Nairit Bandyopadhyay
email: nairit.bandyopadhyay@grenoble-inp.org

Sébastien Riou
email: sebastien.riou@univ-grenoble-alpes.fr

Didier Schwab
email: didier.schwab@univ-grenoble-alpes.fr

EFFECT OF PERSONALIZED CALIBRATION ON GAZE ESTIMATION USING DEEP-LEARNING

Keywords: Gaze tracking, Deep Learning, Personal Calibration

With the increase in computation power and the development of new state-of-the-art deep learning algorithms, appearance-based gaze estimation is becoming more and more popular. It is believed to work well with curated laboratory data sets, however it faces several challenges when deployed in real world scenario. One such challenge is to estimate the gaze of a person about which the Deep Learning model trained for gaze estimation has no knowledge about. To analyse the performance in such scenarios we have tried to simulate a calibration mechanism. In this work we use the MPIIGaze data set. We trained a multi modal convolutional neural network and analysed its performance with and without calibration and this evaluation provides clear insights on how calibration improved the performance of the Deep Learning model in estimating gaze in the wild.

Introduction

Increase in interactions between human and computing devices is leading to the popularity of research in gaze-tracking. Specific domains like Human Computer Interaction and Computer Vision are researching on the various applications of appearance based gaze estimation or tracking. [START_REF] Morimoto | Eye gaze tracking techniques for interactive applications[END_REF]. Machine learning methods can be used to train gaze-estimators, provided we have lots of reliable data, and head pose-independent training data [START_REF] Sugano | Learning-by-Synthesis for Appearance-Based 3D Gaze Estimation[END_REF][START_REF] Schneider | Manifold Alignment for Person Independent Appearance-Based Gaze Estimation[END_REF]. These methods can bring appearance-based methods into a certain state where these methods do not require any user or device-specific training. The quantity of user data is sufficient to provide information to the gaze-estimators. Gaze tracking using monocular cameras in mobile phones, laptops and interactive displays are cost effective because of their widespread availability. While appearance-based gaze estimation performs well with machine learning-based methods, new techniques are still developed. These new techniques are evaluated on curated data sets collected under controlled laboratory conditions [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. As a result there is lack of availability of different eye shapes as well as we have to make assumptions that the head pose is accurate. This often results in problems in object recognition [START_REF] Torralba | Unbiased look at dataset bias[END_REF] and object detection [START_REF] Li | The secrets of salient object segmentation[END_REF].

In our study we have used MPIIGaze data set. The data set and annotations are publicly available online [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF].

Here in this article we are going to see how introduction of small amounts of test data in the training set can significantly increase the accuracy of the model. We are calling this approach as a personal calibration that is done in gaze estimation software -when a user starts using it for the first time. We have divided our work in two parts. First, we train a CNN model for appearance-based gaze estimation. The data set we are using is one order of magnitude larger than existing data sets and has more variation with respect to illumination and appearance [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. Second, we perform experiments on the model with and without calibration to see the effect in performance of the model.

Dataset and its features

We used MPIIGaze dataset that contains a total of 213,659 images from 15 participants. The number of images of each participant varied from 34,745 to 1,498. The data set contains larger variability in illumination and larger appearance variations [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. All these data were collected from the laptop because it can be used for daily recordings and are an important platform of eye-tracking devices [START_REF] Majaranta | Advances in physiological computing[END_REF].

Method

Figure 1 provides a high level view of gaze estimation task using multi modal convolutional neural networks (CNN). A monocular camera is used to capture the image of the face. Then the face detection and facial landmarks detection methods are used to locate landmarks in the image [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. A 3D facial shape model is used to estimate 3D poses of the detected faces and apply the space normalisation technique to crop and warp the head pose and eye images to the normalised training space [START_REF] Sugano | Learning-by-Synthesis for Appearance-Based 3D Gaze Estimation[END_REF][START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. Then we train a convolutional neural network to learn the mapping from the head poses and eye images to 2D gaze-points in the camera coordinate system.

Figure 1: Overview of our method for appearance-based gaze estimation using multi modal convolutional neural networks [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF].

Preprocessing : Face Alignment and 3D Head Pose Estimation

Before using the images to train the model, the images present in the MPIIGaze data set needs to be processed. The user's face is detected in the image using Li et al.'s SURF cascade method [START_REF] Li | Learning surf cascade for fast and accurate object detection[END_REF][START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. Afterwards, Baltrušaitis et al.'s constrained local mode framework to detect facial landmarks [START_REF] Baltrušaitis | Continuous conditional neural fields for structured regression[END_REF][START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. The 3D positions of six facial landmarks (eye and mouth corners, cf. Figure 1) constitutes the facial model. The head coordinate system is defined according to the triangle connecting three midpoints of the eyes and mouth. EPnP algorithm is used to fit the model by estimating the initial solution and further refining the pose via non-linear optimisation [START_REF] Lepetit | Epnp: An accurate o (n) solution to the pnp problem[END_REF][START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. 3D head rotation r is defined as the rotation from the head coordinate system to the camera coordinate system, and the eye position t is defined as the midpoint of eye corners for each eye [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF].

Normalizing the Preprocessed Data

The normalisation is done by two camera matrix transformation operations: scaling and rotation of camera [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. The x axes of the camera coordinate space and that of head coordinate space is made parallel by pointing the camera to one of the facial landmarks -the midpoint of eye corners. After that, the eye images were cropped at a fixed resolution W × H with a fixed focal length f in the normalised camera space, and histogram-equalised (to improve the contrast of the image) to form the input eye image. This results in a set of fixed-resolution eye images e and 2D head angle vectors h [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. In order to reduce the effect of different lighting conditions, so that no extra noise [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. The setting for camera distance d, focal length f and the resolution W ×H remains same [START_REF] Sugano | Learning-by-Synthesis for Appearance-Based 3D Gaze Estimation[END_REF].

Using Multimodal CNN To Learn The Mapping

The CNN learns the mapping from the input features (2D head angle h and eye image e) to 2D points g in the screen normalised space [START_REF] Sugano | Learning-by-Synthesis for Appearance-Based 3D Gaze Estimation[END_REF][START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. Our model uses a pre-activated ResNet like architecture that consists of one convolutional layer containing 3 stages, followed by batch normalisation and a final connected layer. There is a linear regression layer on top of the final connected layer to predict gaze positions g. The head pose information is introduced in our CNN model by concatenating h with the output of the fully connected layer. Our input for the network are the grey-scale eye images e with a fixed size of 60 ×36 pixels [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. The number of parameters for the convolutional layer is 144. For more details on the architecture please refer to Figure 2 The output of the network is a 2D gaze position g that consists of x and y coordinates (normalised) of the 2D screen. Our loss function is the sum of the individual losses that measure the euclidean distance between the predicted g and the actual g. Thus with the neural network we are solving an optimisation problem of minimising this loss function The normalized data set is available online for public use. Hence we did not have to perform the techniques in Section 3.1 and Section 3.2. We mainly worked on Section 3.3.

Experiments

Here we will be discussing about the person-independent gaze estimation task with and without calibration to validate the effectiveness of the proposed CNN based gaze estimation approach with personalized calibration. There are various ways of personal calibration, like fine-tuning the model in target domain [START_REF] Cheng | Appearance-based Gaze Estimation With Deep Learning: A Review and Benchmark[END_REF]. Some researchers have added some target person specific features for gaze tracking which are learned during fine tuning (Linden et al., 2019). We will be retraining the model with the target feature and observe if there are any changes in the performance of the model, with and without personalized calibration. Please refer to this Figure 3 to understand the pipeline of personalized calibration.

Equipment Used

All the training were done using GeForce GTX 1650 Max-Q. The average time to train 1 epoch having 12 steps is around 20 seconds at around 90% GPU utilisation.

Methodology

We conduct the experiments using the MPIIGaze dataset only. We choose 1,500 left eye samples and 1,500 right eye samples from each person so that we can take into account the sample number bias in the data set. Since one participant has only 1,448 images, we randomly over sampled the data to get 3,000. So for each person, we now have 3000 images [START_REF] Zhang | Appearance-based gaze estimation in the wild[END_REF]. We split these 3000 images into 10 partitions, each containing 10% i.e 300 images for calibration.

We used leave-one-out evaluation strategy. This means, if we want to evaluate the performance of a model on the data of person X, then we train a model using data of the remaining persons and some small percentage of data of person X.

Here we have used 10% of the test data for calibration. Then we test the resulting model A with the remaining 90% data of the person X. Again, we also have to train a model B that is without calibration and test with the same 90% test data that we used to test model A. This will now give us the difference in estimation error with and without calibration. For each person, we perform this experiment 10 times (because we have 10 partitions) so as to get the mean estimation error and see which partition provides best calibration for each person.

We trained every model for 40 epochs because we found that around 40 epochs, the training loss and validation loss converges and also the loss function is minimum at that time.

Results

As described in Section 5.1, the experiments were conducted. We had to conduct 150 experiments and in each experiment we had to create two models -with and without calibration. For each experiment we obtain two error values. It is important to note that the error value here is unit less because it is a relative error. Due to differences in screen size for each participant, the ground truth pixel coordinates values were normalized by dividing the x and y pixel coordinate by width and height of the screen in pixels respectively so that the pixel coordinate values remain between 0 and 1. Hence the error value also remains in between 0 and 1. For training and testing of model, such normalization was necessary, or else the model was not getting trained due to some out of range errors. However for real time setup where the model infers the gaze position on the screen, in that case the x,y coordinate values can be multiplied with dimensions of the screen so as to get the exact pixel position. The results obtained showed an interesting observation that calibration significantly reduces the mean error compared to without calibration. This is because the model now has some prior information about the person (See Figure 4).

Let's look at the results of p00. The mean error is 0.25 before calibration. It means this 0.25 is the difference in the As mentioned before, it has no unit because the pixel coordinates were normalised. After calibration, it is now 0.20. There has been a significant improvement in case of p00 where the error decreased by almost 20%.

Limitations

One of the major limitation is the process of head-pose estimation in real time. Since we used the normalized dataset of MPIIGaze, we did not have to perform the head-pose estimation. However if someone decides to create a new dataset of their own, then they have to perform this head-pose estimation. Sometimes the detection of facial landmarks and obtaining eye-images are also challenging because of lighting conditions and presence of accessories like when a person wears glasses. We did not have to do this as we used the clean normalized MPIIGaze dataset.

Future Work

By using only 300 extra images (10% of the test data) we have seen significant decrease in estimation errors. So our next work will be to create a real-time software with this calibration method. Before the software starts gaze-tracking, it will capture images of the face of the person using webcam. Most modern day webcams can shoot 30 times in one second. So we can capture around 450 images in 15 seconds by which we can calibrate a pre-trained model. After that we can start using that software to get a predicted 2-D position on the screen. We have already started creating such software which is still in development stage. As of now, it does not have the calibration feature and also we have not tested it rigorously. It can do the tasks of facial landmark detection and predicting the x,y coordinates of the estimated eye-gaze on a 2D screen. Please refer to Figure 5 which shows our software working in real time.

Apart from this, we can also try using other models like UNet etc. to see if it improves the performance of learning by the model. Thus, there are multiple scope of future works based on this work.

Conclusion

Appearance-based gaze estimation methods have so far been evaluated exclusively under controlled laboratory conditions. In this work, we present an extensive study on appearance-based gaze estimation along with a calibration technique. We used a data set that contains images which has wide variations. Our CNN-based estimation model significantly shows improvement in performance when calibration is performed. This work provides a critical insight on addressing the performance challenges of Deep Learning models in daily-life gaze interaction.

Figure 2 :

 2 Figure 2: ResNet Model Summary

Figure 3 :

 3 Figure 3: Higher level representation of personal calibration method for Deep Learning technique(Cheng et al., 2021)

Figure 4 :

 4 Figure 4: Leave-one-person-out evaluation on MPIIGaze. Mean estimation errors are plotted for each person for the proposed method of personalised calibration. Error bars indicate standard deviations when the experiments are conducted with and without calibration.

  Figure 5: Real Time 2D Gaze Points Estimation


Raphaël Bailly
email: rbailly@univ-paris1.fr

Laurent Leblond
email: laurent.leblond1@stellantis.com

Kata Gábor
email: kata.gabor@inalco.fr

Syntax and Geometry of Information

This paper presents an information-theoretical model of syntactic generalization. We study syntactic generalization from the perspective of the capacity to disentangle semantic and structural information, emulating the human capacity to assign a grammaticality judgment to semantically nonsensical sentences. In order to isolate the structure, we propose to represent the probability distribution behind a corpus as the product of the probability of a semantic context and the probability of a structure, the latter being independent of the former. We further elaborate the notion of abstraction as a relaxation of the property of independence. It is based on the measure of structural and contextual information for a given representation. We test abstraction as an optimization objective on the task of inducing syntactic categories from natural language data and show that it significantly outperforms alternative methods. Furthermore, we find that when syntax-unaware optimization objectives succeed in the task, their success is mainly due to an implicit disentanglement process rather than to the model structure. On the other hand, syntactic categories can be deduced in a principled way from the independence between structure and context.

Introduction

In the context of both human learning and statistical machine learning, what distinguishes generalization from memorization is the process of deliberately ignoring a part of the input information. In machine learning, the generalization will be guided towards some specific direction by the learning hypothesis: the model structure and the choice of regularization impacts the nature of the information loss. We can talk about syntactic generalization from textual data when the information pertaining to sentence structure tend to be preserved by the model and the information pertaining to other aspects of the text tend to be ignored. Syntactic generalization is of great interest because the human capacity to assign an abstract structure to utterances is a prerequisite to creatively combine constituents and understand novel sentences [START_REF] Frege | Über Sinn und Bedeitung[END_REF]. Knowledge of syntax can boost the robustness of NLP applications with respect to unseen data, in particular when there is a distribution shift [START_REF] He | Enhancing generalization in natural language inference by syntax[END_REF][START_REF] Wu | Improving the robustness of deep reading comprehension models by leveraging syntax prior[END_REF]. In a broader perspective, understanding syntactic generalization informs the discussion on the learnability of syntax from unlabelled text without any built-in grammatical knowledge or inductive bias [START_REF] Gold | Language identification in the limit[END_REF][START_REF] Clark | Unsupervised learning and grammar induction[END_REF][START_REF] Bailly | Emergence of syntax needs minimal supervision[END_REF]. Finally, studying syntactic generalization in large language models (LLMs) sheds light on whether and to what extent these models emulate human functioning with respect to linguistic competence.

The prevailing formalization of syntax is by means of algebraic compositional rules operating on a finite set of discrete categories (parts of speech). Language models can acquire syntactic knowledge when they are given specific supervision or bias [START_REF] Dyer | Recurrent neural network grammars[END_REF][START_REF] Shen | Explicitly modeling syntax in language model improves generalization[END_REF][START_REF] Sartran | Transformer grammars: Augmenting transformer language models with syntactic inductive biases at scale[END_REF]. Whether unsupervised settings can lead to syntactic generalization and under which conditions is still unknown. Current LLMs use distributed representations that cannot be unequivocally mapped to a set of categories, let alone syntactically meaningful categories. The question whether their representations encode syntactic information and how to uncover it is actively investigated today [START_REF] Hu | A systematic assessment of syntactic generalization in neural language models[END_REF][START_REF] Marvin | Targeted syntactic evaluation of language models[END_REF]. The majority of works in the topic of syntactic generalization in language models adopt an empirical approach, such as probing or analysis of a model with a comparison to actual or expected human performance on linguistically motivated tasks. In contrast, we present a theoretical approach to formalize syntactic generalization in an information theoretical framework.

Statistical learning can be formulated as the minimization of KL-divergence -a measure of information loss -subject to constraints. The constraints on model expressivity ensure that generalization takes place by eliminating the information resulting from sampling noise. We claim that the training objective of maximum likelihood estimation by nature does not incentivize models to syntactic generalization. In the case of syntactic generalization, the information loss needs to be directed to non-structural information, which is only remotely related to the elimination of sampling noise. First, a corpus is not randomly sampled from the set of grammatical sentences. Word co-occurrences in a corpus are indeed influenced by different factors such as semantics and pragmatics. The process of abstracting away from these factors is arguably different from the concept of generalization in machine learning, as the acquisition of syntactic knowledge always involves a shift of distribution [START_REF] Hupkes | State-of-the-art generalisation research in NLP: a taxonomy and review[END_REF]. Second, the target of generalization is the capacity to recognize the set of grammatical sentences: well-formedness is inherently a binary notion rather than a probabilistic one. These considerations motivate our proposition to decompose a corpus distribution as a factor of semantic/pragmatic context and a factor of structure representing well-formedness. In what follows, we reinterpret syntactic generalization based on the separation of structural and semantic information, and we show that our approach outperforms concurrent methods on unsupervised POS induction. We also define the notion of abstraction, an optimization objective specifically conceived for disentangling semantic information and syntactic well-formedness.

Related work

Generative linguists agree on the nativist argument that learners cannot converge on the same syntax unless some of their linguistic knowledge is innate [START_REF] Baker | Syntactic theory and the projection problem[END_REF][START_REF] Chomsky | Aspects of the Theory of Syntax[END_REF][START_REF] Chomsky | Reflections on language[END_REF], which makes the complete unsupervised learning of syntax impossible. Therefore, theoretical linguistics demonstrated little interest in machine learning and the interaction between the two fields is limited [START_REF] Lappin | Machine learning theory and practice as a source of insight into universal grammar[END_REF][START_REF] Linzen | Syntactic structure from deep learning[END_REF]. 1With the recent advent of large language models [START_REF] Devlin | BERT: Pre-training of deep bidirectional transformers for language understanding[END_REF][START_REF] Peters | Deep contextualized word representations[END_REF][START_REF] Radford | Language models are unsupervised multitask learners[END_REF] it has become relevant to test their linguistic competence [START_REF] Linzen | Assessing the ability of LSTMs to learn syntax-sensitive dependencies[END_REF][START_REF] Belinkov | Analysis methods in neural language processing: A survey[END_REF][START_REF] Baroni | Linguistic generalization and compositionality in modern artificial neural networks[END_REF]. Researchers in NLP thus turned to linguistic theory to create probing tasks [START_REF] Alain | Understanding intermediate layers using linear classifier probes[END_REF][START_REF] Giulianelli | Under the hood: Using diagnostic classifiers to investigate and improve how language models track agreement information[END_REF] or test sets targeted at specific linguistic knowledge [START_REF] Linzen | Assessing the ability of LSTMs to learn syntax-sensitive dependencies[END_REF]. Linguistic challenges like long-distance agreement [START_REF] Linzen | Assessing the ability of LSTMs to learn syntax-sensitive dependencies[END_REF], hierarchical syntax [START_REF] Lin | Open sesame: Getting inside BERT's linguistic knowledge[END_REF][START_REF] Dyer | Recurrent neural network grammars[END_REF][START_REF] Conneau | What you can cram into a single $&!#* vector: Probing sentence embeddings for linguistic properties[END_REF][START_REF] Hupkes | Visualisation and 'diagnostic classifiers' reveal how recurrent and recursive neural networks process hierarchical structure[END_REF], parts of speech [START_REF] Saphra | Language models learn POS first[END_REF][START_REF] Kim | Testing for grammatical category abstraction in neural language models[END_REF], or morphology [START_REF] Belinkov | What do neural machine translation models learn about morphology[END_REF][START_REF] Peters | Deep contextualized word representations[END_REF] have been applied to probe the latest language models with contrasting results. Recently, probing classifiers have also been subjected to methodological criticism. Models can succeed on some test tasks by learning shallow heuristics [START_REF] Mccoy | Right for the wrong reasons: Diagnosing syntactic heuristics in natural language inference[END_REF][START_REF] Poliak | Hypothesis only baselines in natural language inference[END_REF]. It was also argued that the presence of sufficient information to learn a given task does not entail alone that models rely on it [START_REF] Ravichander | Probing the probing paradigm: Does probing accuracy entail task relevance[END_REF]; [START_REF] Hewitt | Designing and interpreting probes with control tasks[END_REF]; [START_REF] Xu | A theory of usable information under computational constraints[END_REF]. On rarer occasions, studies aimed to test the capacity of language models to predict grammaticality judgments. Out of distribution testing systematically shows that the performance drops when the test data contains natural or artificial examples which are deliberately different from the training examples (Lake and [START_REF] Brenden | Generalization without systematicity: On the compositional skills of sequence-to-sequence recurrent networks[END_REF][START_REF] Marvin | Targeted syntactic evaluation of language models[END_REF][START_REF] Absar | RNN simulations of grammaticality judgments on long-distance dependencies[END_REF][START_REF] Van Schijndel | Quantity doesn't buy quality syntax with neural language models[END_REF][START_REF] Hall | Do syntactic probes probe syntax? Experiments with Jabberwocky Probing[END_REF].

Another branch of model analysis and interpretation studies are concerned with the nature of the generalization that takes place, with a particular accent on the notion of compositionality [START_REF] Loula | Rearranging the familiar: Testing compositional generalization in recurrent networks[END_REF][START_REF] Baroni | Linguistic generalization and compositionality in modern artificial neural networks[END_REF][START_REF] Valvoda | Benchmarking compositionality with formal languages[END_REF]. Among others, [START_REF] Fodor | Compositionality Papers[END_REF] and [START_REF] Kottur | Natural language does not emerge 'naturally' in multi-agent dialog[END_REF] claim that syntactic compositionality [START_REF] Chomsky | Syntactic Structures[END_REF][START_REF] Chomsky | Aspects of the Theory of Syntax[END_REF] is a prerequisite to learn to generalize to complex unseen input. In empirical studies, [START_REF] Gulordava | Colorless green recurrent networks dream hierarchically[END_REF] and [START_REF] Lakretz | The emergence of number and syntax units in LSTM language models[END_REF] report a lack of compositionality in the models they analyse, despite their impressive performance. In contrast, [START_REF] Bastings | Jump to better conclusions: SCAN both left and right[END_REF] and [START_REF] Valvoda | Benchmarking compositionality with formal languages[END_REF] find that some compositional relations can be learned by neural sequenceto-sequence models. [START_REF] Chaabouni | Compositionality and generalization in emergent languages[END_REF] argue that there is no correlation between the compositionality of an emergent language and its ability to generalize.

The problem of conflation between semantic and syntactic information in language models has been identified (Maudslay and Cotterell, 2021) as a factor hindering syntactic generalization. A new line of research is concerned with disentangling syntactic and semantic information in representations [START_REF] Felhi | Disentangling semantics in language through vaes and a certain architectural choice[END_REF][START_REF] Huang | Disentangling semantics and syntax in sentence embeddings with pre-trained language models[END_REF] by adversarial training or syntactic supervision. In order to incite syntactic generalization in models, [START_REF] Shen | Explicitly modeling syntax in language model improves generalization[END_REF] and [START_REF] Dyer | Recurrent neural network grammars[END_REF] propose to integrate explicit syntactic information for language modelling. [START_REF] Hu | A systematic assessment of syntactic generalization in neural language models[END_REF] show that there is a trade-off between the general language modelling objective and syntax-specific performance.

Some recent work relies on information theory to improve our understanding of the syntactic knowledge in LMs. [START_REF] Pimentel | Information-theoretic probing for linguistic structure[END_REF] reformulates probing as approximating mutual information between a linguistic property and a contextual representation. Subsequently, [START_REF] Pimentel | A bayesian framework for information-theoretic probing[END_REF] introduced Bayesian Mutual Information, a definition that allows information gain through processing. [START_REF] Voita | Information-theoretic probing with minimum description length[END_REF] use Minimum Description Length to measure regularity in LM representations with respect to the labels to be predicted in a linguistic probe. Our work builds on the propositions formulated in [START_REF] Bailly | Emergence of syntax needs minimal supervision[END_REF] who address the problem of the learnability of grammar by separating syntactic and semantic information in a corpus.

Syntactic Representation

Autonomy of Syntax

The concept of generalization we introduce is based on the autonomy of syntax [START_REF] Chomsky | Syntactic Structures[END_REF][START_REF] Chomsky | Some concepts and consequences of the theory of government and binding[END_REF][START_REF] Adger | The autonomy of syntax[END_REF] reinterpreted in terms of statistical independence. In the process of linguistic generalization, learners need to abstract away from semantic, pragmatic and idiosyncratic lexical information in the input they are exposed to. With a string prediction task and likelihood maximization as a training objective, models have no incentive to abstract away from these features. One can expect a statistical learner to ignore sampling noise, but the above features are relevant to learn the distribution behind a corpus. This insight motivates our proposition of statistical abstraction, a training objective that focuses on certain aspects of the input while deliberately ignoring others.

We want our learner to concentrate on the structure and ignore the factors we call context, i.e. all the aspects that are unrelated to well-formedness. We do so by creating two representations of the input: one of them structured, the other having structural information removed but co-occurrence relations conserved.

Let us consider a small artificial example for illustration. Our observation is a corpus with the two sentences below: cats eat mice men build houses A valid syntactic generalization would recognize the sentence cats build mice as grammatical. In order to do so, we consider p(cats eat mice) as a factor of the probability of the co-occurrence of its words in the same context : p({cats, eat, mice}) and a factor of the probability of the words to appear in a given structure : p(cats eat mice|{cats, eat, mice})

A syntactic representation with a desirable degree of generalization would identify the distributional classes {cats, men}, {build, eat}, {mice, houses}.

This set of distributional classes can be seen as a function f that associates a word (e.g cats) with its class ({cats, men}). Our goal is to study the properties of such a function so that it can be considered as achieving syntactic generalization, for instance: p(cats eat mice|{cats, eat, mice}) can be deduced from

p(f(cats) f(eat) f(mice)|{f(cats), f(eat), f(mice)})

Properties of a Syntactic Partition

We define the probability distribution that predicts the grammaticality of sequences, learned from observation. In order to do so, we first define a partition of words into abstract categories. This mapping, together with the category sequences found in the corpus, will allow us to induce the grammar.

Behind the corpus data there is a probability distribution p(w 1 w 2 . . . w n ). This distribution can be written as a product of two factors. First, the unstructured data, i.e. the probability of the elements of the vocabulary to occur in the same sequence without considering their order. Second, the probability of these elements to be observed in a particular structure. The contextual information is related to the former, and the structural information to the latter.

Let us see a probabilistic interpretation. Let A be the vocabulary, one defines the set A + = A * \ε where ε is the empty sequence. w = w 1 . . . w n ∈ A n a sequence (of words) of length n and p({w 1 , . . . , w n }) the probability of observing these elements in the same sequence, in any order. A trivial decomposition of p(w 1 w 2 w 3 ) would be

p({w 1 , w 2 , w 3 })p(w 1 w 2 w 3 |{w 1 , w 2 , w 3 })
However, we want structural information to be independent of the context. The decomposition above does not suppose the autonomy of structure. We propose to transform the above distribution with a mapping f , which will induce a partition over the elements of the vocabulary. In what follows, we examine which properties of this mapping will ensure that the categories of the resulting partition do not contain contextual information, while still preserving the information necessary to predict grammaticality. Let A be the vocabulary, one defines the set A + = A * \ε where ε is the empty sequence.

w = w 1 . . . w n ∈ A n |w| = length of w f (w) = f (w 1 ) . . . f (w n ) f [w] = {w ′ ∈ A + | f (w ′ ) = f (w)} for σ ∈ S n , σ(w) = w σ(1) . . . w σ(n) ⟨⟨W ⟩⟩ = ∪ σ∈Sn,w∈W {σ(w)} µ(w) = card({σ ∈ S n | w = σ(w)})
Let f (w) denote the sequence of categories resulting from the mapping of a word sequence, and f [w] the set of sequences that map to f (w). W denotes a set of sequences w. In the case of a singleton we will denote ⟨⟨w⟩⟩ = ⟨⟨{w}⟩⟩. The contextual information will be modeled through the probability p(⟨⟨w⟩⟩), where one can see the object ⟨⟨w⟩⟩ as a bag of words, from which the information of the structure (order) has been erased. A syntactically relevant representation needs to meet two criteria: it has to allow to recover the structure, i.e. the ordering of the bag of words, and it needs to be independent of contextual information. The first criterion is defined below as factorization, the second as minimality.

Factorization. One will say that a mapping f factorises a distribution p if the order of a bagof-words {w i } drawn from p can be entirely deduced from the knowledge of the corresponding categories. Definition 1. Let p be a distribution over A + , and f : A → B be a mapping. The distribution p is factorised by f if there exists a mapping λ f (⟨⟨w⟩⟩)

such that ∀w ∈ A + p(w | ⟨⟨w⟩⟩) = λ f (⟨⟨w⟩⟩) p(f [w] | ⟨⟨f [w]⟩⟩) in that case, one has λ f (⟨⟨w⟩⟩) = µ(w)
µ(f (w)) . In the case where f factorises p, one will say that context and structure are independent conditionally to f . Independence. As the property of factorization does not guarantee the complete independence of structure and context (for instance the identity always factorises p), we need to limit the information carried by f to its minimal value in order to reach this independence. From f [w] one can deduce, at the minimum, the length of w. The purpose of minimality is to ensure that knowing f [w] provides no further information for finding w: Definition 2. Let p be a distribution over A + and let f : A → B be a mapping. We will say that f is (information)-minimal for p if

∀w ∈ A + , p(w | f [w]) = p(w | A |w| )
We will say that context and structure are independent in p if there exists an informationminimal factorization of p.

Induced grammar

From a probability distribution p and a mapping f , it is possible to induce a syntax based on the observed patterns: a sequence is structurally correct if its pattern corresponds to an observed pattern. Definition 3. Let p be a distribution over A + and let f : A → B be a mapping. One denotes the syntax induced by p and f by

w ∈ G(p, f ) ⇔ p(f [w]) > 0
One has for instance G(p, id) = supp(p): this representation is a memorization with no generalization.

Minimal syntax. A syntax induced by minimal factorization of p will be called minimal syntax. The set of all minimal syntaxes will be denoted G * (p).

It can be shown that the intersection of all minimal syntaxes of p is a minimal syntax of p:

G * (p) = ∩ f ∈G * (p) G(p, f ) ∈ G * (p)
Hence, if the independence between context and structure holds, there exists a canonical way to define the set of well-structured sequences which is different from the support of p.

Example 1. Let us consider the first example above: let p be the distribution defined by p(cats eat mice) = 1 2 p(men build houses) = 1 2 then the mapping f defined by

f (cats) = f (men) = b 0 f (eat) = f (build) = b 1 f (mice) = f (houses) = b 2
is a minimal factorization of p. The minimal syntax G * (p) is the set cats eat mice cats eat houses cats build mice cats build houses men eat mice men eat houses men build mice men build houses

Geometry of Information

Using information theoretical tools, we transform the criteria above into metrics and define an information space which allows to track the amount of contextual and structural information in a partition, as well as the direction of generalization during a training process.

The concept of minimal factorization provides the formal definition of minimal syntax; however, the conditions of factorization (Definition 1) and minimality (Definition 2) are restrictive. In natural language corpora, a perfect independence between semantic context and grammaticality cannot be expected. Syntax and semantics do interface in natural language, semantic acceptability interacts with grammaticality and depending on how one deals with this interface, either the assumption of perfect independence or the precise retrieval of the distribution underlying the corpus may not be met. This motivates our methodology for relaxing both conditions in a way that gives an equivalent but quantifiable formulation for each criterion in terms of information. We thus provide a method to measure the amount of structural information present in a partition, hence relaxing the factorization criterion. We also define contextual information, which relaxes the minimality requirement.

Structural information

Let

H(p ∥ q) = - w∈A + p(w) log(q(w))
be the cross entropy of the distribution q with respect to the distribution p.

For a distribution p over A + , we will consider the distance (in terms of cross-entropy) between p and the class of factorised distributions. Definition 4. Let p be a distribution over A + and let f be a mapping. One denotes:

F f = {q | q is factorised by f }
and one defines the projection of p conditionally to f by

p |f = arg min q∈F f H(p ∥ q)
The structural information of f with respect to p is given by

i s (p ∥ f ) = H(p ∥ p |z ) -H(p ∥ p |f )
where z is the null mapping.

The set F f represents the set of distributions for which the knowledge of f is sufficient to recover the order of a sequence. The structural information is minimal for z, and maximal for the identity (see Appendix):

i s (p ∥ z) = 0 ≤ i s (p ∥ f ) ≤ i s (p ∥ id)
The link between structural information and factorization is given by: Lemma 1. Let p be a distribution over A + and let f : A → B be a mapping. One has

i s (p ∥ f ) is maximal ⇔ f factorises p

Contextual information

An optimal syntactic representation is one that fulfills the independence requirement: the probability of a sequence of categories does not provide information about which actual words are likely to appear in the sentence. The contextual information will measure the amount of lexical or semantic information that is present in a representation.

Let

H(p) = H(p ∥ p)
be the Shannon entropy. Let p be a distribution over A + and let f : A → B be a mapping. One will denote p • f -1 the distribution on B + induced by f . One has p

• f -1 (f (w)) = p(f [w]).
Definition 5. The contextual information of f with respect to p is given by

i c (p ∥ f ) = H(p • f -1 ) -H(p • z -1 )
where z is the null mapping.

From standard properties of Shannon entropy, i c (p ∥ f ) is minimal for z, and maximal for the identity (see Appendix):

i c (p ∥ z) = 0 ≤ i c (p ∥ f ) ≤ i c (p ∥ id) The maximum value of i c (p ∥ f ) is reached for H(p • f -1 ) = H(p).
The link between contextual information and information-minimality is given by: Lemma 2.

i c (p ∥ f ) = 0 ⇔ f is minimal for p

Representation of a mapping in the information space

Let us now consider how to represent geomtrically the two types of information in a partition. For a given distribution p, any mapping f will be represented in R 2 by its coordinates

x f = i s (p ∥ f ), y f = i c (p ∥ f )
Example 2. Let us consider the same distribution as in Example 1. Fig. 1 represents all possible mappings g by the point with coordinates

(i s (p ∥ g), i c (p ∥ g)).
Details are in the Appendix. One can check that the minimal factorization is in (1, 0), and the second closest mapping to a minimal factorization is {cats, mice, men, houses}{eat, build}.

Abstraction

Abstraction relaxes the definition of a minimal factorization of p in terms of a solution to an optimization problem. For a given probability distribution p and a mapping f , the abstraction measures the distance between f and the position of a minimal factorization of p in the information space: Definition 6. Let p be a distribution over A + and let f :

A → B be a mapping. Let d be a distance on R 2 . Let t f = (i s (p ∥ f ), i c (p ∥ f )) and t * = (i s (p ∥ id), 0)). The abstraction (w.r.t. d) is defined as α d (p ∥ f ) = e -d(t f ,t * )
One has α d (p ∥ f ) ≤ 1, with the maximum value reached iff f is a minimal factorization of p. For a mapping f , maximizing abstraction can be considered as a relaxation of the property of being a minimal factorization.

Minimal Syntax Identification

We prove here that abstraction can be used to identify the set of minimal syntaxes of p from a sample.

Consistency of the plug-in estimator of abstraction. In the case where the set of of possible sequences is infinite, it is not possible to ensure a convergence rate of the abstraction (cf. [START_REF] Antos | Convergence properties of functional estimates for discrete distributions[END_REF]). Nevertheless, it is possible to show the following consistency result: Proposition 1. Let p be a distribution over A + , and let d be a distance on R 2 . Let pN be the empirical distribution derived from an i.i.d. sample of size N drawn from p.

The plug-in estimator for the abstraction α d (p||f ) is consistent:

α d (p N ∥ f ) N →∞ -→ α d (p ∥ f ) a.s.
As a consequence, when the vocabulary A is finite, abstraction can be used to isolate the set of minimal factorizations of p.

Corollary 1. Let p be a distribution over A + , with |A| < ∞. Let d be a distance measure on R 2 . Let pN be the empirical distribution derived from an i.i.d. sample of size N drawn from p.

Then one has:

lim N →∞ P [G(p, f * d (p N )) ∈ G * (p)] = 1
where f * d (p N ) maximizes abstraction for pN .

Experiments

We test abstraction as an optimization objective for learning syntactic representations, when the representation takes the form of a mapping into discrete syntactic categories. The results are evaluated on an unsupervised POS induction task. While our understanding of a syntactic category may not perfectly overlap with actual parts of speech (the latter being defined on the basis of a mixture of criteria instead of pure syntax, and are usually more coarse-grained than real distributional categories), this task will allow a good comparison with concurrent models on a gold standard.

In NLP, part-of-speech categories are usually a part of a probabilistic model; typically a parameter which will be tuned during learning. For instance, if the model is an HMM, its hidden states correspond to POS categories. If the model is a PCFG, categories will correspond to non-terminals. We call this approach -when POS categories are deduced from a given model structure as a parameter -the model-specific approach. In the experiments, we compare the model-specific approach with our hypothesis: that POS categories can be deduced from the independence of structure and context. We consider the task of unsupervised POS induction, and compare the accuracy of the abstraction maximization criterion with model-specific crossentropy minimization.

The corpus we use comes from Wikipedia in simplified English, contains 430k sentences, 8M tokens, and was POS tagged by the Stanford POS tagger [START_REF] Toutanova | Feature-rich part-ofspeech tagging with a cyclic dependency network[END_REF]. To create the target partition, words (a vocabulary of 6044 elements) were assigned to their most frequent POS. There are 36 POS categories.

The target partition in the information space

We created (Fig. 2) the information space for the Wikipedia corpus with the coordinates indicating structural and contextual information. We represented the target partition (36 categories, correct mapping), and located randomly generated modifications of this partition obtained by changing 1) the assignment of words to the target POS categories (in red) and 2) the number of categories between 2 and 2000 (partitions with > 36 categories are in yellow, partitions with < 36 categories in green) by merging or splitting existing categories. First, it can be observed that any random modification of the target partition (whether it increases or decreases the information) comes at the expense of the abstraction objective. This distinctive position of the syntactic partition could not be visualized in one dimension, suggesting the relevance of the coordinates in the information space in identifying it.

Second, with a strict constraint on the number of categories, the representation of the noisy target (in red) indicates a negative correlation between contextual information and structural information: a trade-off induced by the limitation of information capacity. The choice of normalised d || 2 distance for abstraction is driven by the shape of random partitions in the information space (in blue).

POS induction

We compare abstraction and likelihood maximization as training objectives for unsupervised POS induction. The most efficient POS induction methods at present are mainly -if not exclusively -based on models derived from HMM [START_REF] Peter | Class-based n-gram models of natural language[END_REF][START_REF] Merialdo | Tagging English text with a probabilistic model[END_REF][START_REF] Lin | Unsupervised POS induction with word embeddings[END_REF][START_REF] Stratos | Unsupervised part-of-speech tagging with anchor hidden Markov models[END_REF][START_REF] Ke | Unsupervised neural hidden Markov models[END_REF][START_REF] He | Unsupervised learning of syntactic structure with invertible neural projections[END_REF]. We experiment with different variations of the model by [START_REF] Peter | Class-based n-gram models of natural language[END_REF], because the method is purely distributional, involves discrete embeddings and is still competitive (cf. [START_REF] Stratos | Unsupervised part-of-speech tagging with anchor hidden Markov models[END_REF][START_REF] Christodoulopoulos | Two decades of unsupervised POS induction: How far have we come[END_REF]).

As we cannot perform a brute-force search for the best possible partitions for our criteria, we replaced it by a local measure of the performance : for every single word, provided that all other words are correctly classified, we checked whether the criterion would attribute the correct POS category. Accuracy indicates the rate of correctly classified words.

Tested models We will call plain model the general form of a distribution p factorised by a mapping f :

p(f [w])p(⟨⟨w⟩⟩ | ⟨⟨f [w]⟩⟩) µ(w) µ(f (w))
We can add model-specific constraints:

(MK): Markov constraint for p(f [w]) = p(f (w 1 )) n i=2 p(f (w i ) | f (w i-1 )) (CI): contextual independence constraint for p(⟨⟨w⟩⟩ | ⟨⟨f [w]⟩⟩) = n i=1 p(w i | f (w i )) µ(f (w)) µ(w)
We will consider the normalised α || 2 abstraction maximization objective, and the likelihood maximization objective (with a constraint on the number of categories) for the plain model alone, with contextual independence (CI) constraint, with Markov (MK) constraint, or with both constraints (MK) + (CI) (Brown clustering criterion).

The results are shown in Figure 3. They indicate that the abstraction criterion significantly outperforms likelihood maximization for any model considered. This reinforces our hypothesis that syntactic categories emerge naturally from the criterion of independence between structure and context, without any assumption about the structure of the model.

The second important finding concerns the phenomenon we call implicit disentanglement. By definition, if we estimate the parameters of a distribution q |f with likelihood maximization, we maximize structural information (i.e. the partition f tends towards the right-most solutions in the information space). However, contextual information will still be present. Syntactic generalization may occur when the encoding capacity of the model is bounded (e.g. by limiting the number of categories), inducing a trade-off between structural and contextual information. A way to estimate the role of implicit disentanglement is to consider the confusion matrix of correctly classified or misclassified words for abstraction maximization classifier (a) and a likelihood maximization classifier (b), and decompose (b) into a convex combination of (a) and an independent classification process (c).

With the confusion matrix for Brown clustering criterion: M = b b a 0.202 0.049 a 0.212 0.537 one obtains that a proportion F = 0.667 of the correct classification of (b) is imputable to (a), and at most 33.3% of correct classification by (b) can be considered as independent from implicit disentanglement. This factor F is known in literature as certainty factor (see [START_REF] Tan | Selecting the right interestingness measure for association patterns[END_REF], Appendix for details)

The accuracy of maximum likelihood with the plain model (no constraint) is a good example of implicit disentanglement : it can only be the result of the limitation on the number of categories. The hatched part in Figure 3 represents the fraction of correct classification due to implicit disentanglement in max-likelihood classifiers. These results indicate that the impact of model structure in the ability to infer syntactic categories (and, more broadly, in syntactic generalization capacity) is over-estimated: parameter tuning seems far less efficient than the application of the principle of independence between context and structure.

Conclusion

As to our current knowledge, language models do not have a convincing performance on modelling grammaticality: despite their impressive results on downstream tasks, they are not good at syntactic generalization unless syntactic knowledge is somehow injected in the system. Moreover, there is a trade-off in large LMs between syntactic generalization and language modelling performance.

We suggest a measurable interpretation of syntactic generalization and show results that align with the observations reported by many authors: training on a natural language corpus (e.g. using language models) results in memorization of semantics and entanglement with syntactic information. This motivates our proposition of abstraction, a new training objective for syntactic generalization without supervision. We prove the statistical consistency of abstraction in the task of grammar identification. Empirical results on an unsupervised POS induction task show that abstraction considerably outperforms concurrent models trained with a likelihood estimation objective, without making any assumptions about the structure of the model.

Limitations

The contribution of this paper is mainly theoretical. Like most of the POS identification algorithms, the optimization of a criterion among the space of all partitions requires the use of heuristics, and finding the optimum is never guaranteed. Additional work is required before a generalization model that is efficient in practice can be obtained.

Acknowledgement

We would like to thank Guillaume Wisniewski and the anonymous reviewers for their valuable comments.

Remark 1. One has

⟨⟨σ(w)⟩⟩ = ⟨⟨w⟩⟩ ⟨⟨f [σ(w])⟩⟩ = ⟨⟨f [w]⟩⟩ f (σ(w)) = σ(f (w)) p(f [w]) = p • f -1 (f (w)) p(⟨⟨f [w]⟩⟩) = p • f -1 (⟨⟨f (w)⟩⟩) Remark 2. One has p(⟨⟨w⟩⟩) = 1 µ(w) σ∈Sn p(σ(w)) Remark 3. One has p(⟨⟨f [w]⟩⟩) = 1 µ(f (w)) σ∈Sn p(f [σ(w)])
Proof. By Remark 2 applied to p • f -1 and Remark 1.

Remark 4. One has

⟨⟨⟨⟨w⟩⟩⟩⟩ = ⟨⟨w⟩⟩ f [f [w]] = f [w] ⟨⟨σ(w)⟩⟩ = ⟨⟨w⟩⟩ ⟨⟨f [σ(w)]⟩⟩ = ⟨⟨f [w]⟩⟩ Proof. The second equality comes from f -1 • f • f -1 • f = f -1 • f Lemma 3.
Let p be a distribution over A + , and f : A → B be a mapping. One supposes that there exists a mapping λ f (⟨⟨w⟩⟩) such that ∀w ∈ A

+ p(w | ⟨⟨w⟩⟩) = λ f (⟨⟨w⟩⟩) p(f [w] | ⟨⟨f [w]⟩⟩) then one has λ f (⟨⟨w⟩⟩) = µ(w) µ(f (w)) .
Proof. One has

∈ A + , p(w | f [w]) = p(w | A |w| ) ⇔ ∀w ∈ A + , p(f [w]) = p(A |w| ) or 0
Proof. Suppose the left hand side of the equivalence, then, for any w ∈ A + , either p(f [w]) = 0 or there exists

w ′ ∈ f [w] such that p(w ′ ) > 0 then p(f [w]) = p(f [w ′ ]) = p(A |w| ).
On the other way, either p(w) = 0 (and the equality holds) or p(w) > 0 implying p(f

[w]) > 0 hence p(f [w]) = p(A |w| ) implying in turn p(w | f [w]) = p(w | A |w| ).
Computing Example 1. Let w 1 be the sentence "cats eat rats" and let w 2 be the sentence "men build houses". one has ⟨⟨w 1 ⟩⟩ = {cats eat mice, cats mice eat, . . . , mice eat cats} ⟨⟨w 2 ⟩⟩ = {men build houses, men houses build, . . . , houses build men}

f (w 1 ) = b 0 b 1 b 2 f (w 2 ) = b 0 b 1 b 2 f [w 1 ] =
{cats eat mice, cats build mice, . . . , men eat mice, men build mice} ⟨⟨f [w 1 ]⟩⟩ = {cats eat mice, cats build mice, . . . , build mice men, men houses eat} One has

p |f (w 1 ) = p(⟨⟨w 1 ⟩⟩)p(f [w 1 ] | ⟨⟨(f [w 1 ]⟩⟩) µ(w 1 ) µ(f [w 1 ]) = 1 2 • 1 • 1 1 hence f factorises p. One has p(w 1 | f [w 1 ]) = 1 2 1 = p(w 1 | A 3 )
hence f is information-minimal for p.

The Lemma 5 gives a formula for the projection p |f of p conditionally to f . Lemma 5. Let p be a probability distribution, and let f be a mapping. Let us define π(p, f ) by

π(p, f )(w) = p(⟨⟨w⟩⟩)p(f [w] | ⟨⟨f [w]⟩⟩) µ(w) µ(f (w)) then one has p |f = π(p, f )
One needs a few steps in order to prove Lemma 5. Lemma 6. Let p be a distribution over A + and f : A → B be a mapping. Then

1. π(p, f )(⟨⟨w⟩⟩) = p(⟨⟨w⟩⟩) 2. π(p, f )(f [w]) = p(f [w]) Proof. 1: one has π(p, f )(⟨⟨w⟩⟩) = 1 µ(w) σ∈Sn π(p, f )(σ(w)) with π(p, f )(σ(w)) = p(⟨⟨w⟩⟩)p(f [σ(w)]) p(⟨⟨f [w]⟩⟩) µ(w) µ(f (w)) and σ∈Sn p(f [σ(w)]) p(⟨⟨f [w]⟩⟩)µ(f (w)) = 1 hence the result. 2: one has π(p, f )(f [w]) = p(⟨⟨f [w]⟩⟩)p(f [f [w]] | ⟨⟨f [f [w]]⟩⟩) µ(f (w)) µ(f (w)) = p(⟨⟨f [w]⟩⟩)p(f [w] | ⟨⟨f [w]⟩⟩) µ(f (w)) µ(f (w)) = p(f [w])
Corollary 3. Let p be a distribution over A + and f : A → B be a mapping. Then

1. π(p, f ) is a probability distribution. 2. π(π(p, f ), f ) = π(p, f ).
Proof.

(1): The ⟨⟨w⟩⟩ form a partition of A + hence

π(p, f )(A + ) = ⟨⟨w⟩⟩ π(p, f )(⟨⟨w⟩⟩) = ⟨⟨w⟩⟩ p(⟨⟨w⟩⟩) = 1
(2): Since π(p, f ) is only computed from p(⟨⟨w⟩⟩) and p(f [w]), with the Lemma 6 one has the conclusion.

Remark 5. In particular, one can give another definition of the set

F f = {q | q is factorised by f } as F f = {π(q, f ) | q is a distribution}
Lemma 7. Let p and q be two probability distributions over A + , and let f : A → B be a mapping. Then one has:

H(p||π(q, f )) ≥ H(p||π(p, f ))
with equality iff π(q, f ) = π(p, f ).

Proof. The inequality is equivalent to

w∈A + p(w) log( q(⟨⟨w⟩⟩) p(⟨⟨w⟩⟩) ) + log( q(f [w])p(⟨⟨f [w]⟩⟩) p(f [w])q(⟨⟨f [w]⟩⟩) ) ≤ 0
By Jensen's inequality and concavity of the log, and summing over ⟨⟨w⟩⟩ one has w∈A + p(w) log( q(⟨⟨w⟩⟩) p(⟨⟨w⟩⟩) ) = ⟨⟨w⟩⟩ p(⟨⟨w⟩⟩) log( q(⟨⟨w⟩⟩) p(⟨⟨w⟩⟩) ) ≤ 0 with equality iff ∀w, q(⟨⟨w⟩⟩) = p(⟨⟨w⟩⟩).

By summing over ⟨⟨f [w]⟩⟩, one has

f [w] q(f [w])p(⟨⟨f [w]⟩⟩) q(⟨⟨f [w]⟩⟩) = ⟨⟨f[w]⟩⟩ q(⟨⟨f [w]⟩⟩)p(⟨⟨f [w]⟩⟩) q(⟨⟨f [w]⟩⟩) = 1
and by Jensen's inequality and concavity of the log, and summing over f [w] one has

w∈A + p(w) log( q(f [w])p(⟨⟨f [w]⟩⟩) p(f [w])q(⟨⟨f [w]⟩⟩) ) = f [w] p(f [w]) log( q(f [w])p(⟨⟨f [w]⟩⟩) p(f [w])q(⟨⟨f [w]⟩⟩) ) ≤ 0 with equality iff ∀w, q(f [w] | ⟨⟨f [w]⟩⟩) = p(f [w] | ⟨⟨f [w]⟩⟩).
Because the value of π(p, f ) only depends on p(⟨⟨w⟩⟩) and p(f [w] | ⟨⟨f [w]⟩⟩), the equality holds in the statement iff π(q, f ) = π(p, f ).

Proof of Lemma 5. One applies Remark 5 and Lemma 7, and one gets the result.

Separating structure from data.

Our goal is to isolate structural information form contextual information for an observation (a 1 , . . . , a n ).

For any permutation σ ∈ S n , the tuple

(a σ(1) , . . . , a σ(n) ) = (a ′ 1 , . . . , a ′ n ) satisfies a relation (a 1 , . . . , a n ) = (a ′ σ -1 (1) , . . . , a ′ σ -1 (n) )
which will be denoted σ -1 .

Definition 7. For an observation X = (a 1 , . . . , a n ), and a permutation σ ∈ S n , let us denote σ(X) = (a σ(1) , . . . , a σ(n) )

For any probability distribution p over A + , one will define

p 2 (X = (a 1 , . . . , a n ), Y = σ) = 1 n! p(σ(X)) Lemma 8. One has p(w) = σ∈Sn p 2 (X = σ -1 (w), Y = σ) p(⟨⟨w⟩⟩) = |w|! p 2 (X = w) µ(w)
where |w| is the length of w.

Proof. The first statement is just straightforward from the definition of p 2 . In particular, one has

p(w) = |w|! p 2 (Y = id, X = w) One has p(⟨⟨w⟩⟩) = 1 µ(w) σ∈Sn p(σ(w)) = 1 µ(w) σ∈Sn,ρ∈Sn p 2 (X = ρ -1 (σ(w)), Y = ρ) = σ∈Sn p 2 (X = σ(w)) µ(w)
and, with the fact that

∀σ ∈ S n , p 2 (X = w) = p 2 (X = σ(w))
one has the result.

Definition 8. Let us define

p 2|f (X = w, Y = σ) = p 2 (X = w)p 2 (Y = σ|f (X) = f (w)) Lemma 9. One has 1. p 2|f = p |f 2 2. H(p 2 ||p |f 2 ) = H p 2 (Y |f (X)) + H p 2 (X) 3. H(p||p |f ) = H(p 2 ||p 2|f ) -E p (log(|w|!)) Proof. 1. One has p |f 2 (X = w, Y = σ) = 1 n! p |f (σ(w)) = 1 n! p |f (⟨⟨σ(w)⟩⟩)p |f (f [σ(w)]) p |f (⟨⟨f [σ(w)]⟩⟩) µ(σ(w)) µ(f (σ(w)))
with, by Lemma 6 and Lemma 8,

p |f (⟨⟨σ(w)⟩⟩) = p(⟨⟨w⟩⟩) = n! p 2 (X = w) µ(w) and p |f (f [σ(w)]) = p(f [σ(w)]) = n!p 2 (f (X) = f (w), Y = σ) and p |f (⟨⟨f [σ(w)]⟩⟩) = p(⟨⟨f [w]⟩⟩) = n! p 2 (f (X) = f (w)) µ(f (w)
and one has the result. The second statement is an application of the definition of p 2|f together with statement 1.

The third statement comes from the fact that

p(w) = σ∈Sn p 2 (X = σ -1 (w), Y = σ) p |f (w) = |w|!p |f 2 (X = σ -1 (w), Y = σ)
one has

H(p||p |f ) = - w p(w) log(p( |f (w)) = - w,σ p 2 (X = σ -1 (w), Y = σ) log(|w|!p |f 2 (X = σ -1 (w), Y = σ)) = H(p 2 ||p 2|f ) -E p (log(|w|!))
Lemma 10. Let p be a probability distribution, and let f and g be two mappings. One has

i s (p, g • f ) ≤ i s (p, f ) Proof. From Lemma 9, one has i s (p||f ) = H p 2 (Y |z(X)) -H p 2 (Y |f (X))
and one has

H p 2 (Y |f (X)) ≤ H p 2 (Y |g • f (X)) hence i s (p||f ) ≥ i s (p||g • f )
Lemma 11. Let p be a probability distribution, and let f and g be two mappings. One has

i s (p, g • f ) ≤ i s (p, f ) Proof. With the fact that H p (f (w)) ≥ H p (g • f (w))
one has the result.

Details of Example 2 . The two sentences are strictly equivalent, thus we will only compute the values for, say, u = cats eat mice . One has p(⟨⟨u⟩⟩) = p({cats, eat, mice} = 1 2 . Let z : A → {a} be a null mapping. By Lemma 5, one has One has p • h -1 (ccc) = 1 2 , thus

p |z (u) = p(⟨⟨u⟩⟩) p(z[u]) p(⟨⟨z[u]⟩⟩) µ(u) µ(z(u))
H(p • h -1 ) = log(2), i c (p ∥ h) = log(2)
One can check that the gain of information from z to g is purely structural, while the gain from z to h is purely contextual.

Proof of Proposition 1. From Lemma 9, one has

i s (p||f ) = H p 2 (Y |z(X)) -H p 2 (Y |f (X))
hence i s is a sum of entropies, as well as i c .

The Corollary 1 in [START_REF] Antos | Convergence properties of functional estimates for discrete distributions[END_REF] states that for a countable support target distribution p with entropy H and its M.L.E. p n with entropy Ĥn , the plugin estimator satisfies lim n→∞ Ĥn = H a.s. and this directly implies the conclusion.

Lemma 12. Let p be a probability distribution, having a minimal factorization f . Then the intersection of all minimal syntaxes of p is a minimal syntax of p: • an independent process of POS identification involving HMM constraints has a 40% success rate.

• the rate of correct classification by (b) is 0.586, decomposed in 0.195 of success from an independent process, and 0.391 due to implicit disentanglement.

• 66.7% of the overall success rate can be imputed to implicit disentanglement.

The fraction of success due to implicit disentanglement is exactly the certainty factor F = 0.667 (see [START_REF] Tan | Selecting the right interestingness measure for association patterns[END_REF] where M ⇒ represents a complete implication and M ⊥ ⊥ represents a complete independence, both with same marginals as M . (see [START_REF] Tan | Selecting the right interestingness measure for association patterns[END_REF] for a definition in the context of association rules)

Figure 1 :

 1 Figure 1: Information space: normalised representation of partitions for the distribution p in Example 1.

Figure 3 :

 3 Figure 3: Unsupervised POS induction: abstraction vs likelihood maximization

  ⟨⟨σ(w⟩⟩)) p(f [σ(w)] | ⟨⟨f [σ(w)]⟩⟩) = λ f (⟨⟨(w⟩⟩) p(⟨⟨f [w]⟩⟩) σ∈Sn p(f [σ(w)]) = λ f (⟨⟨(w⟩⟩)µ(f (w))from Remark 1 and Remark 3. With the fact that p(σ(w)) = p(⟨⟨w⟩⟩)p(σ(w) | ⟨⟨σ(w)⟩⟩) Let p be a distribution and f be a mapping. Then f factorises p iffp(w) = p(⟨⟨w⟩⟩(f [w] | ⟨⟨f [w]⟩⟩) µ(w) µ(f (w))Lemma 4. Let p be a distribution and f a mapping. Then one has ∀w

  with p(z[u]) = p(aaa) = 1 and p(⟨⟨z[u]⟩⟩) = p({a, a, a}) = 1, µ(u) = 1 and µ(aaa) = 6, one has p |z (u) = 1 12 and finallyH(p ∥ p |z ) = log(12)One has also p • z -1 (aaa) = 1 thusH(p • z -1 ) = 0 One has p |id (u) = 1 2 thus H(p ∥ p |id ) = log(2), i s (p ∥ id) = log(6) and p • id -1 = p thus H(p • z -1 ) = log(2), i c (p ∥ id) = log(2)Let g be the mappinga = {cats, mice, men, houses}, b = {eat, g[u]) = p(aba) = 1, p(⟨⟨g[u]⟩⟩) = p({a, b, a}) = 1, µ(u) = 1 and µ(g(u)) = µ(aba) = 2 one has p |g (u) = 1 4 and H(p ∥ p |g ) = log(4), i s (p ∥ g) = log(3) One has p • g -1 (aba) = 1, thus H(p • g -1 ) = 0, i c (p ∥ g) = 0Let h be the mapping c = {cats, eat, mice}, d = {men, build, houses} then one hasp |h (u) = p(⟨⟨u⟩⟩) p(h[u]) p(⟨⟨h[u]⟩⟩) µ(u) µ(h(u)) with p(h[u]) = p(ccc) = 1 2 , p(⟨⟨h[u]⟩⟩) = p({c, c, c}) = 1 2 , µ(u) = 1 and µ(h(u)) = µ(ccc) = 6 one has p |h (u) = 1 12 and H(p ∥ p |h ) = log(12), i s (p ∥ h) = 0

G

  * (p) = ∩ f ∈G * (p) G(p, f ) ∈ G * (p)Proof. Let f be an optimal factorizations of p. The property∀w ∈ supp(p), p(w | f [w]) = p(w | A |w| )means that there is only one observed pattern of categories of length |w|, say b 1 . . . b n .For any (i, n) ∈ N 2 , let us define the subsetA (i,n) = {a ∈ A | ∃w ∈ A n , p(w) > 0, w i = a}Each subset A (i,n) is entirely inside a class of the partition induced by f -otherwise there would be at least two patterns for sequences of size n.One builds a partition m of A by merging the subsets A (i,n) with non-empty intersection. Any class of m is entirely included in a class induced by f .The mapping m is minimal by construction, and it is a refinement of f : G(p, f ) ⊆ G(p, f ), and, by Lemma 10, m factorises p.Implicit disentanglement. We consider the following matrix of confusion 251(1 -µ) 0.251µ a 0.749(1 -µ) 0.749µand we write M as a convex combination of M ⇔ and M ⊥ ⊥ :M = λM ⇔ + (1 -λ)M ⊥ ⊥This gives λ = 0.522, µ = 0.408 which leads to the interpretations:

  ) which represents a convex decomposition of M M = F.M ⇒ + (1 -F ).M ⊥ ⊥ with:

Table 1 :

 1 Notations

The nativist claim has however been subjected to criticism from other fields of research[START_REF] Hsu | The logical problem of language acquisition: a probabilistic perspective[END_REF];[START_REF] Yang | One model for the learning of language[END_REF].

Appendix -Proofs and complements

Let p be a probability distribution, and f a mapping. One will denote p • f -1 the probability distribution induced by f .


Marie Potapushkina-Delfosse

In Abou Haidar

COORD.), L'ENSEIGNEMENT L De

L A Prononciation

E N Classe

D E Langue

La marche et le tracé pour entendre et prononcer une langue étrangère

Introduction

« Les enfants sont comme des éponges ! » : qui n'a pas entendu cette comparaison lorsqu'il s'agit de louer la facilité avec laquelle les enfants apprennent les langues ? Or, la plasticité auditive et phonatoire en question ne caractérise que les plus jeunes, se réduisant au fur et à mesure de l'acquisition de la langue maternelle. Un « âge heureux » [START_REF] Guberina | Rôle de la perception auditive dans l'apprentissage précoce des langues[END_REF] se terminant vers les 6 ans, l'enfant entre dans un « âge critique » où la sensibilité aux sonorités peu familières diminue fortement, avant d'atteindre, chez les 10-12 ans, le « crible phonologique » [START_REF] Troubetzkoy | Principes de phonologie[END_REF] qui ne laisse percevoir des langues inconnues qu'à travers le filtre du système phonologique de la L1. Ainsi, contrairement aux plus jeunes qui apprennent la prononciation d'une langue étrangère intuitivement et par imitation [START_REF] Voise | Evaluation des compétences phonologiques en anglais d'enfants d'école maternelle[END_REF], les préadolescents nécessitent une approche plus analytique [START_REF] Voise | Linguistique et didactique de l'anglais à l'école. Eléments d'[END_REF][START_REF] Voise | Approches de l'altérité linguistique et culturelle en classe de langue[END_REF]. Quant à la présente étude, elle est consacrée à une démarche intermédiaire. Celle-ci s'adresse aux 7-8 ans pour à la fois réactiver leur intuition et stimuler leur « kinéflexion » qui représente « l'union du mouvement corporel et de la réflexion » (Lapaire, 2017 : 195). En partant du principe de la « physicalité du langage » (Aden&Aden, 2017), nous avons élaboré une approche visant une adaptation (voire une rééducation) de l'oreille et de l'appareil laryngo-buccal au travers de la motricité globalement corporelle et manuelle. Cette approche s'inscrit dans un paradigme énactif [START_REF] Varela | L'inscription corporelle de l'esprit[END_REF] et fait suite à la proposition d'un « apprentissage des langues par corps » [START_REF] Aden | Apprendre les langues par corps[END_REF].

Au cours des six premiers mois de l'année scolaire 2018-2019, 27 élèves de CE1 et leur enseignante, dans une école publique des Yvelines, ont participé à une recherche-action que l'auteure avait initiée et dont l'objectif consistait à développer une approche holistique d'enseignement/apprentissage d'une langue étrangère (l'anglais en l'occurrence), inspirée par la logique de l'ontogénèse du langage. Pour retracer celle-ci brièvement, faisons-nous éclairer par Fónagy : avant d'être apte à produire des unités verbales, la voix du nouveau-né ne fait qu'accompagner des suites de « mouvements plus ou moins violents, plus ou moins coordonnés » (1983 : 149) accomplis par son corps, le but de ces émissions vocales étant de « réduire momentanément la tension physique » (Ibid.). Bien que le nouveau-né soit encore loin d'accéder à la dimension verbale du langage, ses parents lui parlent. Or, le « parler bébé » diffère de la parole ordinaire en ce qu 'il contient, d'après Gratier (2015), plus de vocalisations non verbales, de sons modulés, d'allitérations, de rimes. Autrement dit, le locuteur qui s'adresse au nourrisson place son effort communicatif dans ce que Fónagy, fondateur de la théorie du double encodage du discours, appelle le « message secondaire ». La dimension musicale de ce dernier actualise le « message primaire » qui est, quant à lui, purement linguistique. La musicalité renforcée de la parole que la mère destine au nouveau-né permet l'établissement d'un accordage affectif très précoce [START_REF] Petitmengin | Le dedans et le dehors. Une exploration de la dynamique pré-réfléchie de l'expérience corporelle[END_REF].

En effet, la musique et la langue possèdent de nombreuses ressemblances. Les deux se servent de la même unité minimale : le son qui peut être décrit en termes de hauteur, d'intensité, de timbre, de durée. Les deux affichent une organisation segmentale qui relève de la combinaison de sons et suprasegmentale se référant aux éléments prosodiques tels que le rythme, l'accentuation, l'intonation. En didactique des langues, ces similitudes ne manquent pas à susciter une interrogation au sujet des liens qui puissent exister entre la pratique musicale et l'apprentissage des langues étrangères. Des expériences conduites par [START_REF] Dodane | La langue en harmonie : influences de la formation musicale sur l'apprentissage précoce d'une langue étrangère[END_REF] auprès d'enfants francophones pratiquant et ne pratiquant pas la musique ont révélé que les premiers tendaient à mieux percevoir certains contrastes vocaliques et à reproduire avec plus de justesse des figures prosodiques propres à l'anglais.

L'origine commune de la musique, du chant en particulier, et du langage verbal s'explique non seulement par le caractère sonore qu'ils partagent. La glotte, génératrice d'une voix parlée et chantée, est un lieu de miniaturisation des mouvements du corps dans sa globalité (Fónagy, 1983), l'oralité n'étant que « la transposition du geste global sur les mécanismes laryngo-buccaux » (Jousse, 1974(Jousse, /2008 : 748) : 748). Un autre lieu de miniaturisation de la kinésie corporelle est, selon Jousse, la main, cet organe si souple et expressif qui parvient à traduire avec fidélité ce que, de manière plus dispendieuse, aurait pu dire le corps entier. Tout comme les organes phonatoires, la main est apprivoisée par le nourrisson dès le début de sa vie, lors de l'installation d'un des premiers réflexes -celui d'agrippement. Pendant longtemps, la main reste pour l'enfant l'outil principal d'exploration du monde, un des moments cruciaux du développement, arrivant le jour où, armée d'un instrument scripteur ou d'un objet banal, elle laisse, sur une surface, une trace que son auteur remarque. Cette découverte déclenche, chez la plupart des enfants, une passion pour le dessin. Avant de devenir intentionnel et verbalement explicable, celui-ci, à l'instar des premières expériences vocales, libère des tensions (Luquet, 1927[START_REF] Luquet | Le dessin enfantin[END_REF]. Des tracés initialement fortuits, faits de lignes droites, brisées, courbes, constituent, pour Kandinsky (1926[START_REF] Kandinsky | Point et ligne sur plan[END_REF], les symboles premiers de la vie et la source de tout art. La ligne, au même titre que le son, est l'incarnation d'une force provenant d'une tension et se concrétisant dans un mouvement directionnel. Cette définition kandinskienne trouve un écho en linguistique cognitive, dans la théorie de [START_REF] Johnson | The Body in the Mind[END_REF] pour qui tout acte humain, qu'il s'agisse d'un simple mouvement de main ou d'une réflexion complexe, possède une trajectoire spatio-temporelle et peut être représenté comme une ligne prenant naissance dans un point A et se concluant dans un point B.

Ingold va encore plus loin dans l'affirmation du caractère profondément anthropologique de l'acte de tracer, en avançant que tous les aspects de la vie quotidienne de l'homme, dont l'exercice du langage verbal ou musical, sont « englobés dans la fabrication de lignes » (2012 : 7).

Le bref développement théorique ci-dessus met en lumière le fait que le langage verbal constitue une forme réduite de l'expression globalement corporelle et possède des caractéristiques communes avec la musique et le geste manuel, notamment celui qui implique le tracé. En nous appuyant sur ce postulat, nous avons émis l'hypothèse suivante : l'enseignement d'une langue étrangère mettant l'accent sur la perception globalement corporelle de la musicalité de celle-ci et guidé par le geste manuel traçant, facilitera le développement de la compétence phonologique, sur les plans de la discrimination auditive et de la production. Afin de tester cette hypothèse, nous avons conçu et mis en oeuvre un dispositif évolutif d'enseignement-apprentissage de l'anglais oral dans une classe rassemblant des élèves de 7 à 8 ans, quasi-débutants. Nos interventions d'une heure environ ont eu lieu à raison de deux fois par semaine, de septembre 2018 à février 2019, l'ensemble des séances étant organisé en quatre séquences. Toutes les séances ainsi que les productions des élèves, réalisées en fin de chaque séquence, ont été filmées, ce qui nous a permis de collecter un vaste corpus vidéo dont certains éléments d'analyse figurent dans ce texte. Lorsque nous citons les élèves ou décrivons leurs attitudes, nous nous référons soit aux films en question soit à notre carnet de bord dans lequel nous avons soigneusement consigné les remarques, les réactions, les comportements des enfants, qui nous paraissaient significatifs, et qui ne sont pas toujours visibles/audibles dans les films.

S'ouvrir à la musicalité de la langue

Ayant préconisé l'entrée dans la langue moyennant l'exploration de sa matière sonore, nous avons décidé de consacrer la première séquence du dispositif à l'initiation à l'écoute musicale par corps. Pour cette séquence préparatoire, nous avons choisi quelques oeuvres minimalistes (Satie, Riley, Reich) caractérisées par une pulsation régulière et la répétition d'un motif qui évolue très progressivement. Ce choix s'explique par le fait que la répétitivité est propre aux comptines, qui feront l'objet des séquences 2 et 3, ainsi qu'au folklore enfantin de manière générale : elle est recherchée par les enfants en tant que « principe de plaisir » généré par la joie de la reconnaissance qui contribue à « l'économie du travail intellectuel » (Vivès, 2002 : 45). En effet, l'écoute musicale que nous avons l'intention de développer n'est pas effectuée par une « tête bien pensante » (Jousse, 1974(Jousse, /2008 : 734) : 734) mais « avec tout (le) corps, des pieds à la tête » (Ibid.). Pour aider les enfants à adopter l'approche visée, nous les invitons à se relier à la musique par la marche, cette activité quotidienne qui rythme notre existence comme le font la respiration et le battement cardiaque. L'enfant a le temps de s'approprier le motif qui se répète à maintes reprises et engendre une composition marchée dont la forme (nombre, vitesse, longueur, direction des pas) se précise avec chaque répétition. Bien souvent, le groupe-classe finit par s'accorder, sans passer par la parole, sur l'agencement le plus juste et termine l'écoute-marche de façon synchrone.

Vient ensuite la proposition d'accueillir la musique manuellement : à l'aide du tracé. Sur de grandes feuilles de papier, étalées au sol ou tapissant les murs, les enfants dessinent le rythme de la musique, des patterns graphiques se cristallisant au gré des répétitions. Le partage des dessins ainsi produits s'effectue de la manière suivante : chaque enfant guide la main d'un camarade sur le tracé qu'il avait réalisé, au rythme de la musique réécoutée, puis se laisse guider par son camarade sur les chemins graphiques créés par celui-ci. Chacun procède à cet exercice plusieurs fois, ce qui lui donne l'occasion d'échanger avec différents partenaires. La séance se termine par un retour verbal sur l'expérience : mise en mots des émotions ressenties, comparaison des formes motrices/graphiques émergées.

Dès le début de la deuxième séquence, nous présentons aux enfants une comptine en anglais et les invitons à l'aborder selon les mêmes principes : écoute-marche, écoute-tracé, retour verbal sur le processus vécu. Lors de la comparaison des productions graphiques, sont recherchés des invariants à partir desquels la classe construira une figure graphique commune que nous appelons schéma rythmo-mimique (SRM) et dont un exemple détaillé sera donné plus loin. Une fois réalisé sur papier, les élèves s'entraînent à reproduire le SRM « en l'air », en suivant de leur main le rythme de la comptine réécoutée. Nous utiliserons désormais les termes « SRM graphique » et « SRM gestuel » pour nous référer respectivement à la réalisation du schéma rythmo-mimique sur papier, à l'aide d'un outil scripteur, et « en l'air », sans outil ni support matériel.

Le sens de la comptine est ensuite bâti collectivement, au cours d'une confrontation des hypothèses émises par les enfants avec l'appui sur le SRM et quelques signes littéraux que nous y ajoutons (images mobiles des personnages de la comptine, objets qui y sont évoqués). Puis le texte est appris, en choeur, à l'aide du SRM gestuel, cette phase collective étant suivie d'un travail individuel ou par petit groupe autour du SRM graphique. Lorsque la comptine est mémorisée, on procède à l'isolation de ses mots-clés et on en découvre quelques autres relevant des mêmes champs lexicaux. À partir de ce bagage commun, sont menées des activités où la manipulation d'éléments langagiers se fait au gré des situations qui se présentent dans le contexte de jeux de société, de jeux collectifs ou encore de jeux de rôles, créés par nos soins autour des structures syntaxiques de la comptine et de son vocabulaire élargi. Tel est le déroulé des séquences 2 et 3 du dispositif.

Processus de transposition rythmo-mimique d'un texte

Présentation de l'extrait choisi et des résultats d'une écoute-marche

Pour donner un exemple de construction d'un schéma rythmo-mimique, nous nous appuierons sur l'extrait du texte qui a fait l'objet de la quatrième séquence du dispositif. Il ne s'agit plus d'une comptine mais du conte britannique traditionnel, Tom and Mr Miacca, dont la prégnance sonore a été augmentée moyennant une mise en forme poétique. L'enregistrement que nous avons utilisé auprès des élèves avait été effectué par une comédienne, locutrice native de l'anglais. Elle avait mis en exergue les émotions des personnages, en exagérant parfois, à notre demande, les caractéristiques prosodiques du texte. Nous avons opté pour un texte long, en fin de dispositif, pour rejoindre le projet annuel de la classe, axé sur l'écoute et la lecture de nombreux contes issus du patrimoine oral de différents pays du monde et sur l'initiation à l'art de conter. Le personnage principal de cette histoire est un petit garçon prénommé Tom qui, malgré les avertissements de sa mère, s'éloigne de la maison, se fait capturer par un ogre mais parvient à se libérer. La trame est dévoilée aux élèves progressivement, un épisode (une strophe) par séance. La strophe dont nous explorerons la transposition rythmo-mimique renvoie aux paroles de la mère de Tom, qu'elle adresse à son fils, au début de l'histoire : Au regard de sa dimension poétique et didactique, le texte est ponctué de plus de silences qu'une production orale réalisée dans une situation naturelle: pour reprendre le vocabulaire musical, les « demi-soupirs » sont symbolisés ici par un trait vertical et les « soupirs » par deux traits, chaque segment rythmique placé entre deux « soupirs » étant numéroté. Au cours de l'écoute-marche, la majorité des élèves marquent les soupirs par des arrêts et suspendent légèrement le pas en réponse aux demi-soupirs. Presque tous réagissent aux phrasés legato (Don't go out of the street ; he's so horrible ; he's so terrible) par des suites de petits pas courus. Nombreux sont ceux qui traduisent les phrasés staccato (He will catch you ; he will cook you, etc.) en une série de pas marquant la cadence. Lorsqu'on passe à l'écoute-tracé, le mouvement de la main s'interrompt en réaction aux silences. Certains éléments segmentaux, tels que la qualité des voyelles, sont mis en évidence par le tracé, ce qui n'était pas le cas lors de l'écoute-marche. À la suite du partage de leurs productions graphiques individuelles, les élèves aboutissent au SRM commun ci-dessous, reflétant les perceptions les plus fréquentes.

Ecoute-tracé : visualisation de figures prosodiques et des caractéristiques de voyelles

Figure 1. Première étape de la construction du schéma rythmo-mimique Nous procéderons à présent à l'analyse du SRM commun de la classe par segments rythmiques, ces derniers n'ayant pas été numérotés dans la version produite par les enfants. La numérotation introduite plus haut, lors de la présentation du texte, est utilisée ici aux seules fins d'une meilleure lisibilité de l'analyse. Chaque chiffre est placé à l'origine du geste graphique auquel il se réfère.

1. Dans les versions individuelles, au mot Tom correspondait le plus souvent un trait court vertical, horizontal ou oblique, renvoyant à la brièveté de la voyelle [ɒ]. En apprenant qu'il s'agit du prénom du personnage principal, les élèves choisissent, de façon unanime, la verticalité symbolisant, pour eux, la station debout d'un être humain.

2. Le phrasé legato (Don't go out of the street) est figuré par une ligne sinueuse à trois ondulations se rapportant respectivement aux trois diphtongues : [əʊ] de don't et de go, [aʊ] de out. Quant au [i:] long de street, un trait droit allongé fait écho à la durée de cette voyelle, sur laquelle insiste la voix de la comédienne. La tendance à représenter les diphtongues et les voyelles longues de cette manière persiste depuis le début du dispositif, ce qui rejoint la perception de Kandinsky (1926[START_REF] Kandinsky | Point et ligne sur plan[END_REF]. Selon lui, les lignes courbes sont issues de deux forces simultanées inégales, description qui convient également aux diphtongues dont les deux sons se prononcent en un seul jet, le premier étant toutefois plus énergique que le second. Concernant la ligne droite, les enfants l'utilisent massivement pour faire figurer des voyelles simples, toujours en accord avec Kandinsky qui voyait dans la ligne droite le résultat de l'action d'une seule force.

A nouveau, une courbe accompagne une diphtongue ([ɛə] dans there).

4. You can est représenté par un seul trait court : les élèves n'y perçoivent donc, à juste titre, qu'un seul mouvement vocal porté par un temps fort correspondant à you. Le [i:] de meet, quant à lui, est transcrit de la même façon que celui de street (trait allongé), la rime induisant une inévitable analogie graphique.

5. Un trait oblique tracé de bas en haut, en réponse à l'intonation montante de la comédienne, correspond au mot MISTER.

6. Un trait oblique, symétrique au précédent, est dirigé du haut vers le bas, suivant l'intonation, cette fois-ci descendante. Il renvoie au mot MIACCA. La tendance à transcrire des éléments disyllabiques par un seul trait, que nous avons déjà remarquée dans le traitement de you can, se confirme ici. En effet, à ce stade du dispositif, la majorité des élèves a abandonné la propension à découper le flux sonore en syllabes, propre à toute oreille francophone habituée au fonctionnement syllabique, pour adopter une saisie auditive prenant en compte l'accentuation. La taille assez importante des traits (segments 5 et 6) s'explique par une charge émotionnelle particulièrement forte dont la locutrice remplit ce passage. Le ton mystérieux qu'elle adopte pour prononcer le mot MISTER et l'expression menaçante avec laquelle elle profère le mot MIACCA conduisent les élèves à appréhender ces segments comme étant le coeur du message, avant même d'apprendre qu'il s'agit du nom de l'ogre. 7-8-9. Les phrasés staccato (He will + verbe + you), dotés d'un rythme accéléré, sont rendus à l'aide de petits traits dont chacun transcrit un mot monosyllabique à voyelle brève, la disposition « en escalier » reflétant l'intonation descendante. 10-11. Une « vague » qui monte exprime l'intonation ascendante que la comédienne attribue au mot so, la courbure embrassant la diphtongue [əʊ]. Les mots horrible et terrible sont matérialisés par des lignes marquées de nombreuses ondulations à faible amplitude, qui traduisent l'effet de rebondissement dû à la succession de consonnes h/t-r-b-l vocalisées par les seuls [e], dans terrible, et [ɒ], dans horrible, l'audibilité du [ɪ] étant très réduite dans les deux mots.

Ce tracé est aussi pourvu d'un aspect mimique : la faible mais fréquente ondulation de la ligne évoque le tremblement qui fait référence, en français aussi bien qu'en anglais, à la peur véhiculée par la voix de la comédienne.

12. Un tracé ininterrompu visualise l'expression be careful, la diphtongue [ɛə] générant une courbe à l'instar des diphtongues précédentes.

On peut remarquer que l'écoute incorporée, impliquant d'abord la marche puis le tracé, centre l'attention des élèves sur des aspects prosodiques et vocaliques du texte, la qualité des consonnes ne transparaissant presque pas dans les productions issues de ces expériences. En effet, le mouvement du corps suit le mouvement du flux langagier. Or, ce dernier émane du souffle associé au passage libre de l'air dans la cavité buccale et les fausses nasales, qui soutient la prosodie et constitue le mécanisme de la prononciation des voyelles. La production des consonnes vient, au contraire, obstruer le passage de l'air en provoquant des arrêts momentanés du mouvement. C'est donc après la création d'un schéma rythmo-mimique intrinsèquement moteur et vocal que nous orientons l'écoute des élèves vers le repérage de la qualité des consonnes et la recherche des gestes buccaux permettant de les reproduire avec exactitude.

Visualisation du paysage consonantique

Au stade du dispositif que nous évoquons dans cette partie de l'analyse, les élèves sont déjà familiers avec un certain nombre de consonnes spécifiquement anglaises et ont élaboré un code commun pour la notation de celles-ci. Ce code se base à la fois sur la convention alphabétique et la représentation métaphorique de « ce qui se passe dans la bouche » quand telle ou telle consonne est prononcée. Des écoutes supplémentaires du texte sont requises pour emmener les élèves à identifier les consonnes spécifiques qui apparaissent à des moments clés du texte (syllabes initiales et parfois finales de certains groupes rythmiques). dit sombre, que l'on entend dans les mots horrible, terrible et careful, est signalé par la lettre l entourée d'un cercle qui renvoie à la sensation d'« avaler sa langue », provoquée par la prononciation de cette consonne.

De la figuration graphique au mime rythmé

Une fois le sens linguistique dévoilé, les enfants proposent d'ajouter au SRM graphique quelques signes figuratifs qui, nous le verrons, renforceront l'aspect mimique du SRM gestuel : 2. Des maisons apparaissent sur la ligne horizontale correspondant au mot street.

5-6. Deux grands yeux effrayants sont ajoutés aux traits obliques qui évoquent l'ogre.

7. Une main symbolisant le geste de saisir complète la série de petits traits se rapportant à He will catch you.

8. Le dessin d'une casserole vient compléter l'énoncé He will cook you. 9. Le dessin d'une bouche ouverte, faisant entrevoir quelques dents, illustre l'énoncé He will eat you.

Lorsque l'on passe de ce SRM graphique à son pendant gestuel, la linéarité géométrique des tracés est maintenue pour tous les segments sauf certains parmi ceux qui ont été enrichis par les signes figuratifs. Ainsi les enfants proposent de former un cercle avec les doigts d'une main autour d'un oeil, en prononçant MISTER (5), et un autre cercle, autour de l'autre oeil, en disant MIACCA (6), le tout étant agrémenté d'une expression faciale censée inspirer la peur. En ce qui concerne les énoncés construits selon le modèle he will + verbe + you, ils sont accompagnés de gestes suivants : les deux mains miment le fait de saisir quelque chose (7), une main effectue une rotation rappelant le fait de mélanger le contenu d'une casserole (8), les doigts des deux mains s'ouvrent et se referment alternativement imitant la bouche de l'ogre qui mange (9). Tous ces gestes s'effectuent dans un rythme saccadé transcrit, dès la première étape de la construction du SRM, par une série de petits traits en escalier.

Conclusion et perspectives

L'analyse des vidéos des productions d'élèves, réalisées en fin de chaque séquence tout au long du projet, montre que l'ancrage corporel de l'approche expérimentée facilite la perception, la conscientisation et la (re)production des éléments phonologiques de la langue anglaise, qui généralement constituent des obstacles pour des apprenants francophones, même très jeunes. Il s'agit des intonations modelées par des montées et des descentes qu'effectue la voix, des intensités qui se manifestent par des syllabes plus ou moins fortes, des durées variant en fonction du type de voyelles (brèves ou longues), des jeux vocaux imposés par la prononciation des diphtongues, des gestes buccaux inhabituels conditionnant la prononciation de certaines consonnes. Les élèves d' « âge critique » ont réussi à se soustraire à l'influence de leur langue première pour apprendre à accueillir et à reproduire les spécificités sonores de l'anglais grâce à un travail basé sur la transmodalité perceptive [START_REF] Cytowic | [END_REF] propre à l'être humain. En effet, la qualité phonologique des productions finales a été assurée par le fait que les élèves ont pu adopter des postures de chorégraphe et de « plasticien du son » [START_REF] Girault | La place du corps dans l'apprentissage sonore[END_REF] en amont de celle de locuteur. Cela invite à repenser la transversalité de l'enseignement des LVE à l'école. À cette transversalité qui est habituellement d'ordre thématique, il serait pertinent d'attribuer des dimensions perceptive et productive afin de permettre aux élèves des passages souples et fréquents d'un mode perceptuel (audition, vision, toucher…) à un autre ainsi qu'une alternance entre différents langages expressifs (mouvement corporel, dessin, production en différentes langues…) En didactique des langues, l'écoute-découverte est souvent assimilée à l'appropriation d'un concept extérieur (nouveaux mots dont la signification est dans un premier temps inconnue), conduisant à la construction d'un concept intérieur (intégration de nouvelles combinaisons de signifiés et de signifiants dans le bagage langagier de l'élève). Or, nous proposons un point de départ plus lointain qu'une unité conceptuelle : partir non pas du mot, mais du son, en l'abordant dans sa réalité physique avant de mettre en lumière sa fonction sémantique. En traitant les sonorités étrangères à l'aide de ses divers canaux sensoriels, l'enfant vit une expérience phénoménologique unique. Il ne « subit » pas la langue s'impose à lui mais la voit résulter de sa biographie corporelle.

Notons que la compétence d'écoute ne se limite pas à l'apprentissage des langues : dans un sens large, elle est un des piliers du savoir-être-élève et, plus tard, du savoir-être-citoyen. Qu'il s'agisse d'une écoute interpersonnelle, introspective ou apprenante (en lien avec les disciplines scolaires), il convient de réfléchir à la place de la pluralité sensorielle dans son développement à l'école.

Figure 2 .

 2 Figure 2. Deuxième étape de la construction du SRM 1. Le [t] de Tom est désigné par la lettre t soulignée d'un trait zigzag, signe que les élèves ont choisi pour symboliser cette consonne « brouillée » -c'est ainsi qu'ils ont pris l'habitude de nommer le « bruit » du [t] alvéolaire anglais, différent du [t] dental pratiqué en français. 2. Le [d] alvéolaire de don't est figuré par un signe analogue.3. Le [θ] interdental de est désignée par la lettre z « serrée » entre deux traits horizontaux, symbole trouvé par les élèves afin d'indiquer que, pour prononcer cette consonne, l'on procède « comme si on voulait dire [z] mais en plaçant la langue entre les dents de devant ».

Figure 3 .

 3 Figure 3. Troisième étape de la construction du schéma rythmo-mimique 1. Un garçon est dessiné à côté du trait vertical représentant Tom.

La voix et la main comme outils d'expression incorporée : point de départ d'une recherche-action