounted since the sampling starting point of this study to the end point of the quantification of the extracts. And despite the obtained results contributed for a better understanding of the siloxanes levels, sources and trends in the studied coastal environments, other matrices (such as sediment, or possibly other biota samples with a higher trophic level) should be considered in the future for a more consistent indicator of the presence and behaviour of VMSs in the unique and delicate ecosystems that constitute coastal areas.

Fig. 1 .

 1 Fig. 1. Geographical locations of the different sampling sites.

  2367 F. Rocha et al. / Science of the Total Environment 650 (2019) 2364-2373

Fig. 2 .

 2 Fig. 2. Concentrations of VMSs in samples of Fucus vesiculosus (n = 2) from 5 sites. Error bars represent the standard deviation of total concentrations per month.

Fig. 4 .

 4 Fig. 4. Concentrations of VMSs in samples of Ulva lactuca (n = 2) from the beach of Conchinha. Error bars represent the standard deviation of total concentrations per month.

Fig. 5 .

 5 Fig. 5. Concentrations of VMSs in summer and winter samples of Posidonia oceanica (n = 2) from 7 sites in the Mediterranean coastline of the Region of Murcia, Spain. Error bars represent the standard deviation of the total VMSs concentrations.

Fig. 6 .

 6 Fig. 6. Concentrations of VMSs in samples of Posidonia oceanica (n = 2) from 5 sites in the Mediterranean coastline of Marseille, South France. Error bars represent the standard deviation of the total VMSs concentrations.

F. Rocha et al. / Science of the Total Environment 650 (2019) 2364-2373

Acknowledgments

This work was financially supported by projects: (i) POCI-01-0145-FEDER-006939 (LEPABE UID/EQU/00511/2013), funded by the European Regional Development Fund (ERDF), through COMPETE2020 -Programa Operacional Competitividade e Internacionalização (POCI) and by national funds, through FCT -Fundação para a Ciência e a Tecnologia; (ii) NORTE-01-0145-FEDER-000005 -LEPABE-2-ECO-INNO-VATION, supported by North Portugal Regional Operational Programme (NORTE 2020), under the Portugal 2020 Partnership Agreement, through the ERDF; (iii) Investigator FCT contract IF/01101/2014 (Nuno Ratola); (iv) Post-Doctoral grant SFRH/BPD/76974/2011 (Vera Homem); (v) Labex OT-Med (no. ANR-11-LABX-0061) -ECOMARE, funded by the 2371 F. Rocha et al. / Science of the Total Environment 650 (2019) 2364-2373

French Government "Investissements d'Avenir" (ANR) through the A*MIDEX project (no ANR-11-IDEX-0001-02). The authors wish to thank the people that helped in the sampling campaigns: Dr. Pedro Jiménez-Guerrero, Dr. Raquel Sousa, Daniela Capela, Ana Ratola, Idalete Ratola, and Melanie Ourgaud and Sandrine Ruitton for collecting the Posidonia samples in the Marseille area.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi. org/10.1016/j.scitotenv.2018.10.012.