Konstantinos Skianis
email: kskianis@lix.polytechnique.fr

Fragkiskos D Malliaros
email: fragkiskos.me@gmail.com

Michalis Vazirgiannis

Fusing Document, Collection and Label Graph-based Representations with Word Embeddings for Text Classification

Contrary to the traditional Bag-of-Words approach, we consider the Graph-of-Words (GoW) model in which each document is represented by a graph that encodes relationships between the different terms. Based on this formulation, the importance of a term is determined by weighting the corresponding node in the document, collection and label graphs, using node centrality criteria. We also introduce novel graph-based weighting schemes by enriching graphs with wordembedding similarities, in order to reward or penalize semantic relationships. Our methods produce more discriminative feature weights for text categorization, outperforming existing frequency-based criteria. Code and data are available online 1 .

Introduction

With the rapid growth of the social media and networking platforms, the available textual resources have been increased. Text categorization or classification (TC) refers to the supervised learning task of assigning a document to a set of two or more predefined categories (or classes) [START_REF] Sebastiani | Machine learning in automated text categorization[END_REF]. Well-known applications of TC include sentiment analysis, spam detection and news classification.

In the TC pipeline, each document is modeled using the so-called Vector Space Model (Baeza-Yates and [START_REF] Baeza-Yates | Modern Information Retrieval[END_REF]. The main issue here is how to find appropriate weights regarding the importance of each term in a document. Typically, the Bag-of-Words (BoW) model is applied and a document is represented as a multiset of its terms, disregarding co-occurence between 1 Code and data: github.com/y3nk0/Graph-Based-TC the terms; using this model, the importance of a term in a document is mainly determined by the frequency of the term. Although several variants and extensions of this modeling approach have been proposed (e.g., the n-gram model (Baeza-Yates and Ribeiro-Neto, 1999)), the main weakness comes from the underlying term independence assumption, where the order of the terms is also completely disregarded.

After the introduction of deep learning models for TC [START_REF] Blunsom | A convolutional neural network for modelling sentences[END_REF][START_REF] Kim | Convolutional neural networks for sentence classification[END_REF], recent work by [START_REF] Johnson | Effective use of word order for text categorization with convolutional neural networks[END_REF] shows how we could effectively use the order of words with CNNs [START_REF] Lecun | Convolutional networks for images, speech, and time series[END_REF]. In many cases though, space and time limitations may arise due to complex neural network architectures. As stated in work by [START_REF] Joulin | Bag of tricks for efficient text classification[END_REF], computation can still be expensive and prohibitive.

In this paper, we explore fast term weighting criteria for TC that go beyond the term independence assumption. The notion of dependencies between terms is introduced via a Graph-of-Words (GoW) representation model. Under this model, each term is represented as a node in the graph and the edges capture co-occurrence relationships of terms with a specified distance in the document. We implicitly consider information about n-grams in the document as well as the collection of documents -expressed by paths in the graph -without increasing the dimensionality of the problem. Furthermore, we introduce word-embedding similarities as weights in the GoW approach, in order to further boost the performance of our methods. Finally, we successfully mix document, collection and label GoWs along with word vector similarities into a single powerful graph-based framework. An overview of our approach is shown in Fig. 1.

Related Work

Term weighting schemes. A core aspect in the Vector Space Model for document representation, is how to determine the importance of a term within a document. Many criteria have been introduced with the most prominent ones being TF, TF-IDF [START_REF] Salton | Termweighting approaches in automatic text retrieval[END_REF][START_REF] Singhal | Pivoted document length normalization[END_REF][START_REF] Baeza-Yates | Modern Information Retrieval[END_REF][START_REF] Robertson | Understanding inverse document frequency: On theoretical arguments for idf[END_REF] and Okapi BM25 [START_REF] Stephen E Robertson | Okapi at trec-3[END_REF], while some recent ones include N-gram IDF [START_REF] Shirakawa | N-gram IDF: A global term weighting scheme based on information distance[END_REF]. [START_REF] Lan | A comprehensive comparative study on term weighting schemes for text categorization with support vector machines[END_REF] conducted a comparative study of frequency-based term weighting criteria for text categorization; one of their outcomes was that, in many cases, the IDF factor is not significant for the categorization task, leading to no improvement of the performance. It is interesting to point out here that, more specialized approaches have been proposed for specific classification tasks, such as the Delta TF-IDF method that constitutes an extension of TF-IDF for sentiment analysis [START_REF] Martineau | Delta tfidf: An improved feature space for sentiment analysis[END_REF]. However, most of the previously proposed frequency-based weights consider the document as a Bag-of-Words; that way, any structural information about the ordering or in general, syntactic relationship of the terms, is ignored by the weighting process.

Text categorization. A number of diverse approaches have been proposed for TC [START_REF] Joachims | Text categorization with suport vector machines: Learning with many relevant features[END_REF][START_REF] Mccallum | A comparison of event models for naive bayes text classification[END_REF][START_REF] Nigam | Text classification from labeled and unlabeled documents using EM[END_REF][START_REF] Sebastiani | Machine learning in automated text categorization[END_REF][START_REF] Kim | Some effective techniques for naive bayes text classification[END_REF]. The first step of TC concerns the feature extraction task, i.e., which features will be used to represent the textual content. Typically, the straightforward Bag-of-Words approach is adopted, where every document is represented by a feature vector that contains boolean or weighted representation of unigrams or n-grams in general. In the case of weighted feature vectors, various term weighting schemes have been used, with the most wellknown ones being TF (Term Frequency), TF-IDF (Term Frequency -Inverse Document Frequency). Although these weighting schemes were initially introduced in the NLP and IR fields, they have also been applied in the TC task. [START_REF] Paltoglou | A study of information retrieval weighting schemes for sentiment analysis[END_REF] reported that, in the case of sentiment analysis, extensions of the TF-IDF weighting schemes introduced in the IR field, can further improve the classification accuracy. A comprehensive review of this area is offered in the article by [START_REF] Sebastiani | Machine learning in automated text categorization[END_REF].

Deep Learning for TC. With the rise of deep learning models, CNNs were applied for text classification [START_REF] Blunsom | A convolutional neural network for modelling sentences[END_REF][START_REF] Kim | Convolutional neural networks for sentence classification[END_REF][START_REF] Johnson | Effective use of word order for text categorization with convolutional neural networks[END_REF]. Next, [START_REF] Zhang | Character-level convolutional networks for text classification[END_REF] presented Character-level CNNs for the task of TC. Finally, [START_REF] Joulin | Bag of tricks for efficient text classification[END_REF] proposed a novel text classifier which achieves equivalent performance to state-of-the-art TC models, with faster learning times. Our work does not focus on the classifier part, as the aforementioned methods, but on the extraction of better features.

Graph-based text categorization. In the related literature, most of the graph-based method for TC, rely on graph mining algorithms that are applied to extract frequent subgraphs, which are then used to produce feature vectors for classification [START_REF] Deshpande | Frequent substructure-based approaches for classifying chemical compounds[END_REF][START_REF] Jiang | Text classification using graph mining-based feature extraction[END_REF][START_REF] Rousseau | Text categorization as a graph classification problem[END_REF][START_REF] Nikolentzos | Shortest-path graph kernels for document similarity[END_REF]. The basic shortcoming of those methods stems from the computational complexity of the frequent subgraph mining algorithm. Furthermore, most of these methods require from the user to set the support parameter, which concerns the frequency of appearance of a subgraph. Close to our work are the approaches followed by [START_REF] Hassan | Random-walk term weighting for improved text classification[END_REF] and [START_REF] Fragkiskos | Graph-based term weighting for text categorization[END_REF]; they explored how random walks and other graph centrality criteria can be applied to determine the importance of a term.

Graph-based text mining, NLP and IR. Representing documents as graphs is a well-known approach in NLP and IR. TextRank algorithm, proposed by [START_REF] Mihalcea | Textrank: Bringing order into text[END_REF], was among the first works that considered a random walk model similar to PageRank, over a graph representation of the document, in order to extract representative keywords and sentences. Later, sev- eral methods for those tasks were followed [START_REF] Erkan | Lexrank: Graph-based lexical centrality as salience in text summarization[END_REF][START_REF] Litvak | Graph-based keyword extraction for single-document summarization[END_REF][START_REF] Boudin | A comparison of centrality measures for graph-based keyphrase extraction[END_REF][START_REF] Lahiri | Keyword and keyphrase extraction using centrality measures on collocation networks[END_REF][START_REF] Rousseau | Text categorization as a graph classification problem[END_REF]. Another domain where graph-based term weighting schemes have been applied is the one of ad hoc Information Retrieval [START_REF] Franc | Graph-of-word and TW-IDF: new approach to ad hoc IR[END_REF]). An interesting survey can be found in the work of [START_REF] Blanco | Graph-based term weighting for information retrieval[END_REF] for a detailed description of graph-based methods in the text domain.

Preliminaries and Background

Let D = {d 1 , d 2 , . . . , d m } be a collection of documents and let C = {c 1 , c 2 , . . . , c |C| } be the set of predefined categories. Text categorization is considered the task of assigning a boolean value to each pair (d i , c i ) ∈ D × C, i.e., assigning each document to one or more categories [START_REF] Sebastiani | Machine learning in automated text categorization[END_REF]. The main point here is how to find appropriate weights for the terms within a document. As we will present below shortly, our approach utilizes network centrality criteria.

Node Centrality Criteria. Centrality 2 represents a central notion in graph theory and network analysis in general; it constitutes of measures that capture the relative importance of the node in the graph based on specific criteria [START_REF] Newman | Networks: An Introduction[END_REF]. One important characteristic of the centrality measures is that they consider either local information 2 en.wikipedia.org/wiki/Centrality.

of the graph (e.g., degree centrality, in-degree/outdegree centrality in directed networks, weighted degree in weighted graphs, clustering coefficient) [START_REF] Newman | Networks: An Introduction[END_REF], or more global information -in the sense that the importance of a node is determined by the properties of the node globally in the graph (e.g., PageRank, closeness). Let G = (V, E) be a graph (directed or undirected), and let |V |, |E| be the number of nodes and edges respectively. Next, we define basic centrality criteria that are used in the proposed methodology.

Degree centrality. The degree centrality is one of the simplest local node importance criteria, which captures the number of neighbors that each node has. Let N (i) be the set of nodes connected to node i. Then, the degree centrality can be derived based on the following formula: degree centrality

(i) = |N (i)| |V |-1 . Closeness centrality.
Let dist(i, j) be the shortest path distance between nodes i and j. The closeness centrality of a node i is defined as the inverse of the average shortest path distance from the node to any other node in the graph: closeness

(i) = |V |-1 j∈V dist(i,j) .
PageRank centrality. PageRank counts the number and quality of edges to a node to determine a rough estimate of how important the node is:

PR(i) = 1-α |V | + α ∀(j,i)∈E PR(j) out-deg(j)
, where α is the teleportation probability and out-deg(i) denotes the out degree on node i.

In this section, we present the components of the proposed graph-based framework for TC.

Graph Construction

We model documents as graphs that capture dependencies between terms. More precisely, each document d ∈ D is represented by a graph

G d = (V d , E d ),
where the nodes correspond to the terms t of the document and the edges capture co-occurrence relationships between terms within a fixed-size sliding window of size w. That is, for all the terms that co-occur within the window, we add edges between the corresponding nodes of the graph. Note that, the windows are overlapping starting from the first term of the document; at each step, we simply remove the first term of the window and add the new one from the document. As graphs constitute rich modeling structures, several parameters about the construction phase need to be specified, including the directionality of the edges, the addition of edge weights, well as the size w of the sliding window. Fig. 2 gives a toy example of the construction of GoW for a collection composed by two documents.

To summarize, the key point of the graph-based representation for TC is the fact that it deals with the term independence assumption. Even if we consider the n-gram model, still information about the relationship between two different n-grams is not fully captured -as happens in the case of graphs. This has also been noted in other application domains (e.g., IR [START_REF] Franc | Graph-of-word and TW-IDF: new approach to ad hoc IR[END_REF]).

Term Weighting

Having the graph, the importance of a term in a document can be inferred by the importance of the corresponding node in the graph. In the previous section, we presented local and global centrality criteria that have been widely used for graph mining and network analysis purposes; here, we propose that those criteria can also be used for weighting terms in the TC task. That way, similar to TF, we can define the Term Weight (TW) weighting scheme as TW(t, d) = centrality(t, d), where centrality(t, d) corresponds to the score of term (node) t in the graph representation G d of document d. The interesting point here is that TW can be used along with any centrality criterion in the graph, local or global.

Furthermore, we can extend this weighting scheme by considering information about the inverse document frequency (IDF factor) of the term t in the collection D. That way, we can derive the TW-IDF model as follows:

TW-IDF(t, d) = TW(t, d) × IDF(t, D). (1)
In fact, TW and TW-IDF constitute suites for graph-based term-weighting schemes and thus, can be applied in any text analytics task. Some of them have already been explored in graphbased IR [START_REF] Franc | Graph-of-word and TW-IDF: new approach to ad hoc IR[END_REF] and keyword extraction [START_REF] Mihalcea | Textrank: Bringing order into text[END_REF][START_REF] Rousseau | Text categorization as a graph classification problem[END_REF].

The proposed weights are inferred from the interconnection of features (i.e., terms) -as suggested by the graph -and therefore information about n-grams is implicitly captured. That way, the feature space of the learning problem is kept to the one defined by the unique unigrams of our collection (instead of using simultaneously as features all the possible unigrams, bigrams, 3-grams, etc.), but the produced term weights incorporate n-gram information through the graph-based representation.

Inverse Collection Weight (ICW)

In this paragraph, we introduce the concept of Inverse Collection Weight (ICW) -a graph-based criterion to penalize the weight of terms that are "important" across the whole collection of documents. The main concept behind ICW is the collection level graph G -an extension of the Graphof-Words in the collection of documents D.

Definition 1 (Collection Level Graph G) Let {G 1 , G 2 , . . . , G d } |D| be the set of graphs that correspond to all documents d ∈ D. The collection level graph G is defined as the union of graphs G 1 ∪ G 2 ∪ . . . ∪ G d over all documents in the collection.

The union of two graphs G = (V G , E G ) and H = (V H , E H ) is defined as the union of their node and edge sets, i.e., G∪H = (V G ∪V H , E G ∪E H ). The number of nodes in graph G is equal to the number of unique terms in the collection, while the number of edges is equal to the number of unique edges over all document-level graphs (see also Fig. 2).

This graph captures the overall dependencies between the terms of the collection; the relative overall importance of a term in the collection will be proportional to the importance of the corresponding node in G. Following similar methodological arguments as used for IDF [START_REF] Robertson | Understanding inverse document frequency: On theoretical arguments for idf[END_REF], we define a probability distribution over the nodes of G (or equivalently, the unique terms of D), with respect to a centrality (term-weighting in our case) criterion; then, the probability of node (term) t will be:

Pr(t) = TW(t, D) v∈D TW(v, D)
.

(2)

Note that, in Eq. ( 2), we use D instead of G; we consider that the space defined by the document collection D is equivalent to the one defined by graph G with respect to the unique terms of the collection. This way, the notion of TW(t, D) used here is consistent with what was described earlier.

Based on this, we define the ICW measure as:

ICW(t, D) = max v∈D TW(v, D) TW(t, D) . ( 3
)
Instead of selecting the maximum centrality in the collection level (Eq. ( 3)), the sum of all centralities also yields good results. ICW shares common intuition with the inverse total term frequency described in [START_REF] Robertson | Understanding inverse document frequency: On theoretical arguments for idf[END_REF]. In fact, it can be considered as an extension of the total collection frequency of a term, to the graph-based document representation. Furthermore, similar to TW, it can be used along with any node centrality criterion.

Using ICW as a graph-based collection-level term penalization factor, we derive a new class of term-to-document weighting mechanism, namely TW-ICW. This weighting scheme is derived combining different local (i.e., document-level) and global (i.e., collection-level) criteria as follows:

TW-ICW(t, d) = TW(t, d) × log(ICW(t, D)).
In the case of TW and ICW, any centrality criterion can be applied. However, the computational complexity is a crucial factor that should be taken into account. Nevertheless, as we have noticed from the experimental evaluation, even using simple and easy-to-compute local criteria (e.g., degree), we achieve good classification performance. 2016) introduced supervised term weighting (TW-CRC) as a method to integrate class information with graphs. Similarly, we create a graph for each class (label), where we add all words of documents belonging to the respective class as nodes and their co-occurrence as edges. Our weighting scheme is a variant of TW-CRC; we define LW for a term t as:

Label Graphs

Shanavas et al. (

LW(t) = max(deg(t, L)) max(avg(deg(t, L)), min(deg(L))
,

where the maximum degree of term t in all label graphs (L) is divided by the max of two values: the average degree of the term in all label graphs (except the one having the max degree) and the min degree of all the terms in all the label graphs.

Then, we obtain ICW-LW as follows:

ICW-LW(t, d) = log(ICW(t, D) × LW(t)),
and multiply it with TW(t, d) to get TW-ICW-LW. Notice that, supervised frequency-based methods have also been proposed in previous work [START_REF] Debole | Supervised term weighting for automated text categorization[END_REF][START_REF] Huynh | A new term ranking method based on relation extraction and graph model for text classification[END_REF].

Edge Weighting using Word Embeddings

With our proposed framework, we can now use word embeddings [START_REF] Bengio | A neural probabilistic language model[END_REF] in order to extract similarities between terms. Our goal is to integrate these similarities in the graph representation as weights on the edge between two words.

The key idea behind our approach is that we want to reward semantically close words in the graphdocument level (TW) and penalize them in the collection level (ICW).

The most commonly used similarity between two words t 1 and t 2 in the word-embedding space is cosine similarity, which ranges between -1 and 1. In order to have a valid distance metric, we need to bound this between 0 and 1. We use the angular similarity to represent the weight of an edge between two words, and since the vector elements may be positive or negative, the formula becomes:

weight(t 1 , t 2 ) = 1 - arccos(sim(t 1 , t 2 )) π . (4)
The best performance was given by using Google's pre-trained word embeddings [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF] and not by learning them by the datasets. Since the words included in the pretrained version of word2vec are case sensitive and not stemmed, we did not apply any of these transformations. For words that do not appear in word2vec, we add a small value as similarity. Other distances (e.g. inverse euclidean, fractional) did not yield any further improvement. A similar approach for generic keyphrase extraction can be tracked in work by [START_REF] Wang | Corpus-independent generic keyphrase extraction using word embedding vectors[END_REF].

Providing more information in the weights, like number of co-occurrences between words, did not yield better results.

Classification Algorithms

Since the goal of this paper is to introduce new term weighting schemes, we rely on widely used classification algorithms. Specifically, we have used linear SVMs, due to their superior performance in TC [START_REF] Joachims | Text categorization with suport vector machines: Learning with many relevant features[END_REF]. Furthermore, as discussed in [START_REF] Leopold | Text categorization with support vector machines. How to represent texts in input space?[END_REF], the choice of the kernel function of SVM is not very crucial, compared to the significance of the term weighting schemes.

Experiments

We have evaluated our method on six freely available standard TC datasets, covering multi-class document categorization, sentiment analysis and subjectivity. Specifically: (1) 20NG 3 : newsgroup documents belonging to 20 categories, (2) REUTERS 3 : 8 categories of Reuters-21578, (3) WEBKB 3 : 4 most frequent categories of webpages from Computer Science departments, (4) IMDB [START_REF] Pang | A sentimental education: Sentiment analysis using subjectivity summarization based on minimum cuts[END_REF]): positive and negative movie reviews; (5) AMAZON [START_REF] Blitzer | Biographies, bollywood, boom-boxes and blenders: Domain adaptation for sentiment classification[END_REF]: product reviews acquired from Amazon over four different sub-collections; (6) SUBJEC-TIVITY [START_REF] Pang | A sentimental education: Sentiment analysis using subjectivity summarization based on minimum cuts[END_REF]: contains subjective sentences gathered from Rotten Tomatoes and objective sentences gathered from IMDB. A summary of the datasets can be found in Table 1.

In the experiments, linear SVMs were used with grid search cross-validation for tuning the C parameter. We also examined logistic regression, and observed similar performance. In the text 3 web.ist.utl.pt/acardoso/datasets/ preprocessing step, we have removed stopwords. No stemming or lowercase transformation was applied in order to match the words in word2vec.

For evaluation we use macro-average F1 score and classification accuracy on the test sets; that way, we deal with the skewed class size distribution of some datasets [START_REF] Sebastiani | Machine learning in automated text categorization[END_REF]. For the notation of the proposed schemes, we use TW (centrality measure) (e.g., TW (degree)) to indicate the centrality and TW-ICW (centrality at G, centrality at G) (e.g., TW-ICW (degree, degree)) for the local and collection-level graphs respectively. In TW-IDF (w2v), we compute the weighted degree centrality on the document level, with word-embedding similarities as weights. Similarly, in TW-ICW (w2v) we compute both weighted centralities for document and collection graphs. Finally, we denote as TW-ICW-LW the blending of TW, ICW and label graphs (LW). In label graphs we only make use of the degree centrality, since it is fast and performs best.

Results

Table 2 presents the results concerning the categorization performance of the proposed schemes for the six datasets. As discussed previously, the size of the window considered to create the graphs is one of the model's parameters. From the extensive experimental evaluation that we have performed, we have concluded that small window sizes give the most persistent results across various datasets and weighting schemes. For completeness in the presentation, we report results for two window sizes. In order to capture more information, we need larger window sizes for small datasets (e.g. SUBJECTIVITY). Also, since for the baseline methods (TF, TF binary, TF-IDF, w2v, TF-IDF-w2v) there is no notion of window size, the results for w = {2, 3} are the same. We have also examined several centrality criteria (using both undirected and directed graphs); undirected giving better results.

Comparing TF to the graph-based ones, namely TW (degree), in almost all cases TW gives higher F1 and accuracy results. Similar observations can be made in the case where the IDF penalization is applied. In most of the datasets, the TW-IDF (degree) scheme performs quite well. The interesting point here, which is confirmed by the related literature [START_REF] Lan | A comprehensive comparative study on term weighting schemes for text categorization with support vector machines[END_REF], is that TF-IDF is in general inferior to TF in TC. However, when the IDF penalization factor is applied on the TW term-to-document weighting, a powerful mechanism is derived. In the case of purely graph-based schemes, we can observe that some of them produce very good classification results. In almost all cases, TW-ICW-LW (degree or closeness) achieve the best performance.

Significant improvement is observed by adding the w2v similarities as weights in the document, collection level and label graphs in almost all datasets. In fact, we have obtained better results in 20NG (TW-ICW (w2v)), WebKB (TW-ICW (w2v)) and Reuters (TW-IDF(w2v)), by boosting semantically close words in the document level and penalizing them in the collection level.

TF n-gram binary scheme (TF binary) has also been examined, i.e., all the possible n-grams of the collection with binary weights (up to 6-grams in our experiments). For comparison reasons, the size of the unigram feature space considered by our framework is equal to the unique terms in the collections and much smaller compared to the n-grams ones. Moreover, graph-based weighting is able to outperform TF (binary) in all datasets.

We clearly see that by fusing document, collection and label graphs we obtain the best results in almost in 5 out of 6 datasets. Label graphs information consist a powerful weighting method, when combined with our proposed collection level graph approach. Adding word2vec similarities as weights, when label graphs are used, does not improve the accuracy. This implies that important terms concerning different labels can be close in the word vector space. Choosing closeness in the document level GoW yields the best performance in 3 datasets. Closeness can only have an affect in larger document lengths and when used along with label graphs.

To further investigate the effectiveness of our approach, we have compared our results with current state-of-the-art graph-based and non graphbased methods. In Table 3 we compare against CNN for text classification ,without pre-trained word vectors [START_REF] Kim | Convolutional neural networks for sentence classification[END_REF], FastText [START_REF] Joulin | Bag of tricks for efficient text classification[END_REF], TextRank [START_REF] Mihalcea | Textrank: Bringing order into text[END_REF], Word Attraction weights based on word2vec similarities [START_REF] Wang | Corpus-independent generic keyphrase extraction using word embedding vectors[END_REF] and Supervised Term Weighting (TW-CRC) by [START_REF] Shanavas | Centrality-based approach for supervised term weighting[END_REF]. Our work produces comparable to state-of-the-art results. Since the implementation of most models is our own, their performance is not optimal.

Selecting the window size w is also important. As we observed, the maximum accuracy is achieved while using small window sizes. In any case, even if larger values of w were able to get slightly better results, a smaller window size would be preferable, due to the overall overhead that could be introduced (increase of the density of the graph). Figure 3 depicts the F1 score and accuracy on the WEBKB, REUTERS and SUBJEC-TIVITY datasets, using the TW, TW-ICW and TW-ICW-LW(deg) schemes for various window sizes. We notice also that larger sliding windows are only improving accuracy in datasets with small document length (e.g. SUBJECTIVITY).

Conclusion & Future Work

In this paper, we proposed a graph-based framework for TC. By treating the term weighting task as a node ranking problem of interconnected features defined by a graph, we were able to determine the importance of a term using node centrality criteria. Building on this formulation, we introduced simple-yet-effective weighting schemes at the collection and label level, in order to penalize globally important terms (as analogous to "globally frequent terms") and reward locally important terms respectively. We also incorporate additional word-embedding information as weights in the graph-based representations.

Our proposed methods could also be applied in IR. In fact, document-level graph-based term weighting has already been applied there, so it would be interesting to examine the performance of the proposed collection-level (ICW) penalization mechanism. In the unsupervised scenario, where label information is not available, community detection algorithms may be applied to identify clusters of words or documents in collection graphs. Graph-based representations of text could also be fitted into deep learning architectures following the idea of [START_REF] Lei | Molding cnns for text: non-linear, non-consecutive convolutions[END_REF]. Lastly, one could examine a Graph-of-documents approach, in which we create a graph, where nodes represent documents and edges correspond to similarity between them. In this case, graph kernels could be utilized for graph comparison and/or Word Mover's distance [START_REF] Kusner | From word embeddings to document distances[END_REF] between two documents as weights.

Figure 1 :

 1 Figure 1: Blending different types of GoWs and word vector similarities in one framework.

Figure 2 :

 2 Figure 2: Example of document, collection-level and label GoWs for a collection composed by two documents and window size w = 3. Weights on the edges of G d 1 and G d 2 correspond to the similarity of two terms in the vector space. Here, Label GoWs are the same with Document GoWs (one document per class-label).

Figure 3 :

 3 Figure 3: F1 score (left) and accuracy (right) of TW, TW-ICW and TW-ICW-LW (all degree) on REUTERS, WEBKB and SUBJECTIVITY, for window size w = {2, . . . , 10}.

Table 1 :

 1 Datasets' statistics: #ICW shows the number of edges in the collection-level graph; #w2v: number of words that exist in pre-trained vectors.

		Train	Test	Voc	Avg #w2v #ICW
	IMDB	1,340	660 32,844 343 27,462 352K
	WEBKB	2,803 1,396 23,206 179 20,990 273K
	20NG	11,293 7,528 62,752 155 54,892 1.7M
	AMAZON 5,359 2,640 19,980 65 19,646 274K
	REUTERS 5,485 2,189 11,965 66	9,218 163K
	SUBJ.	6,694 3,293 8,639	11	8,097	58K

Table 2 :

 2 Macro-F1 and accuracy for window size w. Bold shows the best performance on each window size and blue the best overall on each dataset. * indicates statistical significance of improvement over TF at p < 0.05 using micro sign test. MAX and SUM state the best nominator for ICW in Eq. (3).

			20NG (MAX)			IMDB (SUM)			SUBJECTIVITY (MAX)
	Methods	w = 3	w = 4	w = 2	w = 3	w = 6	w = 7
		F1	Acc	F1	Acc	F1	Acc	F1	Acc	F1	Acc	F1	Acc
	TF	80.88 81.55	-		84.23 84.24	-		88.42 88.43	-
	w2v	74.43 75.75	-		82.57 82.57	-		87.67 87.67	-
	TF-binary (ngrams)	81.64 82.11*	-		83.02 83.03	-		87.51 87.51	-
	TW (degree)	82.37 83.00*	82.21 82.83*	84.82 84.84	84.67 84.69	88.33 88.33	89.00 89.00*
	TW (w2v)	81.88 82.51*	82.21 82.87*	84.66 84.69	84.52 84.54	87.75 87.57	87.66 87.67
	TF-IDF	82.44 83.01*	-		83.33 83.33	-		89.06 89.06*	-
	TF-IDF-w2v	82.52 83.09*	-		82.87 82.87	-		89.91 89.91*	-
	TW-IDF (degree)	84.75 85.47*	84.80 85.46*	82.86 82.87	83.02 83.03	89.33 89.34*	89.33 89.34*
	TW-IDF (w2v)	84.66 85.32	84.46 85.13	83.47 83.48	83.31 83.33	86.42 86.42	86.51 86.51
	TW-ICW (deg, deg)	85.24 85.80*	85.41 86.05*	84.98 85.00	85.13 85.15	89.30 89.31*	89.61 89.61*
	TW-ICW (w2v)	85.33 85.93*	85.29 85.90*	85.12 85.15	84.82 84.84	89.61 89.61*	87.30 87.30
	TW-ICW-LW (deg)	85.01 85.66*	85.02 85.66*	85.73 85.75	85.28 85.30	90.12 90.13*	90.27 90.28*
	TW-ICW-LW (w2v)	82.56 83.11*	82.24 82.81*	85.29 85.30	84.39 84.39	87.70 87.70	87.70 87.70
	TW-ICW-LW (pgr)	83.92 84.66	83.80 84.54	84.97 85.00	85.73 85.75	86.60 86.60	86.45 86.45
	TW-ICW-LW (cl)	84.61 85.22	84.71 85.27	87.27 87.27* 86.06 86.06	89.97 89.97*	90.09 90.10*
			AMAZON (MAX)			WEBKB (SUM)			REUTERS (MAX)
	Methods	w = 2	w = 3	w = 2	w = 3	w = 2	w = 3
		F1	Acc	F1	Acc	F1	Acc	F1	Acc	F1	Acc	F1	Acc
	TF	80.68 80.68	-		90.31 91.91	-		91.51 96.34	-
	w2v	79.05 79.05	-		84.54 86.58	-		91.35 96.84	-
	TF-binary (ngrams)	79.84 79.84	-		91.22 92.85	-		86.33 95.34	-
	TW (degree)	80.07 80.07			91.69 92.64	91.45 92.49	93.58 97.53*	93.08 97.25*
	TW (w2v)	80.07 80.07	79.54 79.54	91.70 92.64	91.00 92.06	93.09 97.35*	93.43 97.25*
	TF-IDF	80.26 80.26	-		87.79 89.89	-		91.89 96.71	-
	TF-IDF-w2v	80.49 80.49	-		88.18 90.18	-		91.33 96.80	-
	TW-IDF (degree)	81.47 81.47*	81.55 81.55*	90.38 91.70	90.47 91.84	93.80 97.30*	93.13 97.35*
	TW-IDF (w2v)	79.61 79.62	77.60 77.61	90.81 92.20	90.60 91.91	93.38 97.44*	93.87 97.44*
	TW-ICW (deg, deg)	82.08 82.08*	82.02 82.02*	91.72 92.78	91.42 92.49	92.91 97.35	93.59 97.39*
	TW-ICW (w2v)	80.86 80.87*	78.82 78.82	91.58 92.64	91.84 92.85	93.57 97.30*	92.96 97.25
	TW-ICW-LW (deg)	82.72 82.72*	82.91 82.91*	91.86 92.92	91.95 92.92	93.88 97.53*	93.48 97.35*
	TW-ICW-LW (w2v)	80.56 80.56	78.32 78.33	90.74 91.99	90.01 91.34	92.51 96.89	92.14 96.98
	TW-ICW-LW (pgr)	82.23 82.23*	82.46 82.46*	91.18 92.20	92.23 93.07	93.38 97.35*	93.37 97.35*
	TW-ICW-LW (cl)	82.90 82.91*	83.02 83.03*	92.72 93.57*	92.86 93.57*	93.12 97.25	92.87 97.21

Table 3 :

 3 Comparison in accuracy(%) to state-of-the-art deep learning and graph-based approaches.

		0.94		0.98
		0.93		0.96
	F1 Score	0.90 0.91 0.92	Accuracy	0.92 0.94
		0.89		0.90
		0.88		0.88
		0.87
		1 2 3 4 5 6 7 8 9 10 11 Window Size w		1 2 3 4 5 6 7 8 9 10 11 Window Size w
		Reuters: TW	WebKB: TW	Subj: TW
		Reuters: TW-ICW	WebKB: TW-ICW	Subj: TW-ICW
		Reuters: TW-ICW-LW	WebKB: TW-ICW-LW	Subj: TW-ICW-LW


Caroline Bouissou
email: carolinebouissou@gmail.com

Michel Petitjean
email: petitjean.chiral@gmail.com

Giovanni Fusco

Alice Guyon

Ulrich Kuhl

Uwe Meierhenrich

Frédéric Patras

Asymmetric Exchanges

Keywords: arts, symmetry, asymmetry measures, chirality, most scalene triangle 1

We emphasize the asymmetric character of exchanges between artists and scientists or mathematicians in the framework of multidisciplinary collaborations. As an example, we look at the perception of asymmetry in an artwork presented at the First European Asymmetry Symposium (FEAS), Nice, 15-16 March 2018, and its potential connection with the unsolved mathematical problem of maximal or asymptotically maximal asymmetric figures and distributions.

I INTRODUCTION

As stated by [START_REF] Poey | Strategies to integrate the arts in science[END_REF], active involvement in arts can help learners in science and other fields explore different perspectives and internalize new ideas and ways of thinking. But sciences cannot be reduced to their experimental aspects: each time an equation is written to provide a model describing some natural phenomena, in physics, chemistry, biology, earth sciences, geography, econometrics, human sciences, etc., it means that the scientists are using mathematical tools. Thus mathematics cannot be dissociated from sciences. And indeed, there are strategies to integrate the arts in mathematic [START_REF] Dacey | Strategies to integrate the arts in mathematics[END_REF]. [START_REF] Fenyvesi | Bridging art and mathematics: introduction[END_REF] noticed that both mathematics and arts are conceptual and symbolic 'languages' that humans have used in their attempts to depict their empirical perceptions and visions. Mathematical sculptors invoke mathematics not only for its elegant abstractions but also in hopes of speaking to the ways math underlies the world, often in hidden ways, as noticed by [START_REF] Omes | Armed with a knack for patterns and symmetry, mathematical sculptors create compelling forms[END_REF]. In fact it was even recently asked if artists lead the way in mathematics: see [START_REF] Adams | Did artists lead the way in mathematics? The Conversation[END_REF].

Then it is impossible to miss the major role of symmetry in arts, science and mathematics. This is not a new topic. [START_REF] Shubnikov | Symmetry in science and art[END_REF] noticed that the perception of symmetry in art and in nature has been appreciated since antiquity, a remarkable example being Islamic art: see [START_REF] Rózsa | Symmetry in Muslim arts[END_REF], [START_REF] Tennant | Medieval Islamic architecture, quasicrystals, and Penrose and girih tiles: Questions from the classroom[END_REF]. This perception of symmetry is illustrated by a number of images in the books of [START_REF] Hahn | Symmetrie als Entwicklungsprinzip in Natur und Kunst[END_REF][START_REF] Hahn | Symmetry as a developmental principle in nature and art[END_REF] and of [START_REF] Darvas | Symmetry[END_REF]. [START_REF] Jablan | Theory of symmetry and ornament[END_REF][START_REF] Jablan | Symmetry, ornament and modularity[END_REF] showed the major role of symmetry in ornaments. [START_REF] Gerdes | Le cercle et le carré. Créativité géométrique, artistique et symbolique de vannières et vanniers d'Afrique, d'Amérique[END_REF] gave numerous examples of geometric creativity of handmade basket makers all around the world. In his book, [START_REF] Kappraff | Connections: The geometric bridge between art and science[END_REF] wrote that design science is a subject which may be considered to be a geometric bridge between art and science and devoted a whole chapter to symmetry. The role of symmetry has been investigated in literature, e.g. by [START_REF] Pavlović | On symmetry and asymmetry in literature[END_REF] and in poetry by [START_REF] Bonch-Osmolovskaya | Creation ideas expressed through combinatorial literary forms in Elena Katsyuba's poetry[END_REF], and in music by [START_REF] Banney | Symmetry reduction in tonal and atonal music -Towards a thermodynamic model of music[END_REF], while numerous other contributors could have been cited in these fields. [START_REF] Stylianou | Symmetry in mathematics and art: an exploration of an art venue for mathematics learning[END_REF] noticed that familiarity with symmetry concepts allowed arts students to use abstract mathematical concepts as a tool in their art creation. Symmetry has also been tagged as connecting art and ethnography, linking geometric symmetry with interculturality: see [START_REF] Marchis | Symmetry and interculturality[END_REF].

The relations between arts and beauty have been outlined through symmetry by [START_REF] Weyl | Symmetry[END_REF]. It has been discussed by [START_REF] Mcmanus | Symmetry and asymmetry in aesthetics and the arts[END_REF] and by [START_REF] Zaidel | Asymmetry and symmetry in the beauty of human faces[END_REF]. As noticed by [START_REF] Molnar | Symmetry-making and -breaking in visual art[END_REF], symmetry has always been considered an important factor in visual aesthetics. Quoting the French philosopher Montesquieu (1757) "Une des principales causes des plaisirs de notre âme lorsqu'elle voit des objets c'est la facilité qu'elle a de les apercevoir et la raison qui fait que la symétrie plait à l'âme c'est qu'elle soulage", which could be translated in English by "One of the main causes of the pleasures of our soul when it sees objects is the ease with which they see them and the reason that symmetry pleases the soul is that it relieves". However, in a recent study [START_REF] Weichselbaum | Implicit and explicit evaluation of visual symmetry as a function of art expertise[END_REF] showed that with higher art expertise, the ratings for the beauty of asymmetrical patterns significantly increased, while participants to the questionnaire still preferred symmetrical over asymmetrical patterns. But perceptual symmetry is not always identical to the symmetry defined by the mathematicians. This can be illustrated by the work Escher (1959), one of the most famous artists having produced artworks considered to be based on symmetry: see [START_REF] Schattschneider | [END_REF].

The readers willing to read more about connections of arts with mathematics and sciences can find in annex A at the end of this paper a short list of journals and newsletters publishing papers in this field. However, while many artists play with symmetry, not all of them think to symmetry as a way to exchange ideas with scientists and mathematicians. Furthermore, these exchanges appear to be asymmetric in the sense that what is sent may differ from what is received, while both parties get benefits from these exchanges. In the next section we present artworks illustrating such facts.

II FIRST EUROPEAN ASYMMETRY SYMPOSIUM (FEAS): ARTS-SCIENCES CROSSED PERSPECTIVES

FEAS, March 2018

Complex asymmetric systems such as the origin and evolution of asymmetric life, asymmetric amplification, asymmetric structures, asymmetry in economy and art -to name a few -are far from being understood and expressed and we expect that fundamental answers can be found by a transdisciplinary approach only that systematically complements knowledge of traditional individual disciplines. FEAS (2018) is the first European symposium of its kind. It was intended to attract leading international scientists and researchers and to advance our trans-disciplinary understanding in asymmetry-related phenomena. As noticed in the FEAS [START_REF] Editorial | Asymmetry symposium unites economists, physicists and artists[END_REF] For the FEAS Caroline Bouissou composed a purified installation shown in figure 1 with recognizable everyday life elements (glasses and glass surfaces). The perception of asymmetry in "Reflections 2018" and its potential connection with the unsolved mathematical problem of maximal or asymptotically maximal asymmetric figures and distributions constituted an echo to the research of Michel Petitjean. At first glance, "Reflections 2018" evokes mirror symmetry: there are glasses under glasses, separated by a glass (the mirror). Looking closer, it can be seen that there are true reflexions and false reflexions: glass is a matter distributed unsymmetrically around the mirror, so that the whole mass distribution is asymmetric. The installation is based on the contradiction of the symmetry desire provoked by a gap, a problem, an asymmetry. It is easier to identify a symmetric shape than an asymmetric one. Shapes corresponding one to another are more comfortable to get. This perception is perturbated at the exact moment when a shape that should correspond to another does not correspond anymore: rupture. All elements of the installation are transparent. Glasses and glass are all made of transparent glass and elements confuse with the glass surface. As with a water surface there is a reflexion but here this reflexion is materially represented: objects are the reflection of their identical copy. Thus the artist evokes asymmetry (see also annex B). As mentionned in the FEAS Editorial (2018): More surprisingly, perhaps, asymmetry could be fundamental to aesthetics, too -perhaps reflecting what the art historian Martin Kemp calls a 'structural intuition' that lets us discern vitality in organic form; by contrast, geometric perfection creates a sense of sterility."

In the next chapter we analyse what could mean "most asymmetric", in a mathematical sense. Some of the results are surprising, and it is concluded in section IV that the approach of asymmetry done by Caroline Bouissou was pertinent. In annex B are presented additional artworks which are not in the core of this study, but which are connected to it. We plan to develop in a future work a pluridisciplinary reading of these additional artworks.

Art-mathematics exchange

We present two researches stemming from two different fields which found here a meeting point. Terms, purposes and reasonings, methods and media of application differ from one to the other. In research-creation the gesture accompanies the thought and follow a chain of explorations with aesthetic, formal and abstract experiments from which the work of art is the result submitted to the projections, feelings and interpretations of visitors. The scientific research is based on experiments answering protocols and results into reports mostly addressed to specialists. Here there is an exchange more than a collaboration, it is one of the points the authors try to develop. Reading the work of art under a mathematical prism adds to it a new possible lecture and makes it the vehicle to communicate a mathematical research. Glasses reflect in glass surface. Real and false reflections get superimposed, producing an offset. Each element has many dimensions of symmetry, the artist plays with the title and the description of the installation itself: "Reflection" being both the mirror reflection and the thought reflection. The objet and the surface of reflection have the same name "glass". Here the offset disposition of glasses is deeply moving the all symmetric composition into an asymmetric composition.

III MATHEMATICAL ASPECTS

Terminology

The word symmetry is usually applied to a large diversity of situations, not all of them being assumed to receive a mathematical meaning. A mathematical symmetry may be used as a way to modelize some physical situation where it is wished to recognize some symmetry. [START_REF] Weyl | Symmetry[END_REF] is often cited to have provided a mathematical definition of symmetry, while this latter is an intuitive definition rather than a mathematical one. [START_REF] Rosen | Symmetry rules. How science and nature are founded on symmetry[END_REF] considered that symmetry is "immunity to a possible change": this is an intuitive definition, too. We may retain the definition of [START_REF] Petitjean | A Definition of symmetry[END_REF], which is an unifying definition covering the cases of functions, geometric figures with or without colors, probability distributions, matrices, graphs, text strings, etc. Practically, all these cases can be handled through functions. The details are out of the scope of the present paper: the readers, including those who have only basic mathematical skills, are invited to look at the original publication of [START_REF] Petitjean | A Definition of symmetry[END_REF].

For clarity, we recall that two kinds of symmetry exist in the Euclidean space: direct symmetry and indirect symmetry. Roughly speaking, an object has a direct symmetry when it is recognized to be identical to itself after having been submitted to some combination of translations and rotations, the net result of this combination being assumed to be non null (i.e. the null result is the case of neither translation nor rotation). Still roughly speaking, an object has indirect symmetry when a mirror image of this object is recognized to be identical to the original object.

When the the object has no indirect symmetry, it is declared to be chiral, and when it has indirect symmetry, it is declared to be achiral: this is what stated Lord Kelvin (1904). It should be mentioned that an older analysis of mirror symmetry was done more than two centuries ago by [START_REF] Kant | Von dem ersten Grunde des Unterschiedes der Gegenden im Raume. Wöchentliche Königsbergische Frag-und Anzeigungs-Nachrichten[END_REF][START_REF] Kant | De mundi sensibilis atque intelligibilis forma et principiis[END_REF][START_REF] Kant | Prolegomemna zu einer jeden künftigen Metaphysik die als Wissenschaft wird auftreten können[END_REF]) while Kant's interpretation of mirror symmetry seems to be still discussed: see [START_REF] Hoefer | Kant's hands and Earman's pions: Chirality arguments for substantival space[END_REF]. In fact, it appears that chirality can be defined in any metric space, on not only in Euclidean spaces: see the full mathematical definition of [START_REF] Petitjean | Chirality in metric spaces[END_REF]; a more recent version of this paper is available in open access: see [START_REF] Petitjean | Chirality in metric spaces[END_REF].

There are chiral objects which have direct symmetry, and there achiral objects (i.e. having indirect symmetry) which have no direct symmetry. Of course, there are objects having both types of symmetries together, such as a cube.

Asymmetry is defined as the full absence of symmetry, as stated by [START_REF] Saha | Dissymmetry and asymmetry[END_REF], [START_REF] Hargittai | Symmetry through the Eyes of a Chemist[END_REF], and by [START_REF] Gal | Louis Pasteur, language, and molecular chirality. I. Background and dissymmetry[END_REF]. The word dissymetry seemed to be used for objects with a lack of some symmetry element, while in practice dissymetry appears to be an obsolete word for chirality: see [START_REF] Eliel | Infelicitous stereochemical nomenclature[END_REF], [START_REF] Mislow | Stereochemical terminology and its discontents[END_REF], [START_REF] Gal | Louis Pasteur, language, and molecular chirality. I. Background and dissymmetry[END_REF], and Saha and Chakraborty (2012).

Measuring asymmetry

Stating that an object is symmetric or not refers to an ideal situation. An object can be nearly symmetric, thus arising the problem of how to measure quantitatively its degree of asymmetry. Many attempts to build such measures were performed in various areas of science since the end of 19 th century: see the review of [START_REF] Petitjean | Chirality and symmetry measures: a transdisciplinary review[END_REF]. Most of these measures dealt with chirality and were shown to offer a serious lack of general applicability. The mathematical analysis of asymmetry measures being out of the scope of this present paper, we retain the following measures of asymmetry:

• DSI, the Direct Symmetry Index defined by [START_REF] Petitjean | On the root mean square quantitative chirality and quantitative symmetry measures[END_REF], which is a scale and isometry insenstive quantity taking values in the interval [0; 1], the value 0 being reached if and only if the object has a direct symmetry. DSI applies to finite set of points, with or without color constraints. • χ, the chiral index of [START_REF] Petitjean | About second kind continuous chirality measures. 1. Planar sets[END_REF][START_REF] Petitjean | On the root mean square quantitative chirality and quantitative symmetry measures[END_REF][START_REF] Petitjean | Chiral mixtures[END_REF], which is a scale and isometry insenstive quantity taking values in the interval [0; 1], the value 0 being reached if and only if the object has an indirect symmetry, i.e. meaning that the object is achiral. χ applies to discrete and continuous probability distributions, with or without color constraints. E.g., a finite set of n points is viewed as a sample of size n issued from a probability distribution, continuous or not. The chiral index measures a normalized distance between mirror images rather than the deviation from an idealized achiral object, because this latter is not ensured to exist: see [START_REF] Petitjean | À propos de la référence achirale[END_REF]. Once defined asymmetry measures, the next problem which arises is to characterize the most asymmetrical objects, if existing. The calculation has been made for various classes of objects. We give in table 1 the examples of the most asymmetric triangles.

In the case the colors are discarded, the chiral index is an asymmetry coefficient applicable to probability distributions of random vectors taking values in the d-dimensional space R d , d ≥ 1. When d = 1, at the difference of the widely used skewness introduced by Pearson (1895), the chiral index is null if and only if the distribution is achiral. As mentioned by [START_REF] Petitjean | About second kind continuous chirality measures. 1. Planar sets[END_REF] and in Petitjean (2003, section 2.9), the chiral index of a sample of n real values is easy to compute: sort the values in increasing order, then in decreasing order, and compute the correlation coefficient r m between these two ordered samples: χ = (1 + r m )/2.

Seeking the most asymmetric objects

Let us come back to the problem of characterizing the most asymmetric distributions (in the sense of indirect symmetry, i.e. mirror symmetry). The simplest example is the one of three equiprobable points on the real line. As calculated in [START_REF] Petitjean | The chiral index: Applications to multivariate distributions and to 3D molecular graphs[END_REF], χ = (1 -α) 2 /4(1 + α + α 2 ), where α is ratio of the lengths of the two adjacent segments defined by the three points. It can be shown that the most asymmetric distribution is reached when α = 0, i.e. when two of the three points have the same abscissa, which in fact means that we have one point with a weight of 1/3 and one point with a weight of 2/3: the maximal chiral index is here equal to 1/4. In this situation, we should speak about about the probability distribution of a random variable taking two values, one with a probability p=1/3 and the other one with a probability q=2/3 (thus p+q=1). Such a probability distribution is usually called a Bernoulli distribution with parameter p = 1/3 (see any textbook on statistics).

Going further in our quest of the most asymmetric distribution, it was shown by [START_REF] Petitjean | Chiral mixtures[END_REF] that, for a random variable, the upper bound of χ is 1/2, and it can be asymptotically approached for the Bernoulli distribution with the parameter p tending either to 0 or to 1. Now let us look at the characterization of the maximally chiral (asymptotically or not) distributions in the plane when d ≥ 2. The calculations are by for more difficult. When d = 2, [START_REF] Coppersmith | About the optimal density associated to the chiral index of a sample from a bivariate distribution[END_REF] show that the upper bound of χ fall on the interval [1 -1/π; 1 -1/2π] and gave an awkward example of a family of distributions for which χ approaches asymptotically the limiting value 1 -1/π, which they conjectured to be optimal.

But this is still an open problem. In higher-dimensional spaces (i.e. d ≥ 3), [START_REF] Petitjean | About the Upper Bound of the Chiral Index of Multivariate Distributions[END_REF] shown that the upper bound of χ is somewhere in [1/2; 1], but characterizing the most chiral distributions remains an open problem.

Number of The most asymmetric triangle

The most chiral triangle equivalent vertices (minimal direct symmety) (minimal indirect symmetry)

None Does not exist χ = 1 Equilateral triangle 2 DSI = 1 χ = 1 - √ 2/2 ≈ 0.292893 3 DSI = 1 - √ 2/2 ≈ 0.292893 χ = 1 -2 √ 5/5 ≈ 0.105573
Table 1: The most asymmetric triangles, according to [START_REF] Petitjean | About second kind continuous chirality measures. 1. Planar sets[END_REF][START_REF] Petitjean | On the root mean square quantitative chirality and quantitative symmetry measures[END_REF] However, in the three-dimensional space the maximum of the chiral index can be computed for specifice classes of objects. An example is the most chiral disphenoid, for which the chiral index χ = 3(13 -6 √ 2)/97 ≈ 0.139630: see [START_REF] Petitjean | The most chiral disphenoid[END_REF]. The triangular faces of this disphenoid are congruent to a scalene triangle which can be used for a funny application: anybody can remember how it was difficult for the teacher to draw a scalene triangle on the blackboard because it may be either too close to a right triangle or too close to an isosceles triangle; thus we propose to retain the latter scalene triangle because, as a face of the most chiral disphenoid, it is the farthest possible to the right triangle and to the isosceles triangle which both induce an achiral disphenoid. Of course, if an obtuse triangle is needed, no disphenoid can be built and thus we rather recommend the most chiral triangle in light blue presented at the right bottom of table 1. Both triangles are displayed in figure 2. Other most scalene triangles were proposed by [START_REF] Robin | The most scalene triangle[END_REF], and by [START_REF] Bowden | Spherical geometry and the least symmetric triangle[END_REF], but their builduing process cannot be generalized to other least symmetric figures.

Other examples of chiral three-dimensional objects are molecules: see [START_REF] Iupac | Basic terminology of stereochemistry[END_REF]. A commonly used molecular model is such that the atoms are represented by the nodes of a graph and the chemical bonds are represented by the edges connecting the atoms. This graph is realized in the three-dimensional space by attributing measured or computed coordinates to the atoms, considered to be ponctual. Thus the chiral index can be computed for any molecular conformer. Looking for extreme chirality molecules, Alan Schwartz designed the chiralanes and chirolanes families of molecules: see Schwartz (2004a). The most remarkable chiralane, shown in figure 3, has a chiral index close to 1. Seeking extreme chirality molecules has a potential application in physics: [START_REF] Schwartz | Affine vs. metric gravitation parity test[END_REF][START_REF] Schwartz | Gravitation symmetry test[END_REF] suggested that the Equivalence Principle (i.e. all substances fall at equal speeds independently of composition and internal structure) could be falsified for enantiomeric pairs.

IV DISCUSSION AND CONCLUSION Looking for the least symmetric figures or distributions is a topic neglected in the literature. It has been suggested by [START_REF] Field | Symmetry in chaos: a search for pattern in mathematics, art and nature[END_REF] that chaos could be the antithesis of symmetry because chaos is regarded to be chracterized by unpredictability and complexity, and because chaos is defined in the dictionary as "any condition or place of total disorder or confusion". This is unclear for the mathematician: in turn disorder needs to be defined, and then how to define the maximal disorder? Then, it is sometimes assumed that an other way to look at the opposite of symmetry could be to operate at random. Consider a rectangular domain of the plane and generate a sequence of samples of increasing sizes following the uniform law over this rectangular domain. In virtue of a theorem of [START_REF] Petitjean | Chiral mixtures[END_REF], the sequence of chiral indices of these samples converges to the chiral index of the parent population, which is zero: in this situation, "more random there is, more symmetry there is". So, which sample issued from this parent population could be expected to have the highest chiral index?

Looking at figure 1 and to its explanation given at the end of section II, we understand now why the location of the glasses should have been neither chaotic nor at random: none of these choices would have been pertinent to evoke asymmetry as best as possible. What took to the scientist several years of hard mathematical work was immediate evidence for the artist. Following the intuition of Caroline Bouissou, we think that there is no perfectly asymmetric object or figure (in the sense of mirror symmetry), without colors. Thus we emit the conjecture that the upper bound of the chiral index will never reach the value 1 for an unit mass distribution in the ddimensional space, d ≥ 3 (i.e. for a probability distribution of a random vector), because we think that no chiral distribution can approach the limiting value 1, even asymptotically. This latter is reachable only when color constraints are added (see section 3.2). Here again the artist was right: the glasses of figure 1 have no color, they are transparent! This shows why a collaboration with artists can be enlighting for the scientists. Such a collaboration is an asymmetric exchange: the scientist takes benefits from the creativity of the artist while the artist takes benefits from the mathematical results proved by the scientist.

B ANNEX 2

Symmetry and Asymmetry artworks below were all realized by Caroline Bouissou.

C ANNEX 3

The most scalene triangle of [START_REF] Petitjean | The most chiral disphenoid[END_REF] is congruent to the one of which the coordinates of the vertices p 1 , p 2 and p 3 are given below. These coordinates satisfy to: p 1 + p 2 + p 3 = 0.

We The angles at these vertices are respectively about 35.075, 60.475 and 84.450 degrees.

D ACKNOWLEDGEMENTS

We thank a lot the reviewers for having carefully read our manuscript and for having suggested several useful corrections.

Figure 1 :

 1 Figure 1: Reflections 2018. Glass and glasses: artwork designed by Caroline Bouissou. Photographs: © Caroline Bouissou. On the top, version presented at FEAS (2018). On the bottom: a variant. Installations by Caroline Bouissou. The installation may vary, the elements and the dimensions can change, the principle is the same: reflections, false reflections. Some of the glasses are disposed symmetrically, some are disposed asymmetrically. They maintain the glass surface between them. Glasses reflect in glass surface. Real and false reflections get superimposed, producing an offset. Each element has many dimensions of symmetry, the artist plays with the title and the description of the installation itself: "Reflection" being both the mirror reflection and the thought reflection. The objet and the surface of reflection have the same name "glass". Here the offset disposition of glasses is deeply moving the all symmetric composition into an asymmetric composition.

  , in function of the number of equivalent vertices: two vertices are equivalent when they have the same color. The triangles are represented by their colored vertices. The colors filling the interior of the triangles have no meaning: they were added for clarity. Top: the unequivalence of all vertices precludes the existence of any direct symmetry, while the the most chiral triangle with three unequivalent vertices (in yellow) is equilateral. Mid: the most asymmetric triangle with two equivalent vertices is degenerate: its aligned vertices are at abscissas proportional to (-1 -√ 3)/2, (-1 + √ 3)/2, 1; the most chiral triangle with two equivalent vertices (in dark blue) has sidelengths ratios 1 -√ 6/4 : 1 : 1 + √ 6/4. Bottom: the most asymmetric triangle with three equivalent vertices (in light red) has angles: π/4, π/8, 5π/8; the most chiral triangle with three equivalent vertices (in light blue) has sidelengths ratios 1

Figure 2 :

 2 Figure 2: Comparison of the two most scalene triangles. On the left: the obtuse triangle presented at the right bottom of table 1. On the right: the triangular face of the most chiral disphenoid defined by Petitjean (2015); the coordinates are given in annex C; the sidelengths ratios are 1 : 3 -√ 2/2 : √ 3. Strangely, squaring these ratios defines the sidelengths of an obtuse triangle shown to be the least symmetric one in the sense of Bowden et al (2018, see figure 7).

Figure 3 :

 3 Figure3: The two mirror images of the[6.6]chiralane, designed by Alan Schwartz. The chiral index of this compound is χ = 0.9824 and the one of its carbon skeleton is χ = 1.000; image and data from[START_REF] Schwartz | 6Chiralane: a remarkably symmetric chiral molecule[END_REF].

Figure 4

 4 Figure 4 Orchids and 5 Home war deal with left and right chirality introducing a human paradox of world understanding.

Figure 6

 6 Figure 6 Ambiguous landscape and 7 Observatoire show a landscape work dealing with a fragmentation of composition where symmetry and assymetry play as a dynamic.

Figure 4 :

 4 Figure 4: Orchids 2017: drawing serie by Caroline Bouissou. Photograph: © Caroline Bouissou. Drawings with red pen realized with the left hand and drawings wih the blue pen realized with the right hand. Two sketchbooks. 200 drawings.

Figure 5 :

 5 Figure 5: Home war 2004: video by Caroline Bouissou. Photograph: © Caroline Bouissou. Bilateral war committing only a single body and building a drawing. On the kitchen table, the ongoing war creates a drawing. Each adversary belongs to the same body. The drawing constructs itself, but what is more important is the process.

Figure 6 :

 6 Figure 6: Ambiguous landscapes 2015-... Photographs by © Caroline Bouissou. Photographs layers cut onto geometrical patterns (2015-ongoing). The addition of patterns layers creates a complex image.

Figure 7 :

 7 Figure 7: Observatoire 2015 (continued on next page): kinetic sculpture by Caroline Bouissou. Photograph: © Caroline Bouissou.

Figure 7 :

 7 Figure 7: continued. Observatoire 2015: kinetic sculpture by Caroline Bouissou. Photograph: © Caroline Bouissou. 78.74 × 157.48 × 0.59 inches submarine iron sheet folded and cut where are fixed three activatable wheels. The pattern composed by geometrical forms (geometricization of natural shapes) plays with the garden we see thought. Wheels movements modulate the viewer perception when patterns compositions overlap.

ISSN: 2430-3038, JIMIS, Creative Commons Volume: 4 -Year: 2018 , DOI: 10.18713/JIMIS-230718-4-1

J. of Interd. Method. and Issues in Science Open-access journal: jimis.episciences.org

A ANNEX 1 There are many journals and newsletters where symmetry papers dealing with arts and mathematics or sciences can be read. We report in table 2 some examples of such journals and newsletters. There are many other ones but we warn the readers that several of them are classified as predatory: see [START_REF] Beall | The open-access movement is not really about open access[END_REF][START_REF] Beall | Essential information about predatory publishers and journals[END_REF]; [START_REF] Sorokowski | Predatory journals recruit fake editor[END_REF] 8 Journal of the International Society for the Arts, Sciences and Technology (ISAST): see [START_REF] Malina | Leonardo[END_REF]. 9 Related to architecture and thus related to arts; see [START_REF] Williams | [END_REF].

10 Publishes sometimes special issues related to arts: see [START_REF] Sasaki | Symmetry and beauty[END_REF][START_REF] Tyler | Symmetry processing in perception and art[END_REF].

11 Founded in 2001 by [START_REF] Nagy | Symmetry: Art and Science[END_REF] and presented as the continuation of "Symmetry: Culture and Science" while the latter is still published; has published the proceedings of several symmetry conferences organized by ISIS-Symmetry since 2001.

12 It was suggested to use SIS as an alternative abbreviation of the name of the Society: see ISIS (2016). 13 The oldest journal fully devoted to all aspects of symmetry, founded by [START_REF] Darvas | Symmetry: Culture and Science[END_REF]