Yvan Caballero

Cédric Duvail

Sébastien Chazot

Frédéric Bouchette

Octavian Dobricean

Yann Balouin

Serge Berne

François Bourrin

Cédric Champollion

Valentin Dall'alba

Benoît Dewandel

Patrice Garin

Benoît Issautier

Bernard Ladouche

Sandra Lanini

Johanna Lofi

Marielle Montginoul

Noémie Neverre

Philippe Pezard

Philippe Renard

Transdisciplinary characterisation of a complex coastal aquifer, for a sustainable exploitation of its groundwater resources in a Mediterranean context. The DEM'EAUX ROUSSILLON project

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

More than 80 million m3 per year are pumped into the Roussillon plain coastal aquifer, covering 850 km² and located between the Pyrenean massif to the west and the Mediterranean Sea to the east, south of France. This is a multilayer aquifer of more than 350 m thick, made up of sandy layers embedded in low-permeability clayey material from the Pliocene and topped by alluvial formations from the Quaternary. Its groundwater resource is primarily used for the supply of drinking water, but also contributes to the irrigation of some 13,000 hectares.

For more than 40 years, this aquifer has been undergoing a general decline in its piezometric level due to pumping and water demand is expected to increase (growing irrigation areas and climatic demand). Moreover, given its flat topography, the Roussillon plain is likely to suffer sea water intrusions and marine submersion, due to the sea level rise, which could reach 1 m by 2100. This context shaped the Dem'Eaux Roussillon project, which brought together nearly ten partners from the Occitanie region (research units, consultancies and local authorities). Its objective was to characterise the behaviour of the groundwater resource in this aquifer, in order to be able to project its future situation, in the context of climate change, rising sea levels (risk of saline intrusion) and changes in water use. A detailed characterisation of the geological reservoir highlighted the need to consider the offshore extension of this coastal aquifer. The analysis of the piezometric evolution at the scale of the Roussillon plain over the last 50 years allowed the spatialized characterization of the hydrodynamic parameters and the understanding of the vertical drainage processes that control the hydraulic equilibrium between the Quaternary and the Pliocene water tables. Two high-resolution hydro-geophysical observatories have been set up to quantify these processes and improve understanding of saline intrusions processes. Finally, a conceptual model presenting the main features of the main processes controlling the groundwater evolution and the sea water intrusion risk was obtained ready to launch a numerical modelling work.

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Background

Staphylococcus lugdunensis is a virulent coagulase-negative staphylococcal species 1 . The ability to form biofilm appears to be one of its main virulence factors 2 , and is promoted by iron-limitation condition 3 . Nevertheless, little is known about biofilm-associated genotypes. In this context, the objective of this study was to compare the impact of iron restriction on fitness and biofilm formation of strains belonging to all clonal complexes (CCs) defined by Multilocus sequence typing 4 to date.

Materials

Clonal complex CC1 CC2 CC3 CC4 CC5 CC6 CC7

Results

Figure 3: Biofilm formation of 49 S. lugdunensis strains after a 24h incubation at 37°C in rich medium (TSB + 1% glucose) according to the CC (OD mean values). Error bars represent the standard deviation within each CC.

Figure 4: Biofilm formation of 34 S. lugdunensis strains after a 24h incubation at 37°C in rich medium (TSB + 1% glucose [full bars]) or iron-restricted medium (RPMI + 1% casamino acids [hatched bars]) according to the CC (OD mean values). Error bars represent the standard deviation within each CC.

• No difference in growth either according to the strains or according to the CCs, whatever the medium (Table 2) • All strains were biofilm producers (DO 590nm > 0,1) in rich medium after 24h

• CC2 strains produced significantly more biofilm than CC4 (p= 0.0203), CC5 (p= 0.0215) and singleton STs (p= 0.0045) strains (Figure 3)

Conclusions

• This study shows for the first time that the biofilm-forming ability of S. lugdunensis strains is strongly correlated with CC affiliation.

• In addition, two different CC-dependent behaviors are observed in an iron-restricted environment. These results are to be confirmed with a larger number of strains.

Figure 1 :Figure 2 :•

 12 Figure 1 : Diagram of a growth curve

Table 1 : S. lugdunensis strains categorized by clonal complexes and geographical origins.

 1

*Singleton Sequence Type

(ST)


Yaru Wu

Mathilde Hutin

Ioana Vasilescu

Lori Lamel

Martine Adda-Decker

Liesbeth Degand

Fine phonetic details for DM disambiguation: a corpus-based investigation

In this study we examine phonetic variation of discourse markers in French, using for this purpose the 4-hour richly annotated LOCAS-F corpus. Both linguistic factors and stylistic variables are considered: speech style, part-of-speech category, mean phone duration and vowel formant distributions with respect to the word status. The results show that the use of discourse markers increases with the degree of spontaneity of the speech. Coordinating conjunctions are the part-of-speech which is most frequently used for discourse markers. Moreover, the mean phone duration tends to be shorter and the vowel space more centralized when words are employed as discourse markers, suggesting that discourse markers undergo hypoarticulation and, more generally, reduction.

Introduction

Languages are known to be ambiguous, and the primary sources of ambiguity in the lexicon are polysemy and homophony, both resulting in similar phonological forms referring to different objects.

Polysemy refers to similar phonological forms with different, yet related meanings. For example, in French, /kafe/ refers to the coffee plant, to the drink made of this plant and to the place where this beverage is served. So-called "word sense disambiguation", i.e. "the ability to computationally determine which sense of a word is activated by its use in a particular context" [START_REF] Navigli | Word sense disambiguation: A survey[END_REF]) is a well-known challenge for word recognition systems [START_REF] Schütze | Automatic word sense discrimination[END_REF][START_REF] Jurafsky | Speech and Language Processing: An Introduction to Natural Language Processing, Computational Linguistics, and Speech Recognition[END_REF]. It is solved as best as possible with various methodologies that nevertheless all still need to mature [START_REF] Singh | Techniques for Disambiguation of Polysemy Words: A Review[END_REF].

Homophony refers to two identical phonological forms that have different meanings, and sometimes even belong to different grammatical categories. For instance, in French, the sequence /sÃ/ can stand either for the numeral cent, "hundred", for the noun sang, "blood", or the conjugated verb sens/sent, "(I, you, he/she/it) smell(s)". Homophones or near homophones, sometimes called "phonological neighbors", represent approximately 4% of the words in the world's languages (according to [START_REF] Dautriche | Weaving an ambiguous lexicon[END_REF]'s study on 67 languages). Moreover, studies have shown that the acquisition of a novel word is harder when the new entry has a homophone in the existing lexicon of the learner [START_REF] Swingley | Lexical competition in young children's word learning[END_REF], especially if the new entry and the existing homophone belong to the same grammatical category [START_REF] Dautriche | Learning novel phonological neighbors: Syntactic category matters[END_REF], and that the identification of a new homophone is more difficult when the semantic context it appears in is close to the lexical field of the existing homophone [START_REF] Cole | Perceptibility of phonetic features in fluent speech[END_REF]. In sum: phonological, grammatical and semantic proximity greatly complexify homophone disambiguation.

This paper proposes to add to the literature on disambiguation by investigating a question that seems to have been less studied: that of achieving disambiguation by modeling fine-grained phonetic details. We know that phonetic variation depends on word frequency [START_REF] Pierrehumbert | Word-specific phonetics[END_REF][START_REF] Phillips | Word Frequency and the actuation of sound change[END_REF]. For instance, it has been shown that word-frequency influences word-duration, allowing homophonous nouns and verbs in English to be distinguished [START_REF] Lohmann | Phonetic effects of grammatical category: How category-specific prosodic phrasing and lexical frequency impact the duration of nouns and verbs[END_REF], and that more frequent words appear to have more centralized vowels due to shortening [START_REF] Dinkin | The real effect of word frequency on phonetic variation[END_REF]. Phonetic variation also depends on grammatical function in that function words are shorter, acoustically poorer and more prone to reduction [START_REF] Adda-Decker | Quantifying temporal speech reduction in french using forced speech alignment[END_REF][START_REF] Ernestus | An introduction to reduced pronunciation variants[END_REF]. Finally, we know that phonetic variation depends on pragmatic usage, since fine prosodic cues can help disambiguate the function of words such as discourse markers (see [START_REF] Didirková | Naïve annotations of french et and alors: comparison with experts and effect of implicitation[END_REF][START_REF] Didirková | Impact of prosody on the perception and interpretation of discourse relations: Studies on "et" and "alors" in spoken french[END_REF] in French, [START_REF] Lee | Correlation between prosody and pragmatics: case study of discourse markers in French and English[END_REF] in French and English). For example, the French discourse marker alors, "then, so" will display major differences in the presence/absence of preceding pauses, word duration and pitch reset as a function of whether it conveys consequence (the basic meaning) or topic-shift and specification (less predictable relations) [START_REF] Didirková | Naïve annotations of french et and alors: comparison with experts and effect of implicitation[END_REF].

Our working hypothesis is that, since fine phonetic characteristics are word-specific, and thus will not apply equally to similar phonological forms with different meanings and/or functions (and therefore use), they can be exploited to disambiguate problematic phonological sequences, whether they be homophones or polysemic words. To test this hypothesis, we take into account a well-known case of ambiguity: discourse markers (henceforth DMs). DMs are words or expressions such as well, you know, I mean, that are highly frequent in language use and have been shown to play an essential role as fluency devices [START_REF] Crible | Discourse Markers and (Dis) fluency: Forms and functions across languages and registers[END_REF] and in discourse planning and communication management [START_REF] Levelt | Speaking: From Intention to Articulation[END_REF][START_REF] Hasselgren | Learner corpora and language testing[END_REF]. They are especially interesting to study because they usually emerge from other parts of speech with which they then co-exist in the language [START_REF] Degand | Alors between discourse and grammar: the role of syntactic position[END_REF]. For example, Fr. alors, "then, so", has been grammaticalized from Lat. illa hora, "at this hour", to become a temporal adverb as in J'avais alors 16 ans, "I was 16 at the time" and is now in parallel used as a DM, for instance to mark a boundary-opening as in Alors, comment c ¸a va?, "So, how are you?".

Following the cited literature, we aim at observing how DMs are realized in terms of their mean phone duration per word (suggesting local speech rate) and their vowels' pronunciations. We also specify how DM use correlates with speech style, which has been less investigated. To that extent, we use a 4-hour long French corpus that is richly annotated in part-of-speech and other metadata to show whether different grammatical functions of similar phonological forms bring about different phonetic behaviors. In the following, we will mainly compare the behavior of DMs (that we call "DM uses", e.g., opening boundary alors) with (i) the behavior of all the other words in the corpus (henceforth "Others", e.g., table, "table" or manger, "eat"), and (ii) the behavior of the same word-types (e.g., alors) not used as DMs (that we call "Non-DM uses"; e.g., temporal adverb alors) [START_REF] Bolly | Towards a model for discourse marker annotation in spoken french: From potential to feature-based discourse markers. Pragmatic Markers[END_REF].

In the remainder of this paper, we first describe our corpus (Section Corpus) and methodology (Section Methodology). Results are presented in Section Results regarding the relation between speech style and part-of-speech and DM usage, and the effect of DM usage on mean phone duration and vowel realization by comparing DMs ("DM uses") as opposed to all other words ("Others") and similar word-types not used as DMs ("Non-DM uses").

Corpus

For the present study, we chose to study the 4h long LOCAS-F corpus because of its already fine-grained manual annotations for parts-of-speech and DMs. The LOCAS-F corpus was put together by the linguistic research team at the University of Louvain [START_REF] Degand | Unités discursives de base et leur périphérie gauche dans locas-f, un corpus oral multigenres annoté[END_REF]. It is composed of 42 sound tracks of 3 to 5 minutes audio files, including in total 48 speakers. Multiple social practices were included in this corpus of primarily Belgian and metropolitan French, such that several speech styles are represented. This corpus will thus allow us to study DMs with regards to parts-of-speech as a function of phonetic cues as well as stylistic variables.

Methodology

Thirteen large part-of-speech categories were identified manually by specialists, all of which are considered in our analyses. In the section dedicated to partof-speech, only the nine parts-of-speech that are relevant to DMs are presented (see table 1 for the complete names of the parts of speech). The recordings were categorized into three speech styles according to the degree of preparation: (formal) prepared speech, (less formal) semi-prepared speech and (informal) unprepared speech. More details on the speech styles can be found in Degand et al. (2014a). Concerning the section on vowel space, measurements of the first and second formants were extracted using PRAAT [START_REF] Boersma | Praat: doing phonetics by computer[END_REF]. Given that /o/ and /O/ were manually annotated with similar standards in the LOCAS-F corpus, we decided to group the two phonemes into the same group ("o-O") in our vowel analyses. Vowel spaces are illustrated for discourse markers ("DM uses") vs all other words ("Others") and then for discourse markers ("DM uses") vs word-types that can fulfill the function of DM but are not employed as such ("Non-DM uses"). Due to limited space, we did not present vowel spaces as a function of speech style. Outliers were mostly observed at the bottom left corner of the vowel spaces and were excluded using a "y <= -0.5x + 1850" filter.

Statistical analyses were carried out on whether or not the word-token is a DM for each part-ofspeech and speech style using generalized linear models (GLM) in R (R Core Team, 2013). The model was used to test the effect of part-of-speech and speech style. ADJ, PRO and DET were grouped together, given that we observe almost no "DM uses" for these parts of speech. This grouping left us 7 categories for part-of-speech. The fixed effects considered were: part-of-speech (reference: CON) and speech style (reference: prepared speech).

Results

In this section, we investigate the DMs' distribution and phonetic properties with respect to speech style, part-of-speech categories, mean phone duration and vowel space.

Speech style

Table 2 presents the occurrences and percentages of discourse markers ("DM uses") vs all other word-tokens ("Others") for each speech style. These results show that discourse markers ("DM uses") are observed more in unprepared speech (7%) than in prepared speech (2%). The more prepared the speech is, the less discourse markers are observed, which is consistent with the literature as more spontaneous settings entail the use of such items for dialog management or planning purposes. Table 3 shows the occurrences and percentages of word-types when they correspond to "DM uses" vs "Non-DM uses", for each speech style. Among wordtypes that can fulfill a DM function, 8% of occurrences are used as discourse markers in prepared speech and more than 20% of the occurrences (24.49%) are used as discourse markers in unprepared speech. These results suggest that words that can be used as DMs are more likely to fulfill this function in less prepared speech. Part-of-speech and Speech style Figure 1 illustrates the rate of discourse markers ("DM uses") for each part-of-speech and each speech style, according to the classes specified in Table 1. Similar patterns are found for the three investigated speech styles, however, the less formal the speech style is, the more DMs are observed for each part-of-speech. Categories that are most often used as DMs are Coordinating conjunctions (CON; e.g., et, "and"), followed by adverbs (ADV; e.g., alors, "so"), interjec-Figure 1: Percentage of discourse markers ("DM uses") for each part-of-speech in LOCAS-F corpus and for each speech style. tions (ITJ; e.g., euh, "uh") and prepositions (PRP; e.g., pour, "for"). Interestingly, pronouns (moi, "me") and determiners (DET; le, "the" or mon, "my") are barely ever used as DMs in our corpus. The GLM results show that it is less likely to observe DMs in any of the other categories than in CON (p < 0.001) and that it is more likely to observe DMs in less prepared speech than in more prepared speech (p < 0.001 for all pairwise comparison).

Mean phone duration per word

Figure 2 gives an overview of mean phone duration per word for "DM uses" vs "Others". The highest bar is located slightly more to the left for "DM uses" than for "Others". This suggests that the local speech rate during the production of DM is slightly higher than for other words. A similar tendency is observed for "DM uses" vs "Non-DM uses" (Figure 3), suggesting that the local speech rate is higher for "DM uses" than for "Non-DM uses". While Figure 2 could be affected by the influence of word frequency (given that word-types that can correspond to DM realization concern words frequently used in speech and thus produced with higher local speech rate), analyses in Figure 3 which controlled for a frequency related impact, suggest that local speech rate is higher for "DM uses" than for "Non-DM uses".

Vowel space

Figure 4 illustrates the vowel space for DMs ("DM uses", left panel) vs other words ("Others", right panel) for female (F, top panel) and male speakers (M, bottom panel). The vowel space for other words is much larger than that for DMs, suggesting that vowels tend to be hypo-articulated for "DM uses". It is worth noting that DMs comprise only limited wordtypes, and thus limited vowel identities (i.e. /i/, /y/, /e/, /E/, a, /u/, /o-O/) were involved in the vowel space for DMs. Moreover, the larger vowel space for the other words could also be due to the large range of word frequencies for words engaged in this category. In order to have a better understanding of DM vowel space, we ran supplementary analyses on "DM uses" vs "Non-DM uses". Figure 5 illustrates the vowel space for DMs ("DM uses", left) and DM word-types that are not employed as DMs ("Non-DM uses", right) for female (F, top) and male speakers (M, bottom). The comparison thus allows us to investigate the same words used as DMs or not, and consequently comparable vocalic realizations. This comparison also allows us to control for word-frequency related variation. Similar to what is demonstrated in Figure 4, the group "DM uses" shows a smaller vowel space, suggesting that the acoustical realization of "DM uses" is more prone to reduction phenomena than in other situations ("Non-DM uses").

Conclusions

This study aims to characterize phonetic variation of discourse markers ("DM uses") compared to all other word-tokens ("Others") and to word-types similar to discourse markers yet not used as discourse markers ("Non-DM uses") -thus raising a question of ambiguity -using the 4-hour richly annotated LOCAS-F corpus in French. Different linguistic and stylistic parameters were investigated: speech style, part-of-speech, mean phone duration per word and vowel space. Results show that discourse markers ("DM uses") are observed more in less prepared speech. The partof-speech that frequently fulfills the function of discourse marker is coordinating conjunction, followed by adverbs, interjections and prepositions. The distribution of mean phone duration per word, suggesting local speech rate, shows that mean phone duration tends to be shorter for "DM uses" than for "Others" and for "Non-DM uses". Vowel space is smaller for "DM uses" than for "Others" and for "Non-DM uses", suggesting that discourse markers undergo hypo-articulation, and thus reduction, compared to other usages. Overall, our results encourage for further investigations of patterns of phonetic variation as cues for disambiguation in connected speech.

Figure 2 :

 2 Figure 2: Mean phone duration per word for discourse markers ("DM uses") vs other words ("Others").

Figure 3 :

 3 Figure 3: Mean phone duration per word for discourse markers ("DM uses") vs DM words that are not employed as DMs ("Non-DM uses").

Figure 4 :

 4 Figure 4: Vowel space for DMs ("DM uses", left panel) and other words ("Others", right panel) for female (F, top panel) and male speakers (M, bottom panel).

Figure 5 :

 5 Figure 5: Vowel space for DMs ("DM uses", left panel) and DM words that are not used as DMs ("Non-DM uses", right panel) for female (F, top panel) and male speakers (M, bottom panel).

Table 1 :

 1 Details of part-of-speech used in 1.

	Abbreviation Complete category name
	ADJ	Adjective
	ADV	Adverb
	CON	Coordinating conjunction
	DET	Determiner
	ITJ	Interjection
	NOM	Noun
	PRO	Pronoun
	PRP	Preposition
	VER	Verb

Table 2 :

 2 Occurrences and rates of DM uses vs Others for the three investigated speech styles.

		Prepared Semi-prepared Unprepared
	DM uses	252 (2%)	440 (5%)	1321 (7%)
	Others	12009 (98%)	8270 (95%)	16472 (93%)
	Total	12261	8710	17793

Table 3 :

 3 Occurrences and rates of DM uses vs Non-DM uses for the three investigated speech styles.

		Prepared Semi-prepared Unprepared
	DM uses	252 (8%)	440 (18%)	1321 (24%)
	Non-DM uses	2729 (92%)	1981 (82%)	4073 (76%)
	Total	2981	2421	5394

Acknowledgements

This research was supported by DATAIA/MSH Paris-Saclay "Excellence" grant and by IdEX U. de Paris, ANR-18-IDEX-0001, "Emergence" grant.


Luc-Etienne Pommé

Romain Bourqui

Romain Giot

David Auber

Relative Confusion Matrix: Efficient Comparison of Decision Models

Keywords: Confusion Matrix, Models comparison, Evaluation

Current machine learning and deep learning approaches are cutting-edge methods for solving classification tasks. Comparing the performances of classification models has become a prominent task since the outbreak of these techniques. The performance of such classification models is measured by the ratio between the correctly predicted samples and the others. The most widely used visualization to represent this information is the Confusion matrix. Yet, if this technique is suited to apprehend one model performances, very few works use this representation to compare models. In that paper, we present the Relative Confusion Matrix (RCM), a new matrix visualization that leverages Confusion matrices and a color encoding to expose the class-wise differences of performances between two models. We conduct a user evaluation to compare RCM with two confusion matrix variants. Our results show that RCM encoding leads to a more efficient comparison of two models than existing approaches.

INTRODUCTION

Machine Learning (ML) and Deep Learning (DL) algorithms are widely used to solve detection [START_REF] Ren | Faster R-CNN: towards realtime object detection with region proposal networks[END_REF], segmentation [START_REF] Girshick | Rich feature hierarchies for accurate object detection and semantic segmentation[END_REF] or classification [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF] problems. In the literature, instances of theses algorithms are called models. Prior to be used, these models must be trained to setup their hyperparameters. Then these models can be benchmarked to measure their accuracy and behavior. The focus of that paper is to present a comparison and a user evaluation of different visualization techniques which enable to compare results obtained by two different trained models on the same classification task.

A classification task consists in assigning to an input data (for instance, an image) a label called class (for instance "cat") from a bounded dictionary (ie. alphabet). One calls a prediction the output class. To benchmark models, one computes predictions on an input dataset for which classes are already known (ground truth) and then predictions are compared to ground truths.

The performance of a model can be evaluated at different levels. [START_REF] Gleicher | Considerations for visualizing comparison[END_REF] At a global-level, measures such as the accuracy (good predictions / number of samples) provide a score for the entire test set. [START_REF] Hinterreiter | ConfusionFlow: A model-agnostic visualization for temporal analysis of classifier confusion[END_REF] At a class-level, measures are used to assess the performance of the model through the prism of a specific class. For instance, the recall of a class indicates whether the samples of that class were correctly classified.

A lot of works [START_REF] Hinterreiter | ConfusionFlow: A model-agnostic visualization for temporal analysis of classifier confusion[END_REF], [START_REF] Talbot | EnsembleMatrix: interactive visualization to support machine learning with multiple classifiers[END_REF], [START_REF] Alsallakh | Visual methods for analyzing probabilistic classification data[END_REF], [START_REF] Ren | Squares: Supporting interactive performance analysis for multiclass classifiers[END_REF] leverage class-level measures for models analysis and rely on a visualisation called confusion matrix (CM) [START_REF] Powers | Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation[END_REF].

A CM is a matrix where the rows represent the ground truth of input samples and the columns represent the predictions of a model, in the same order. The cell (row, col) counts the percentage of samples of ground truth row predicted as class col. Standard representation of CM fill each cell using a gradient of colors mapped to the percentages. Some variants display the percentages in each cell. A dark color is tied to a high percentage while a light color is tied to a low percentage.

Confusion matrices were designed to compare the class performance within a model. However, when designing a new model, a common approach consists in training several model and then selecting the best one. In such a case, confusion matrices may not be sufficient to efficiently compare the performance of two (or more) models. While the accuracy can provide some cues about the overall behaviors of the compared models, it does not support the comparison of models at a class level. In particular, when comparing two models it is important to assess (1) whether one of the models achieves better performance for all classes or (2) one model is better for a set of classes but worst for another one. Identifying these classes can also be important to the application domain; some classes may be more crucial than others.

In this work, we study different strategies to compare two models that solve the same classification task. The contribution of this paper is twofold. First, we present the Relative Confusion Matrix (RCM) which allows a class-level performance comparison between two models. Last, we conducted a user evaluation to compare RCM with two existing visualization methods to compare two models at a class-level.

In the next section, we provide a brief state of the art of model comparison techniques. Then, we present RCM and how it differs from other visualizations. Then, we describe the user evaluation protocol and the results of the evaluation. Finally, we discuss the results and draw conclusions.

RELATED WORK

In this paper, we are interested in the visual comparison of two models performances. This section discusses about existing techniques for that purpose.

To visually compare quantitative information, it is well established that position-based [START_REF] Mackinlay | Automating the Design of Graphical Presentations of Relational Information[END_REF] visualizations like bar charts are the most efficient approaches. They are particularly adapted to compare one or two variable over a few models [START_REF] Purchase | Experimental Human-Computer Interaction: A Practical Guide with Visual Examples[END_REF], for example to compare a user response time or an accuracy error over several views during a user evaluation. For binary classification models, the Receiver Operating Characteristic (ROC) [START_REF] Fawcett | An introduction to ROC analysis[END_REF] curve in a form of a line chart can be drawn. The curve gives information about how much the model benefits, making correct predictions for a class, compared to how much it costs, incorrectly classifying samples from another class into that class. However, it is less suitable for problems with more than two classes due to overlaps. In Squares [START_REF] Ren | Squares: Supporting interactive performance analysis for multiclass classifiers[END_REF], Ren et al. compare multiple classification models in rows using an horizontal histogram per class to show the distribution of samples in each class.

Alsallakh et al. [START_REF] Alsallakh | Visual methods for analyzing probabilistic classification data[END_REF] spread the classes along a confusion wheel to visualize more information about the content of the classes. In particular, for each class, they use colors to differentiate common performance measures like true positives (TP), false positives (FP), false negatives (FN) and true negatives (TN) in histograms of classification probabilities. In this representation, each class along the wheel requires enough space to display a curved histogram. This may be problematic when comparing two spatially distant classes, especially when the number of classes increases.

Most of the time, to compare the performance of a model at a class-level, a confusion matrix is used [START_REF] Talbot | EnsembleMatrix: interactive visualization to support machine learning with multiple classifiers[END_REF], [START_REF] Amershi | Modeltracker: Redesigning performance analysis tools for machine learning[END_REF]. CM have the advantage of being model-agnostic. Instead of filling the cells of a CM using a gradient of colors, NEO [START_REF] Görtler | Neo: Generalizing Confusion Matrix Visualization to Hierarchical and Multi-Output Labels[END_REF] encodes the percentages in the cells with rectangles of different sizes.

As Hinterreiter et al. [START_REF] Hinterreiter | ConfusionFlow: A model-agnostic visualization for temporal analysis of classifier confusion[END_REF] stated, very few works address the problem of comparing two models at a class level. In general, this comparison is implicit. For example, Li et al. [START_REF] Li | CN-NPruner: Pruning Convolutional Neural Networks with Visual Analytics[END_REF] display one CM per neural network obtained through iterations of pruning. The CM of two distinct models are not displayed at a time, which may require memory attention from the user to compare both models.

For comparison problems in general, Gleicher [START_REF] Gleicher | Considerations for visualizing comparison[END_REF] suggested three ways to design components. The first and most common way to compare objects is the juxtaposition of instances of the same visualization. Talbot et al. [START_REF] Talbot | EnsembleMatrix: interactive visualization to support machine learning with multiple classifiers[END_REF] compared multiple classifiers by juxtaposing confusion matrices. The second way is the superimposition of two instances of the same visualization. Basak et al. [START_REF] Basak | Weighted graph comparison techniques for brain connectivity analysis[END_REF] make the parallel between superimposed directed weighted graphs and superimposed matrices. In each cell of their CM, they display two colored concentric rectangles to encode the result of two models on that cell. The third way uses a special encoding to combine the information of the two objects. In this last category, ConfusionFlow [START_REF] Hinterreiter | ConfusionFlow: A model-agnostic visualization for temporal analysis of classifier confusion[END_REF] suggest to compare multiple confusion matrices in a single visualization using lines in each cell. The points of the lines in each cell estimate the variation of the percentage of samples in that cell for all models. This representation allows a global comparison of several models at a class-level.

DESIGN CHOICES

In this section, we present the three visualizations we decided to compare in our user evaluation. First, we describe our contribution: the Relative Confusion Matrix (RCM). This matrix is designed specifically to compare two models that solve the same classification task, and more precisely, to show where a model B improves or deteriorates the performance of a model A. From now on, we consider both model process the same input data and have the same number of output classes.

The design of the RCM is very similar to the design of standard confusion matrices: it is a square table where rows represent all possible ground truths and columns represent all possible predictions, sorted with the same order.

Two different models may have different percentages in every cell. The goal of RCM is to highlight these differences between two models A and B. Each cell of RCM contains the cell-by-cell absolute percentages difference between B and A.

To ease the interpretation of RCM, a gradient of colors is used to fill in the cells. A dark color is used when the difference is high. A light color is used when the difference is small. In addition, if there is some difference between the two models on a cell, a + (resp. -) is displayed on that cell to indicate that B is better than A (resp. A better than B). A blank cell indicates there is no difference between A and B on that cell.

We chose + andto indicate which model is the best on a cell. It is particularly interesting in a context of B being a variant of A. Then, the symbol answers the question of B improving or deteriorating the performance of the cell compared to A (Fig. 3). A or B could have been a reasonable choice but these letters are written with more pixels than simple + and -, and they contain loops. Especially when the number of classes is high, the color perception of the cells could be altered with letters. Moreover, we did not use more than one gradient of colors (e.g to differentiate diagonal cells from the other cells or to differentiate cells showing improvements from cells showing deterioration) for RCM to avoid comparisons between the lightness of two or more different colors.

In our evaluation, we compared RCM with two matrixbased representations: the Twin Confusion Matrix (TCM), which is a special case of the EnsembleMatrix [START_REF] Talbot | EnsembleMatrix: interactive visualization to support machine learning with multiple classifiers[END_REF] and the Combined Confusion Matrix (CCM) which is based on Basak et al. [START_REF] Basak | Weighted graph comparison techniques for brain connectivity analysis[END_REF] matrix. This way, we compare visualizations of the three categories: juxtaposition, superimposition and special encoding [START_REF] Gleicher | Considerations for visualizing comparison[END_REF]. We call TCM the juxtaposition of two confusion matrices, one for A, the other for B. We consider this representation as the baseline of our comparisons. The CCM superimposes the matrices of both A and B in a single matrix. Each cell contains two nested colored rectangles indicating the performance of each model on that cell.

The goal of the evaluation is to study three hypothesis: first, we make the assumption that comparing two models with a superimposed visualization or a visualization that encodes the difference is better than a side-by-side visualization (H vis ). Second, we want to study, for the comparison of two models, if symbols are better suited than colors (H symb ). Third, we make the assumption that finding the darkest color is easier than comparing contrasts (H dark ).

USER EVALUATION

In this section, we describe the protocol we established to build our evaluation. It is based on the Purchase protocol [START_REF] Purchase | Experimental Human-Computer Interaction: A Practical Guide with Visual Examples[END_REF]. We define four tasks to solve on these visualizations, how data have been generated, and how the sessions were organized.

A. Visualizations

This evaluation targets the comparison of two models on three visualizations (TCM, CCM and RCM). This means we are not interested in how a model performs alone, but relatively to another. Thus, none of the tasks targets the performance of a single model. We focus on the functionalities all visualizations share.

B. Tasks

Our tasks are inspired from those defined for Squares [START_REF] Ren | Squares: Supporting interactive performance analysis for multiclass classifiers[END_REF] to compare models. For example, one of their tasks consists in detecting the classifier with the largest number of errors. Another of their tasks consists in comparing distributions of classes. We have retained the two ideas of giving the best model under a given condition (T1 and T2) and comparing two distributions (T3 and T4), which led to the following tasks, formulated as questions:

• T1: Which model does correctly classify more samples for the highlighted class? • T2: Which model is the best on the highlighted cell? • T3: On which class is there the most difference of correct predictions between the two models? • T4: Among incorrect predictions, which pair (GT, prediction) shows the greatest difference between the two models? For T1 (resp. T2), given a diagonal cell (resp. non-diagonal cell) the best model on that cell has to be selected. In both cases, the given cell is chosen randomly and displayed with a distinct color directly on the matrix or on the matrices. Both the line and the column of the chosen cell are framed with that color. To solve T3 (resp. T4), the diagonal cell (resp. non-diagonal cell) that show the greatest difference between models A and B has to be selected. The difference between T1 and T2 (resp. T3 and T4) is that the reasoning is inverted. Indeed, diagonal cells refer to correct predictions while nondiagonal cells refer to incorrect predictions.

C. Dataset

We randomly generated data (matrices) to ensure generalizability of the results [START_REF] Basak | Weighted graph comparison techniques for brain connectivity analysis[END_REF]. For each trial, we generated a pair of matrices of same size, one for A and one for B. We generated two pairs of matrices per task (T1-T4), per visualization (TCM, CCM, RCM) per size (2x2, 10x10 or 30x30). We added two constraints in our generator that aims to simulate models with relatively average or high performance, which is often the case with real models. First, we arbitrarily fixed the same density at 0.3 for all matrices, which means only 30% of the nondiagonal cells of each individual matrix contain at least one sample. Thus, the percentage of nondiagonal cells that are different between A and B are in [30%, 60%]. Our generator imposes as a second constraint that all classes of each model have a minimal recall of 0.3. This means for each class, at least 30% of samples of that class are correctly classified, on the diagonal. For T1 and T2, the highlighted cells are chosen randomly with a uniform probability distribution. For all matrices created for a specific task, we know the answer by construction. For each trial, we verified the uniqueness of the answer. We generated several additional pairs of matrices with various sizes for the tutorial, and for training [START_REF] Purchase | Experimental Human-Computer Interaction: A Practical Guide with Visual Examples[END_REF].

D. Protocol

During the whole evaluation, all the participants are supervised by the same person. Each participant starts the evaluation with a tutorial that explains what confusion matrices are, how TCM, CCM and RCM are built, and the four tasks to solve. Fig. 1 and Fig. 2 are parts of the tutorial and are accompanied by textual explanations and examples. For each task in the tutorial, the participant is given one answered example per visualization followed by one example per visualization to train. The participant can see the correction for all examples, retry or come back at any point in the tutorial, and ask as many questions as she wants to the supervisor until the end of the tutorial. Once the participants have finished the tutorial, they are all given the same instructions and indications for the next part of the evaluation. The instructions insisted on not answering randomly or too fast, but correctly as fast as possible.

For the whole evaluation, the resolution of all matrices in pixels are the same. 2x2 matrices are displayed in 300x350, 10x10 matrices are displayed in 400x450, and 30x30 matrices are displayed in 500x550.

During the evaluation, the trials are grouped by task since we do not measure how participants adapt to a task, but how they solve it. For generalizability, two latin squares are used to ensure the participants do not face neither the same trials in the same order nor the same tasks in the same order. However, all participants are given one minute to answer each trial, with three seconds break between each trial. A break is scheduled between each task. At the end, the participants are asked to give their opinions on the visualizations and tasks through a short survey.

RESULTS

In this section, we present the results of our user evaluation. We analyze the results under three conditions which are the three visualizations (TCM, CCM, RCM).

Participants. 26 volunteers performed the evaluation in total with a mean age of 31.5 years. We had to not consider 3 of them due to their lack of focus and because some trials were answered randomly according to both the participants and the supervisor. For each participant, we collected both the answer to compute the Error Percentage (EP), and the mean Response Time (RT) per trial. Some participants did not answer within the time limit for specific trials. We decided to not include their answers to these trials in the response time measurements. In the following, we report the statistical analysis and results.

Significance per visualization. We start with a high-level comparison to detect if there is a global significant difference between the visualizations for both EP and RT. For at least one condition, the data are not normally distributed for both EP and RT. We chose a non-parametric statistical approach, using the Kruskal-Wallis independent measures test, with a significance threshold of 0.05. This test indicates an overall significant difference on the mean RT (p-value RT = 4.9 * 10 -8 < 0.05). However, the same test on the EP does not reveal any significant difference (p-value EP = 0.17 > 0.05). For further posthoc pairwise comparison on RT, we use the Conover test with corrected p-values which reveals a significant difference between all pairs of conditions. In Fig. 4, a red bar indicates a significant pairwise difference to all other conditions. The mean response times of all conditions reveal that globally, RCM is the most efficient visualization. We conduct further experiments to analyze the reason of these differences. Significance per task, per visualization. By aggregating data per task and per visualization before computing the mean RT and Error, we aims to highlight what visualizations characteristics may be more efficient than others. At this level, a Kruskal-Wallis test reveals an overall significant difference for both EP and the mean RT per task. Fig. 5 shows the posthoc pairwise comparisons between each visualization. From this figure, we can notice several interesting results. For T1 and T3, the only significant differences are between TCM and the other visualizations. TCM has the worst response times and the worst error percentages. The juxtaposition of the two matrices of A and B forces the participants to memorize colors for comparisons of two distant cells which increases the response time of each trial and the risk of errors. Interestingly, the pairwise tests do not reveal any significant difference neither for EP nor RT when comparing all nondiagonal cells to detect the greatest difference of incorrect predictions (T4). Though, we observe a large standard deviation for this task that has to be explored with further investigation. The second significant difference lies in T2. There is a significant mean RT difference between CCM and RCM, RCM being more efficient. However, CCM is more confusing to participants than other visualizations. Participants answer relatively fast but tend to be misled by the color inversion of CCM between diagonal and nondiagonal cells, until they realize their mistake.

Significance per task, per matrix size, per visualization. From these results, we investigate further to see whether the size of matrices impact the performances on the different visualizations. These measures are shown in Fig. 6. In this figure, the same phenomenon described before appears on CCM with T2. The EP difference is more relevant as the matrices size grows. The examples of CCM with a size of 30x30 have indeed a small visible difference between light colors. With comparable matrices size, RCM outperforms both TCM and CCM: participants make less errors within less time. As shown in Fig. 6, statistical tests also reveal participants answered significantly faster for RCM without committing significantly more or less mistakes. The only case CCM outperforms RCM is for the comparison of all nondiagonal cells to find the greatest difference (T4) on large matrices (30x30). The reason of RCM being less efficient than CCM could be the density of black symbols in the matrices that affect the light colors perception. The last important remark is that TCM almost never outperforms the other visualizations.

Participants preference. All participants rated the three visualizations between 0 and 4, 4 being the best mark. We collected their rates for each individual task, and globally. We analyzed them with the Kruskal-Wallis test. Globally, the overall difference between the three visualizations is significant. Posthoc comparisons using the Conover test reveal a significant difference between all pairs after p-values correction. In general, users preferred RCM to CCM to TCM. This tendency is exactly the same for each individual task (see Fig 7).

DISCUSSION AND CONCLUSION

The user evaluation we conducted reveals that most users are not comfortable with manipulating side-by-side visualizations to compare models. Such comparisons are relatively slow and require from the user to memorize at least four different colors at a time. As we expected, the spatial distance between the elements to compare makes the tasks hard to solve. This validates our hypothesis H vis .

Surprisingly, the symbols do not make the tasks T1 and T2 faster to solve. After surveying our participants, it appeared that, unlike for the two other visualizations (TCM and CCM), they needed to check systematically the legend on the left to remember the meaning of the + and -symbols. This validation step they performed only for RCM artificially increased their response time. We strongly believe that an expert of the three visualizations that would be able to learn the meaning of the symbols without constantly using the legend, would answer significantly more efficiently on RCM (H symb ).

The results on task T3 validated half of the hypothesis H dark . Indeed, participants found the darkest diagonal cell of RCM faster than the greatest contrast of TCM or CCM. However, when the matrix is large and dense, the symbols that better highlight the small differences tend to disturb the perception of the colors in the cells. Some participants mentioned this problem for large matrices. To solve this problem, we suggest to make RCM interactive by adding a slider that determines the minimum required amount of differences in a cell, from which symbols + andneed to be displayed. Otherwise, this threshold could control the percentage of cells that show the greatest difference between the two models in which symbols have to be displayed.

Participants also mentioned the neighborhood of a cell can falsify the perception of its color(s), especially for TCM and CCM, when the aforesaid cell is close to the diagonal dark cells.

In the end, we validate the effectiveness of the Relative Confusion Matrix to compare two classification models, up to 30 classes. To address more classes and reduce the number of rows and columns of the confusion matrices, we suggest to establish a hierarchy that groups similar classes if it does not exist, and apply hierarchical interactions, as described by Görtler et. al. [START_REF] Görtler | Neo: Generalizing Confusion Matrix Visualization to Hierarchical and Multi-Output Labels[END_REF]. To enhance the most important symbols for large and dense matrices, we suggest to add an interaction like a slider to threshold the display of symbols.

Eventually, one may not want to compare only two classification models. To adapt our visualization to more classification models, we can imagine to designate each model with a unique symbol, that can be written with as few pixels as possible. Instead of displaying + andsymbols in the cells, we could display the symbol representing the best model in this cell. Then, the color of the cells could represent how much this model is better in this cell than the other.

Fig. 1 :Fig. 3 :

 13 Fig. 1: Example of Twin Confusion Matrix (TCM) representation

Fig. 4 :Fig. 5 :

 45 Fig.4: Mean Error Percentages and Mean Response Times with standard deviation bars per visualization. Statistical tests showed that the visualization had a significant effect on the MRT. A bar with a red color indicates that the posthoc pairwise comparisons of this visualization against all others is significant (p-value < 0.05) according to Conover test[START_REF] Conover | On multiple-comparisons procedures[END_REF].

Fig. 6 :

 6 Fig. 6: Mean Error Percentages and Mean Response Times with standard deviation bars per task, per matrix size, and per visualization. A red bar indicates that the posthoc pairwise comparisons of this visualization against all others is significant. An arc connecting two blue bars indicates a significant posthoc comparison between the two visualizations.

Fig. 7 :

 7 Fig. 7: Mean Ratings per visualization, and per task per visualization with standard deviation bars. All bars are red, indicating that all the rating differences between pairs of visualizations are significant. The means and standard deviation indicate RCM is the most preferred visualization.

ACKNOLEDGMENT

We thank the Nouvelle-Aquitaine Region, Bordeaux Métropole and SUEZ, le LyRE for mainly funding and supporting this work through the Convention N°AAPR2020-2019-8171810.


Hugo Tessier

Vincent Gripon

Mathieu Léonardon

Matthieu Arzel

David Bertrand

Thomas Hannagan

Investigating the Not-So-Obvious Effects of Structured Pruning

Structured pruning is a popular method to reduce the cost of convolutional neural networks. However, depending on the architecture, pruning introduces dimensional discrepancies which prevent the actual reduction of pruned networks and mask their true complexity. Most papers in the literature overlook these issues. We propose a method that systematically solves them and generate an operational network. We show through experiments the gap between the theoretical pruning ratio and the actual complexity revealed by our method.

Introduction

Deep convolutional neural networks are at the state of the art in many domains, such as computer vision. However, their cost in energy, memory and latency is prohibitive on embedded hardware, and this is why many works focus on reducing their cost to fit targets with limited resources [START_REF] Chen | Deep convolutional neural network on ios mobile devices[END_REF]. The field of deep neural networks compression counts multiple types of method, such as quantization [START_REF] Courbariaux | Binaryconnect: Training deep neural networks with binary weights during propagations[END_REF] or distillation [START_REF] Hinton | Distilling the knowledge in a neural network[END_REF]. The one we focus on in this article is pruning (Han et al., 2015b), that involves removing unnecessary weights from a network.

To focus on the theoretical approach of studying the impact of removing weights from the network's function on its accuracy, many papers only remove weights by putting their value to zero, which does not reduce the cost of networks. Leveraging pruning to get gains on hardware is actually not a trivial task. Pruning isolated weights (Han et al., 2015b) ("non-structured pruning") produces sparse matrices, that are difficult to accelerate [START_REF] Ma | Non-structured dnn weight pruning-is it beneficial in any platform[END_REF]. Pruning entire convolution filters (a.k.a. "structured pruning") is more easily exploitable, but the input and output dimensions of layers are altered, which can induce many problems in net-Proceedings of the 39 th International Conference on Machine Learning, Baltimore, Maryland, USA, PMLR 162, 2022. Copyright 2022 by the author(s).

works, especially those including long-range dependencies between layers [START_REF] He | Deep residual learning for image recognition[END_REF].

In this paper we propose a solution to reduce effectively the size of networks using structured pruning. Our method is generic, automatic and reliably produces an effectively pruned network. We demonstrate its ability to operate on networks of any complexity by applying it on both a standard classification network [START_REF] He | Deep residual learning for image recognition[END_REF] on the Ima-geNet ILSVRC2012 dataset [START_REF] Russakovsky | ImageNet Large Scale Visual Recognition Challenge[END_REF] and on a more complex semantic segmentation network [START_REF] Sun | High-resolution representations for labeling pixels and regions[END_REF] trained on CityScapes [START_REF] Cordts | The cityscapes dataset for semantic urban scene understanding[END_REF]. Our experiments show that not taking into account the aforementioned problems tends to distort the apparent trade-off between the computational complexity and accuracy, which harms the relevance of the results. Therefore, our method is a useful tool for a more reliable study of pruning.

Related Works

When pruning a network, three aspects have to be tackled: 1) what kinds of parts to prune, 2) how to identify those to prune and 3) to prune them. We already mentioned how the first aspect can be divided into non-structured and structured pruning. The second issue can be solved using various types of pruning criteria. In the case of non-structured pruning, the magnitude of weights (Han et al., 2015b) or their gradients [START_REF] Molchanov | Pruning convolutional neural networks for resource efficient inference[END_REF] are two popular criteria. In the case of structured pruning, these criteria can be extended to either their norm over a filter [START_REF] Li | Pruning filters for efficient convnets[END_REF] or a proxy that accounts for the whole filter's importance, for example the multiplicative learned weight included in batchnormalization layers [START_REF] Liu | Learning efficient convolutional networks through network slimming[END_REF]. These criteria can be applied in two different ways: either they are used to identify the same (or a pre-determined) amount of weights/filters to remove in all layers (local pruning) or the target is set globally and the criterion is applied to all layers, possibly unevenly, at the same time (global pruning).

Concerning the third issue, many popular methods apply a simple framework (Han et al., 2015a): training the network, pruning a given proportion of weights by masking them away, fine-tuning the network and repeating the last two steps multiple times until a target pruning rate is reached. Other methods can involve a more progressive approach (He Figure 1. Pruning filters (greyed-out columns) means producing fewer output channels: the input dimensions of the following layers must be adapted (greyed out rows). However, the number of channels a layer gets as an input is not easy to predict: because of the addition at the end of each residual block, the respective index of each channel must be conserved in the destination tensor. Summing together the outputs of two pruned layers can produce a tensor whose number of channels is different from that of any of the two summed tensors, as the addition must be replaced with a mix of concatenations and additions (which we call an indexation-addition operation). Therefore, the dimensions of tensors and of the layers can only be deduced by taking into account all dependencies in the network at once. et al., 2018) that can include a regrowing mechanism [START_REF] Mocanu | Scalable training of artificial neural networks with adaptive sparse connectivity inspired by network science[END_REF]. Some techniques propose a more continuous way to prune weights, for example the application of a penalty on them during training [START_REF] Tessier | Rethinking weight decay for efficient neural network pruning[END_REF].

Method

In Section 2, we explained what is structured pruning. Pruning filters can introduce some dimensional discrepancies, especially in architectures such as ResNets [START_REF] He | Deep residual learning for image recognition[END_REF], that must be tackled to be able to allow proper inference. Figure 1 and its caption sum up the different problems that can occur and that our method is designed to solve.

These problems are often overlooked in the literature, even though most papers deal with ResNet-based architectures that are subject to all of them. Some expertise allows manually figuring out dependencies in such networks, but the complexity can get out of hand in the case of networks such as HRNets [START_REF] Sun | High-resolution representations for labeling pixels and regions[END_REF]. Missing any of these problems makes the networks impossible either to run efficiently or to run at all. Moreover, avoiding these problems by only putting pruned weights to zero, and thus not taking into account any of these dependencies, leads to evaluate incorrectly the actual runtime complexity of pruned networks, which harms the reliability of their study. This is the reason why we propose a method that can automatically and reliably produce pruned networks in which of all these problems have been solved.

Automatic Adaptation of Networks

The first step of our method involves identifying all the parts of the network that are disconnected when removing filters by taking into account all the dependencies.

To identify all the parameters whose contribution in a network's function is null, one can use their gradients over, for example, a mini-batch from the training data. Indeed, provided this mini-batch is a satisfying approximation of the network's domain of definition, a null gradient means that the network's function is null relatively to the involved weights, or at least constant in the case of biases. However, for our use-case, this is insufficient: not only does it not allow removing disconnected biases that still produce constant outputs that somehow contribute to the function, but it may also identify some isolated weights as pruned in a non-structured way, which we do not want to consider. This is why we instead operate on an architectural abstraction of the network, which is a copy of it that received modifications that are illustrated and described in Figure 2 and its caption. These modifications allow a single input, filled with non-null values of the same sign, to be enough to identify all disconnected weights. Indeed, this network behaves like a purely linear and positive function and any null gradient in its parameters can only be due to a null function that can be removed. Weights, identified as disconnected in this copy network, are then removed from the original network, effectively reducing its size.

Automatic Indexation in Additions

Deep convolutional networks can contain operations that create constraints on the tensors they take as an input. In the case of networks such as ResNets [START_REF] He | Deep residual learning for image recognition[END_REF] or HRNets [START_REF] Sun | High-resolution representations for labeling pixels and regions[END_REF], the addition at the end of every residual block needs its two input tensors to be of the exact same dimensions, which may not be the case anymore after pruning. We can tackle this problem by replacing additions with a generalized operator able to handle missing filters in any of its inputs. To this mean, we replace additions with a new indexation-addition operation, with a and b the tensors to sum, that contain respectively n a and n b channels, i a and i b two lists of indices and the output tensor c, that contains n c channels, defined in Equation ( 1).

If n a = n b , i a = [1, 2, . . . , n a ] and i b = 1, 2, . . . , n b , this indexation-addition operation is purely equivalent to an element-wise addition. Properly parameterized by adequate i a and i b , this operation allows leveraging any type of filter pruning. It is however necessary to find the right i a and i b and we provide a solution in Section 3.2. Figure 1 illustrates how our solution relates to the problems mentioned in its caption and provides a simple way to view how it can behave like a mix of additions and concatenations.

∀k ∈ 1; n c , c k = a i a k , if i a k ∈ 1; n a ∅, otherwise (1)
+ b i b k , if i b k ∈ 1; n b ∅, otherwise
To deduce automatically the right i a and i b , we add another modification to the copy network described in Section 3.1: we apply an identity convolution to the two tensors before summing them together. This identity convolution has weights of shape n × n × 1 × 1 (with n the number of channels in the input tensor, 1 × 1 because we consider 2d inputs) whose values equates that of an identity matrix.

The gradient of the weights of this identity convolution allows deducing the corresponding list of indices. Indeed, once the null rows and columns of its weights are removed, the output dimensions are the same for both tensors to be summed while the input dimension matches that of the input tensors after pruning. The zero and non-zero remaining coefficients allow deducing how to map the input to the corresponding output channels.

Summary

Algorithm 1 sums up our overall method:

Algorithm 1 Summary of the Method 1: train the network N 2: generate the pruning mask m that masks out filters 3: create a copy N ′ of the network 4: remove all the biases b from N ′ 5: remove all the activation functions and normalizations from N ′ 6: replace the weights w of N ′ by m 7: insert the identity convolutions where needed in N ′ 8: generate an input tensor x, of adequate size, filled with ones and run N ′ (x) 9: compute δN ′ δw (x) 10: generate the new pruning mask m ′ that masks away all the weights whose gradients are null in N ′ 11: apply m ′ to N and mask away biases whose weights are pruned 12: deduce from the mask of the identity convolutions the right i a and i b to replace additions with indexationaddition operations where needed This method produces pruned networks in which interdependant weights are guaranteed to be pruned altogether, which prevents any dimensional mismatch between layers.

Our method also adapts the addition operations so that they can accommodate tensors of different number of channels. Therefore, all the filters and kernels are guaranteed to contribute to the network's function, which allows a more reliable measurement of the actual pruning rate and associated complexity of pruned networks.

Experiments

In our experiments, we will detail the impact of our method on both the accuracy of the network and the evaluation of its compression rate. Our source code is available at: https://github.com/HugoTessier-lab/ Neural-Network-Shrinking.git.

Training conditions

ImageNet We trained ResNet-50 [START_REF] He | Deep residual learning for image recognition[END_REF] on the ImageNet ILSVRC2012 image classification dataset [START_REF] Russakovsky | ImageNet Large Scale Visual Recognition Challenge[END_REF] for 90 epochs with a batch-size of 170 and a learning rate of 0.01 reduced by 10 every 30 epochs. We used the SGD optimizer with weight decay set to 1•10 -4 and momentum set to 0.9.

Cityscapes We trained the HRNet-48 network [START_REF] Sun | High-resolution representations for labeling pixels and regions[END_REF] on the Cityscapes semantic segmentation dataset [START_REF] Cordts | The cityscapes dataset for semantic urban scene understanding[END_REF] for 200 epochs with a batch size of 10 and a learning rate of 0.01 reduced by (1current epoch epochs

) 2 at each epoch. We used the RMI loss [START_REF] Zhao | Region mutual information loss for semantic segmentation[END_REF] and the SGD optimizer with weight decay set to 5 • 10 -4 and momentum set to 0.9. During training, images are randomly cropped and resized, with a scale of [0.5, 2], to 3 × 512 × 1024. Data augmentation involves random flips, random Gaussian blur and color jittering.

Pruning We prune networks following the method of Liu et al. [START_REF] Liu | Learning efficient convolutional networks through network slimming[END_REF]: pruning is divided in three iterations, with a linearly growing proportion of removed filters until the final pruning rate is matched. At each iteration, filters are masked out depending on the magnitude of the weight of their batch-normalization layer. After each iteration, ResNet-50 fine-tuned during 10 epochs and HRNet-48 during 20 epochs. The method of Liu et al. [START_REF] Liu | Learning efficient convolutional networks through network slimming[END_REF] also implies penalizing weights of batch-normalization layers with a smooth-L 1 norm, with an importance factor of λ = 10 -5 for ResNet-50 and λ = 10 -6 for HRNet-48.

Impact on Accuracy and Compression Rate

In our experiments, we mostly reported no difference in accuracy before and after applying our method, as it can be seen horizontally in Figure 3. This implies that the parameters removed by our method, that did not have a null contribution to the function, such as the remaining biases mentioned in Section 3.1, might have had a negligible impact on the network's accuracy. The only outliers are points where accuracy is already severely decreased, for example the accuracy of ResNet-50 pruned at 60% that goes from 66.1% to 63.5%, while the baseline is at 75.7%. accuracy and two different types of pruning rate: one defined as the proportion of removed filters, which the target criterion, widespread in the literature, that we used when pruning the networks, and one defined as the exact count of remaining parameters in the network once our method has been applied after pruning. We see that using the percentage of removed filters is not a reliable indicator of the actual compression rate of the network. The actual trade-off is more advantageous once our method has been applied to both purge the network from useless weights and get a reliable estimation of all eliminated weights. In Figure 4, we compare the compression rate (i.e. 100% 100%-pruning rate% ) in terms of removed filters or removed parameters, i.e. before and after our method. The relationship between the two measures seems to differ greatly depending on the involved architecture, and we expect it to depend on the pruning criterion too. Moreover, in either cases, the relationship is non-linear, which highlights that the rate of remaining parameters in our operational models is hardly predictable from the initial rate of pruned filters.

Conclusion

We have proposed an efficient and generic way to leverage any type of filter pruning in deep convolutional neural networks. Indeed, even though removing filters in a network can induce a certain array of problems that can even prevent inference, our solution is able to tackle them and generate operational pruned networks that can be used for a more reliable study of the impact of pruning.

Figure 2 .

 2 Figure 2. Illustration of the proposed method to identify disconnected weights, with the original network on the left and the modified version on the right. (A) Input tensors are filled with positive values to avoid unwanted zeros, (B) weights of layers are replaced with their mask, therefore (C) the output only contain zeros if a filter is pruned. (D) Normalization and biases are removed to preserve zeros and (E) activation functions are removed not to add extra ones. Final output (F) allows deducing pruned filters.

Figure 3

 3 Figure3features another, more significant, difference between the two curves, as they show the trade-off between

Figure 4 .

 4 Figure4. For ResNet-50 on ImageNet (left) or HRNet-48 on Cityscapes (right): relationship between the estimated compression rate in terms of pruned filters (x-axis) and remaining parameters after reducing the network using our method (y-axis). The dashed lines provide a y = x reference.

Stellantis, Vélizy-Villacoublay, France

IMT Atlantique, Lab-STICC, UMR CNRS 6285, F-29238 Brest, France. Correspondence to: Hugo Tessier <hugo.tessier@imt-atlantique.fr>.