Christos Papageorgakis
email: christos.papageorgakis@inria.fr

Sebastian Hitziger
email: sebastian.hitziger@gmx.de

Théodore Papadopoulo
email: theodore.papadopoulo@inria.fr

Dictionary Learning for multidimensional data

L'électro-encéphalographie (EEG) et la magnéto-encéphalographie mesurent les activités respectivement électriques et magnétiques dues au fonctionnement du cerveau à partir de capteurs (électrodes et magnétométres) placés à la surface du scalp. Les signaux EEG ou MEG (conjointement M/EEG) mesurent la même activité biologique: celle des neurones pyramidaux situés dans le cortex. Ces signaux sont fondamentalement vectoriels (à chaque instant on dispose d'un vecteur de mesures). Ils sont, par contre, trés bruités. Pour pallier à cela, les mesures sont traditionnellement acquises de fac ¸on répétée (plusieurs "essais") puis moyennées. Ce procédé n'est pas optimal à cause de la variabilité des signaux d'un essai à l'autre. La méthode "jitter-adaptive dictionary learning" (JADL) [1] permit découverte via les données des composantes prototypiques qui se répétent à travers les essais. Cette méthode est, à l'heure actuelle, limitée à un seul signal et n'exploite pas la dimension vectorielle des signaux mesurés. Ce papier propose donc une adaptation vectorielle de la méthode JADL. Cette nouvelle méthode est testée sur des données synthétiques comme sur des données MEG réelles. Les résultats obtenus semblent prometteurs et indiquent que de meilleures performances peuvent être obtenues par rapport à la méthode mono-dimensionnelle.

Introduction

Biological signals show important variability and diversity. Furthermore, they are often very noisy. This is particularly true of magnetic or electrical signals such as those obtained from magneto-or electro-encephalography (M/EEG). Both MEG and EEG have the same biological origin, the activity generated by displacement of charges of billions of synchronously active cells existing within the brain, called pyramidal neurons. The sets of synchronously active neurons, generating the magnetic fields and electric potentials recorded by M/EEG measures, are called generators or sources of the brain activity.

While the source activity is spread spontaneously within the head, the electric potentials and magnetic fields that reach the surface of the head, are the result of the linear mixture of the individual sources' activities. The propagations of the source activity is instantaneous, leading to M/EEG measurements that reflect the sources synchronously, in means of well-aligned features in the recorded data. The mixture of the sources in the recorded data is explained by the following matrix expression:

M = GS , (1)
where M ∈ R C×T is the measurement matrix either MEG or EEG, G ∈ R C×S is the lead-field matrix (or gain matrix), S ∈ R S×T is the sources matrix. C, S and T are the numbers of channels, sources and time samples respectively. Practically, the lead field matrix is a linear operator that maps source activations, to the estimated M/EEG measurements at sensors locations.

The M/EEG measurements are very noisy and inherently multi-dimensional, i.e. provide a vector of measurements at each single time instant. The components of each vector correspond to M/EEG channels. Traditionally, researchers, acquire the M/EEG measurements over multiple repetitions, called trials, and average them to improve the signal to noise ratio. But this degrades the shapes and timings of the activities and hides their inherent variability. To cope with this problem, various methods have been developed such as the differentially Variable Component Analysis method (dVCA) [START_REF] Knuth | Differentially variable component analysis: Identifying multiple evoked components using trial-to-trial variability[END_REF], that relies on trial-to-trial variability in response amplitude and latency to identify multiple components from multi-channel recordings. Other methods are the Multichannel matching pursuit (MMP) [START_REF] Durka | Multi-channel matching pursuit and EEG inverse solutions[END_REF] and the Consensus Matching Pursuit (CMP) [START_REF] Bénar | Consensus matching pursuit for multi-trial EEG signals[END_REF] which aim at extracting signal patterns from raw (i.e. non averaged) signals while still accounting for repetitions (events of interest over all the trials). However, these methods need a previously defined dictionary.

To remove this constraint, the JADL method [START_REF] Hitziger | Jitter-adaptive dictionary learningapplication to multi-trial neuroelectric signals[END_REF] has been developed. It is a data-based method, that learns a dictionary from a set of example signals. This method has the advantage of better modelling prototypical brain activity, and also allows some variability (time jitter). Yet, JADL method is currently limited to a single channel, which restricts its capacity with very noisy data such as M/EEG.

The goal of this work is to extend the jitter adaptive dictionary learning method to handle multi-dimensional measurements such as M/EEG and to study the improvements this brings to the detection of some brain activity.

Dictionary learning and the JADL model

The dictionary learning problem generally aims at decomposing a given set of signals as a weighted sum of basic elements, called atoms. The method finds the atom shapes and weights assuming that the signal is represented as a sparse combination of atoms. The set of the decomposed basic elements (atoms) is the learned dictionary over the given signals, while the coefficients can be used to reconstruct the given set of signals using the learned dictionary and weights.

Jitter-adaptive dictionary learning (JADL) [START_REF] Hitziger | Jitter-adaptive dictionary learningapplication to multi-trial neuroelectric signals[END_REF] is a dictionary learning framework that is designed to compensate for variations in latency and phase of atoms. The JADL model supposes that atoms present in a signal can suffer from unknown time delays, which will be referred to as jitter. Atoms learned by JADL are defined on the entire signal domain and are supposed, but not restricted, to shift only up to a small fraction of the signal length and adapt their positions across trials. This is typically the case of multi-trial M/EEG signals. It is expected that any independent source in the measurements is learned by an atom.

The model hypothesizes that multiple recordings {x j } M j=1 of one electrode (a line of matrix M in Eq. 1), can be generated by a dictionary D = {d i } K i=1 with few atoms K in the following way: Given a finite set of time shift operations ∆, for every j there exist coefficients a ij ∈ R and shift operators δ ij ∈ ∆, such that

x j = K i=1 a ij δ ij (d i ) . (2)
One also assumes that ∆ contains only small shifts relative to the size of the time window. The JADL problem can be formulated as:

min di,aij ,δij M j=1 1 2 x j - K i=1 a ij δ ij (d i ) 2 2 + λ a j 1 , (3)
s.t. d i 2 = 1, δ ij ∈ ∆, i = 1, . . . , K, j = 1, . . . , M. (4)
The algorithm solving Eq. 3, is based on an implementation in [START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF] for common dictionary learning adopted to deal with jitters [START_REF] Hitziger | Jitter-adaptive dictionary learningapplication to multi-trial neuroelectric signals[END_REF], which iteratively alternates between (i) sparse coding (finding the coefficients {a ij } and the jitters {δ ij }) and (ii) dictionary update (finding the shapes {d i }).

(i) Sparse coding This part is solved by rewriting the problem into a form similar to the Lasso problem, which allows to solve it using a modification of least angle regression (LARS) [START_REF] Efron | Least angle regression[END_REF]. First, the "unrolled" version of the dictionary has to be defined. An "unrolled" version of the dictionary contains all allowed shifts of all its atoms; is given by D s = {δ(d) : d ∈ D, δ ∈ ∆}. It can be represented as a matrix D s of dimension T ×KL, where L = |∆| is the number of allowed shifts. The problem is then written as follows:

a s j ← argmin 1 2 x j -D s a s j 2 2 + λ a s j 1 , (5)
s.t. a s,i j 0 ≤ 1, i = 1, . . . , K. (6)
where a s j is the corresponding vector of coefficients for x j . The constraint ensures that once an atom of D s is chosen, all its shifts are forbidden. This choice selects the proper shift as it is encoded in D s .

(ii) Dictionary update Block coordinate descent is used to iteratively solve the constrained minimization problem

d k = argmin d k M j=1 1 2 x j - K i=1 a ij δ ij (d i ) 2 2
, s.t. d k 2 = 1 (7) for each atom d k . This can be solved in two steps, the solution of the unconstrained problem followed by normalization. This is summarized by:

d k = M j=1 a kj δ -1 kj x j - K i=1,i =k a ij δ ij (d i ) , (8)
d k = d k d k 2 . ( 9
)
δ -1 is the opposite shift of δ.

Our modified model

A simple way to extend the usage of the JADL implementation would be, to reshape the input multi-dimensional data to a shape compatible with the JADL implementation. Such a data transformation could be achieved by simply stacking the recordings from the different channels as trials or multiplexing the signals from different channels at the same time point, leading to one mixed signal per trial. As a consequence, in the first case (stacking), the dictionary would be learned over measurements that are assumed to have the same waveforms but can have different jitters and coefficients over the different channel, while in the second case (multiplexing), the dictionary would be learned over measurements that are assumed to have the same jitters, but can have different waveforms and coefficients over the different channel measurements. Those techniques would not be optimal as the multi-dimensional signals should share the same waveform and jitters, but have different coefficients over the channels.

To better account the nature of the M/EEG signals, we developed a novel model that learns a single dictionary over multidimensional recording that have the same waveforms and jitters, but different coefficients over the channels. Significant modifications are applied to the original JADL framework, especially in the (i) sparse coding step in order to handle multidimensional data.

The least angle regression algorithm (LARS) used for solving (i) is modified as:

1. Atom Selection: The best shifted versions of the atoms contained in the extended dictionary D s are selected, over all the channels, leading to a compressed dictionary D. The selection of the atoms is performed using the following criterion:

d s j = argmax d s j ∈D s C c=1 s c • d s j , (10)
where s c is the signal of channel c and d s j is the j-th atom of the extended dictionary D s . 2. Standard LARS sparse coding over the channels for the current atom set: During this step the multi-dimensional coefficients a ijc are computed using the compressed dictionary D selected at step 1 and the multi-channel signals for the given trial. The (ii) dictionary update problem is also slightly modified to treat the measurements corresponding to the different channels as additional trial. The dictionary update problem is then written similarly to Eq. (8) as:

d k = M j=1 C c=1 a kjc δ -1 kj x jc - K i=1,i =k a ijc δ ij (d i ) , (11)
with δ -1 the opposite shift of δ and the normalization of Eq. ( 9).

In the implementation of our model, the initial dictionary contains atoms generated by random values independent from the signals. As a consequence, an atom learned by the dictionary update process can appears in any latency l in the dictionary. In order to be able to account for all the allowed latencies of the window ∆, a centering of the window should be applied, to realign ∆ with respect to the latency l.

Results on lead field synthetic data

We created synthetic measurements using a dictionary of K = 3 synthetic atoms (a spike, and two oscillatory signals, see Fig. 1). We then selected K places as active brain locations (each place extends over 3 source points). Each location is associated with a specific atom of the dictionary. Each source point in a location receives a signal generated by introducing a random jitter (shift) to the atom, drawn from the set ∆ of contiguous allowed shifts (a window of [-51, 51] shifts). Therefore, 9 source signals are generated for the K groups of 3 sources. These signals are then combined with a leadfield matrix G computed from real EEG measurements [START_REF] Henson | Effective Connectivity between OFA and FFA during face perception: DCM of evoked MEG, EEG and fMRI[END_REF] as in Eq. ( 1).

Performing the above procedure for several trials corresponding to new random jitters to the dictionary of K = 3 synthetic atoms, leads to the multidimensional M/EEG measurements of size (T × M × C). The generated data contain measurements from C = 6 channels, M = 200 trials and T = 515 time samples. Additional data are also generated with various amplitudes of white Gaussian noise.

A comparison between the original and our multi-dimensional JADL model

In order to assess on the developed model's performance (multi-channel approach) and the improvements that it brings compared to the original JADL framework (single-channel approach), a comparison between the two implementations is made. Both algorithms are executed with the same signals, initial random dictionary and latency parameters (the same range of shifts that was used to generate the M/EEG measurements).

The multi-channel algorithm is executed using all the channels from the input data, while the single-channel algorithm is executed several times, each time using a different channel. Note that the results using the single-channel algorithm depend on how the atoms are represented in that channel.

A goodness of fit metric, is used to evaluate the quality of the learned dictionary computing the correlation between the generated and reconstructed atoms taking into account possible shifts of the atoms.

The results of our multi-channel algorithm (Fig. 1), look promising as the learned dictionary fits very well to the one used to generated the synthetic data. As the input signals are generated by introducing random jitters to the generated dictionary's atoms, the learned atoms can suffer by small jitters depending on the distribution of the picked jitters.

The results of our multi-channel approach look similar to the one obtained by the single-channel algorithm (Fig. 2) when the best channel is used, but when a medium or the worse channel is used, the results become worst and there are cases where the algorithm is unable to recover correctly all the atoms of the dictionary used to generate the signals. In addition, the goodness of fitness metric, showed a small but superior performance for the multi-channel approach giving the coefficients vector of 0.995, 0.996 and 0.995 instead of 0.992, 0.977 and 0.964 for the single-channel approach using the best channel and 0.939, 0.512, 0.512 using the worst channel. In general, finding the best channels is 1) source dependent and 2) highly non-trivial as it depends on both actual sources and leadfields. The robustness of our algorithm was also tested on the synthetic data contaminated by several noise levels. The algorithm was able to recover correctly the shape of all the atoms even with an SNR of 0.001, whereas it failed to recover all the atoms with SNR of 0.0002 and smaller values.

Results on real data

The performance of the multi-dimensional approach is tested using real MEG and EEG data of 200 channels, 63 trials and 541 time samples. An input parameter of 103 latencies has been provided to the algorithm. Note that, with real data, there is no groundtruth to compare to the obtained results. Most of the input parameters (the number of atoms and the range of jitters) are tuned by "trial and error".

The learned atoms by the multi-dimensional approach (Fig. 3) appear less noisy compared to the single-channel approach, with waveforms that seem to reveal more information for the underling brain activity (last row of atoms in Fig. 3).

Conclusions

The results obtained using the proposed model, look promising, showing superior performance compared to the original single-channel JADL framework. In particular, the learned atoms appear less noisy.

As the multi-dimensional JADL approach uses all the channels from the input data, it relies on more information. There is also no need for a prior selection of the "best" channel, which gives an additional advantage to the multi-channel approach, compared to the original JADL implementation.

Figure 1 -

 1 Figure 1 -The generated (blue) and learned (red) dictonary using our model with no noise (left). The learned dictionary on contaminated signals by noise of SN R : 0.021 (right).

Figure 2 -

 2 Figure 2 -The learned dictionaries by the single-channel method: using the best (left) and the worst (right) channel. Wrong recovered components are marked by the red ellipses.

Figure 3 -

 3 Figure 3 -The single-channel JADL method (left) and the proposed multi-channel method (right).

Table 1 -

 1 Robustness to various noise levels

	SN R	SN R dB Atom1 Atom2 Atom3
	0.804	-0.944	0.998	0.999	0.997
	0.021	-16.700	0.993	0.973	0.983
	0.001	-29.240	0.954	0.821	0.892
	0.0002	-36.107	0.826	0.585	0.462


domain: info scco

text:
Enrico Santus
email: enricosantus@sutd.edu.sg

Emmanuele Chersoni
email: emmanuelechersoni@gmail.com

Alessandro Lenci
email: alessandro.lenci@unipi.itphilippe.blache@univ-amu.fr

Philippe Blache

Measuring Thematic Fit with Distributional Feature Overlap

In this paper, we introduce a new distributional method for modeling predicateargument thematic fit judgments. We use a syntax-based DSM to build a prototypical representation of verb-specific roles: for every verb, we extract the most salient second order contexts for each of its roles (i.e. the most salient dimensions of typical role fillers), and then we compute thematic fit as a weighted overlap between the top features of candidate fillers and role prototypes. Our experiments show that our method consistently outperforms a baseline re-implementing a state-of-theart system, and achieves better or comparable results to those reported in the literature for the other unsupervised systems. Moreover, it provides an explicit representation of the features characterizing verbspecific semantic roles.

Introduction

Several psycholinguistic studies in the last two decades have brought extensive evidence that humans activate a rich array of event knowledge during sentence processing: verbs (e.g. arrest) activate expectations about their typical arguments (e.g. cop, thief ) [START_REF] Mcrae | Modeling the Influence of Thematic Fit (and Other Constraints) in Online Sentence Comprehension[END_REF][START_REF] Altmann | Incremental Interpretation at Verbs: Restricting the Domain of Subsequent Reference[END_REF][START_REF] Ferretti | Integrating Verbs, Situation Schemas, and Thematic Role Concepts[END_REF][START_REF] Mcrae | A basis for generating expectancies for verbs from nouns[END_REF][START_REF] Hare | Activating Event Knowledge[END_REF][START_REF] Matsuki | Event-based Plausibility Immediately Influences On-line Language Comprehension[END_REF], and nouns activate other nouns typically co-occurring in the same events [START_REF] Kamide | The Time-course of Prediction in Incremental Sentence Processing: Evidence from Anticipatory Eye Movements[END_REF][START_REF] Bicknell | Effects of Event Knowledge in Processing Verbal Arguments[END_REF]. Subjects are able to determine the plausibility of a noun for a given argument role and quickly use this knowledge to anticipate upcoming linguistic input [START_REF] Mcrae | People Use their Knowledge of Common Events to Understand Language, and Do So as Quickly as Possible[END_REF]. This phenomenon is referred to in the literature as thematic fit. Thematic fit estimation has been extensively used in sentence comprehension studies on constraint-based models, mainly as a predictor variable allowing to disambiguate between possible structural analyses. 1 More in general, thematic fit is considered as a key factor in a variety of studies concerned with structural ambiguity [START_REF] Vandekerckhove | A Robust and Extensible Exemplar-Based Model of Thematic Fit[END_REF].

Starting from the work of [START_REF] Erk | A flexible, corpus-driven model of regular and inverse selectional preferences[END_REF], several distributional semantic methods have been proposed to compute the extent to which nouns fulfill the requirements of verb-specific thematic roles, and their performances have been evaluated against human-generated judgments [START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF][START_REF] Lenci | Composing and Updating Verb Argument Expectations: A Distributional Semantic Model[END_REF][START_REF] Sayeed | Combining Unsupervised Syntactic and Semantic Models of Thematic Fit[END_REF][START_REF] Sayeed | An Exploration of Semantic Features in an Unsupervised Thematic Fit Evaluation Framework[END_REF][START_REF] Sayeed | Thematic Fit Evaluation: an Aspect of Selectional Preferences[END_REF]Greenberg et al., 2015a,b). Most research on thematic fit estimation has focused on count-based vector representations (as distinguished from prediction-based vectors).2 Indeed, in their comparison between highdimensional explicit vectors and low-dimensional neural embeddings, [START_REF] Baroni | Dont Count, Predict! A Systematic Comparison of Context-counting vs. Context-predicting Semantic Vectors[END_REF] found that thematic fit estimation is the only benchmark on which prediction models are lagging behind stateof-the-art performance. This is consistent with [START_REF] Sayeed | Thematic Fit Evaluation: an Aspect of Selectional Preferences[END_REF]'s observation that "thematic fit modeling is particularly sensitive to linguistic detail and interpretability of the vector space".

The present work sets itself among the unsupervised approaches to thematic fit estimation. By relying on explicit and interpretable count-based vector representations, we propose a simple, cognitively-inspired, and efficient thematic fit model using information extracted from dependency-parsed corpora. The key features of our proposal are a) prototypical representations of verb-specific thematic roles, based on feature weighting and filtering of second order contexts (i.e. contexts that are salient for many of the typical fillers of a given verb-specific thematic role), and b) a similarity measure which computes the Weighted Overlap (W O) between prototypes and candidate fillers. 32 Related Work [START_REF] Erk | A flexible, corpus-driven model of regular and inverse selectional preferences[END_REF] were, at the best of our knowledge, the first authors to measure the correlation between human-elicited thematic fit ratings and the scores assigned by a syntax-based Distributional Semantic Model (DSM). More specifically, their gold standard consisted of the human judgments collected by [START_REF] Mcrae | Modeling the Influence of Thematic Fit (and Other Constraints) in Online Sentence Comprehension[END_REF] and [START_REF] Padó | The Integration of Syntax and Semantic Plausibility in a Wide-coverage Model of Human Sentence Processing[END_REF]. The plausibility of each verb-filler pair was computed as the similarity between new candidate nouns and previously attested exemplars for each specific verb-role pairing (as already proposed in [START_REF] Erk | A Simple, Similarity-based Model for Selectional Preferences[END_REF]). [START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF] evaluated their Distributional Memory (henceforth DM)4 framework on the same datasets, adopting an approach to the task that has become dominant in the literature: for each verb role, they built a prototype vector by averaging the dependency-based vectors of its most typical fillers. The higher the similarity of a noun with a role prototype, the higher its plausibility as a filler for that role. [START_REF] Lenci | Composing and Updating Verb Argument Expectations: A Distributional Semantic Model[END_REF] has later extended the model to account for the dynamic update of the expectations on an argument, depending on how another role is filled. By using the same DM tensor, this study tested an additive and a multiplicative model [START_REF] Mitchell | Composition in Distributional Models of Semantics[END_REF] to compose and update the expectations on the patient filler of the subject-verb-object triples of the Bicknell dataset [START_REF] Bicknell | Effects of Event Knowledge in Processing Verbal Arguments[END_REF].

The thematic fit models proposed by [START_REF] Sayeed | Combining Unsupervised Syntactic and Semantic Models of Thematic Fit[END_REF] and [START_REF] Sayeed | An Exploration of Semantic Features in an Unsupervised Thematic Fit Evaluation Framework[END_REF] are similar to Baroni and Lenci's, but their DSMs were built by using the roles assigned by the SENNA semantic role labeler [START_REF] Collobert | Natural Language Processing (Almost) from Scratch[END_REF] to define the feature space. These authors argued that the prototype-based method with dependencies works well when applied to the agent and to the patient role (which are almost always syntactically realized as subjects and objects), but that it might be problematic to apply it to different roles, such as instruments and locations, as the construction of the prototype would have to rely on prepositional complements as typical fillers, and the meaning of prepositions can be ambiguous. Comparing their results with [START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF], the authors showed that their system outperforms the syntax-based model DepDM and almost matches the scores of the best performing TypeDM, which uses hand-crafted rules. Moreover, they were the first to evaluate thematic role plausibility for roles other than agent and patient, as they computed the scores also for the instruments and for the locations of the Ferretti datasets [START_REF] Ferretti | Integrating Verbs, Situation Schemas, and Thematic Role Concepts[END_REF]. Greenberg et al. (2015a,b) further developed the TypeDM and the role-based models, investigating the effects of verb polysemy on human thematic fit judgments and introducing a hierarchical agglomerative clustering algorithm into the prototype creation process. Their goal was to cluster together typical fillers into multiple prototypes, corresponding to different verb senses, and their results showed constant improvements of the performance of the DM-based model.

Finally, [START_REF] Tilk | Event Participant Modelling with Neural Networks[END_REF] presented two neural network architectures for generating probability distributions over selectional preferences for each thematic role. Their models took advantage of supervised training on two role-labeled corpora to optimize the distributional representation for thematic fit modeling, and managed to obtain significant improvements over the other systems on almost all the evaluation datasets. They also evaluated their model on the task of composing and updating verb argument expectations, obtaining a performance comparable to [START_REF] Lenci | Composing and Updating Verb Argument Expectations: A Distributional Semantic Model[END_REF].

Methodology

As pointed out by [START_REF] Sayeed | Thematic Fit Evaluation: an Aspect of Selectional Preferences[END_REF], most works on unsupervised thematic fit estimation vary in the method adopted for constructing the prototypes. The semantic role prototype is usually a vector, obtained by averaging the most typical fillers, and plausibility of new fillers depends on their similarity to the prototype, assessed by means of vector cosine (the standard similarity measure for DSMs; see [START_REF] Peter | From Frequency to Meaning: Vector Space Models of Semantics[END_REF]).

Its merits notwithstanding, we argue that this method is not optimal for characterizing roles. Distributional vectors are typically built as out-of-context representations, and they conflate different senses. By building the prototype as the centroid of a cluster of vectors and measuring then the thematic fit with vector cosine, the plausibility score is inevitably affected by many contexts that are irrelevant for the specific verb-argument combination.5 This is likely to be one of the main reasons behind the difficulties of modeling roles other than agent and patient with syntax-based DSMs. We claim that improving the prototype representation might lead to a better characterization of thematic roles, and to a better treatment of polysemy.

When a verb and an argument are composed, humans are intuitively able to select only the part of the potential meaning of the words that is relevant for the concept being expressed (e.g. in The player hit the ball, humans would certainly exclude from the meaning of ball semantic dimensions that are strictly related to its dancing sense). In other words, not all the features of the semantic representations are active, and the composition process makes some features more 'prominent', while moving others to the background. 6Although we are not aware of experimental works specifically dedicated to verb-argument composition, a similar idea has been supported in studies on conceptual combinations [START_REF] Hampton | Conceptual Combination[END_REF][START_REF] Hampton | Typicality, Graded Membership, and Vagueness[END_REF]: when a head and a modifier are combined, their interaction affects the saliency of the features in the original concepts. For example, in racing car, the most salient properties would be those related to SPEED, whereas in family car SPACE properties would probably be more prominent. [START_REF] Yeh | The Situated Nature of Concepts[END_REF] used a property priming experiment to show how the concept features activated during language comprehension vary across the background situations described by the sentence they occur in. When concepts are combined in a sentence, the features that are relevant for the specific combination are activated and are then easier to verify for human subjects.

The same could be true for linguisticallyderived properties of lexical meaning: [START_REF] Simmons | fMRI Evidence for Word Association and Situated Simulation in Conceptual Processing[END_REF] brought neuroimaging evidence of the early activation of word association areas during property generation tasks, and [START_REF] Santos | Property Generation Reflects Word Association and Situated Simulation[END_REF] showed that word associates are often among the properties generated for a given concept. Such findings suggest that, while we combine concepts, both embodied simulations and word distributions influence property salience [START_REF] Barsalou | Language and Simulation in Conceptual Processing. Symbols, embodiment, and meaning[END_REF].

Our model makes the following assumptions:

• the composition between a verb role representation and an argument shares the same cognitive mechanism underlying conceptual combinations;

• at least part of semantic representations is derived from, and/or mirrored in, linguistic data. 7 Consistently, the process of selecting the relevant features of the concepts being composed corresponds to modify the salience of the dimensions of distributional vectors;

• thematic fit computation is carried out on the basis of the activation and selection of salient features of a verb thematic role prototype and of the candidate argument filler vectors.

We rely on syntax-based DSMs, using dependency relations to approximate verb-specific roles and to identify their most typical fillers: for agents/patients, we extract the most frequent subjects/objects, for instruments we use the prepositional complements introduced by with, and for locations those introduced by either on, at or in.

Assuming that the linguistic features of distributional vectors correspond to the properties of conceptual composition processes, a candidate filler can be represented as a sorted distributional vector of the filler term, in which the most salient contexts occupy the top positions. Similarly, the abstract representation of a verb-specific role is a sorted prototype-vector, whose features derive from the sum of the most typical filler vectors for that verb-specific role.

Differently from Baroni and Lenci, the core and novel aspect of our proposal, described in the following subsections, is that we do not simply measure the correlation between all the features of candidate and prototype vectors (as vector cosine would do on unsorted vectors), but rather we rank and filter the features, computing the weighted overlap with a rank-based similarity measure inspired by AP Syn, a recent proposal by Santus et al. (2016a,b,c) which has shown interesting results in synonymy detection and similarity estimation. As we will show in the next sections, the new metric assigns high scores to candidate fillers sharing many salient contexts with the verb-specific role prototype.

Typical Fillers

The first step of our method consists in identifying the typical fillers of a verb-specific role. Following [START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF], we weighted the raw cooccurrences between verbs, syntactic relations and fillers in the TypeDM tensor of DM with Positive Local Mutual Information (PLMI; [START_REF] Evert | The Statistics of Word Cooccurrences: Word Pairs and Collocations[END_REF]).

Given the co-occurrence count O vrf of the verb v, a syntactic relation r and the filler f , we computed the expected count E vrf under the assumption of statistical independence:

P LM I(v, r, f ) = log O v,r,f E v,r,f * O v,r,f (1)
From the ranked list of (v,r,f) tuples, for each slot, we selected as typical fillers the top k lexemes with the highest PLMI scores (see examples in Table 1, Typical Fillers column). In our experiments, we report results for k = {10, 30, 50}.

Role Prototype Vectors

To represent the typical fillers, the candidate fillers and the verb-specific role prototypes (which are obtained by summing their typical filler vectors), we built a syntax-based DSM. This includes relation:word contexts, like sbj:dog, obj:apple, etc.. Contexts were weighted with Positive Pointwise Mutual Information (PPMI; [START_REF] Church | Word Association Norms, Mutual Information, and Lexicography[END_REF], [START_REF] Bullinaria | Extracting Semantic Representations from Word Cooccurrence Statistics: Stop-lists, Stemming, and SVD[END_REF], [START_REF] Levy | Improving Distributional Similarity with Lessons Learned from Word Embeddings[END_REF]). Given a context c and a word w, the PPMI is defined as follows:

P P M I(w, c) = max(P M I(w, c), 0)
(2)

P M I(w, c) = log P (w, c) P (w)P (c) = log |w, c|D |w||c| (3)
where w is the target word, c is the given context, P(w,c) is the probability of co-occurrence, and D is the collection of observed word-context pairs.8

The context c of the prototype vector P representing a thematic role has a value corresponding to the sum of the values of c for each of the k typical fillers used to build P . The contexts of P are then sorted according to their weight. Desirably, the highest-ranking contexts for a role prototype will be those that are more strongly associated with many of its typical fillers. Such second order contexts correspond to the most salient features of the verb-specific thematic role, as they are salient for many role fillers (some examples are reported in Table 1, Top Second Order Contexts column).

In summary, we built centroid vectors for our verb-specific thematic roles by means of second order contexts, which are first order dependencybased contexts of the most typical fillers of a verbspecific role. Since we are interested only in the most salient contexts, we ranked the centroid contexts according to their PPMI score, and we took the resulting rank as a distributional characterization of the thematic roles.

Filtering the Contexts

Filtering the prototype dimensions according to syntactic criteria might be useful to improve our role representations. It is, indeed, reasonable to hypothesize that predicates co-occurring with the typical patients of a verb are more relevant for the characterization of its patient role than -let's sayprepositional complements, as they correspond to other actions that are typically performed on the same patients.

Imagine that apple, pizza, cake etc. are among the most salient fillers for the OBJ slot of to eat, and that OBJ-1:slice-v, OBJ-1:devour-v, SBJ:kidn, INSTRUMENT:fork-n, LOCATION:table-n are some of the most salient contexts of the prototype. 9 Things that are typically sliced and/or devoured are more likely to be good fillers for the patient role to eat than things that are simply located on a table or that are patients of actions performed by kids. To test this hypothesis, we evaluated the performance of the system in three different settings, each of which selecting: formance we will not discuss it further. Santus et al. (2016c) previously showed that their rank-based measure performs worse on PLMI-weighted vectors, as they are biased towards frequent contexts. 9 Our DSM also makes use of inverse syntactic dependencies: target SYN-1 context means that target is linked to context by the dependency relation SYN (e.g. meal OBJ-1 devour means that meal is OBJ of devour).

Typical Fillers

Top Second Order Contexts subject: cure-v treatment-n, drug-n, resin-n, doctor-n, surgery-n, medicinen, therapy-n, antibiotic-n, dose-n, operation-n, water-n... obj-1:prescribe-v, sbj-1:prescribe-v, sbj-1:prevent-v, sbj-1:contraindicate-v, [...] object: abandon-v plan-n, idea-n, project-n, attempt-n, position-n, principle-n, policy-n, ship-n, practice-n, hope-n, fort-n, claim-n... obj-1:revive-v, obj-1:defend-v, obj-1:renouncev, obj-1:espouse-v, sbj-1:entail-v... instrument: eat-v bread-n, hand-n, spoon-n, sauce-n, relish-n, fork-n, fingern, meal-n, knife-n, friend-n, chopstick-n, rice-n, food-n... obj-1:flavour-v, obj-1:taste-v, obj-1:spoon-v, sbj-1:taste-v, obj-1:slice-v in:bowl-n... location: walk-v in:direction-n, at:time, at:pace-n, on:path-n, at:night, on:side-n, at:end, on:beach-n, on:leg, in:area, in:way... obj-1:wander-v, obj-1:stroll-v, obj-1:litter-v, obj-1:sweep-v, sbj-1:slope-v, obj-1:tread-v...

Table 1: Typical fillers and top second order contexts for several verb-specific roles.

• only predicates in a subject/object relation (SO setting);

• only prepositional complements (PREP setting);

• both of them (ALL setting).

Computing the Thematic Fit

Our hypothesis is that fillers whose salienceranked vector has a large overlap with the prototype representation should have a high thematic fit. Such overlap should take into account not only the number of shared features, but also their respective ranks in the salience-ranked vectors.

When the prototype has been computed and the candidate filler vector has also been sorted, we can measure the Weighted Overlap by adapting AP Syn (Santus et al., 2016a,b,c) to our needs:

W O(wx, wy) = ∀f (x [1:N ] ∩y [1:N ] ) 1 avg(rx(f ), ry(f )) (4)
where for every feature f in the intersection between the top N features of the sorted vectors x, x [1:N ] , and y, y [1:N ] , we sum 1 divided by the average rank of the shared feature in x and y, r x (f ) and r y (f ) (N is a tunable parameter). This measure assigns the maximum score to vectors sharing exactly the same dimensions, in the same salience ranking. The lower the rank of a shared context in the sorted vector, the smaller its contribution to the thematic fit score. If the feature set intersection is empty, the score will be 0.

Differently from cosine similarity, which conflates multiple senses, measuring the Weighted Overlap between prototype and candidate filler can improve the estimation of the thematic fit by favoring the appropriate word senses: for example, for a verb-argument pair like embracev-communism-n, communism-n is likely to intersect and to increase the saliency (through the average rank) only of the second-order features of embrace-v referring to its abstract sense. 2 for the coverage of each system for the datasets).

Metrics. Performance is evaluated as the Spearman correlation between the scores of the systems and the human plausibility judgments.

Fillers. In order to make our results more comparable with previous studies, the typical fillers for each verb role were extracted from the TypeDM tensor of the Distributional Memory framework (see Section 3.1). 10 Those were the same fillers used by [START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF] and Greenberg et al. (2015b). DSM. Distributional information is derived from the concatenation of two corpora: the British National Corpus [START_REF] Leech | 100 Million Words of English: the British National Corpus (BNC)[END_REF] and Ukwac [START_REF] Baroni | The WaCky wide web: a Collection of Very Large Linguistically Processed Web-crawled Corpora[END_REF]. Both were parsed with the Maltparser [START_REF] Nivre | Maltparser: A Language-independent System for Data-driven Dependency Parsing[END_REF]. From this concatenation, we built a dependency-based DSMs, weighted with PPMI, containing 20,145 targets (i.e. nouns and verbs with frequency above 1000) and 94,860 contexts. The syntactic relations taken into account were: sbj, sbj-1, obj, obj-1, at-1, in-1, on-1, with-1. Settings. To prove our hypotheses and verify the consistency of the system, we tested a large range of settings, varying: • the number of fillers used to build the prototype, with the most typical values in the literature ranging between 10 and 50. We report the results for 10, 30 and 50 fillers

• the types of the dependency relations used for calculating the overlap: we report results for the SO, PREP and ALL settings;

• the value of N , that is the number of top contexts that we take into account when computing the weighted overlap. Table 3 reports the scores for our best setting, while the performances for other values of N are discussed in the Section 5.

Baseline and State of the Art. As a baseline, we use the thematic fit model by [START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF], with no ranking of the features of the prototypes and with vector cosine as a similarity metric. 11 Results are reported for 10, 30 and 50 fillers. For reference, we also report the results of state-of-the-art models, both the unsupervised [START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF][START_REF] Sayeed | Combining Unsupervised Syntactic and Semantic Models of Thematic Fit[END_REF]Greenberg et al., 2015b) and the supervised ones [START_REF] Tilk | Event Participant Modelling with Neural Networks[END_REF].

Results

Table 3 describes the performance of the best setting (weight: PPMI; N=2000). In the first three rows, the table shows the scores obtained by our system varying the types of dependency contexts (i.e. ALL, SO, PREP) and the number of fillers considered for the prototype (i.e. 10, 30 and 50). The other rows respectively show i) the scores obtained by calculating the vector cosine between the role prototype vector (i.e. the vector obtained by summing the most typical fillers, with no salience ranking of the dimensions) and the candidate filler vector and ii) the scores reported in the literature for the best unsupervised and supervised models. At a glance, our best scores always outperform the reimplementation of Baroni and Lenci, being mostly competitive with the state of the art models. More precisely, for agents and patients the performance is close to the reported scores for DM, when only predicates are used in the W O calculation, as hypothesized in Section 3.3. The neural network of Tilk and colleagues retains a significant advantage on our models only for the McRae dataset. Our system, however, shows a remarkable improvements on the Ferretti's datasets, and specifically on Ferretti-Instruments, when only complements are used (see Section 3.3), outperforming even the supervised and more complex model by [START_REF] Tilk | Event Participant Modelling with Neural Networks[END_REF], which has access to semantic roles information. Compared to the other unsupervised models, our system has a statistically significant advantage over [START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF] on the locations dataset and over [START_REF] Sayeed | Combining Unsupervised Syntactic and Semantic Models of Thematic Fit[END_REF] on the locations and on the instruments dataset (p < 0.05). 12At the best of our knowledge, the result for the 1). Unlike all the other datasets, instrument prototypes built with more fillers tend to be more noisy and therefore to pull down both the vector cosine and W O performance (this is partially true also for locations, where the performances -for cosine and W O with a lower number of contexts -drop with more than 30 fillers: see Figure 1). Systems based on semantic role labeling have an advantage in this sense, as they do not have to deal with prepositional ambiguity.

Our results show that, by weighting and filtering the features of the role prototype, dependency-based approaches can be successful in modeling roles other than agent and patient, eventually dealing also with the ambiguity of prepositional phrases.

Settings. Apart from the above-mentioned exceptions, the best scores are obtained building the prototypes with a higher number of fillers, typically with 50, and calculating the W O only with a syntactically-filtered set of contexts. More specifically, Padó and McRae benefit from the calculation of W O using only second order subject-object predicates (i.e. SO), while Ferretti-Instruments and Ferretti-Locations benefit from the exclusive use of prepositional complements (i.e. PREP). On the other hand, the opposite setting (e.g. SO for Ferretti-Instruments and Ferretti-Locations and PREP for Padó and McRae) leads to much lower scores, whereas the full vectors (i.e. ALL) tend to have a stable-but-not-excellent performances on all datasets.

As briefly mentioned above, in our experiments, we tested both PPMI and PLMI as weighting measures. Table 3 only reports PPMI scores because it performs more regularly than PLMI, whose behaviour is often unpredictable.

A parameter that has an impact on the performance of our system is the value of N , which is the number of second order contexts that are considered when calculating the W O. We have noticed that the performance of W O is directly related to the growth of N , and this can be noticed in Figure 1, where W O is plotted for the different values of N with every combination of dataset and number of fillers. For space reasons, the plot only contains the performance for the best type of second order contexts for each dataset (i.e. SO for Padó and McRae and COMP for Ferretti-Locations and Ferretti-Instruments). As it can be seen in Figure 1, the scores of W O tend to grow with the growth of N in all datasets. Interestingly, they are largely above the competitive baseline in most of the cases, the only exceptions being Padó (where a large N is necessary to outperform the baseline) and Ferretti-Locations with 10 fillers (prepositional ambiguity might have caused the introduction of noisy fillers among the top ones).

Agent & Patient. In order to further evaluate our system, we have split Padó and McRae datasets into agent and patient subsets. Figure 2 describes the performance of W O and vector cosine baseline while varying N and the number of fillers. The plot shows a clearly better performance of W O for the agent role (i.e. subject), especially when N is equal or over 1000 (note that the value of N has little impact in the agent subset of the McRae dataset). Such advantage, however, is reduced for the patient role (i.e. object). This is particularly interesting because we do not observe large drops in performance for the vector cosine between agent and patient role (except for Padó, k = 10). The drop is particularly noticeable in Padó, a dataset which has several non-constraining verbs (especially for the patient role: a similar observation was also made by [START_REF] Tilk | Event Participant Modelling with Neural Networks[END_REF]). As the constraints on the typical fillers of such verbs are very loose, we hypothesize that it is more difficult to find a set of salient features that are shared by many typical fillers. Therefore, estimations based on the whole vectors turn out to be more reliable. This can be confirmed by looking at the worst correlated words reported in Lexemes column, in Table 4.

Error Analysis

We performed an error analysis to verify -for the best settings of W O in each dataset -the correlation between vector cosine and W O scores (see Table 5), and the peculiarities of the entries with the strongest and the weakest correlation (see Table 4).

We found that W O and vector cosine always have a high correlation (i.e. above 0.80), with the highest correlations reported for McRae and Ferretti-Instruments. Looking at Table 4 we can also observe that:

• the average gold value of the 35 most (4.65) and least (4.56) correlated items does not substantially differ from the average gold value calculated on the full datasets (4.31), meaning that the distribution of likely and unlikely fillers among the best and worst correlated items is similar to the one in the datasets (i.e. no bias can be identified);

• both measures have difficulties on the same test items (probably because of loose semantic constraints), but report their best performances on different pairs (see Overlap and Lexemes columns);

• syntactically, vector cosine correlates better with objects, while W O is more balanced between objects and subjects, often showing a preference for the latter (see the distribution in Syntax column).

Conclusions

In this paper, we have introduced an unsupervised distributional method for modeling predicateargument thematic fit judgments which works purely on syntactic information. The method, inspired by cognitive and psycholinguistic findings, consists in: i) extracting and filtering the most salient second order contexts for each verb-specific role, i.e. the most salient semantic dimensions of typical verb-specific role fillers; and then ii) estimating the thematic fit as a weighted overlap between the top features of the candidate fillers and of the prototypes. Once tested on some popular datasets of thematic fit judgments, our method consistently outperforms a baseline re-implementing the thematic fit model of [START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF] and proves to be competitive with state of the art models. It even registered the best performance on the Ferretti-Instruments dataset and it is the second best on the Ferretti-Locations, which were known to be particularly hard to model for dependency-based approaches.

Our method is simple, economic and efficient, it works purely on syntactic dependencies (so it does not require a role-labeled corpus) and achieves good results even with no supervised training. Finally, it offers linguistically and cognitively grounded insights on the process of prototype creation and contextual feature salience, preparing the ground for further speculations and optimizations. For example, future work might aim at identifying strategies for tuning the parameter N to account for the different degrees of selectivity of each verb-specific role. Another possible extension would be the inclusion of a mechanism for updating the role prototypes depending on how the other roles are filled, which would be the key for a more realistic and dynamic model of thematic fit expectations [START_REF] Lenci | Composing and Updating Verb Argument Expectations: A Distributional Semantic Model[END_REF].

Figure 1 :

 1 Figure 1: Results for Padó, McRae and Ferretti, Instruments and Locations, with W O (respectively SO and PREP) computed on PPMI matrix, varying the number of fillers (i.e. 10, 30 and 50) and the value of N(i.e. 500, 1000, 1500 and 2000). A baseline reimplementing[START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF] -with 10, 30 and 50 fillers -is also reported in every test for comparison.

Figure 2 :

 2 Figure 2: Results for the agent and patient roles in Padó and McRae, with W O (SO) computed on PPMI matrix, varying the number of fillers (i.e. 10, 30 and 50) and the value of N(i.e. 500, 1000, 1500 and 2000). A baseline reimplementing[START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF] -with 10, 30 and 50 fillers -is also reported in every test for comparison.

Table 2 :

 2 Dataset coverage (%) for all systems.

	Data	Our system	BL2010	SD2014	G2015	T2016
	Padó	96	100	99	100	99
	McRae	100	95	96	95	96
	Instr.	100	93	94	93	94
	Loc.	96	99	100	99	100
	4 Experiments

Datasets. We tested our method on three popular datasets for thematic fit estimation, namely

[START_REF] Mcrae | Modeling the Influence of Thematic Fit (and Other Constraints) in Online Sentence Comprehension[END_REF][START_REF] Ferretti | Integrating Verbs, Situation Schemas, and Thematic Role Concepts[END_REF]

and

[START_REF] Padó | The Integration of Syntax and Semantic Plausibility in a Wide-coverage Model of Human Sentence Processing[END_REF]

. All the datasets contain human plausibility judgments for verb-role-filler triples. McRae and Padó include scores for agent and patient roles, whereas Ferretti includes instruments and locations (see Table

Table 3 :

 3 Results for Padó, McRae and Ferretti, Instruments and Locations, with W O computed on PPMI matrix, varying the number of fillers (i.e. 10, 30 and 50) and the types of dependency contexts (i.e. ALL, SO and PREP). The best results of our system are in bold. A baseline reimplementing Baroni and Lenci (2010) -with 10, 30 and 50 fillers -and state of the art results from previous literature are reported for comparison.

	Weight	N	# Fillers	Padó ALL SO PREP ALL SO PREP ALL SO Mcrae Ferretti -Instruments Ferretti -Locations PREP ALL SO PREP
			10	0.43 0.45	0.26	0.25 0.27	0.19	0.43 0.41	0.46	0.25 0.27	0.28
	PPMI 2000	30	0.47 0.49	0.33	0.26 0.28	0.22	0.42 0.41	0.50	0.28 0.31	0.37
			50	0.46 0.50	0.35	0.27 0.29	0.24	0.39 0.38	0.47	0.28 0.32	0.39
	Vector Cosine (Baseline)	10 30 50	0.43 0.47 0.48		0.25 0.26 0.26		0.42 0.41 0.38		0.29 0.32 0.31
						State of the Art
	Baroni and Lenci (2010)	0.53		0.33		0.36		0.23
	Sayeed and Demberg (2014)	0.56		0.27		0.28		0.13
	Greenberg et al. (2015)	0.53		0.36		0.42		0.29
	Tilk et al. (2016)	0.52		0.38		0.45		0.44

Table 4 :

 4 Average gold values, number of items listed for both metrics, and distribution of syntactic and lexical forms among the 35 best and worst correlated items for every measure in the given datasets.instruments is the best reported until now in the literature. This is particularly interesting because -as pointed out by Sayeed and Demberg (2014)instruments and locations are difficult to model for a dependency-based system, given the ambiguity of prepositional phrases (e.g. with does not only encode instruments, but it can also encode other roles, such as in I ate a pizza with Mark). We think this is the main reason behind the different trend observed for the Instruments datasets with respect to the number of the fillers (see Table3 and Figure

Table 5 :

 5 Correlation between W O and vector cosine in W O best settings for all datasets

	Dataset	Correlation
	McRae	0.88
	Padó	0.81
	Ferretti -Instruments	0.90
	Ferretti -Locations	0.83

For an overview on constraint-based models, seeMac- Donald and Seidenberg (2006).

 2 We adopt the terminology from[START_REF] Baroni | Dont Count, Predict! A Systematic Comparison of Context-counting vs. Context-predicting Semantic Vectors[END_REF].

Code: https://github.com/esantus/Thematic Fit

In this paper, we will make reference to two different models of DM: DepDM and TypeDM. DepDM counts the frequency of dependency links between words (e.g. read, obj, book), while TypeDM uses the variety of surface forms that express the link between words, rather than the link itself.

For an overview on the limitations of vector cosine, see:[START_REF] Li | Distance Weighted Cosine Similarity Measure for Text Classification[END_REF];[START_REF] Dinu | Improving Zero-shot Learning by Mitigating the Hubness Problem[END_REF];[START_REF] Schnabel | Evaluation Methods for Unsupervised Word Embeddings[END_REF];[START_REF] Faruqui | Problems With Evaluation of Word Embeddings Using Word Similarity Tasks[END_REF];Santus et al. (2016a).

An early proposal going in this direction is the predication theory by[START_REF] Kintsch | Predication[END_REF], which exploited Latent Semantic Analysis to select only the vector features that are appropriate for predicate-argument composition.

See also the so-called 'strong version' of the Distributional Hypothesis[START_REF] Miller | Contextual Correlates of Semantic Similarity[END_REF][START_REF] Lenci | Distributional Semantics in Linguistic and Cognitive Research[END_REF].

A variant of this DSM weighted with PLMI (which is simply the PPMI multiplied by the word-context frequency) was also built, but because of its lower and inconsistent per-

This baseline is equivalent to the approach of[START_REF] Baroni | Distributional Memory: A General Framework for Corpus-based Semantics[END_REF], except for the fact that it is applied on a standard dependency-based DSM and not on TypeDM, which combines dependency links and handcrafted lexico-syntactic patterns: see Section 2.

p-values computed with Fisher's r-to-z transformation.

Acknowledgments

We would like to thank the anonymous reviewers for their helpful suggestions.

This work has been carried out thanks to the support of the A*MIDEX grant (nANR-11-IDEX-0001-02) funded by the French Government "Investissements d'Avenir" program.