Laura Cruz
email: lcruzq@unsa.edu.pe

José Ochoa
email: jeochoa@ucsp.edu.pe

Mathieu Roche
email: mathieu.roche@cirad.fr

Pascal Poncelet
email: pascal.poncelet@lirmm.fr

Dictionary-Based Sentiment Analysis Applied to a Specific Domain

Keywords: Text Mining, Web Mining, Sentiment analysis

The web and social media have been growing exponentially in recent years. We now have access to documents bearing opinions expressed on a broad range of topics. This constitutes a rich resource for natural language processing tasks, particularly for sentiment analysis. Nevertheless, sentiment analysis is usually difficult because expressed sentiments are usually topic-oriented. In this paper, we propose to automatically construct a sentiment dictionary using relevant terms obtained from web pages for a specific domain. This dictionary is initially built by querying the web with a combination of opinion terms, as well as terms of the domain. In order to select only relevant terms we apply two measures AcroDefMI3 and TrueSkill. Experiments conducted on different domains highlight that our automatic approach performs better for specific cases.

Introduction

The web and social media have been growing exponentially in recent years, which constitutes a rich resource for sentiment analysis tasks. For instance, social networking sites enable users to express their thoughts and opinions about products [START_REF] Amine | Detecting opinions in tweets[END_REF] and companies are increasingly taking these opinions into account to make better decisions [START_REF] Marrese-Taylor | Identifying customer preferences about tourism products using an aspectbased opinion mining approach[END_REF]. Sentiment analysis currently involves a process to identify the sentiment orientation of opinions. The latter are highly unstructured by nature, thus requiring the application of Natural Language Processing (NLP) techniques [START_REF] Varghese | Aspect based sentiment analysis using support vector machine classifier[END_REF].

Obviously, documents may include opinions about several topics, but terms5 used to express opinions are usually specific and highly correlated to a particular domain [START_REF] Duthil | Locating complex named entities in web text[END_REF]. For instance, the sentence "The fruit is organic" would be very unusual in movie domain and then irrelevant in this case. However, it is obviously useful in agricultural domain. Both machine learning and dictionary-based approaches have been proposed in the literature to tackle these issues. For instance, a machine learning method applying text categorization techniques was proposed in [START_REF] Pang | A sentimental education: Sentiment analysis using subjectivity summarization based on minimum cuts[END_REF]. By this method, graphs, minimum cut formulation, context, and domain were considered to extract subjective portions of documents. Some dictionary-based approaches are currently available for general applications (e.g. SentiWordNet 6 ). They are not really appropriate for specific domains and new approaches have been developed to automatically learn the dictionary. These methods generally assume that positive (resp. negative) adjectives or verbs appear more frequently near a positive (resp. negative) seed term [START_REF] Harb | Web opinion mining: How to extract opinions from blogs[END_REF]. For instance, in [START_REF] Peter | Thumbs up or thumbs down?: Semantic orientation applied to unsupervised classification of reviews[END_REF][START_REF] Wang | Modifying so-pmi for japanese weblog opinion mining by using a balancing factor and detecting neutral expressions[END_REF], the authors propose an unsupervised learning algorithm for getting a dictionary in order to classify reviews considering seed terms to calculate the semantic orientation of phrases.

In this paper, we propose a new approach to automatically learn expressed opinions. We first focus on a new method for selecting relevant candidate terms from a set of documents. As many candidate terms may be extracted we propose to use two different but complementary measures to select the most representative ones: AcroDef M I3 and TrueSkill. Furthermore, in order to highlight the fact that of our approach is well useful for extracting terms for a specific domain we compare our proposal to the well-known SentiWordNet.

The paper is organized as follows. Our approach is presented in Section 2. The experimental setup is described in Section 3. In Section 4, we present and discuss the obtained results. Concluding remarks are presented in Section 5.

Our approach

The main process of our proposal is depicted in Figure 1 which involves the following steps:

1. First, a huge corpus for a specific domain is created by querying the web in order to get positive and negative documents relative to this domain. 2. Some pre-processing methods are performed over the documents in order to get the language of the document, remove tags, scripts, images, etc. 3. This step forms the core of the process. It focuses on the selection of terms that could be classified as positive (resp. negative) for the domain. To do, so first, a part-of-speech tagging is performed on documents in order to focus on nouns and adjectives since they are well relevant for extracting opinions 7 .

In order to find such terms (adjectives or nouns) as in [START_REF] Harb | Web opinion mining: How to extract opinions from blogs[END_REF], we follow the following hypothesis: the closer a positive (resp. negative) adjective/noun to another positive (resp. negative) adjective/noun, the more positive (resp. negative) it is. Accordingly, to this, we apply a window size algorithm for selecting the relevant terms closest to given seed terms. Finally, as many candidate terms may be generated, an efficient filtering approach is applied by using AcroDef M I3 and TrueSkill to select the most relevant positive and negative terms. Finally based on the results, two lexicons of positive and negative terms are generated. Fig. 1: Lexicons are automatically obtained from web pages for a specific domain filtered by seed terms. Extraction of relevant terms are then obtained by evaluating the relevance of candidate terms that are obtained after the analysis of the documents.

Basically, our approach could be used in many different domains. So in order to highlight its generality, experiments have been conducted on four different domains. They will be described in the experimental section. In the next sections, we describe more in detail the different steps.

Corpus Acquisition

It is now well admitted that some terms can be positive, neutral, or negative depending of the domain. Nevertheless, some terms are positive or negative irrespective of the domain (e.g. good). The main idea of our approach is thus to start the process by considering adjectives which are positive or negative in all domains. These terms will be considered as seed terms. We thus select the two following seed sets: P = {good, nice, excellent, positive, fortunate, correct, superior }, Q = {bad, nasty, poor, negative, unfortunate, wrong, inferior }. From these sets, we ensure a positive (resp. negative) web page retrieved from the other web pages related to a given domain. The following query illustrates an example of what is generated to get only positive documents about Genetic Modified Organism (GMO):

+GMO +good -bad -nasty -poor -negative -unfortunate -wrong -inferior where + and -mean that the document must have (+) or not (-) a given term. At the end, we are thus provided with positive and negative web pages denoted by corpus + , corpus -. Each corpus is splitted by the term used in the query. For instance, by considering the previous example we have in corpus + a set of document focusing mainly on good, i.e. no other positive terms are within documents, and more importantly not having a negative term (e.g -bad -nasty, and so on).

In the next section, we focus on the terms that are close to the seed terms by considering POS-Tagging as well as a window size algorithm.

Term extraction

First of all HTML tags, scripts, blank spaces and stop words are removed from web pages. We apply a part-of-speech tagger (in our case we have experimented TreeTagger8 ) to keep only adjectives and nouns. To be relevant with the previous hypothesis that an opinion candidate term is close to a seed term, a window size algorithm has been used. It aims at finding terms in both left and right sides of a seed term given a distance k. This distance is then the number of left (resp. right) terms of a given seed term. By varying k we are this able to better extract the most correlated opinion terms. For instance, by applying TreeTagger to retrieve adjectives (i.e. JJ) and nouns (i.e. NNS) as illustrated in Figure 2.

In Figure 2, the 'good ' term is a positive seed term and its nearest adjective is 'safe' given a k = 1 distance. Likewise, 'scientific' and 'studies' terms are retrieved with distance of k = 2. In addition, 'safe' is a positive candidate term because it occurs close to the positive seed term (i.e. good ). In this sense, we can have a set of opinion terms that can be candidates to be included into the resulting dictionaries.

Scientific studies have frequently found that GMO's are safe to eat and even good. Fig. 2: An example of applying a window size algorithm on the good seed term.

To get the correlation score and the usefulness on our specific domain (here GMO in the example) of each extracted term two approaches have been used: AcroDef M I3 and TrueSkill and this is described in the next section.

Candidate Term Selection

From the set of candidate terms, we thus have to filter the most relevant ones: the positive (resp. negative) terms that are very specific to a domain. In order to select the relevant candidate opinion terms, we propose to adapt the statistical measure AcroDef M I3 [START_REF] Roche | Acrodef: A quality measure for discriminating expansions of ambiguous acronyms[END_REF][START_REF] Roche | A web-mining approach to disambiguate biomedical acronym expansions[END_REF] (see Algorithm 1 where we illustrate only for positive terms, the process for the negative terms is similar) as well as a probabilistic measure based on TrueSkill [START_REF] Herbrich | Trueskill(tm): A bayesian skill rating system[END_REF][START_REF] Guo | Scorebased bayesian skill learning[END_REF] (see Algorithm 2).

The AcroDef M I3 measure: To filter associations extracted at the previous step, we use a ranking function in order to delete the irrelevant adjectives associations placed at the end of a list. Several quality measures in the literature are based on ranking functions. They are brought out of various fields: Association rules extraction [START_REF] Guillet | Quality Measures in Data Mining[END_REF], terminology extraction [START_REF] Daille | Study and implementation of combined techniques for automatic extraction of terminology[END_REF], and so forth. One of the most commonly used measures to compute a sort of relationship between the terms, called co-occurrence, is Church's Mutual Information (M I). The formula is the following [START_REF] Church | Word association norms, mutual information, and lexicography[END_REF]:

M I(x, y) = log nb(x, y) nb(x)nb(y) (1)
This measure tends to extract rare and specific co-occurrences according to [START_REF] Daille | Study and implementation of combined techniques for automatic extraction of terminology[END_REF].

The Cubic Mutual Information (M I3) is an empirical measure based on M I that enhances the impact of frequent co-occurrences. This measure defined by formula (2) gives interesting results [START_REF] Downey | Locating complex named entities in web text[END_REF][START_REF] Vivaldi | Improving term extraction by system combination using boosting[END_REF].

M I3(x, y) = log nb(x, y) 3 nb(x)nb(y) (2)
Like many other studies based on web resources, the nb function used by the M I and M I3 measures represents the number of web pages provided by a search engine.

Our approach relies on the dependence calculation of two terms, i.e. seed terms (st), and candidate term (ct). This is based on the number of pages given by a search engine with the queries 'st and ct' and 'ct and st'. This dependence is computed in a given domain D (for instance D = {GM O}). Then we apply AcroDef M I3 (formula (3)) described in [START_REF] Roche | Acrodef: A quality measure for discriminating expansions of ambiguous acronyms[END_REF]:

AcroDef M I3 (st, ct) = log
(nb(st and ct and D) + nb(ct and st and D)) 3 nb(st and D) × nb(ct and D)

The selection of terms is based on the application of Algorithm 1.

Algorithm 1 Term selection algorithm using AcroDef M I3

Require: corpus, seed terms = P, terms of the domain Ensure: correlation score values for each term 1: for each corpus do 2: terms + = window size(corpus + , P ) 3:

for term in terms + do 4:

given each seed term and terms of the domain compute the correlation score: 5:

score ← max(AcroDefMI3)

The TrueSkill measure: Unlike AcroDef M I3 , in TrueSkill, terms are extracted for each positive (resp. negative) page against k random negative (resp. positive) pages, and then the scores for each term are computed. Therefore, after having this outcome TrueSkill can give a score for each term of the positive page. This score depends on how many times it appears in the positive page so that it increases or decreases if it is also found on a negative page. The principle of TrueSkill is illustrated in Figure 3.

In Figure 3, we have S = {s 1,1 , s 1,2 , ..., s 1,n } and S = {s 2,1 , s 2,2 , ..., s 2,n } where s are the learned score value for each term in positive and negative web pages. p is the performance value for each term, which depends mainly of previously score s of the term; t is the sum of total performance for each term in the corpus. As TrueSkill learns s according its outcome of matches, we set a high punctuation for corpus + , and less punctuation for corpus -. This process is detailed in Algorithm 2.

Finally, TrueSkill gives a score to each term of the corpus in a match, and those values are updated for each match. On one hand, if a term is often found in a corpus -its value tends to decrease. On the other hand, if it is in a corpus +

d 1 t 2 t 1 p 2,2 p 2,1 p 1,2 p 1,1 s 2,2 s 2,1 s 1,2 s 1,1 I(d1 > ε) I(d1 = t1 -t2) I(t2 = p2,1 + p2,2) I(t1 = p1,1 + p1,2)
N (p2,2; s2,2; β 2 ) N (p2,1; s2,1; β 2 ) N (p1,2; s1,2; β 2 ) N (p1,1; s1,1; β 2 )

N (s2,2, µ2,2; σ 2 2,2 ) N (s2,1, µ2,1; σ 2 2,1 ) N (s1,2, µ1,2; σ 2 1,2 ) N (s1,1, µ1,1; σ 2 1,1 ) 1 2
Fig. 3: TrueSkill measure provides a score for each term selected given the positive and negative corpora. its value will increase. If the term is found in both corpora it tends to be constant. The velocity of the score increases or decreases depending on the term combination in each corpus and the number of matches.

Experiments

In order to evaluate our approach, experiments over four datasets were conducted. First we focused on both measures AcroDef M I3 and TrueSkill to evaluate their efficiency for pruning candidate terms. Second we evaluated the opinion classification task with both measures. Finally in order to really evaluate our automatic obtained dictionaries, classification are also compared with a general

Algorithm 2 Term selection algorithm using TrueSkill

Require: corpus, seed terms(P, Q) Ensure: correlation score values for each term 1: k = 10 number of matches for each corpus.

2: for each corpus do 3: terms + = window size(corpus + , P ) 4:

for k random corpus -do 5:

terms -= window size(corpus -, Q) 6:

given each term compute the correlation score: 7:

score ← T rueSkill(terms

+ , terms -, t = [1, 2])
dictionary. SentiW ordnet is a lexical resource for opinion mining, mainly it comprises 21479 adjectives and 117798 nouns, and assigns three sentiment scores to each word, i.e. positive, negative, and neutral.

Datasets

In order to show that our approach is generic, we use four datasets on very different domains: agriculture, movie, kitchen, and book:

-On the agricultural domain, tweets have been collected, and have been manually labeled. We obtained a corpus of 183 tweets, i.e. 72 positive and 111 negative tweets. -Available resources9 of the movie domain were introduced in [START_REF] Pang | Thumbs up?: Sentiment classification using machine learning techniques[END_REF] with 1000 positive and 1000 negative opinion documents. -Finally, the kitchen and book domains 10 introduced in [START_REF] Blitzer | Biographies, bollywood, boomboxes and blenders: Domain adaptation for sentiment classification[END_REF] have both 1000 positive and 1000 negative opinions.

Table 1 shows the number of candidate terms related to each domain after applying the window size algorithm with k = 1, and for each seed term we get the first 20 web pages.

Results of AcroDef M I3 and TrueSkill

In the following W S stands for the terms extracted after the window size algorithm. We thus compared M I3: seed terms with W S followed by AcroDef M I3 and T S: seed terms with W S followed by TrueSkill.

Figure 4 shows the normalized scores of all measures over each term by using the min-max scale algorithm. The window score is based on the frequency of a given term after applying the window size algorithm. As expected we thus have a high number of terms with low score. We can notice that terms have the more distributed score after applying AcroDef M I3 and TrueSkill. Fig. 4: Lexicons for each domain with their normalized score.

Classification

As the context of SentiW ordN et is not exactly the same as ours, the neutral class is considered as follows. For a term, we compute the difference between its positive and negative score and if the result is greater than zero we assign the term as positive otherwise as negative.

We have positive and negative lexicons (dictionaries) for each dataset (i.e. agriculture, movie), as shown in Table 1. In order to validate the algorithms we calculate F-Score values for each domain. Figures 5,[START_REF] Duthil | Locating complex named entities in web text[END_REF][START_REF] Guillet | Quality Measures in Data Mining[END_REF][START_REF] Guo | Scorebased bayesian skill learning[END_REF] show the F-Score values using the built lexicons with our approach and SentiWordNet. On our experiments, the F-Score is evaluated using the lexicons with a score greater than a given threshold.

Discussion of the results

In order to evaluate F-Score results, Table 2 shows the high values obtained for each dataset when the inferred lexicons for each domain are considered. For predicting negative elements, the F-Score values of TrueSkill are 0.66 and 0.62 for movie and book domains respectively, and 0.55 for agriculture domain based on AcroDef M I3 . To sum up, our approach performs better with F-Score results than Senti-WordNet for negative reviews. However, when positive reviews are considered, SentiWordNet performs better.

Figure 4 shows that kitchen domain is more generic than other domains due to the high number of terms (≈ 70) with a high score (≈ 0.9) obtained with AcroDef M I3 . This could explain that SentiWordNet performs better than TrueSkill and AcroDef M I3 for positive and negative reviews for this domain.

Conclusion

In this paper, we proposed a dictionary-based algorithm for sentiment analysis. Our approaches used AcroDef M I3 and TrueSkill to compute an association score between each term and its sentiment orientation (i.e. positive, negative). The extraction of these new terms related to each domain is obtained using the window size algorithm. This enables us to automatically create dictionaries that have been proved useful to identify positive and negative documents of specific domains.

In future work, we plan to extend our approach to other languages (e.g. French and Spanish), and we would like to study the behavior of our methods with other domains by using multi-word terms in our lexicons.

Fig. 5 :

 5 Fig. 5: F-Score results for agriculture tweet classification.

Fig. 6 :

 6 Fig. 6: F-Score results for movie review classification.

Fig. 7 :

 7 Fig. 7: F-Score results for kitchen review classification.

Fig. 8 :

 8 Fig. 8: F-Score results for book review classification.

Table 1 :

 1 Total of inferred lexicon terms by domain.

	Lexicon	Agriculture Movie P N P	N	Kitchen P N	Book P	N
	Adjective	146	83	104	72	157	26	168	87
	Noun	334	207	247	169	335	81	330	197

Table 2 :

 2 Top F-score result of each domain classification, TrueSkill (TS) improves the results of SentiW ordN et (SWN) in some cases.

					Agriculture Movie			Kitchen		Book
					P		N		P	N			P	N		P		N
			Approaches SWN MI3	SWN TS			SWN SWN SWN TS
			F-Score		0.62	0.55 0.66	0.66 0.70	0.54	0.67	0.62
	0.8 1								MI3 TS SWN	0.8 1							MI3 TS SWN
	0.6									0.6
	0.4									0.4
	0.2									0.2
	0	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	0	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
					Threshold										Threshold
					(a) Positive							(b) Negative

In this paper, we use term in order to characterize linguistic features.

http://sentiwordnet.isti.cnr.it

For simplicity, in this paper, we only report experiments that have been conducted on nouns and adjectives. Other experiments have been done by using adverbs and verbs.

http://www.cis.uni-muenchen.de/∼schmid/tools/TreeTagger/

http://www.cs.cornell.edu/People/pabo/movie-review-data/

http://textmining.biz/Projects/Songes

Acknowledgement

This work has been supported and funded by FONDECYT and SONGES project 11 (FEDER and Occitanie).


Mathieu Chassé
email: mathieu.chasse@normalesup.org

William L Griffin

Olivier Alard

Suzanne Y O'reilly

Georges Calas

Insights into the Mantle Geochemistry of Scandium from a Meta-Analysis of Garnet Data

Keywords: Crystal chemistry, Partitioning, Rare-earth elements, Melt extraction, Metasomatism, Volatile-rich fluids

The meta-analysis of about 13,000 analyses of scandium content in garnet grains shows that, below the spinel-garnet transition, this phase carries about three-quarters of the Sc budget of the mantle, indicating its control on Sc mobility. The Sc content of garnets in mafic rocks is low, due to a dilution effect resulting from their high modal content in garnet. Garnets from ultramafic rocks exhibit a wider range of Sc concentrations. We assess the relative influence of thermobarometry, crystal chemistry and fluid-related events on the distribution of Sc in garnet from such rocks to improve the tracking of geochemical processes in the mantle. Pressure and temperature of equilibration in the mantle are second-order factors influencing the Sc content of garnet, while crystal-chemistry, in particular Cr Cr+Al and Ca Ca+M g , is the main

Introduction

Scandium is a common trace element in the silicate Earth, with an average concentration of ca 16.4 ppm [START_REF] Palme | Cosmochemical Estimates of Mantle Composition[END_REF]. It is the lightest first-row transition element, and is sometimes considered as a rare-earth element (REE; [START_REF] Connelly | Nomenclature of Inorganic Chemistry: IUPAC Recomendations[END_REF]. Occurring only in the trivalent state, Sc is not affected by redox conditions and shows a lithophile behaviour [START_REF] Samson | Scandium[END_REF]. During mantle melting, Sc is mildly incompatible, being enriched in the continental crust, arc magmas and mid-ocean ridge basalts (MORB) relative to the primitive mantle, but is believed not to be mobilised in other mantle-related fluids [START_REF] Lee | Similar V/Sc Systematics in MORB and Arc Basalts: Implications for the Oxygen Fu-gacities of Their Mantle Source Regions[END_REF].

Based on the Geochemistry of Rocks of the Oceans and Continents (GEOROC) database [START_REF] Sarbas | The GEOROC Database As Part of a Growing Geoinformatics Network[END_REF], we show that Sc concentration varies by one order of magnitude between Sc-poor ultramafic rocks and Sc-rich mafic rocks (Fig. 1a). This selective concentration of Sc results from wide variations of its compatibility in rock-forming minerals (Fig. 1b). Typical Sc concentrations in garnets are seven times higher than the estimate for the silicate Earth, well above any other mineral group. The crystal chemistry and trace-element patterns of mantle garnets record a great variety of geochemical processes [START_REF] Baxter | Garnet: Common Mineral, Uncommonly Useful[END_REF], associated with mantle geochemistry [START_REF] Wood | Garnet in the Earth's Mantle[END_REF], metamorphism or tectonics [START_REF] Caddick | Garnet: Witness to the Evolution of Destructive Plate Boundaries[END_REF]. Garnet from the subcontinental lithospheric mantle is a key indicator of fluid-related events in the mantle (Griffin et al., 1999a) which may control Sc mobility between the shallow and deep regions of the upper mantle.

Trace-element concentration in garnet depends on site preference and crystal-chemical mechanisms of incorporation as functions of pressure, temperature and the chemical composition of the melt and solid phases present during garnet formation. Despite the ability of garnet to incorporate significant amounts of Sc, such information is limited in natural contexts. The discovery of a Sc-bearing garnet, eringaite (Ca 3 Sc 2 (SiO 4 ) 3 ; [START_REF] Galuskina | Eringaite, Ca 3 Sc 2 (SiO 4 ) 3 , a New Mineral of the Garnet Group[END_REF], the latest of the only eight natural Sc silicate minerals discovered on Earth, demonstrates the affinity of Sc for the six-fold-coordinated Y site of calcic garnets, already known in synthetic garnets by the existence of a solid solution between andradite (Ca 3 Fe 2 (SiO 4 ) 3 ) and eringaite [START_REF] Quartieri | Site Preference and Local Geometry of Sc in Garnets: Part II. the Crystal-Chemistry of Octahedral Sc in the Andradite -Ca 3 Sc 2 Si 3 O 12 Join[END_REF]. However, in synthetic pyrope-grossular (Mg 3 Al 2 (SiO 4 ) 3 )-Ca 3 Al 2 (SiO 4 ) 3 ) solid solutions [START_REF] Oberti | Site Preference and Local Geometry of Sc in Garnets: Part I. Multifarious Mechanisms in the Pyrope-Grossular Join[END_REF][START_REF] Kim | Scandium-45 NMR of Pyrope-Grossular Garnets: Resolution of Multiple Scandium Sites and Comparison With X-Ray Diffraction and X-Ray Absorption Spectroscopy[END_REF], the pyrope end-member preferentially incorporates Sc in the eight-fold-coordinated X site. Natural compositions of mantle garnets studied here are in the range (Mg 1-x-y Ca x Fe y ) 3 (Al 1-z Cr z ) 2 (SiO 4 ) 3 , with x in the range [0.1, 0.25], y in the range [0.1, 0.25] and z in the range [0, 0.25], corresponding to an intermediate member of the pyrope50-almandine25-uvarovite25 [(Mg 3 Al 2 (SiO 4 ) 3 )50-(Fe 3 Al 2 (SiO 4 ) 3 )25-(Ca 3 Cr 2 (SiO 4 ) 3 )25] solid solution. The term garnet is used hereafter to cover this variability, of prime importance to understand Sc incorporation in this phase.

The objective of this paper is to study the influence of garnet on the geochemical behaviour of Sc, by gathering about 13,000 analyses of Sc content in mantle-related garnet representing about 600 different samples from 97 studies published over the last 50 years. Our statistical approach demonstrates the influence of host-rock composition, crystal chemistry, thermobarometric conditions and fluid-related events on Sc content in mantle-related garnet (hereafter [Sc] grt1 ). This discussion demonstrates that Sc may be used to trace geochemical events, including partial melting and metasomatism, occurring below the spinel-garnet transition zone.

Database and Methods

Constitution of the Database of Garnet Analyses

The database used for this work contains about 13,000 analyses on garnet grains, mostly obtained by laser ablation-inductively coupled plasma mass spectrometry. Before 2000, some data were obtained by X-ray fluorescence, mass spectrometry and instrumental neutron activation analysis on separated garnets or using proton microprobe on garnet grains. All variables analysed in at least one publication are provided (Table A.1). More than 5500 analyses come from GEMOC/CCFS (see Griffin et al., 1999a[START_REF] Griffin | Cr-Pyrope Garnets in the Lithospheric Mantle 2. Compositional Populations and Their Distribution in Time and Space[END_REF]. The other 7500 garnet analyses were gathered from the GEOROC database [START_REF] Sarbas | The GEOROC Database As Part of a Growing Geoinformatics Network[END_REF]. Most analysed grains come from the subcontinental lithospheric mantle and were extracted from ultramafic xenoliths or concentrates from kimberlites. A minor part consists of eclogitic garnets. The analysed minerals come from 165 localities worldwide (Table A.2). Apart from garnet xenocrysts and megacrysts, for which the lithotype of the host-rock is unknown or not relevant, six categories have been selected to obtain statistically representative subsets: eclogite, pyroxenite, lherzolite, peridotite, harzburgite and wehrlite. The peridotite category is used when the host-rock was not described more precisely. A few lithotypes, do not fall into any of these categories and have been left out. When the major-element chemistry of garnet was available, garnet xenocrysts have been subclassified [START_REF] Grütter | An Updated Classification Scheme for Mantle-Derived Garnet, for Use by Diamond Explorers[END_REF][START_REF] Schulze | A Classification Scheme for Mantle-Derived Garnets in Kimberlite: a Tool for Investigating the Mantle and Exploring for Diamonds[END_REF].

Only variables contained in at least 70 % of the records have been used, leaving Si, Al, Cr, Fe, Mn, Mg and Ca as major-and minor elements and Ce, Nd, Sm, Eu, Gd, Dy, Ho, Er, Yb, Y and Sc as trace elements. Despite appearing in only 55 % of the records, Ti analyses have been kept to classify garnet xenocrysts. The database has been analysed using the R software (version 3.3.2, R Core Team, 2016). Details on the data analysis and the statistical treatment of the database are given in Appendix A.

Modelling the Influence of Mantle Melting on Scandium Concentrations

To assess the validity of our analysis on the influence of partial melting on Sc behaviour, we used a model based on the experimental determination of Sc concentration during the melting of a near-pyrolitic mantle in the garnet stability field from 3 GPa to 7 GPa [START_REF] Walter | Melting of Garnet Peridotite and the Origin of Komatiite and Depleted Lithosphere[END_REF]. For each melting experiment, the modal proportions are calculated by mass balance between the composition of the coexisting phases and the bulk composition of the system at equilibrium [START_REF] Walter | Melting of Garnet Peridotite and the Origin of Komatiite and Depleted Lithosphere[END_REF].

Mass balance calculations provide Sc concentration in the melt and in the residue according to the following formula:

[Sc] melt = [Sc] 0 D res/melt Sc • (1 -F) + F (1) [Sc] res = D res/melt Sc • [Sc] 0 D res/melt Sc • (1 -F) + F (2)
where [Sc] grt = D grt/melt Sc

• [Sc] melt (3)
The starting Sc concentration ([Sc] 0 ) corresponds to a pyrolitic mantle (ca 16.4 ppm, [START_REF] Palme | Cosmochemical Estimates of Mantle Composition[END_REF]. The partition coefficients of Sc between the residue and the melt are obtained by summing Sc partition coefficients between each phase and the melt [START_REF] Davis | Experimentally Determined Mineral/Melt Partitioning of First-Row Transition Elements (FRTE) During Partial Melting of Peridotite at 3 GPa[END_REF], weighted by the modal proportion of the phase considered. To our knowledge, this is the only study reporting together the partition coefficients of Sc for olivine, garnet and pyroxenes, applicable to partial melting of the mantle in the stability field of garnet.

Variations in the range of deviations indicated in this study do not influence the results.

Results

Scandium Distribution in Garnet and its Host-Rocks

Garnet is the main rock-forming mineral incorporating Sc (Fig. 1b and Table A.3). Scandium concentrations in host-rocks are statistically higher in mafic than in ultramafic rocks (Fig. 1a, Tables A.3 and A.4). The Sc content in the host-rock decreases with the depletion in fusible components, but such differences are considered with care as they are not statistically significant (p-value > 0.05). A positive correlation exists between the Sc concentration in the host-rock and the proportion of Sc incorporated in garnet, indicating that ca 75 % of the Sc content of the host-rock is incorporated in garnet (Fig. 2a). Where studies from the dataset report the average Sc concentration in garnet, in their host-rock and the modal garnet content of the hostrock, we have used these data to estimate the proportion of Sc incorporated by garnet for each lithotype; values are in the range of 65 % to 80 % (Fig. 2b).

The bimodal distribution of [Sc] grt is related to the nature of the hostrocks (Fig. 3). The low-[Sc] grt population, ca 50 ppm median, corresponds to mafic rocks: eclogites and pyroxenites. The main population, with a median

[Sc] grt of ca 120 ppm corresponds to peridotitic rocks.

The nature of the host-rocks (Fig. 4 and Table A.5) and the class of the garnet xenocrysts [START_REF] Schulze | A Classification Scheme for Mantle-Derived Garnets in Kimberlite: a Tool for Investigating the Mantle and Exploring for Diamonds[END_REF][START_REF] Grütter | An Updated Classification Scheme for Mantle-Derived Garnet, for Use by Diamond Explorers[END_REF] show that Sc concentration in peridotitic garnet increases with the degree of melt depletion in the host-rock, confirming the compatibility of Sc in mantle-derived garnet. The statistical significance of these observations is confirmed by ANOVA tests

(Table A.6).
Garnets from mafic and ultramafic host-rocks form two distinct populations in terms of [Sc] grt . For later discussion, the dataset considered contains only analyses of garnets from peridotitic host-rocks, according to the description of the host-rock and to the classification by [START_REF] Schulze | A Classification Scheme for Mantle-Derived Garnets in Kimberlite: a Tool for Investigating the Mantle and Exploring for Diamonds[END_REF] in the case of xenocrysts.

Relationships between Scandium Concentration and Thermobarometric

Parameters of the Mantle

Garnets from ultramafic rocks equilibrated with mantle olivine, enabling the use of the nickel thermometer (T Ni , [START_REF] Ryan | Garnet Geotherms: Pressure-Temperature Data from Cr-Pyrope Garnet Xenocrysts in Volcanic Rocks[END_REF] to investigate the role of temperature on the distribution of [Sc] grt . No correlation exists between T Ni and [Sc] grt (Fig. 5a, Table A.7, p-value > 0.05). On the subset of data for which the thermobarometer of [START_REF] Brey | Geothermobarometry in Four-Phase Lherzolites II. New Thermobarometers, and Practical Assessment of Existing Thermobarometers[END_REF] has been used to estimate the temperature (T BKN ) and pressure (P BKN ), there is no correlation with [Sc] grt (Figs. 5b andc, Table A.7, p-value > 0.05).

Relationships between Scandium and Major-Element Concentrations

Scandium concentration in garnet is correlated with garnet major-element chemistry (Fig. 6, Table A.8

Relationships between Scandium and Trace-Element Concentrations

Scandium concentration in garnet is positively correlated with the concentration of light rare-earth elements (LREE, e.g., r = 0.28 for Nd with a p-value <0.05, Table A.9) and becomes negative for heavy rare-earth elements and Y (HREE, e.g., r = -0.20 for Yb with a p-value <0.05, Table A.9, Fig. 8).

Plots of [Y]

grt versus [Zr] grt provide a signature of melt depletion and metasomatism (Griffin et al., 1999b). Such plots (Fig. 9a), on which [Sc] grt has been reported, confirm that garnets from depleted rocks are enriched in Sc compared to those in fertile ones, while indicators of metasomatism by silicate melt are weakly correlated with [Sc] grt .

The variation of (Ce/Yb) N versus (Zr/Sm) N can be used to distinguish metasomatic processes related to volatile-rich fluids or to silicate melts [START_REF] Aulbach | Genesis and Evolution of the Lithospheric Mantle Beneath the Buffalo Head Terrane, Alberta (Canada)[END_REF]. In our dataset, high [Sc] grt is associated with high (Ce/Yb) N , indicating that volatile-rich metasomatism positively affects [Sc] grt (Fig. 9b).

On the contrary, no correlation is visible between [Sc] grt and (Zr/Sm) N , confirming that metasomatism by silicate melts has a limited influence on [Sc] grt .

Evolution of Scandium Concentration During Melting

The evolution of the Sc content in the residual rock, in the melt and in the garnets is estimated from melting experiments on fertile peridotite [START_REF] Walter | Melting of Garnet Peridotite and the Origin of Komatiite and Depleted Lithosphere[END_REF], as explained in the "Database and Methods" section. In the presence of garnet and with increasing degree of partial melting, Sc concentration increases in the melt and decreases in the residue, whatever the pressure (Figs. 10a andb). The comparison with the modal garnet content (Fig. 10c) indicates that low modal proportions, which may result from melting, increase the Sc concentration in the melt and decrease Sc concentration in the residue. The calculated values of [Sc] grt are consistent with the concentrations of the dataset, confirming the validity of this approach. With increasing degree of partial melting, [Sc] grt increases by up to 100 ppm over a range of partial melting representative of the mantle (Fig. 10d).

Discussion

Garnet, the Main Scandium Carrier in the Deep Lithospheric Mantle

At the depths considered in the dataset, between 70 km and 200 km, garnet is the main host for Sc. It accounts for ca 75 % of the Sc budget of the deep lithospheric mantle (Fig. 2a, regression curve), in agreement with the proportion of Sc incorporated in various lithotypes of mantle-rocks (Fig. 1b).

Considering the median [Sc] grt from mafic and ultramafic rocks and estimates of the modal content of garnet [START_REF] Perrillat | Phase Transformations of Subducted Basaltic Crust in the Upmost Lower Mantle[END_REF][START_REF] Stixrude | Influence of Phase Transformations on Lateral Heterogeneity and Dynamics in Earth's Mantle[END_REF][START_REF] Pearson | Mantle Samples Included in Volcanic Rocks: Xenoliths and Diamonds[END_REF], our estimate of the abundance of Sc in the lithospheric mantle is consistent with those based on other methods (Table 1; Palme et al., 2014;[START_REF] Pearson | Mantle Samples Included in Volcanic Rocks: Xenoliths and Diamonds[END_REF][START_REF] White | Natural Weathering Rates of Silicate Minerals[END_REF], confirming that regardless of the context, garnet incorporates three-quarters of the Sc budget of the subcontinental lithospheric mantle. Such a constant high proportion, independent of the context, indicates that, in the presence of garnet, the mobility of Sc is controlled by this phase.

Dilution Effect

The low [Sc] grt in mafic host-rocks compared to ultramafic ones cannot be attributed to a compositional effect, as the median Sc content of mafic rocks (ca 40 ppm, Fig. 1c) is higher than that of ultramafic rocks (ca 10 ppm). This could reflect a lower compatibility of Sc in garnet from mafic parent rocks.

However, experimentally-determined partition coefficients are inconsistent with this hypothesis: D grt/melt Sc = 8.3 for eclogitic garnet and D grt/melt Sc = 6.0 for peridotitic garnet [START_REF] Pertermann | Experimental Determination of Trace Element Partitioning Between Garnet and Silica-Rich Liquid During Anhydrous Partial Melting of MORB-Like Eclogite[END_REF]Davis et al., 2013, respectively). This difference must reflect a dilution effect due to the high modal garnet content of mafic rocks.

The rest of this discussion will focus on Sc concentration in garnets from ultramafic host-rocks, which constitute the major part of the dataset. The lower garnet contents of these rocks makes them insensitive to dilution effects.

Impact of Thermobarometric Conditions

Models of the dependence of D experimentally, but experimental studies on garnet crystal-chemistry provide likely explanations on the influence of Cr. When the uvarovite component is limited, the composition is close to pyrope and Sc is mainly incorporated in the X site of garnet [START_REF] Oberti | Site Preference and Local Geometry of Sc in Garnets: Part I. Multifarious Mechanisms in the Pyrope-Grossular Join[END_REF]. When the composition shifts toward the grossular end-member, with Cr content remaining low, the relaxation of the garnet structure by substitution of Mg 2+ (r( VIII Mg 2+ ) = 0.89 A, [START_REF] Shannon | Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides[END_REF] for the larger cation Ca 2+ (r( VIII Ca 2+ ) = 1.12 A) results in an enlargement of both X and Y sites [START_REF] Oberti | Site Preference and Local Geometry of Sc in Garnets: Part I. Multifarious Mechanisms in the Pyrope-Grossular Join[END_REF][START_REF] Kim | Scandium-45 NMR of Pyrope-Grossular Garnets: Resolution of Multiple Scandium Sites and Comparison With X-Ray Diffraction and X-Ray Absorption Spectroscopy[END_REF] and increases the incorporation of Sc in these sites. However, in grossular, the size of the Y octahedron is constrained by the sole presence of the small Al 3+ cation (r( VI Al 3+ ) = 0.54 A), limiting the incorporation of Sc in the Y site [START_REF] Oberti | Site Preference and Local Geometry of Sc in Garnets: Part I. Multifarious Mechanisms in the Pyrope-Grossular Join[END_REF]. When Sc substitutes for the larger Fe 3+ ion in the high-spin state (Lin and Wheat, 2012, r( VI Fe 3+ , high spin) = 0.65 A) along the join andradite-eringaite, this barrier is removed, and Sc is incorporated exclusively in the Y site [START_REF] Quartieri | Site Preference and Local Geometry of Sc in Garnets: Part II. the Crystal-Chemistry of Octahedral Sc in the Andradite -Ca 3 Sc 2 Si 3 O 12 Join[END_REF]. Iron(III) has an ionic radius of similar to Cr 3+ (r( VI Cr 3+ ) = 0.62 A). By analogy, when the uvarovite component is significant, the less compact structure of Cr-Ca-rich garnet thus will favour the incorporation of Sc in the Y site. Large Y octahedra occupied by Cr and locally related to X dodecahedra occupied by Ca should coexist with small Y octahedra occupied by Al close to X dodecahedra occupied by Mg. Because Ca can reach 1.5 apfu, the result should be a highly strained structure. In these compositions, Sc might predominently enter a highly deformed Y site close to X sites occupied by Mg, helping to reduce the strain because this X site is large compared to the one in pyrope.

This incorporation of Sc in the Y site of Ca-Cr-rich garnet is consistent with lattice-strain theory [START_REF] Wood | Trace Element Partitioning: the Influences of Ionic Radius, Cation Charge, Pressure, and Temperature[END_REF], which relates the charge and radius of an ion to its partition coefficient in the crystallographic site of a given mineral species at a given temperature and pressure. From this theory and the experimental determination of trace-element partition coefficients, Onuma diagrams [START_REF] Onuma | Trace Element Partition Between Two Pyroxenes and the Host Lava[END_REF]) have been retrieved (Fig. 11)

for mantle-related garnet end-members at thermobarometric conditions representative of the mantle [START_REF] Corgne | Trace Element Partitioning Between Majoritic Garnet and Silicate Melt at 25 GPa[END_REF][START_REF] Dasgupta | Trace Element Partitioning Between Garnet Lherzolite and Carbonatite at 6.6 and 8.6 GPa With Applications to the Geochemistry of the Mantle and of Mantle-Derived Melts[END_REF][START_REF] Corgne | Trace Element Partitioning Between Majoritic Garnet and Silicate Melt at 10-17 GPa: Implications for Deep Mantle Processes[END_REF]. Scandium is less compatible in the X or Y sites of the pyrope end-member than in the grossular end-member (Fig. 11a). The Onuma diagram expected from the interpretation given above is presented for the Y site for the uvarovite end-member as experimental data are not available. As explained above, incorporation of Cr and Ca in the garnet structure expands the Y site, increasing the compatibility of Sc in this site (Fig. 11b). This has a prominent effect on the variability of [Sc] grt . As Sc compatibility in the Y site is low in pyrope, even a small expansion of the Y site can dramatically increase Sc compatibility.

This interpretation is consistent with the relationships between [Sc] grt and

[REE] grt . Scandium is positively correlated with LREE and negatively with HREE (Fig. 8). The small ionic radius of Sc should make it more comparable to a HREE in terms of compatibility in the X site of garnet (Fig. 11a).

However, the decrease in [Sc]

Impact of Metasomatic Processes

Metasomatism of previously depleted rocks by silicate melts is not associated with variations in [Sc] grt (Fig. 9a). According to our melting model, Sc is mildly compatible in garnet-bearing mantle rocks. The Sc content of silicate melts is close from that of a garnet peridotite and [Sc] grt is not influenced by silicate-melt metasomatism. However, depleted garnets can reach high [Sc] grt . A slight decrease in [Sc] grt exists along the silicate-melt metasomatism trend (from bright red to dark purple spots, Fig. 9a). In that case, the difference in Sc concentration between the garnet and the metasomatic melt is great enough to lead to a decrease in [Sc] grt .

Metasomatism by volatile-rich fluids is correlated with higher [Sc] grt (Fig. 9b), contrasting with the low mobility expected for Sc in fluids from the shallow lithospheric mantle [START_REF] Lee | Similar V/Sc Systematics in MORB and Arc Basalts: Implications for the Oxygen Fu-gacities of Their Mantle Source Regions[END_REF]. This behaviour is expected for strongly-incompatible elements [START_REF] Mckenzie | Some Remarks on the Movement of Small Melt Fractions in the Mantle[END_REF][START_REF] Bedini | Evolution of LILE-Enriched Small Melt Fractions in the Lithospheric Mantle: a Case Study from the East African Rift[END_REF] This hypothesis is consistent with the occurrence of several Sc-rich carbonatites (up to 190 ppm Sc in whole-rock analyses, see e.g. [START_REF] Gramaccioli | The Formation of Scandium Minerals As an Example of the Role of Complexes in the Geochemistry of Rare Earths and HFS Elements[END_REF]. However, few data are available on Sc-complexes in fluids except for hydrothermal fluids [START_REF] Wood | The Aqueous Geochemistry of Gallium, Germanium, Indium and Scandium[END_REF]. Experiments indicate that D grt/fluid Sc increases from silicate to carbonate-rich melts (Fig. 11a, [START_REF] Dasgupta | Trace Element Partitioning Between Garnet Lherzolite and Carbonatite at 6.6 and 8.6 GPa With Applications to the Geochemistry of the Mantle and of Mantle-Derived Melts[END_REF]. In the presence of garnet, carbonate-rich melts should not transport significant amounts of Sc compared to other melts. (ii) The second mechanism relies on the differences of compatibility between fertile and depleted garnet, which is not modified during metasomatism by volatile-rich fluids, as the fluids produced involve low volumes of melting (< 1 %, [START_REF] Mckenzie | Some Remarks on the Movement of Small Melt Fractions in the Mantle[END_REF]. Volatile-rich fluids may be generated at the base of the lithosphere [START_REF] Mckenzie | Some Remarks on the Movement of Small Melt Fractions in the Mantle[END_REF][START_REF] Bedini | Evolution of LILE-Enriched Small Melt Fractions in the Lithospheric Mantle: a Case Study from the East African Rift[END_REF]. In this context, the deeper lithosphere is more fertile than in the upper parts, and consequently is richer in Sc (Fig. 10). Scandium is also mildly compatible due to the lower D The second stage is attributed to metasomatism by a kimberlitic melt, a product of small-degree partial melting (Fig. 12a, arrow 2). This event is associated with a decrease of [Sc] grt , from ca 150 ppm to ca 100 ppm, typical of lherzolitic garnet, and a decrease in [Ca] grt and [Cr] grt , decreasing Sc compatibility in garnet. The third stage of metasomatism is associated with the percolation of extensive amounts of basaltic fluids (Fig. 12a, arrow 3).

It is also associated with the strongest decrease of [Sc] grt , from ca 150 ppm to ca 50 ppm. This metasomatic event led to complete re-equilibration of garnet and increased modal garnet content, explaining the decrease in [Sc] grt .

Peridotite xenoliths from the Western Kaapvaal Craton show three groups of garnets: low-and high-Cr lherzolitic garnets and harzburgitic garnets, characterised by the shape of their REE patterns and distinct La versus Cr/Ca ratio diagrams [START_REF] Gibson | Melt Depletion and Enrichment Beneath the Western Kaapvaal Craton: Evidence from Finsch Peridotite Xenoliths[END_REF]. Figure 12b shows these three groups in the Sc versus Cr/Ca plot. According to [START_REF] Gibson | Melt Depletion and Enrichment Beneath the Western Kaapvaal Craton: Evidence from Finsch Peridotite Xenoliths[END_REF], the high content of LREE, such as La, in harzburgitic garnet results from an enlargement of the X site. This is linked with the distortion of the crystal network caused by the substitution of Cr for Al, similar to the enhanced incorporation of Sc in the Cr-substituted Y site of harzburgitic garnet. This explains why La and Sc behave similarly with respect to Cr/Ca ratio, even though, Sc is closer in ionic radius to HREE than to LREE. This analogy between HREE and Sc leads us to suggest that the sinuous REE patterns of harzburgitic garnets reflect a partial incorporation of the heaviest REE in the Y site of Cr-rich harzburgitic garnet. In this case, the incorporation of both LREE and HREE in garnet is favoured by increasing [Cr] grt while such effects are limited for the middle-REE, between Gd and Er, which are not affected by the enlargement of the X site nor by the possibility of incorporation in the Y site. This would explain the low concentration of middle-REE in REE patterns of harzburgitic garnet observed by [START_REF] Gibson | Melt Depletion and Enrichment Beneath the Western Kaapvaal Craton: Evidence from Finsch Peridotite Xenoliths[END_REF].

Impact of metasomatism. Metasomatic events were proposed in the lithospheric mantle beneath the Canadian Shield, based on the geochemistry of xenoliths and xenocrysts from the Buffalo Hills kimberlite [START_REF] Aulbach | Genesis and Evolution of the Lithospheric Mantle Beneath the Buffalo Head Terrane, Alberta (Canada)[END_REF]. The enrichment of incompatible elements in garnet and clinopyroxene was used to argue for an early stage of volatile-rich fluid metasomatism.

In the Y-Zr diagram (Fig. 12c), most samples show high [Sc] grt , between ca 150 ppm and ca 250 ppm, regardless of the degree of depletion in terms of major elements. Volatile-rich metasomatism increases [Sc] grt , explaining the disconnect between the degree of depletion and [Sc] grt . The Y-Zr diagram also suggests the effects of silicate-melt metasomatism. Samples affected by this type of metasomatism exhibit a decrease in [Sc] grt (blue spots, Fig. 12c).

This confirms that, when garnets are originally Sc-rich, silicate melt metasomatism can be seen in variations of [Sc] grt .

Metasomatic processes were investigated in the lithospheric mantle beneath the Siberian craton [START_REF] Agashev | Metasomatism in Lithospheric Mantle Roots: Constraints from Whole-Rock and Mineral Chemical Composition of Deformed Peridotite Xenoliths from Kimberlite Pipe Udachnaya[END_REF]. The difference between the concentration of incompatible elements calculated from modal mineral- This trend is compatible with the influence of volatile-rich metasomatism on

[Sc] grt , showing that the signature of this metasomatic stage is preserved by [Sc] grt despite later stages of silicate metasomatism and Fe-rich metasomatism described by [START_REF] Agashev | Metasomatism in Lithospheric Mantle Roots: Constraints from Whole-Rock and Mineral Chemical Composition of Deformed Peridotite Xenoliths from Kimberlite Pipe Udachnaya[END_REF].

Conclusions

This meta-analysis and statistical dissection of a large database of garnet compositions provides new information on the geochemistry of Sc in the deep lithospheric mantle and a basis for using [Sc] grt as a tracer of mantle processes:

1. Garnet accounts for about 75 % of the Sc budget of the deep lithospheric mantle, regardless of the context, thus controlling the mobility of Sc in the mantle. In well-constrained contexts, studying Sc concentration will help to understand the conditions of generation of magmas, as well as the nature of their residues.

Declaration of Interest

None. [START_REF] Van Achterbergh | Metasomatism in Mantle Xenoliths from the Letlhakane Kimberlites: Estimation of Element Fluxes[END_REF][START_REF] Grégoire | Garnet Lherzolites from the Kaapvaal Craton (South Africa): Trace Element Evidence for a Metasomatic History[END_REF][START_REF] Ionov | Chemical Variations in Peridotite Xenoliths from Vitim, Siberia: Inferences for REE and Hf Behaviour in the Garnet-Facies Upper Mantle[END_REF][START_REF] Schmickler | Eclogite Xenoliths from the Kuruman Kimberlites, South Africa: Geochemical Fingerprinting of Deep Subduction and Cumulate Processes[END_REF][START_REF] Jacob | Lu-Hf and Geochemical Systematics of Recycled Ancient Oceanic Crust: Evidence from Roberts Victor Eclogites[END_REF][START_REF] Ulianov | Aluminous Websterite and Granulite Xenoliths from the Chyulu Hills Volcanic Field, Kenya: Gabbro-Troctolitic Cumulates Subjected to Lithospheric Foundering[END_REF][START_REF] Appleyard | Petrology and Geochemistry of Eclogite Xenoliths from the Rietfontein Kimberlite, Northern Cape, South Africa[END_REF][START_REF] Gibson | Melt Depletion and Enrichment Beneath the Western Kaapvaal Craton: Evidence from Finsch Peridotite Xenoliths[END_REF][START_REF] Aulbach | Eclogite Xenoliths from the Lace Kimberlite, Kaapvaal Craton: from Convecting Mantle Source to Palaeo-Ocean Floor and Back[END_REF]. Values of Sc concentration are sometimes absent for olivine and orthopyroxene grains, they are replaced by the median value calculated from the other studies reporting this content.

Figure Captions

  [Sc] 0 the concentration of Sc in the initial system, [Sc] res the Sc concentration in the residue, [Sc] melt the Sc concentration in the melt, F the mass fraction of melt and D res/melt Sc the partition coefficient of Sc between the residue and the melt. As the melt is in equilibrium with the different phases during batch melting experiments, [Sc] grt has been obtained as follows:

  ). The strongest positive correlation exists between[Sc] grt and[Cr] grt (r = 0.43, p-value < 0.05) while a negative correlation exists between[Sc] grt and[Al] grt (r = -0.35, p-value < 0.05). A weaker positive correlation exists between[Sc] grt and [Ca] grt (r = 0.28, p-value < 0.05), while a negative correlation is found between[Sc] grt and [Mg] grt (r = -0.20, p-value < 0.05). Chromium and Al occupy the Y site of garnet while Ca and Mg occupy the X site. These correlations confirm the influence of crystal chemistry on the incorporation of Sc in natural garnets shown in experimental conditions by[START_REF] Oberti | Site Preference and Local Geometry of Sc in Garnets: Part I. Multifarious Mechanisms in the Pyrope-Grossular Join[END_REF];[START_REF] Quartieri | Site Preference and Local Geometry of Sc in Garnets: Part II. the Crystal-Chemistry of Octahedral Sc in the Andradite -Ca 3 Sc 2 Si 3 O 12 Join[END_REF] and[START_REF] Kim | Scandium-45 NMR of Pyrope-Grossular Garnets: Resolution of Multiple Scandium Sites and Comparison With X-Ray Diffraction and X-Ray Absorption Spectroscopy[END_REF]. However, Cr# ( [Cr]grt [Cr]grt+[Al]grt ) and Ca# ( [Ca]grt [Ca]grt+[Mg]grt ) are also indicators of melt extraction from the host-rock. Therefore, correlations between [Sc] grt and [Cr] grt , [Al] grt , [Ca] grt or [Mg] grt may also reflect the degree of depletion of the host-rock (Fig.7a).Variations of[Sc] grt versus [Fe] grt show two main trends (Fig.6), resulting in the apparent absence of statistical significance (p-value > 0.05). A negative correlation exists at high [Fe] grt , and a positive correlation at lower[Fe] grt giving a low overall correlation (r = 0.07). This can be related to the variable speciation of Fe: Fe 2+ or Fe 3+ incorporated in the X or Y site. The fertility of the host-rock may also be involved, as shown by the decrease of[Sc] grt with increasing[Mg]grt[Mg]grt+[Fe]grt (Mg#), an indicator of depletion (Fig.7b).A negative correlation between[Sc] grt and [Si] grt (r = -0.24, p-value < 0.05) is related to the correlations between [Si] grt and [Al] grt , [Cr] grt , [Mg] grt and [Ca] grt . Silicon concentration in garnet is positively correlated with [Al] grt and [Mg] grt and negatively correlated with [Cr] grt and [Ca] grt explaining the anticorrelation with [Sc] grt . The lack of a correlation between [Sc] grt and [Mn] grt (p-value > 0.05) reflects the low [Mn] grt in the garnets studied, hindering any influence on garnet crystal chemistry.

  calibrated to experiments for the X site of garnet[START_REF] Van Westrenen | Quantifying Garnet-Melt Trace Element Partitioning Using Lattice-Strain Theory: New Crystal-Chemical and Thermodynamic Constraints[END_REF]. However, the incorporation of Sc in mantlederived garnets is not correlated with pressure or temperature (Figs.5a, b and c), indicating the predominance of compositional effects.4.4. Impact of Melt ExtractionDepletion of mantle peridotites by removal of mafic melts is correlated with increased [Sc] grt . Lherzolitic garnets (G9) have low [Sc] grt compared to harzburgitic ones (G10; Figs. 4a and b) and [Sc] grt increases with Cr#, a measure of depletion (Fig. 7). According to our estimates based on melting experiments, [Sc] grt increases with melt extraction (Fig. 10d) in agreement with the observations in natural garnets. Nonetheless, at constant Cr#, [Sc] grt decreases with increasing Mg# or decreasing Ca# (Fig. 7), two indicators of depletion of the host-rock. Variations of D grt/melt Sc thus cannot be explained by melt-extraction only, and changes in crystal chemistry must be invoked. The correlations between [Sc] grt and Cr# also imply crystal-chemical effects, considering that this parameter reflects the nature of the atoms occupying the Y-site. 4.5. Impact of Garnet Crystal Chemistry Correlations between Sc and major elements (Fig 6) indicate the affinity of Sc for the uvarovite end-member of garnet (Ca 3 Cr 2 (SiO 4 ) 3 ), indicating the significance of Cr in the crystal chemistry of mantle garnet. Although often referred to as a minor element, it can hardly be considered as such when it commonly reaches 0.5 apfu. In Cr-rich systems, the crystal-chemical mechanisms of Sc incorporation are unknown and D grt/melt Sc has not been determined

  grt with increasing [HREE] grt indicates that the compatibility of Sc in garnet falls as the compatibility of HREE increases. This can only be explained by the incorporation of part of the Sc in the Y site of garnet, while other HREE remain incompatible in the Y site of Cr-Ca-rich garnets (Fig. 11b). The shift towards the incorporation of Sc in the Y site of garnet tracks an enlargement of the X site of garnet, increasing the compatibility of LREE and explaining the positive correlation with Sc, while the effect on the compatibility of HREE remains limited. The incorporation of Sc in the Y site may also explain the poor correlation between [Sc] grt and the temperature and pressure of equilibration of garnet; a complex relationship depending on both X and Y sites could blur any existing trend. The variability of [Sc] grt in our database is mainly related to the capacity of the Y site of garnet to accept Sc. The X site may play a lesser role in constraining the partition coefficient of Sc for natural ranges of compositions. However, spectroscopic studies of Sc or experimental data on the partition coefficients of trace elements in the Y site of Cr-Ca-rich garnet are needed to quantify these effects.

  but Sc is compatible in the garnet stability field of the mantle. Two mechanisms could explain the influence of volatile-rich fluids on [Sc] grt . (i) At mantle pressures and temperatures, complexing of Sc by volatile elements transports significant amounts of Sc and drastically decreases its compatibility.

  to depleted ones. In this way, volatile-rich fluids can be enriched in Sc compared to a depleted mantle. When these fluids ascend in the lithosphere and reach depleted parts of the mantle, the high compatibility of Sc in depleted garnets and the relatively high Sc concentration of the fluid combine to increase [Sc] grt . 4.7. Validation of the Meta-Analysis on Selected Case Studies Impact of crystal chemistry. Metasomatic events during the evolution of the subcontinental lithospheric mantle beneath the Siberian craton were inferred from peridotite xenoliths by Howarth et al. (2014). On the basis of the trends observed in the CaO-Cr 2 O 3 diagram, increasing [Ca] grt at constant [Cr] grt is associated with a metasomatic refertilisation of harzburgite to lherzolite (Fig. 12a, arrow 1). Comparison with [Sc] grt shows that harzburgitic garnets have high [Sc] grt and refertilisation is associated with increasing [Sc] grt . Increased [Ca] grt at constant [Cr] grt increases Sc compatibility in garnet explaining the increased [Sc] grt during this first stage of metasomatism.

  ogy and those measured was interpreted as reflecting the presence of cryptic phases induced by volatile-rich fluids during the formation of the garnetbearing xenoliths. In the (Zr/Sm) N -(Ce/Yb) N diagram (Fig.12d), we observe an increase in[Sc] grt associated with increasing (Ce/Yb) N at low (Zr/Sm) N .

2.

  High modal garnet contents in mafic rocks lead to dilution[Sc] grt , leading to a specific geochemical behaviour of Sc in such host-rocks.3. Melt extraction increases[Sc] grt showing the compatibility of this element in the presence of garnet. 4. Crystal chemistry is the major parameter controlling [Sc] grt . In particular, higher Cr# and Ca# in garnet increase the compatibility of Sc in this phase. 5. Metasomatism has limited effect on [Sc] grt except in the specific case of volatile-rich fluids, which can greatly enhance [Sc] grt . These conclusions and the selection of case studies show how Sc geochemistry can help recognise fluid-related processes in the mantle. Scandium is mildly incompatible above the spinel-garnet transition zone (Mallmann and O'Neill, 2009) while this study shows that Sc behaves as a mildly compatible element in the garnet stability field. This unique behaviour results from the ability of this element to enter both X and Y sites of Cr-Ca-rich garnets, at the boundary between trivalent transition elements, which enter the Y site, and REE, which occupy the X site. This behaviour reflects the singular crystal chemistry of garnet from the lithospheric mantle, and emphasises the need for experimental studies on partitioning of trace elements in Cr-Ca-rich garnets. The balance between the variable compatibility of Sc in garnet, depending on Cr# and Ca# and the modal garnet content of the host-rock induces subtle variations in compatibility which can have drastic effects on Sc contents, as shown by the unexpected effect of volatile-rich fluids on [Sc] grt .

Figure 1 :

 1 Figure 1: Distribution of Sc concentration in minerals and mantle-related rocks: (a) box-and-whisker plots of Sc concentration in selected mantlerelated rocks. The grey horizontal line corresponds to 16.4 ppm, the average Sc concentration in the silicate Earth (Palme et al., 2014); (b) box-andwhisker plots of Sc concentration in major rock-forming minerals. The box is limited by the first and third quartiles of the data distribution. The horizontal line represents the median, and the width of the adjacent notches shows the 95 % confidence interval about the median. Where absent, this interval is narrower than the thickness of the line representing the median. The "whiskers" represent the last value before 1.5 times midrange (the arithmetic mean of maximum and minimum values) beyond the first or third quartile.

Figure 2 :

 2 Figure 2: Influence of host-rock parameters on [Sc] grt : (a) scatter plot of Sc concentration in the host-rock versus the estimated Sc content in the host-rock attributable to garnet. This estimate is calculated using the [Sc] grt and the modal content of garnet in the host-rock; (b) average proportion of Sc incorporated in the different mineral phases of garnet-bearing lherzolites, harzburgites, pyroxenites and eclogites (n: number of individual rocks analysed; after the only studies of the database reporting the required information: van Achterbergh et al.[START_REF] Van Achterbergh | Metasomatism in Mantle Xenoliths from the Letlhakane Kimberlites: Estimation of Element Fluxes[END_REF][START_REF] Grégoire | Garnet Lherzolites from the Kaapvaal Craton (South Africa): Trace Element Evidence for a Metasomatic History[END_REF][START_REF] Ionov | Chemical Variations in Peridotite Xenoliths from Vitim, Siberia: Inferences for REE and Hf Behaviour in the Garnet-Facies Upper Mantle[END_REF]

Figure 3 :

 3 Figure 3: Density plot of [Sc] grt and histograms of [Sc] grt from mafic hostrock, fertile ultramafic host-rocks and depleted ultramafic host-rocks.

Figure 4 :

 4 Figure 4: Variations in the distribution of [Sc] grt with the lithology of the host-rock: (a) box-and-whisker plots of [Sc] grt relative to the lithotype of the host-rock described in the original publications; (b) box-and-whisker plots of [Sc] grt in xenocrysts classified according to Grütter et al. (2004) (Xen.: all xenocryst values, G0: unclassified, G1: low-Cr megacrysts, G3: eclogitic, G5: pyroxenitic, G9: lherzolitic, G10: harzburgitic, G11: Ti-metasomatised, G12: wehrlitic). The box is limited by the first and third quartiles of the data distribution. The horizontal line represents the median, and the width of the adjacent notches shows the 95 % confidence interval about the median.Where absent, this interval is narrower than the thickness of the line representing the median. The "whiskers" represent the last value before 1.5 times midrange (the arithmetic mean of maximum and minimum values) beyond the first or third quartile.

Figure 5 :

 5 Figure 5: Influence of temperature and pressure on [Sc] grt : (a) scatter plot of T Ni[START_REF] Ryan | Garnet Geotherms: Pressure-Temperature Data from Cr-Pyrope Garnet Xenocrysts in Volcanic Rocks[END_REF] versus[Sc] grt ; (b) scatter plot of T BKN[START_REF] Brey | Geothermobarometry in Four-Phase Lherzolites II. New Thermobarometers, and Practical Assessment of Existing Thermobarometers[END_REF] versus[Sc] grt ; (c) scatter plot of P BKN[START_REF] Brey | Geothermobarometry in Four-Phase Lherzolites II. New Thermobarometers, and Practical Assessment of Existing Thermobarometers[END_REF] versus[Sc] grt . Estimates of T BKN and P BKN are from the original publications.

Figure 6 :

 6 Figure 6: Correlation matrix and scatter plots for [Sc] grt versus major and minor elements. Positive (respectively negative) correlations are indicated by red (respectively blue) colours and the orientation of the ellipse. The intensity of the correlation is given by the intensity of the colour (see colour bar) and the narrowness of the ellipse.

Figure 7 :

 7 Figure 7: Scatter plots of [Sc] grt relative to the ratios of selected major elements: (a) Cr# versus Ca#; (b) Cr# versus Mg#. The colour of each point depends on the Sc content of the record, from 0 ppm (in blue) to 300 ppm (in red).

Figure 8 :

 8 Figure 8: Correlation matrix and scatter plots for [Sc] grt versus REE. Positive (respectively negative) correlations are indicated by red (respectively blue) colour and the direction of the ellipse. The intensity of the correlation is given by the intensity of the colour (see colour bar) and the narrowness of the ellipse.

Figure 9 :

 9 Figure 9: Scatter plots of Sc relationships to trace-element indicators of fluid-related processes in the mantle: (a) [Y] grt versus [Zr] grt , the fields are after Griffin et al. (1999b); (b) (Ce/Yb) N versus (Zr/Sm) N , ratios have been normalised to the values of[START_REF] Mcdonough | The Composition of the Earth[END_REF]. The metasomatic trends are after[START_REF] Aulbach | Genesis and Evolution of the Lithospheric Mantle Beneath the Buffalo Head Terrane, Alberta (Canada)[END_REF]. The colour of each point depends on the Sc content of the record, from 0 ppm (in blue) to 300 ppm (in red).

Figure 10 :

 10 Figure 10: Influence of partial melting of a fertile ultramafic mantle estimated from Walter (1998) experiments: (a) evolution of the Sc content of the residue; (b) evolution of the Sc content of the melt; (c) evolution of the modal garnet content; (d) evolution of [Sc] grt .

Figure 11 :

 11 Figure 11: Onuma diagrams[START_REF] Onuma | Trace Element Partition Between Two Pyroxenes and the Host Lava[END_REF] showing the partition coefficients of trace elements as a function of their ionic radius: (a) X site of garnet (Prp: pyrope, Grs: grossular); (b) Y site of garnet (Maj: majoritic pyrope, Uv: uvarovite). Diagrams for the X site after van Westrenen and Draper (2007) and[START_REF] Dasgupta | Trace Element Partitioning Between Garnet Lherzolite and Carbonatite at 6.6 and 8.6 GPa With Applications to the Geochemistry of the Mantle and of Mantle-Derived Melts[END_REF]; diagram for the Y site of majoritic pyrope is from[START_REF] Corgne | Trace Element Partitioning Between Majoritic Garnet and Silicate Melt at 25 GPa[END_REF] and an expected curve for the Y site of uvarovite is given to illustrate the influence of Cr and Ca incorporation.

Figure 12 :

 12 Figure 12: Evolution of [Sc] grt with garnet composition in selected case studies: (a) evolution of [Sc] grt (given by the colour of each point) in the CaO-Cr 2 O 3 diagram for garnets from the subcontinental lithospheric mantle beneath the Siberian craton (Hz: harzburgitic garnet, Lz: lherzolitic garnet). The arrows represent three stages of metasomatism, described by Howarth et al. (2014); (b) distinction of garnet groups from the Kaapvaal craton depending on their relative Sc contents and Cr/Ca ratios; data are from Gibson et al. (2008); (c) evolution of [Sc] grt (given by the colour of each point) in the Y-Zr diagram for garnets from the Buffalo Head Terrane (Dep.: depleted); data are Aulbach et al. (2004), and the fields are from Griffin et al. (1999b); (d) evolution of [Sc] grt (given by the colour of each point) in the (Zr/Sm) N -(Ce/Yb) N diagram for garnets from the Siberian craton, data from Agashevet al. (2013). The metasomatic trends are after[START_REF] Aulbach | Genesis and Evolution of the Lithospheric Mantle Beneath the Buffalo Head Terrane, Alberta (Canada)[END_REF].

Table Caption Table 1 :

 Caption1 Comparisons of Sc concentration in the mantle calculated from this study with estimates from the literature for different mantle compositions.The references used for each value are given in the table. The estimated Sc concentration for this study is obtained using the following formula: [Sc] grt * Mod / 0.75, with Mod the modal garnet content,[Sc] grt the median concentration of Sc in the type of garnet considered and 0.75 our assessment of the proportion of the Sc content in the mantle that is incorporated in garnet.

More generally, [X] grt stands for the concentration of the element X in garnet.

Acknowledgements

We thank Mr Pierre Chassé and Dr Max Frenzel for fruitful discussions and help on statistical methods. The authors are indebted to Dr Roberta Oberti, Prof Douglas Smith and Dr Andrew Kerr for their comments and suggestions which helped to improve this manuscript. Mathieu Chassé has been supported by scholarships from the École normale supérieure (ENS) and Macquarie University. Olivier Alard is supported by an Australian Research Council grant (FT150100115). This work was funded by the Institut universitaire de France (IUF) and a Foundation Grant from the ARC Centre of Excellence for Core to Crust Fluid Systems (CCFS). This is contribution 1156 from the ARC Centre of Excellence for Core to Crust Fluid Systems (http://www.ccfs.mq.edu.au) and 1222 from the GEMOC Key Centre (http://www.gemoc.mq.edu.au).


Valentin Vielzeuf

Alexis Lechervy

Stéphane Pateux

Frédéric Jurie

Multi-Level Sensor Fusion with Deep Learning

In the context of deep learning, this article presents an original deep network, namely CentralNet, for the fusion of information coming from different sensors. This approach is designed to efficiently and automatically balance the trade-off between early and late fusion (i.e. between the fusion of low-level vs high-level information). More specifically, at each level of abstraction -the different levels of deep networks -unimodal representations of the data are fed to a central neural network which combines them into a common embedding. In addition, a multiobjective regularization is also introduced, helping to both optimize the central network and the unimodal networks. Experiments on four multimodal datasets not only show state-of-the-art performance, but also demonstrate that CentralNet can actually choose the best possible fusion strategy for a given problem.

Introduction and Related Work

In recent years, the multiplication of sensor technologies has resulted in a massive increase of multimodal data such as videos, RGB+D data, etc.. The fusion of modalities has been successful in various domains, from medical imaging to autonomous driving, including natural language processing or image analysis. The main motivation behind the use of multimodal data is to combine complementary information from different modalities, leading to better decisions than using only one. Many problems that benefit from multiple modalities can be found in the recent deep learning literature such as video classification [START_REF] Wang | Truly multi-modal youtube-8m video classification with video, audio, and text[END_REF], emotion recognition [START_REF] Ringeval | Summary for avec 2017: Real-life depression and affect challenge and workshop[END_REF], or human activity recognition [START_REF] Baradel | Glimpse clouds: Human activity recognition from unstructured feature points[END_REF].

A first basic taxonomy of multimodal fusion [START_REF] Lahat | Multimodal data fusion: an overview of methods, challenges, and prospects[END_REF] distinguishes between the approaches by the level of representation at which the fusion is done, as 1 deep neural networks process the data at different levels of abstraction, the different layers of the networks. A universal optimal level of fusion (early or late) does not exist, as it is task dependent. For example, Simonyan et al. [START_REF] Simonyan | Two-stream convolutional networks for action recognition in videos[END_REF]'s two stream convolutional neural network for human activity recognition is fusing modalities at prediction level (late fusion). In the same way, in audiovisual emotion recognition, better performance is observed using late fusion [START_REF] Ha | Multi-modal emotion recognition using semi-supervised learning and multiple neural networks in the wild[END_REF][START_REF] Vielzeuf | Temporal multimodal fusion for video emotion classification in the wild[END_REF]. In contrast, Arevalo et al. [START_REF] Arevalo | Gated multimodal units for information fusion[END_REF] propose a Gated Multimodal Unit, weighting the modalities features using relatively early fusion information on a textual-visual dataset. Finally, Chen et al. [START_REF] Chen | Multimodal sentiment analysis with word-level fusion and reinforcement learning[END_REF] follow an early fusion hard-gated approach for textual-visual sentiment analysis.

Nevertheless, the early versus late fusion paradigm can be insufficient to describe some approaches. Indeed, Neverova et al. [START_REF] Neverova | Multi-scale deep learning for gesture detection and localization[END_REF] highlight that there are benefits to fuse similar modalities earlier. Another example is the multilayer approach of Yang et al. [START_REF] Yang | Multilayer and multimodal fusion of deep neural networks for video classification[END_REF] performing the fusion on human activity videos with a boosting algorithm applied to all layers. Kang et al. [START_REF] Kang | Contextual region-based convolutional neural network with multilayer fusion for sar ship detection[END_REF] also use a multilayer framework where the layers' representations are aggregated into a unique representation.

Finally, it is worth mentioning some approaches relying on regularization: Andrew et al. [START_REF] Andrew | Deep canonical correlation analysis[END_REF] propose the deep Canonical Correlation Analysis method aiming at maximizing the correlation between the multimodal representations, while Neverova et al. [START_REF] Neverova | Moddrop: adaptive multi-modal gesture recognition[END_REF][START_REF] Li | Modout: Learning to fuse modalities via stochastic regularization[END_REF] propose modDrop and modout regularizations, consisting in dropping modalities during training.

The proposed method borrows from both visions: it is a multilayer framework and uses a constrained multi-objective loss. It builds on existing deep convolutional neural networks designed to process each modality independently. We suggest connecting these networks, at different levels, using an additional central network dedicated to the projection of the features coming from the different modalities into the same common space. In addition, the global loss allows to back propagate some global constraints on each modality, coordinating their representations. The proposed approach aims at automatically identifying an optimal fusion strategy for a given task. The approach is multi-objective in the sense that it simultaneously tries to minimize per modality losses as well as the global loss defined on the joint space. This article is an extension of [START_REF] Vielzeuf | Centralnet: a multilayer approach for multimodal fusion[END_REF].

After presenting our approach in the next section, Section 3 gives an extended experimental validation of the approach showing the advantage of our method on 4 different multimodal tasks.

CentralNet

In this text we use the phrase multimodal fusion to refer to the combination of information provided by different media, under the form of their associated features or the intermediate decisions. More formally, if M 1 and M 2 denote the media and D 1 and D 2 the decisions inferred respectively from M 1 and M 2 , the goal is to make a better decision D 1,2 using both M 1 and M 2 . More than 2 modalities can be used. This paper addresses the case of classification tasks, but any other task, e.g. regression, can be addressed in the same way.

This article focuses on the case of neural nets, for which the data is sequentially processed by a succession of layers. We assume having one neural net per modality, capable of inferring a decision from each modality taken in isolation, and want to combine them. One recurrent question with multimodal fusion is where the fusion has to be done: close to the data (early fusion as in Fig. 1.a), at the decision level (late fusion as in Fig. 1.b) or in between. In the case of neural networks, fusion can be performed at any level between the input and the output of the different unimodal networks.

For simplicity, we assume that the extracted features (at the input of the fusion layers) have the same dimensionality. If it is not the case, the features can be projected, e.g. with 1x1 convolutional layers or zero padded to give them the same size. In practice, the last convolution layers or the first dense layers of separately trained unimodal networks can be used as features.

CentralNet Architecture The architecture of CentralNet [START_REF] Vielzeuf | Centralnet: a multilayer approach for multimodal fusion[END_REF] is a neural network which combines the features issued from different modalities, by taking as input of each one of its layers, a weighted sum of the layers of the corresponding unimodal networks and of its own previous layers. This is illustrated in Figure 1(c). Such fusion layers can be defined by the following equation:

h C i+1 = α C i h C i + n k=1 α M k i h M k i (1)
where n is the number of modalities, α are trainable scalar weights, h M k i is the hidden representation of each modality at layer i, and h C i is the central hidden representation. The resulting representation h C i+1 is then fed to an operating layer cell (which can be a convolutional or a dense layer followed by an activation function).

Regarding the first layer of the central network (i = 0), as we do not have any previous central hidden representation, we only weight and sum the representations of all modalities (M i ) issued from unimodal networks. At the output level, the last weighted sum is done between the unimodal predictions and the central prediction. Then, the output of the central net is used as the final prediction.

Training the CentralNet model All trainable weights of the unimodal networks, the ones of the CentralNet and the fusion parameters α M k i , are optimized jointly by applying a stochastic gradient descent using the Adam approach. The global loss is defined as: loss = loss C + n k=1 loss M k where loss C is the (classification) loss computed from the output of the central

Late Fusion

Prediction model and loss M k the (classification) loss when using only modality k. As already observed by Neverova et al. [START_REF] Neverova | Moddrop: adaptive multi-modal gesture recognition[END_REF], when dealing with multimodal fusion, it is crucial to maintain the performance of each unimodal neural networks. It is the reason why the global loss also includes unimodal losses. It helps generalization by acting as a multi-objective regularization (see Fig. 3). Overall, CentralNet is easy to implement and can be built on top of existing architectures known to be efficient for each modality. We refer to [START_REF] Vielzeuf | Centralnet: a multilayer approach for multimodal fusion[END_REF] for implementation details. The resulting values of the α weights allow some interesting interpretations on where modalities are combined. For instance, getting α M k i values close to 0 for i > 0 is equivalent to early fusion, while having all the α C i close to 0 up to the last weighted sum would be equivalent to late fusion.

1 (b) M 1 M 2 Prediction Pred 1 Pred 2 h M 1 0 h C 0 h M 2 0 h M 1 1 h C 1 h M 2 1 h M 1 2 h C 2 h M 2

Experiments

This section provides extensive evaluations of CentralNet on four multimodal datasets, namely Montalbano [START_REF] Escalera | Chalearn looking at people challenge 2014: Dataset and results[END_REF], MM-IMDb [START_REF] Arevalo | Gated multimodal units for information fusion[END_REF], the 'animal' subset of Audioset [START_REF] Jort F Gemmeke | Audio set: An ontology and human-labeled dataset for audio events[END_REF] and AFEW [START_REF] Dhall | Collecting large, richly annotated facial-expression databases from movies[END_REF]. For each dataset, CentralNet is compared with four different fusion approaches: early fusion (concatenation of low-level features), late fusion (concatenation of the unimodal scores), ModDrop (our own implementation of ModDrop, giving the same results as the one in Neverova et al. [START_REF] Neverova | Moddrop: adaptive multi-modal gesture recognition[END_REF]) and Gated Multimodal Unit (GMU) (re-implemented following Arevalo et al. [START_REF] Arevalo | Gated multimodal units for information fusion[END_REF], our own implementation reaching slightly better performance than the original one). To assess the statistical significance of the results, 64 models with different random initialization are evaluated and we report the average performance and the standard deviation.

Experiments on Montalbano

Montalbano [START_REF] Escalera | Chalearn looking at people challenge 2014: Dataset and results[END_REF] gathers more than 14,000 samples of 20 Italian sign gesture categories. These videos were recorded with a Kinect sensor, capturing audio, skeleton joints, RGB and depth (4 modalities). The task is to recognize the gestures from the video data. The performance is measured as the macro accuracy, which is the average of the accuracy for each class . We used the features given by [START_REF] Neverova | Moddrop: adaptive multi-modal gesture recognition[END_REF]: audio features (size 350), motion capture of the skeleton (size 350), RGB+depth left/right-hand features (size 400). Features are zero-padded to give vectors of size 400, which is possible as done after convolution layers. Fusion architecture includes 1 multilayer perceptron per modality, each having 3 layers of size: 400 × 128, 128 × 42, 42 × 21.

CentralNet connects the 3 layers of the different modalities to the central network.

We use dropout (50% dropping) and batch normalization. The learning rate starts at 0.05 and is multiplied by 0.96 at each epoch, the batch size is of 42, containing two samples of each class, and the model is trained on 100 epochs for all experiments. For ModDrop, the modality drop probability is of 0.5.

The performance obtained with each modality varies from 46% (left hand) to 88% (motion capture of skeletons). The first column of Table 1 shows that both late fusion and early fusion give significant improvement compared to the best modality, thus suggesting some complementarity between the modalities. CentralNet outperforms other approaches. In addition, Fig. 2, gives more information regarding the learned fusion strategy by visualizing the weights proportion at each layer. At the first layer (layer 0), the weights reflect the dimensionality of the modalities. At the next layer, almost no information is taken from the modalities, while at layers 2 and 3, the weight given to each modality and to the central representation are relatively similar. This may be interpreted as a hybrid fusion strategy, mixing "early" and "late" fusions.

Experiments on MM-IMDb

The MM-IMDb dataset [START_REF] Arevalo | Gated multimodal units for information fusion[END_REF] contains 25,959 samples. One sample is composed of a movie plot, its associated poster and its genres (e.g. Drama, Fantasy). The task is to predict a movie's genre based on its plot and on its poster As in [START_REF] Arevalo | Gated multimodal units for information fusion[END_REF][START_REF] Kiela | Efficient large-scale multi-modal classification[END_REF], we measure performance with the micro F1 score. More details about other F1 score measures may be found in our workshop paper [START_REF] Vielzeuf | Centralnet: a multilayer approach for multimodal fusion[END_REF]. For these experiments, we use the features kindly provided by the authors of [START_REF] Arevalo | Gated multimodal units for information fusion[END_REF]. The visual features of size 4096 are extracted from the posters using the VGG-16 [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF] network pre-trained on Imagenet. The 300-d textual ones are computed with a fine-tuned word2vec [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF] encoder.

We build a multilayer perceptron on top of the features of each modality. For both modalities, the network has 3 layers of size input_size × 4096, 4096 × 512 and 512 × 23. The CentralNet architecture is the same, taking 4096-d vectors as inputs, zero-padding the textual features to reach the visual features' size.

We use dropout (50% dropping) and batch normalization. The learning rate is 0.01 and the batch size is 128. For Moddrop, the modality drop probability is of 0.25. The loss of the models is a weighted cross entropy, with a weight of 2.0 on the positive terms to balance precision and recall. More formally, the loss is:

loss = -y log(2σ(pred)) -(1 -y) log(1 -σ(pred)) (2)
with σ(pred) the sigmoid activation of the last output of the network and y the multiclass label. As recommended by Arevalo et al. [START_REF] Arevalo | Gated multimodal units for information fusion[END_REF], we also use early stopping on the validation set. Table 1 reports the performance measured on the test set. First of all, the worst std. dev. we observe is very small. Second, the best modality is text (60.2) and it outperforms image (47.8) by a large margin. Third, we note that the early and late fusion methods bring a large improvement and are not far from the CentralNet, which gives the best performance. Fig. 2 shows that CentralNet gives more weight to the first layers, indicating that an "earlier fusion" strategy is privileged in this case, even if the two modalities contribute significantly at all levels. Fig. 3 shows the evolution of the (normalised to 1) αs during training. We can see that the αs converge to stable values, favoring images at the first 2 layers (early fusion) and slightly favoring text in the last 2 layers (late fusion). We also see that the CentralNet loss makes the convergence faster in addition to making the performance slightly better.

Experiments on AFEW

AFEW [START_REF] Dhall | Collecting large, richly annotated facial-expression databases from movies[END_REF] is a dataset of 1,156 video clips extracted from movies recorded by a standard video-camera. The task is to predict the emotion of a person, among seven classes. The performance is measured as the average of the accuracy on 64 runs (and standard deviation is also provided).

The audio modality is encoded as 1,583-d feature vector, obtained with the OpenSmile toolkit, while the visual modality is following the same approach as [START_REF] Vielzeuf | An occam's razor view on learning audiovisual emotion recognition with small training sets[END_REF] and is a 512-d feature vector. CentralNet is made of a multilayer perceptron with one hidden layer [1583 -> 128 - > 7]. To perform the first weighted sum, the visual modality is zero-padded to the size of the audio modality. The choice of this architecture with a small number of parameters can be explained by the great risk of over-fitting on such a small dataset.

The third column of Table 1 reports the performance measured on the validation set during the different experiments, as the test set is not available. The visual modality achieves an accuracy of 52.5%, while the audio accuracy is of 34.2%. Thus, the visual modality is performing best by a large margin. Early fusion brings no benefits compared to the visual modality, while all other methods are better and more stable (smaller standard deviations). CentralNet brings the largest improvement and prefers a relatively late fusion strategy, as shown in Fig. 2 where the central weights are small in the first layers.

Experiments on Audioset

Audioset [START_REF] Jort F Gemmeke | Audio set: An ontology and human-labeled dataset for audio events[END_REF] is a large audiovisual dataset containing YouTube videos (therefore recorded by different types of video-cameras). We only focus on a subset containing 19,842 videos of 10 seconds of different pets (named "domestic animals, pets" in the ontology of audio set). These videos are annotated with 211 different class labels, which are not only relative to animals but also to context (e.g. "ringtone", "meow"). Therefore, the classification task is multilabel. To take this issue into account, we propose to use the micro F1-score.

Modalities are a spectrogram of the audio data, resized to 512x512 and the video sequence subsampled to 16x256x256 images. An inflated Resnet-50 is applied on the image sequence, in the same fashion as in [START_REF] Baradel | Glimpse clouds: Human activity recognition from unstructured feature points[END_REF]. For audio inputs, we use a model containing six convolutional layers (kernel size is 4 and stride is 2), an average two-dimensional pooling layer (7 by 7) and a fully connected layer. The first convolutional layer is outputting 64 filters, and the number of filters is doubled at each layer until having 2048 filters. The associated 8x8 maps are pooled to 1x1 and the obtained 2048 features are fed to the fully connected layer. Between each layer, we use batch normalization and reLU activation.

The CentralNet architecture is identical to the audio CNN with the first two convolutional layers removed (beginning with 256 filters). The weighted sum of the features, at a given layer, is the average of the features of the inflated Resnet-50 alongside the temporal dimension.

As training is time consuming, we made only one run. The 4th column of Table 1 reports the performance measured on this dataset. ModDrop score is not provided as the authors were not able to make it work on this dataset (convergence issues). Audio and visual models alone respectively achieve F1-scores of 55.2% and 54.8%. Late fusion is outperformed by GMU, itself outperformed by CentralNet. We note that early fusion did not bring any improvements here, mainly because the combined low-level features may not be relevant to each other. This is in line with the Fig. 2, where we observe that the central contribution is ignored in low-level layers, and progressively taken into account when getting closer to prediction level.

Comparison with other fusion methods

Table 1 shows that the proposed method outperforms 5 alternative fusion models (including the very recent approaches of [START_REF] Neverova | Moddrop: adaptive multi-modal gesture recognition[END_REF] and [START_REF] Arevalo | Gated multimodal units for information fusion[END_REF]) on 4 different datasets. The difference is statically significant according the two sample z test, with z values between 12 and 47 for 64 runs. Additional computational cost is marginal, as the number of parameters of the central network is marginal compared to the number of parameters of the unimodal networks, and does not require any additional iterations at training time.

Conclusions

This article, presenting CentralNet and experimentally validating it on 4 datasets, consolidates the idea that the best fusion strategy heavily depends on datasets and tasks. It is underlined by the fact that CentralNet is performing best on all of the 4 datasets. This is explainable by its capacity to automatically find at which layers the multimodal fusion should be performed. Moreover, the interpretation of the different fusion strategies is in line with the observations made in the literature on optimal depths for fusion. Future works will explore more flexible frameworks, using other operations than weighting sums and allowing more than one by one layer combinations.

Figure 1 :

 1 Figure 1: (a) Early fusion method, fusing the low-level representations of the modalities. (b) Late fusion method, fusing the high-level representations of the modalities (or even predictions) obtained for each unimodal neural network. (c) Our CentralNet fusion model, using both unimodal hidden representations and a central joint representation at each layer. The fusion of the unimodal representations is done in this case using a learned weighted sum. For the sake of simplicity, only the overall synoptic views of the architectures are represented. More details are provided in Section 2.

1 Figure 2 :

 12 Figure 2: Visualization of the α i weights after training. They are displayed as the percentage given to each modality and to the central hidden representations across layers and datasets. We observe that the learned fusion strategy is different for each dataset.

Figure 3 :

 3 Figure 3: Evolution of α values (normalized to 1) and F1-scores (on the test set) during training on MM-IMDb

Table 1 :

 1 Overall performance on the different datasets.

		Montalbano [6] MM-IMDb [2] AFEW [5] Animals [7]
	Fusion meth.	Macro accuracy	F1-score	Accuracy	F1-score
	Best moda.	88.00±0.30	60.20±0.2	52.5±0.7	55.2
	Early	97.81±0.05	63.0±0.24	40.1±1.2	50.1
	Late	97.54±0.02	63.5±0.14	55.3±1.2	58.3
	ModDrop [14]	98.19±0.03	62.4±0.15	53.2±0.9
	GMU [2]	97.98±0.04	63.0±0.17	55.1±0.8	58.7
	CentralNet	98.27± 0.03	63.9± 0.12	57.2± 0.4	59.1