Jean Grondin

dir., Religion et vérité. Garth Green

Giulio Jean Mellana

Grondin

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Grondin, Jean et Green, Garth, dir., Religion et vérité. La philosophie de la religion à l'âge séculier, Strasbourg, Presses Universitaires de Strasbourg, coll. « Philosophie de la religion », 2017, 282 pages.

Actes du Colloque 2015 de la Société Francophone de la Philosophie de la Religion, Religion et vérité est un ouvrage qui se propose d'étudier et d'approfondir le sens de la vérité religieuse dans une époque, qui est la nôtre, fortement marquée par ce que les historiens appellent « sécularisation ». C'est pourquoi dès l'introduction1 , Jean Grondin place cet ouvrage dans le tracé de L'âge séculier de Charles Taylor 2 , dont la voix ouvre le volume avec la transcription d'une discussion publique ayant eu lieu le 2 octobre 2015, à l'Université McGill, entre Charles Taylor, Jean Grondin et quelques intervenants 3 . Taylor précise alors que l'expression « âge post-séculier » (Habermas) est à son avis susceptible d'une mécompréhension, car elle donnerait à penser qu'il existe un âge après celui de la sécularité, tandis que la post-sécularité serait à entendre plus précisément comme un changement de regard à l'intérieur du même âge séculier ; un regard, donc, qui cherche à être à la hauteur de la sécularité que nous habitons.

La première section de l'ouvrage collectif, intitulée « La religion comme expérience de vérité », recueille les interventions de Jean Greisch, Jean Grondin, Ghislain Waterlot, Pierre Gisel, Carla Canullo et Yannick Courtel. Greisch approche magistralement le thème par une exégèse de l'Evangile de Jean 4 . Le point de départ est une analyse du « faire la vérité » (Jn 5,21), par quoi Greisch conduit ensuite le lecteur, à travers les questions de la liberté (Jn 8,31-32) et du τέλος eschatologique (Jn 13,1), au verset de Jn 14,6 : « Moi, je suis le Chemin, la Vérité et la Vie », que le philosophe luxembourgeois lit de la manière suivante : « Je suis en chemin, j'ai rendez-vous avec la vérité, je ne suis pas aussi vivant que je pourrais l'être » 5 . Le « rendez-vous avec la vérité » 6 représente le coeur de la plus récente réflexion de Greisch et son noyau original, et l'engage dans une polémique tant avec le « "donjuanisme religieux" des touristes postmodernes » 7 , tant avec la soi-disante post-vérité, mais aussi avec les philosophies de l'adieu à la vérité, telles la « pensée faible » de Gianni Vattimo. En conclusion, Greisch rencontre la voix de Franz Rosenzweig et sa théorie messianique de la connaissance, à qui il emprunte la notion de Bewährung, qu'il propose de traduire par « avération » au lieu de « vérification ». C'est en revanche en s'appuyant sur une signification forte du terme « vérification » que Grondin tâche, par sa contribution 8 , de fournir une méthode pour juger de la vérité propre à la religion. Une telle méthode doit se fonder sur trois vérifications distinctes : 1) de la factualité historique, 2) de la véracité des prophéties, 3) de l'authenticité des prédications. Ensuite, le philosophe québécois montre, en s'appuyant sur l'herméneutique gadamérienne, que le propre de l'expérience de la vérité religieuse est d'être transformatrice. La transformation opérée par, continue-t-il, la vérité-révélation est en accord avec la raison, car « l'effort essentiel de la raison est de reconnaître un sens à notre expérience » 9 , et donc à l'expérience de cet appel à la transformation. Selon une démarche opposée à celle de Grondin et néanmoins complémentaire, le texte de Waterlot nous invite à aborder la question sub specie mysticae 10 . Ainsi, à la vérité d'une telle expérience appartiennent deux caractères remarquables : « 1) le fait que c'est une expérience dont le fond est secret ; 2) le fait que c'est une expérience "impossible" -au regard de notre intelligence »11 . Après avoir montré le propre de l'expérience (mystique) de la divinité, l'auteur peut conclure que « la vérité de l'expérience [religieuse] ne ressemble à aucune autre, mais elle demeure vérité si l'on entend par vérité l'éclat de la présence d'une réalité irréductible »12 . Si Waterlot fait appel au repliement mystique sur soi, Gisel envisage, tout en partant de la situation contemporaine occidentale manifestement multiculturelle et multireligieuse, la nécessité de s'ouvrir à l'autre dans le but de créer un « espace de débat » 13 . Il peut être utile de mettre cette contribution en parallèle avec deux autres publications du même auteur : Traiter du religieux à l'université14 et « Résistances des particularités et pièges de l'universel »15 . Pour sa part, l'italienne Canullo envisage, face à la tradition moderne, une « autre voie possible » pour la conjonction de religion et vérité 16 . Par cette autre voie, elle approfondit étymologiquement la veritas latine (la philosophe se souvient ici de l'ἀλήθεια heideggérienne) pour en découvrir des références linguistiques insoupçonnées au vouloir et au désir. Canullo interprète ce couple conceptuel en terme de libres vouloir et désir, c'est-à-dire comme « assentissement libre »17 , qui donne une véritable chair à la vérité, en la soustrayant du domaine de ce qui est purement formel. La contribution de Courtel est aussi explicitement liée à la pensée de Heidegger18 . L'expression du titre de sa contribution (« Se tenir dans le vrai ») est empruntée à Heidegger (Einführung in die Metaphysik et Die Ursprung des Kunstwerkes) et indique en termes ontologiques ce que la notion de « savoir » exprime ontiquement. La vérité n'est pas de l'ordre de l'étant, mais de l'être lui-même et elle exprime, en ce sens, le domaine de manifestation dans lequel tout étant (y compris nous-mêmes) se trouve et de ce fait est. L'auteur propose ensuite une « variante dans la pensée de l'être comme vérité »19 , qui consiste à relever que tout apparaître est en vérité le se soustraire de l'être lui-même, et que donc « disparition et manifestation »20 font un dans le phénomène de la vérité. Il retrouve une dernière fois Heidegger par la notion de Stimmung, qu'il interprète comme la possibilité d'« une approche émotionnelle du vrai »21 . La deuxième section du recueil, intitulée « La question de la vérité dans les grandes religions », réunit des contributions de Souleymane Bachir Diagne, Anna Ghiglione, Jacob Rogozinski et Danielle Cohen-Levinas. Bachir Diagne nous fait découvrir la figure d'un Descartes ante litteram : Abu Hamid Al-Ghazali (Algazel chez les latins ; 1058-1111) 22 . Le sectarisme religieux de l'époque d'Al-Ghazali jette, dans le propos de Bachir Diagne, une lumière sur notre présent multi-religieux. A travers une lecture aussi coranique, l'auteur fait d'un côté revivre une personnalité majeure de l'histoire de la pensée, et, de l'autre, montre que la vérité, certes à l'origine du pluralisme, ne demeure pas moins l'occasion d'une convergence. En nous déplaçant de l'autre côté de l'Asie, la spécialiste de philosophie chinoise Ghiglione présente de manière synthétique et efficace le statut de (non)religion du confucianisme 23 . Elle réalise également un court excursus historique de la dynastie des Zhou à celle des Han, de Mao à nos jours. De thématique biblique, le très beau texte de Rogozinski traite du sens du sacrifice d'Abraham [START_REF] Rogozinski | De l'Idole au Nom : deux visages de Dieu[END_REF] . L'auteur engage un débat avec plusieurs traditions interprétatives (juives, chrétiennes et musulmanes) pour ensuite retrouver le Schelling de La philosophie de la révélation, qui, par l'interprétation des Noms de Dieu comme auto-révélation de Dieu lui-même, nous donne à comprendre le 'olah d'Abraham comme le passage de l'Idole au Nom, de Elohim à YHVH. Encore sur le monothéisme juif, cette fois lu à travers la voix de Levinas, la contribution de Cohen-Levinas [START_REF] Cohen-Levinas | Religion sans vérité. Quelques réflexions sur le monothéisme juif chez Emmanuel Levinas[END_REF] est un commentaire de « Aimer la Torah plus que Dieu » [START_REF] Levinas | Aimer la Torah plus que Dieu[END_REF] , texte ayant pour thème « la souffrance des innocents », laquelle, en tant que « propre du peuple juif » [START_REF] Cohen-Levinas | Religion sans vérité. Quelques réflexions sur le monothéisme juif chez Emmanuel Levinas[END_REF] , n'est plus l'objet d'une théodicée, mais apparaît comme la trace du passage de Dieu. Selon l'auteure, nous pourrions ainsi éclairer le sens de la mission immémoriale de ce peuple, fondée dans « la dimension prescriptive de la loi, au risque de l'athéisme, » plutôt que dans « l'expérience du divin » 28 . La troisième section intitulée « L'oeuvre de Michel Henry et son apport à la question de la vérité de la religion » se compose des contributions de Garth Green, Jean Leclercq et Roberto Formisano. Green, en soulignant que la relation entre la religion et la vérité est « l'intention ultime de L'essence de la manifestation » et également « la préoccupation constante de C'est moi la vérité », cherche à montrer que bien que cette relation n'y soit pas traitée de la même façon, les deux textes diffèrent surtout au niveau de « la stratégie d'exposition des concepts fondamentaux de la théologie henryenne » [START_REF] Green | Vérité et non vérité dans la philosophie de la religion de Michel Henry[END_REF] . Le texte de Leclercq dégage le statut de la vérité qui est propre au christianisme et qui tient ensemble, de façon unique en son genre dans l'oeuvre henryenne, Vie et Ipséité [START_REF] Leclercq | Vie et méthode. Le tournant épistémologique de Michel Henry[END_REF] . Fruit d'un minutieux travail archivistique, l'apport de Formisano [START_REF] Formisano | Phénoménologie matérielle et philosophie de la religion. Relire Michel Henry à la lumière des inédits[END_REF] vise, à travers la reconstruction des lectures henryennes des décennies 1930 et 1940, à faire ressortir la genèse du problème de la religion chez le phénoménologue. On y retrouve aussi des observations en polémique avec Dominique Janicaud et son Le tournant théologique de la phénoménologie française.

cosmologiques comme « laisser s'imposer l'être ». Vallée tâche quant à lui de réhabiliter le sens d'une vérité personnelle, face à l'impersonnalité des propositions scientifiques 44 . C'est à partir d'un horizon chrétien que la question est abordée, et c'est chez Bonaventure que l'auteur retrouve les outils conceptuels qui conviennent à l'expérience qu'il cherche à décrire : la « nihilité » 45 et l'humilité. L'ouvrage se conclut par un texte écrit par Garth Green 46 , qui laisse l'ouvrage en héritage aux lecteurs, « car la signification d'un événement et celle d'un texte n'appartiennent qu'à l'avenir auquel ils donnent lieu » 47 .

Religion et vérité est d'abord un ouvrage courageux, en ce qu'il tente de montrer, à travers un spectre très large d'approches, l'« incontournabilité » (pour le dire avec Heidegger) de la vérité religieuse. C'est pourquoi il s'agit d'un livre susceptible d'intéresser tant les spécialistes de philosophie de la religion, tant les lecteurs prêts à se questionner sur le sens et le fondement philosophique de leur croyance ou non-croyance. Certaines positions ici exprimées ne manqueront de contrarier certains, mais même les thèses les plus audacieuses laissent transparaître une filigrane philosophique. Finalement, on remarque que l'ouvrage est dédié « à la mémoire des victimes des attentas du 7 janvier et du 13 novembre 2015 » : puisqu'il s'agit des actes d'un colloque ayant eu lieu au Québec, rien n'aurait empêché de dédier cette belle publication (aussi) à la mémoire des victimes de l'attentant du 29 janvier 2017 à la Grand Mosquée de Québec. Peut-être que le délai éditorial ne l'a-t-il pas permis ?

  Vallée, Marc-Antoine, « La vérité en personne(s) », dans Jean Grondin et Garth Green, dir., Religion et vérité. La philosophie de la religion à l'âge séculier, p. 259-265. 45 Ibid., p. 262. 46 Green, Garth, « Religion et vérité : le passé et le futur d'une conjonction », Jean Grondin et Garth Green, dir., Religion et vérité. La philosophie de la religion à l'âge séculier, p. 267-271. 47 Ibid., p. 269.

	Giulio Mellana
	Université Laval, Québec
	Sorbonne Université, Paris

44

Grondin, Jean, « Introduction. Comment penser la vérité de la religion aujourd'hui ? », dans Jean Grondin et Garth Green, dir., Religion et vérité. La philosophie de la religion à l'âge séculier, Strasbourg, Presses Universitaires de Strasbourg, coll. « Philosophie de la religion »,

2017, p. 7-12. 2 Taylor, Charles, L'âge séculier, trad. P. Sadivan, Paris, Éditions du Seuil, 2011, 1339 p. 3 « Qu'est-ce que l'âge séculier ? Discussion avec Charles Taylor », dans Jean Grondin et Garth Green, dir., Religion et vérité. La philosophie de la religion à l'âge séculier, p. 13-20. 4 Greisch, Jean, « "Veritas semper major". La vérité dans l'horizon de l'eschatologie », dans Jean Grondin et Garth Green, dir., Religion et vérité. La philosophie de la religion à l'âge séculier, p. 23-37. 5 Ibid., p. 29. 6 Cf. Greisch, Jean, Rendez-vous avec la vérité, Paris, Hermann, 2017, 376 p. 7 Greisch, Jean, « "Veritas semper major". La vérité dans l'horizon de l'eschatologie », p. 33. 8 Grondin, Jean, « L'expérience de vérité de la religion », dans Jean Grondin et Garth Green, dir., Religion et vérité.La philosophie de la religion à l'âge séculier, p. 39-48. 9 Ibid., p. 45.

Waterlot, Ghislain, « La vérité de l'expérience religieuse est-elle sui generis? », dans Jean Grondin et Garth Green, dir., Religion et vérité. La philosophie de la religion à l'âge séculier, p. 49-56.

Ibid., p. 50.

Ibid., p. 56.

Gisel, Pierre, « Pourquoi et comment reposer la question de la vérité en matières religieuses », dans Jean Grondin et Garth Green, dir., Religion et vérité. La philosophie de la religion à l'âge séculier, p. 57-68.

Gisel, Pierre, Traiter du religieux à l'université, Lausanne, Antipodes, 2011, 174 p.

Gisel, Pierre, « Résistances des particularités et pièges de l'universel. Pour un usage subversif des corps, des traditions et des frontières », dans Jacques Ehrenfreud et Pierre Gisel, dir., Mises en scène de l'humain, Paris, Beauchesne, 2014, 266 p.

Canullo, Carla, « Religion et vérité : l'avenir d'une conjonction », dans Jean Grondin et Garth Green, dir., Religion et vérité. La philosophie de la religion à l'âge séculier, p. 69-76.

Ibid., p. 75.

Courtel, Yannick, « Se tenir dans le vrai », dans Jean Grondin et Garth Green, dir., Religion et vérité. La philosophie de la religion à l'âge séculier, p. 77-85.

Ibid., p. 81.

Ibid., p. 82.

Ibid., p. 82.

Bachir Diagne, Souleymane, « La "vraie" religion et le pluralisme », dans Jean Grondin et Garth Green, dir., Religion et vérité. La philosophie de la religion à l'âge séculier, p. 89-96.


Yannick Le Cacheux

Hervé Le Borgne

Yannick Le

Cacheux Cea

Michel Crucianu

From Classical to Generalized Zero-Shot Learning: a Simple Adaptation Process

come   L'archive ouverte pluridisciplinaire

Introduction

Zero-shot learning (ZSL) [START_REF] Lampert | Learning to detect unseen object classes by between-class attribute transfer[END_REF][START_REF] Larochelle | Zero-data learning of new tasks[END_REF][START_REF] Palatucci | Zero-shot learning with semantic output codes[END_REF] aims to recognize classes for which no training example is available. This is often achieved by relying on additional semantic knowledge, consisting for example in vectors of attributes. During training, a relation between visual features and semantic attributes is learned from training examples belonging to the seen classes, for which both modalities (visual and semantic) are available. This model is then applied in the testing phase on examples from unseen classes, for which no visual instance was available during training. Predictions on these classes can thus be made on the basis of the inferred relation between visual and semantic features.

In classical ZSL, the test set only contains examples from the novel, unseen classes, and these classes alone can be predicted. Although this setting has enabled significant progress in methods linking visual content to semantic information in the last few years [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF], it is hardly realistic. It seems much more reasonable to assume that objects which are to be classified can belong to either a seen class or an unseen class, since in real-life use-cases one could legitimately want to recognize both former and novel classes. This setting is usually referred to as generalized zero-shot learning (GZSL).

However, recent work shows that a direct use of a ZSL model in a GZSL setting usually leads to unsatisfactory results. Indeed, in addition to the number of candidate classes being higher due to the presence of the seen classes among them, most samples from unseen classes are incorrectly classified as belonging to one of the seen classes [START_REF] Chao | An empirical study and analysis of generalized zero-shot learning for object recognition in the wild[END_REF]. Different methods have been proposed to measure this discrepancy, such as the area under the curve representing all the possible trade-offs between the accuracies on samples from seen classes versus samples from unseen classes [START_REF] Chao | An empirical study and analysis of generalized zero-shot learning for object recognition in the wild[END_REF], or their harmonic mean [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF] to penalize models with strong imbalance between the two. While these proposals only measure the extent of the problem, we aim to explicitly address this issue in addition to quantifying its impact.

The main contribution of this paper is a new process for training and evaluating models in a GZSL setting. In accordance with recent studies, we show that the application of a ZSL model "out of the box" gives results that are far from optimal in the GZSL context. We demonstrate how two simple techniques -the calibration of similarities and the use of appropriately balanced regularization -can dramatically improve the performance of most models. The final score for the GZSL task can thus be increased up to a factor of two, with no change regarding the underlying hypotheses of the GZSL task or the data available at any given time, which means that our process is applicable to any ZSL model. We also provide new insights on the reasons why these two techniques are relevant and on the fundamental differences between samples from seen and unseen classes.

We extensively evaluate these techniques on several recent ZSL methods. For sanity-check, we independently reproduce results obtained in the literature before applying our process. We find that some models show a variability in performance with respect to their random initialization, so measures averaged over several runs should be preferred. We eventually find that, with fair comparison under unbiased conditions as enabled by our process, a regularized linear model can give results close to or even better than the state-of-the-art.

Related work

An early rigorous definition and evaluation of GZSL was put forward in [START_REF] Chao | An empirical study and analysis of generalized zero-shot learning for object recognition in the wild[END_REF]. The authors argue that this setting is more realistic than ZSL and highlight the gap between accuracies on seen and unseen classes when labels from all classes can be predicted (denoted respectively A U →C and A S→C , and formally defined in Sec. 3.1). They also introduce the idea of calibration to address this issue and suggest a new metric for GZSL, Accuracy Under Seen-Unseen Curve (AUSUC), which measures the trade-off between the two accuracies but does not directly Figure 1: Illustration of how the regularization parameter λ affects the accuracies on samples from seen and unseen classes A U →Cu and A S→Cs (see Sec. 3.1) as measured on CUB [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF] (left) and AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF] (right). Optimal regularization is not the same in a ZSL setting, where performance is measured by A U →Cu (red dotted line), and in a GZSL setting, where it is measured by the harmonic mean of A U →C and A S→C (black dotted line).

provide the expected performance in real use-cases.

An extensive evaluation of recent ZSL methods with a common protocol is provided in [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF], both in ZSL and GZSL settings. The authors use a different metric for GZSL, the harmonic mean between A U →C and A S→C , which does not directly quantify the trade-off between accuracies but better estimates the practical performance of a given model. However, they do not explicitly address the gap between similarities evaluated on seen and unseen classes [START_REF] Chao | An empirical study and analysis of generalized zero-shot learning for object recognition in the wild[END_REF], which has a significant impact on the final performance as we show in Sec. 4.3.

Further GZSL results based on the harmonic mean metric are provided in [START_REF] Bucher | Generating visual representations for zero-shot classification[END_REF][START_REF] Verma | Generalized zero-shot learning via synthesized examples[END_REF][START_REF] Xian | Feature generating networks for zero-shot learning[END_REF]. All three methods rely on generators of artificial training examples from unseen classes. However, these methods assume that a semantic description of all unseen classes is available during training. This assumption is not necessarily met in practice and makes the inclusion of additional unseen classes more difficult.

Transductive ZSL methods [START_REF] Fu | Transductive multi-view zero-shot learning[END_REF][START_REF] Kodirov | Unsupervised domain adaptation for zero-shot learning[END_REF][START_REF] Rohrbach | Transfer learning in a transductive setting[END_REF]] also assume that additional information, taking the form of unlabeled samples from unseen classes, is available during training. This can naturally lead to improved performance. In this article, we make none of these assumptions and consider that no information regarding unseen classes is available at training time. , as well as K-dimensional semantic class prototypes noted by S tr = (s tr 1 , . . . , s tr |Cs| ) ∈ R |Cs|×K . We seek to learn a function f : R D × R K → R assigning a similarity score to each pair composed of a visual feature vector and a semantic representation so as to minimize the following regularized loss:

1 N tr N tr n=1 |Cs| c=1 L(f (x tr n ; s tr c ), y tr n ) + λΩ[f ] ( 1
)
where L is the loss function and Ω the regularization term weighted by λ.

ŷ = argmax c∈C te f (x; s te c ) (2)
In classical ZSL, performance is measured by the accuracy of unseen classes among unseen classes, noted A U →Cu , while in GZSL we are interested in the accuracy of unseen classes among all classes and the accuracy of seen classes among all classes, noted respectively A U →C and A S→C as in [START_REF] Chao | An empirical study and analysis of generalized zero-shot learning for object recognition in the wild[END_REF]. A S→Cs is similarly defined.

Calibration and GZSL split

As evidenced by [START_REF] Chao | An empirical study and analysis of generalized zero-shot learning for object recognition in the wild[END_REF], when a ZSL model is applied in a GZSL setting, A S→C is usually significantly higher than A U →C . This is because most samples from unseen classes are incorrectly classified into one of the seen classes. To address this, a calibration factor γ is added in [START_REF] Chao | An empirical study and analysis of generalized zero-shot learning for object recognition in the wild[END_REF] to penalize seen classes. Eq. ( 2) then becomes: ŷ = argmax

c∈C te f (x; s te c ) -γ1[c ∈ C s ] (3)
where 1[•] is an indicator function.

The Accuracy Under Seen-Unseen Curve (AUSUC) metric also proposed in [START_REF] Chao | An empirical study and analysis of generalized zero-shot learning for object recognition in the wild[END_REF] is defined as the area under the curve representing A S→C versus A U →C when γ varies from -∞ to +∞, which shows the trade-off between the two.

Instead of computing a metric involving all possible trade-offs between A U →C and A S→C , we look for a single specific value of γ, corresponding to the best compromise between the two as measured by the harmonic mean of A U →C and A S→C [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF]. We propose to determine the optimal value of γ with a crossvalidation specific to GZSL. Usually in machine learning a dataset is divided at random into three parts: a training, a validation and a testing set. In classical ZSL, this splitting process is done with respect to the classes as opposed to the samples: a set of classes is used for training, a disjoint set for validation and a final mutually disjoint set for testing. In GZSL, a fraction (usually 20%) of the samples from the validation and training sets are kept for testing time to be used as test samples from seen classes. We refer to this set as the seen test set. Note that here seen only indicates that these samples belong to seen classes, not that they have been used during training. To be able to cross-validate parameters for GZSL, we further keep an additional 20% of the remaining training set to be used as samples from seen classes when cross-validating parameters; we refer to this set as the seen validation set. Fig. 2 illustrates this partitioning.

To determine the optimal value of γ we first train a model on the GZSL training set. We then use the (GZSL) validation set and the seen validation set to compute the GZSL metric (the harmonic mean) and keep the value γ * that maximizes this metric. The ZSL model is subsequently re-trained on the training, validation and seen validation sets, then class similarities are computed for the test set. The value γ * is subtracted from the similarities of seen classes and the resulting similarities are used to compute the final GZSL score.

Regularization for GZSL

The usual approach to optimize the regularized loss (Eq. ( 1)) in GZSL consists in using the value of λ determined on the ZSL task. We argue here that this is unlikely to be optimal and provide some insight to justify our position. Then, we propose a simple method to determine a better value of λ to improve performance in GZSL.

Figure 1 shows A U →Cu and A S→Cs as a function of λ for a regularized linear model (ridge regression [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF][START_REF] Van Wieringen | Lecture notes on ridge regression[END_REF]), measured on the first validation splits of the proposed splits of [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF] on CUB [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF] and AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF].

In each case, there is a value of λ that maximizes the ZSL score A U →Cu , indicated by the red dotted vertical line, that we note λ * ZSL . The overall ten- dency for A S→Cs is to decrease as λ increases. This is not a concern for the ZSL task, since it only considers samples from unseen classes. However, for the GZSL task, we want the best trade-off between A U →C and A S→C . Note that A U →C ≤ A U →Cu and A S→C ≤ A S→Cs , with equality only if we are able to perfectly distinguish samples from seen and unseen classes. It follows that λ * GZSL , the value of λ that maximizes the GZSL score, is not necessarily the same as λ * ZSL : a small decrease from λ * ZSL can significantly increase A S→Cs while only slightly penalizing A U →Cu . This has a similar impact on the maximum values obtainable by A S→C and A U →C , and can ultimately improve the GZSL score. We quantify in Sec. 4.3 the gains attributed to the use of λ * GZSL . The reason why λ affects A U →Cu and A S→Cs in this way can be explained with the bias-variance decomposition. For regression, we generally assume that we are given a dataset D = (X, t), with samples (x n , t n ) independently drawn from a joint distribution p(x, t), such that p(t|x) = N (t|h(x), σ 2 ), where h is the true dependence. For a prediction function ĥ estimated from D we can then write the expected loss on a new pair (x, t) as:

E D,x,t [(t -ĥ(x)) 2 ] = σ 2 + E D,x [(h(x) -ĥ(x)) 2 ] + var D,x [ ĥ(x)] (4)
where the first term is the intrinsic noise of the dataset, the second is the (squared) bias of the predictor and the third is the variance in the estimation of the predictor. It can be shown [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF][START_REF] Van Wieringen | Lecture notes on ridge regression[END_REF] that for ridge regression the bias increases and the variance decreases with the regularization parameter λ, as illustrated in Fig. 3(b).

In the case of ZSL, x corresponds to visual samples and t to attribute(s) to be estimated from x. The variance comes from both the differences between samples from the same class (intra-class variance) and from the differences between classes (inter-class variance). Intra-class variance is usually significantly smaller than inter-class variance in ZSL. Therefore, most of the variance in Eq. ( 4) can be attributed to the choice of training classes C s . For samples from unseen classes, the bias-variance decomposition applies and there exists a λ corresponding to the best trade-off between the two. This is evidenced in Fig. 3(a), where the red curve shows the Mean Squared Error (MSE) in the predictions of attributes from unseen classes as a function of λ, for a regularized linear model on the first validation split of AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF].

For a sample from a seen class, the variance attributable to the choice of the training classes is much smaller since, by definition, the seen class is present in the training dataset. This allows to better estimate attributes from seen classes and most of the expected error therefore comes from the intrinsic noise and the bias. Thus, the expected error mostly increases with λ, as evidenced by the blue curve in Fig. 3(a). If we plausibly assume that the accuracy of predictions for samples from a given class depends on how well we estimate their attributes, this explains both why predictions are better for samples from seen classes than from unseen classes and why their behavior with respect to λ is different.

We then suggest the following procedure to select the optimal value of λ: we repeat the protocol described in Sec. 3.2 for selecting γ * and we take the value of λ which gives the best result for the harmonic mean between A U →C and A S→C on the validation set after having subtracted γ * from the similarities of seen classes. The rest of the process is identical: we retrain the ZSL model on the training, validation and seen validation sets with the hyperparameter λ * GZSL that we just determined, we compute similarities for the test set, subtract γ * from the similarities of seen classes and compute the resulting GZSL score.

Experimental evaluation 4.1 Methods

We independently reimplemented six methods frequently cited in the literature to evaluate them with our protocol: ALE [START_REF] Akata | Label-embedding for image classification[END_REF], DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF], SJE [START_REF] Akata | Evaluation of output embeddings for fine-grained image classification[END_REF], Sync [START_REF] Changpinyo | Synthesized classifiers for zero-shot learning[END_REF], ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] and SAE [START_REF] Kodirov | Semantic autoencoder for zero-shot learning[END_REF].

In addition, we also evaluate two simple linear models. Linear V → S applies a linear mapping W ∈ R K×D from the visual space V to the semantic space S to minimize standard MSE. With T tr = (s tr y tr 1 , ..., s tr y tr N ) ∈ R N tr ×K the matrix whose rows correspond to the class prototypes associated to each training sample based on its label, the loss function can be formulated as:

1 N tr X tr W -T tr 2 F + λ W 2 F (5)
Linear S → V is based on [START_REF] Shigeto | Ridge regression, hubness, and zero-shot learning[END_REF] where the authors argue that using the semantic space as the embedding space reduces the variance of the projected points and thus aggravates the hubness problem [START_REF] Radovanović | Hubs in space: Popular nearest neighbors in high-dimensional data[END_REF]. They suggest instead to project semantic class prototypes onto the visual space and to compute similarities in this space. Keeping W ∈ R K×D as our linear mapping, we formulate the loss function as:

1 N tr X tr -T tr W 2 F + λ W 2 F ( 6
)
We can easily obtain closed-form solutions for the two models from the objective functions ( 5) and [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF]. For the Linear V → S model we have W = T tr X tr (X tr X tr + λN tr I D ) -1 (7) and for the Linear S → V model W = (T tr T tr + λN tr I K ) -1 T tr X tr (8).

Experimental setting

Datasets

We perform our experiments on two standard datasets for ZSL: Caltech-UCSD-Birds 200-2011 (CUB) [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF] and Animals with Attributes2 1 (AwA2) [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF]. Results for two additional datasets (SUN Attribute Database [START_REF] Patterson | Sun attribute database: Discovering, annotating, and recognizing scene attributes[END_REF] and Attributes Pascal and Yahoo [START_REF] Farhadi | Describing objects by their attributes[END_REF]) are available in the supplementary material. CUB is a fine-grained dataset composed of 11788 pictures of birds from 200 species (black footed albatross, . . . , common yellowthroat). It comes with 312-dimensional binary attributes for each picture, that are averaged by class to obtain semantic class prototypes. AwA2 is a coarse-grained dataset comprising 37322 pictures of 50 animal species (antelope, . . . , zebra). For each class, 85-dimensional attributes are provided.

Splits

The best performing ZSL methods usually rely on visual features obtained with deep neural networks pre-trained on ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF], such as GoogLeNet [START_REF] Szegedy | Going deeper with convolutions[END_REF] or ResNet [START_REF] He | Deep residual learning for image recognition[END_REF]. As evidenced by [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF], this induces a huge bias for ZSL datasets whose classes are not disjoint from categories of ImageNet, as is the case with AwA2, since test classes cannot be considered truly unseen. We therefore adopt the approach of [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF] and use their terms Standard Split (S.S.) for the split widely used in the literature and Proposed Split (P.S.) for the split they introduce. The training and validation splits are further divided for GZSL as described in Sec. 3.2. Statistics regarding the (GZSL) splits are given in the supplementary material.

Settings

Attributes are normalized such that each class prototype has unit 2 norm. We use the 101-layered ResNet [START_REF] He | Deep residual learning for image recognition[END_REF] pre-trained on ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] as visual features extractor, keeping the D = 2048 activations of the last pooling units.

Metrics

For ZSL, we evaluate the accuracy of samples from unseen classes among unseen classes A U →Cu . There are two possible ways to define accuracy: most of the literature uses per sample accuracy, defined as 100 [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF] it is argued that per class accuracy, defined as 100

• 1 N te N te n=1 1[ŷ(x te n ) = y te n ], while in
• 1 |C te | c∈C te 1 |{n|yn=c}| n yn=c 1[ŷ(x te n ) = y te n ],
better takes class imbalance into account.

We report per class accuracy for fair comparison with the extensive results of [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF]. Nonetheless, to enable comparison with the rest of the literature, we also provide per sample accuracy results in Table 3. For GZSL we compute the harmonic mean between A U →C and A S→C , defined as 2•A U →C •A S→C A U →C +A S→C . Accuracy is again assumed to be per class unless otherwise stated.

Results

We first evaluate the performances of the different methods in a classical ZSL setting. Table 1 shows the average per class accuracy measured on testing sets of the Standard Splits (S.S.) and the Proposed Splits (P.S.) [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF] of CUB [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF] and AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF]. We report the average score and the standard deviation over 5 runs with different random initializations. We also report the results from [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF]. We see that some methods such as SJE [START_REF] Akata | Evaluation of output embeddings for fine-grained image classification[END_REF] have high variability with respect to the initialization; for such methods, it is good practice to report average results since a single test run may not be representative of the true performance of the model. On the other hand, methods with closed-form or deterministic solutions such as the Linear V → S , Linear S → V , ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] or SAE [START_REF] Kodirov | Semantic autoencoder for zero-shot learning[END_REF] are not dependent on the initialization and thus have a standard deviation of 0.

Most of the reproduced scores are consistent with [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF], with two notable exceptions: first, a significant increase in performance is observed with SAE [START_REF] Kodirov | Semantic autoencoder for zero-shot learning[END_REF] and can be explained by the fact that similarities are computed in the visual space, with results close to those of the Linear S → V model (results are close to those of Linear V → S when similarities are computed in the semantic space). Second, the score for ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] is significantly lower than reported in [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF]. We found that the use of non-normalized attributes enables to reach performances comparable with [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF], but we could not reproduce the reported results for ES-ZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] with normalized attributes. For the sake of consistency, we chose to report results obtained with normalized attributes. Table 2 shows results for GZSL. We measure the harmonic mean between per class accuracies A U →C and A S→C on the testing set of the Proposed Split [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF]. We evaluate three settings: a ZSL model applied directly in a GZSL setting, i.e. with no calibration and a regularization specific to ZSL (λ * ZSL ) as opposed to GZSL (λ * GZSL ); a ZSL model with calibration and ZSL regularization λ * ZSL ; and a ZSL model with calibration and regularization λ * GZSL specific to the GZSL problem. We report the average score over 5 runs; standard deviations are available in the supplementary material. We also report the results from [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF], which correspond to the setting with no calibration and no λ * GZSL . We can see that the calibration process significantly improves GZSL performance: in our experiments, the average score for all models improves from 28.5 with no calibration to 41.1 with calibration on CUB, and from 28.2 to 56.7 on AwA2. It is worth noting that the lowest score with calibration is close to or higher than the highest score without. The use of a regularization parameter specific to the GZSL task can lead to an additional improvement in performance. In some cases, the optimal λ is the same for the ZSL task and the GZSL task on the validation set, leading to no additional improvement over the score with calibration. However, every time they are different, λ * GZSL is smaller than λ * ZSL , as expected from the results in Sec. 3.3. The only exception is with DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF] on CUB: a λ * GZSL higher than λ * ZSL was selected during crossvalidation, probably due to random noise, resulting in a slightly lower final GZSL score.

Table 3 shows results with improved visual features; each original 256 × 256 image is cropped into ten 224 × 224 images: one in each corner and one in the center for both the original image and its horizontal symmetry. The ResNet features of the resulting images are averaged to obtain a 2048-dimensional vector. We report results for ZSL (A C→Cu , abbreviated Acc.) and GZSL (using the harmonic mean metric, abbreviated H.) on the testing set of the Proposed Split [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF]. In order to facilitate fair comparison with the rest of the literature, both per class (p.c.) and per sample (p.s.) metrics are reported. Additional metrics like AUSUC [START_REF] Chao | An empirical study and analysis of generalized zero-shot learning for object recognition in the wild[END_REF] are available in the supplementary material. Results with 10-cropped visual features are almost always better than the results with standard visual features in Table 2. The per sample metrics are on average not very different from the per class metrics. This is not surprising since classes in both CUB and AwA2 are fairly balanced.

Conclusion

We proposed a simple process for applying ZSL methods in a GZSL setting. This process is based on the empirical observation that ZSL models perform differently on samples from seen and unseen classes. We provided insights about why this should be expected and suggested steps to overcome these problems. Through extensive experiments, we showed that this process enables significant improvements in performance for many existing ZSL methods. Finally, we provided results under optimal conditions for these methods with different metrics to support fair comparison with the rest of the state-of-the-art.

Supplementary Material -From Classical to

Generalized Zero-Shot Learning: a Simple Adaptation Process 6.1 Details regarding the compared methods ALE [START_REF] Akata | Label-embedding for image classification[END_REF], DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF] and SJE [START_REF] Akata | Evaluation of output embeddings for fine-grained image classification[END_REF] employ bilinear similarity functions of the form f (x; s) = s Wx with W ∈ R K×D combined with a hinge rank loss of the form max(0, M +f (x n ; s c )-f (x n ; s yn )) for c = y n , where M is a margin usually set to a fraction of the expected value of f (x; s). For a given visual sample x n , DeViSE simply sums the hinge range losses for all values of c ∈ C s such that c = y n . ALE sums these terms and adds a multiplicative factor to lessen the weight of samples for which many classes contribute to the hinge rank loss. SJE only keeps the loss of the class c corresponding to the maximum of the hinge loss. We add a regularization term of the form λ W 2 F to all these models, • F being the Frobenius norm.

Sync [START_REF] Changpinyo | Synthesized classifiers for zero-shot learning[END_REF] uses "phantom" objects located in both the visual and semantic spaces to generate classifiers for unseen classes. We employ the structured loss described in Sec. 3.2 of [START_REF] Changpinyo | Synthesized classifiers for zero-shot learning[END_REF] as it yields better results. ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] is a simple linear model regularized with respect to the projected visual features of class prototypes, the projected semantic features of the visual samples and the projection matrix. One of its advantages is the existence of a closed form solution. SAE [START_REF] Kodirov | Semantic autoencoder for zero-shot learning[END_REF] learns an autoencoder on visual features aiming to balance a reconstruction loss and a representation loss so that its internal representation corresponds to the semantic attributes of visual features.

Details regarding datasets and splits

In addition to CUB [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF] and AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF], we provide results for two other datasets, the SUN Attribute Database (SUN) [START_REF] Patterson | Sun attribute database: Discovering, annotating, and recognizing scene attributes[END_REF] and Attributes Pascal and Yahoo (aPY) [START_REF] Farhadi | Describing objects by their attributes[END_REF]. SUN is a fine-grained dataset comprising 717 categories of scenes (abbey, . . . , zoo) with 20 images per category, for a total of 14340 images. It comes with 102-dimensional vectors of attributes for each picture; they are averaged by class to obtain semantic class prototypes. aPY is a coarse-grained dataset composed of 15339 images from 3 broad categories (animals, objects and vehicles), further divided into a total of 32 subcategories (aeroplane, . . . , zebra). It is rather imbalanced, with the category person representing a third of all samples. Moreover, with 21 out of 32 classes present in the ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] dataset, it is also highly biased when used in a ZSL context with deep networks pre-trained on ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] as visual features extractors. [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF] also provides Proposed Splits in addition to the Standard Splits for these two other datasets; we use the same protocol as with CUB [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF] and AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF], described in Sec. 3.2 of the main text, to further divide these splits for GZSL.

We thus have a total of four datasets, each with two different testing sets, with three different validation sets for each testing set. Statistics regarding Table 4: Statistics of the dataset splits for each of the four datasets, with two testing splits (Standard Splits or S.S. and Proposed Splits or P.S. as defined in [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF]) per dataset and three validation splits per testing split. Each cell contains a number of samples on the left and the number of corresponding classes on the right. these splits are available in Table 4. For a given dataset and testing set, the average score over the three different validation splits is employed for selecting hyperparameters by cross-validation.

Split

Intra-class and inter-class variances for each dataset are reported in Table 5.

Additional results

Table 6 is an extended version of Table 1 in the main text. It reports the per class accuracy A U →Cu measured on the Standard Splits (S.S.) and the Proposed Splits (P.S.) [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF] of CUB [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF] and AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF], with the addition of the two datasets SUN [START_REF] Patterson | Sun attribute database: Discovering, annotating, and recognizing scene attributes[END_REF] and aPY [START_REF] Farhadi | Describing objects by their attributes[END_REF]. Results are averaged over five runs with different random initializations. Results on SUN [START_REF] Patterson | Sun attribute database: Discovering, annotating, and recognizing scene attributes[END_REF] are mostly consistent with what has been observed on CUB [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF] and AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF]. The addition of a regularization parameter to ALE [START_REF] Akata | Label-embedding for image classification[END_REF], DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF] and SJE [START_REF] Akata | Evaluation of output embeddings for fine-grained image classification[END_REF] enables a slight increase in the ZSL score. The Linear S → V model reaches performances close to or even better than the state-Table 5: Intra-class and inter-class variance for each dataset.

Dataset

Intra-class variance Inter-class variance CUB [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF] 138.0 231.9 AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF] 226.4 379.0 SUN [START_REF] Patterson | Sun attribute database: Discovering, annotating, and recognizing scene attributes[END_REF] 239.9 397.3 aPY [START_REF] Farhadi | Describing objects by their attributes[END_REF] 262.4 370.5

of-the-art. A significant increase in performance is observed with SAE [START_REF] Kodirov | Semantic autoencoder for zero-shot learning[END_REF] and can be explained by the fact that similarities are computed in the visual space, with results close to those of the Linear S → V model (results are close to those of Linear V → S when similarities are computed in the semantic space). The score for ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] is significantly lower than reported in [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF]. We found that the use of non-normalized attributes enables to reach performances comparable with [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF], but we could not reproduce the reported results for ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] with normalized attributes. For the sake of consistency, we chose to report results obtained with normalized attributes. Results on aPY [START_REF] Farhadi | Describing objects by their attributes[END_REF] are more mixed: important differences between reported and reproduced scores exist on at least one split for most methods. This can be explained by the nature of this dataset: the low number of classes, the class imbalance and the high bias due to the presence of most of its classes in ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] imply that the validation sets can be significantly different from the test sets. For example, the average validation score on the three validation splits is 55.2 for DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF], but this score drops to an average of 29.0 on the testing set. It is therefore difficult to find relevant hyperparameters.

Table 7 is an extended version of Table 2 in the main text. It reports the average GZSL score, as measured by the harmonic mean between A U →C and A S→C , for the evaluated methods on the four datasets with and without our process. The results on SUN [START_REF] Patterson | Sun attribute database: Discovering, annotating, and recognizing scene attributes[END_REF] are again mostly consistent with what has been observed on CUB [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF] and AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF], while results on aPY [START_REF] Farhadi | Describing objects by their attributes[END_REF] differ due to the nature of the dataset. In particular, due to the difficulty in selecting good hyperparameters on this dataset, the regularization parameter selected for the GZSL task is not always relevant and does not lead to an increase in the final score.

Table 8 is an extended version of Table 3 in the main text. It reports results for the ZSL and the GZSL tasks when visual features are improved with 10crop. The AUSUC metric [START_REF] Chao | An empirical study and analysis of generalized zero-shot learning for object recognition in the wild[END_REF] has been added for GZSL; the accuracies used for computing the Seen-Unseen Curve are the per sample accuracies as is usually the case. Visual features obtained with 10-crop almost always lead to better results on CUB [START_REF] Wah | The Caltech-UCSD Birds-200-2011 dataset[END_REF], AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF] and SUN [START_REF] Patterson | Sun attribute database: Discovering, annotating, and recognizing scene attributes[END_REF], while again no clear pattern is visible for aPY [START_REF] Farhadi | Describing objects by their attributes[END_REF]. It is worth noting that although the harmonic mean score and the AUSUC score are visibly correlated, a better AUSUC does not necessarily imply a better harmonic mean. For example, among the bilinear compatibility models ALE [START_REF] Akata | Label-embedding for image classification[END_REF], DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF] and SJE [START_REF] Akata | Evaluation of output embeddings for fine-grained image classification[END_REF] on the AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF] dataset, DeViSE has the highest AUSUC score with 0.538 (compared to respectively 0.512 and 0.529 for ALE and SJE) but the lowest harmonic mean score with 55.0 (per class accuracy, compared to respectively 56.9 and 59.4 for ALE and SJE). 14.4 28.9 ± 2.3 58.2 ± 0.9 59.0 ± 1.9 ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] 11.0 39.9 ± 0.0 53.6 ± 0.0 53.7 ± 0.0 Sync [START_REF] Changpinyo | Synthesized classifiers for zero-shot learning[END_REF] 18.0 30.6 ± 0.3 61.0 ± 0.5 61.0 ± 0. [START_REF] Akata | Evaluation of output embeddings for fine-grained image classification[END_REF] 19.8 25.5 ± 0.6 34.9 ± 0.4 35.3 ± 0.4 ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] 15.8 6.7 ± 0.0 11.1 ± 0.0 11.1 ± 0.0 Sync [START_REF] Changpinyo | Synthesized classifiers for zero-shot learning[END_REF] 13.4 20.6 ± 0.8 27.9 ± 1.0 27.9 ± 1.0 SAE [START_REF] Kodirov | Semantic autoencoder for zero-shot learning[END_REF] 11.8 25.1 ± 0.0 34.0 ± 0.0 34. 57.1 ± 0.0 57.2 ± 0.0 47.7 ± 0.0 48.0 ± 0.0 0.315 ± 0.000 ALE [START_REF] Akata | Label-embedding for image classification[END_REF] 57.4 ± 0.3 57.5 ± 0.3 49.2 ± 0.6 49.3 ± 0.6 0.347 ± 0.005 DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF] 52.9 ± 1.0 52.9 ± 1.0 42.4 ± 0.8 42.5 ± 0.7 0.270 ± 0.007 SJE [START_REF] Akata | Evaluation of output embeddings for fine-grained image classification[END_REF] 51.9 ± 1.7 52.1 ± 1.7 46.7 ± 0.9 46.9 ± 1.0 0.308 ± 0.009 ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] 39.0 ± 0.0 38.8 ± 0.0 38.7 ± 0.0 38.6 ± 0.0 0.224 ± 0.000 Sync [START_REF] Changpinyo | Synthesized classifiers for zero-shot learning[END_REF] 57.5 ± 1.8 57.6 ± 1.8 48.9 ± 1.2 49.1 ± 1.2 0.337 ± 0.018 SAE [START_REF] Kodirov | Semantic autoencoder for zero-shot learning[END_REF] 56.1 ± 0.0 56.2 ± 0.0 46.3 ± 0.0 46.6 ± 0.0 0.307 ± 0.000 AwA2 [START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF] Linear V → S 51.0 ± 0.0 43.6 ± 0.0 49.0 ± 0.0 45.5 ± 0.0 0.357 ± 0.000 Linear S → V 70.4 ± 0.0 69.3 ± 0.0 65.1 ± 0.0 68.7 ± 0.0 0.598 ± 0.000 ALE [START_REF] Akata | Label-embedding for image classification[END_REF] 63.0 ± 1.8 61.1 ± 1.7 56.9 ± 1.8 55.5 ± 2.2 0.521 ± 0.014 DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF] 63.1 ± 1.8 62.2 ± 0.6 55.0 ± 1.7 50.6 ± 0.8 0.538 ± 0.005 SJE [START_REF] Akata | Evaluation of output embeddings for fine-grained image classification[END_REF] 63.8 ± 2.0 61.6 ± 2.8 59.4 ± 1.2 57.6 ± 1.8 0.529 ± 0.022 ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] 52.6 ± 0.0 51.9 ± 0.0 54.4 ± 0.0 57.9 ± 0.0 0.434 ± 0.000 Sync [START_REF] Changpinyo | Synthesized classifiers for zero-shot learning[END_REF] 59.3 ± 0.2 56.1 ± 0.4 62.6 ± 0.1 63.2 ± 0.2 0.511 ± 0.003 SAE [START_REF] Kodirov | Semantic autoencoder for zero-shot learning[END_REF] 63.5 ± 0.0 65.4 ± 0.0 62.3 ± 0.0 63.6 ± 0.0 0.585 ± 0.000 SUN [START_REF] Patterson | Sun attribute database: Discovering, annotating, and recognizing scene attributes[END_REF] Linear V → S 48.6 ± 0.0 48.6 ± 0.0 26.7 ± 0.0 25.8 ± 0.0 0.107 ± 0.000 Linear S → V 62.0 ± 0.0 62.0 ± 0.0 36.3 ± 0.0 35.4 ± 0.0 0.184 ± 0.000 ALE [START_REF] Akata | Label-embedding for image classification[END_REF] 62.2 ± 0.2 62.2 ± 0.2 34.9 ± 0.2 34.4 ± 0.3 0.183 ± 0.002 DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF] 61.2 ± 0.3 61.2 ± 0.3 32.5 ± 0.4 32.1 ± 0.4 0.167 ± 0.003 SJE [START_REF] Akata | Evaluation of output embeddings for fine-grained image classification[END_REF] 58.2 ± 0.8 58.4 ± 0.8 36.8 ± 0.6 36.0 ± 0.4 0.194 ± 0.002 ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] 17.2 ± 0.0 17.2 ± 0.0 11.8 ± 0.0 11.6 ± 0.0 0.169 ± 0.000 Sync [START_REF] Changpinyo | Synthesized classifiers for zero-shot learning[END_REF] 56.6 ± 1.8 56.6 ± 1.8 27.9 ± 1.0 27.2 ± 1.1 0.123 ± 0.008 SAE [START_REF] Kodirov | Semantic autoencoder for zero-shot learning[END_REF] 62.7 ± 0.0 62.7 ± 0.0 35.9 ± 0.0 35.1 ± 0.0 0.185 ± 0.000 aPY [START_REF] Farhadi | Describing objects by their attributes[END_REF] Linear V → S 28.6 ± 0.0 12.5 ± 0.0 33.3 ± 0.0 16.2 ± 0.0 0.081 ± 0.000 Linear S → V 40.6 ± 0.0 23.9 ± 0.0 38.4 ± 0.0 21.0 ± 0.0 0.098 ± 0.000 ALE [START_REF] Akata | Label-embedding for image classification[END_REF] 31.3 ± 1.0 16.8 ± 2.7 31.5 ± 0.9 21.1 ± 1.0 0.101 ± 0.005 DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF] 30.9 ± 0.9 16.4 ± 1.6 28.7 ± 1.2 19.8 ± 0.8 0.096 ± 0.005 SJE [START_REF] Akata | Evaluation of output embeddings for fine-grained image classification[END_REF] 34.1 ± 0.8 16.4 ± 1.6 36.9 ± 0.7 23.0 ± 1.2 0.110 ± 0.009 ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF] 12.2 ± 0.0 7.2 ± 0.0 17.1 ± 0.0 11.6 ± 0.0 0.077 ± 0.000 Sync [START_REF] Changpinyo | Synthesized classifiers for zero-shot learning[END_REF] 34.9 ± 0.2 16.8 ± 0.2 37.4 ± 0.1 19.4 ± 0.1 0.102 ± 0.001 SAE [START_REF] Kodirov | Semantic autoencoder for zero-shot learning[END_REF] 29.5 ± 0.0 11.1 ± 0.0 32.7 ± 0.0 13.0 ± 0.0 0.058 ± 0.000

  We denote by C s the set of classes seen during training and by C u the set of unseen classes. We define C = C s ∪ C u , with C s ∩ C u = ∅. During the training phase, we consider N tr training samples consisting of D-dimensional visual features X tr = (x tr 1 , . . . , x tr N tr ) ∈ R N tr ×D and corresponding labels y tr = (y tr 1 , . . . , y tr N tr ) ∈ C s N tr

Figure 2 :

 2 Figure 2: Illustration of the different splits. Each column is a class and each cell is a sample. In this example there are 20 different classes with 10 samples per class. Five classes are used for testing, five other for validation and the remaining ten for training. Among the samples from the validation and training classes, 20% are kept for testing (seen test set) and 20% more samples from training classes are kept for validation (seen validation set).

Figure 3 :

 3 Figure 3: (a) MSE of predicted attributes (averaged over attributes and samples) as a function of the regularization parameter λ; (b) Illustration of the biasvariance decomposition.

  During the testing phase, we consider N te unlabeled visual samples X te = (x te 1 , . . . , x te N te ) ∈ R N te ×D and class prototypes for candidate classes. In a ZSL setting, the candidate classes C te are the unseen classes such that S te ∈ R |Cu|×K . In a GZSL setting, classes to be predicted can be in either C u or C s , such that C te = C and the class prototypes are S te = (s te 1 , . . . , s te |Cs| , s te |Cs|+1 , . . . , s te |Cs|+|Cu| ) ∈ R |C|×K . In both cases, given a function f learned in the training phase, we want to estimate a prediction ŷ for a visual testing sample x such that:

Table 1 :

 1 ZSL score: per-class accuracy A U →Cu , as reported in[START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF] and independently reproduced. S.S.: Standard Split, P.S.: Proposed Split[START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF]. Averaged over 5 runs.

				CUB [6]			AwA2 [7]
	Method	Reported in [4]	Reproduced	Reported in [4]	Reproduced
		S.S.	P.S.	S.S.	P.S.	S.S.	P.S.	S.S.	P.S.
	Linear V → S	n/a	n/a	41.0 ± 0.0 41.8 ± 0.0	n/a	n/a	68.2 ± 0.0 49.7 ± 0.0
	Linear S → V	n/a	n/a	56.0 ± 0.0 53.5 ± 0.0	n/a	n/a	85.5 ± 0.0 68.9 ± 0.0
	ALE [16]	53.2	54.9	54.8 ± 0.8 54.0 ± 1.2	80.3	62.5	80.3 ± 2.2 62.9 ± 2.3
	DeViSE [17]	53.2	52.0	52.5 ± 0.9 52.6 ± 1.3	68.6	59.7	76.6 ± 1.6 62.1 ± 1.6
	SJE [18]	55.3	53.9	53.8 ± 2.3 49.2 ± 1.4	69.5	61.9	80.4 ± 2.9 62.2 ± 1.2
	ESZSL [20]	55.1	53.9	34.9 ± 0.0 34.9 ± 0.0	75.6	58.6	70.5 ± 0.0 50.8 ± 0.0
	Sync [19]	54.1	55.6	56.4 ± 0.9 54.8 ± 0.6	71.2	46.6	65.6 ± 0.8 58.1 ± 0.8
	SAE [21]	33.4	33.3	56.2 ± 0.0 53.3 ± 0.0	80.7	54.1	81.1 ± 0.0 62.8 ± 0.0

Table 2 :

 2 GZSL score (harmonic mean of A U →C and A S→C , per class accuracy) with and without calibration and GZSL regularization. On Proposed Split[START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF], averaged over 5 runs.

	Method	CUB [6] Reported in [4]	Ours	AwA2 [7] Reported in [4]	Ours
	with calibration	-	-		-	-
	with λ * GZSL	-	-	-	-	-	-
	Linear V → S	n/a	18.2 34.3 35.5	n/a	8.3 47.3 48.1
	Linear S → V	n/a	32.5 41.9 43.5	n/a	44.3 62.7 64.0
	ALE [16]	34.4	35.6 45.1 46.2	23.9	26.9 55.8 55.8
	DeViSE [17]	32.8	35.1 43.6 43.4	27.8	17.4 54.6 54.6
	SJE [18]	33.6	29.7 41.2 44.2	14.4	28.9 58.2 59.0
	ESZSL [20]	21.0	17.9 33.7 33.9	11.0	39.9 53.6 53.7
	Sync [19]	19.8	33.2 46.2 47.6	18.0	30.6 61.0 61.0
	SAE [21]	13.6	25.7 43.1 43.1	2.2	29.5 60.2 60.2
	Average	25.9	28.5 41.1 42.2	16.2	28.2 56.7 57.1

Table 3 :

 3

			CUB [6]				AwA2 [7]
	Method	ZSL		GZSL	ZSL		GZSL
		Acc. p.c. Acc. p.s. H. p.c. H. p.s. Acc. p.c. Acc. p.s. H. p.c. H. p.s.
	Linear V → S	45.6	45.6	39.8	39.8	51.0	43.6	49.0	45.6
	Linear S → V	57.1	57.2	47.7	48.0	70.4	69.3	65.1	68.7
	ALE [16]	57.4	57.5	49.2	49.3	63.0	61.1	56.9	55.5
	DeViSE [17]	52.9	52.9	42.4	42.5	63.1	62.2	55.0	50.6
	SJE [18]	51.9	52.1	46.7	46.9	63.8	61.6	59.4	57.6
	ESZSL [20]	39.0	38.8	38.7	38.6	52.6	51.9	54.4	57.9
	Sync [19]	57.5	57.6	48.9	49.1	59.3	56.1	62.6	63.2
	SAE [21]	56.1	56.2	46.3	46.6	63.5	65.4	62.3	63.6

ZSL and GZSL scores with 10-crop features, evaluated with per class (p.c.) and per sample (p.s.) accuracies. With calibration and λ * GZSL . On P.S.

[START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF]

, averaged over 5 runs.

Table 6 :

 6 Previously reported and independently reproduced ZSL per-class accuracy. S.S.: Standard Split, P.S.: Proposed Split[START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF]. Averaged over 5 runs.

	Method	Reported in [4] S.S. P.S.	Reproduced S.S. P.S.
			CUB [6]
	Linear V → S	n/a	n/a	41.0 ± 0.0 41.8 ± 0.0
	Linear S → V	n/a	n/a	56.0 ± 0.0 53.5 ± 0.0
	ALE [16]	53.2	54.9	54.8 ± 0.8 54.0 ± 1.2
	DeViSE [17] 53.2	52.0	52.5 ± 0.9 52.6 ± 1.3
	SJE [18]	55.3	53.9	53.8 ± 2.3 49.2 ± 1.4
	ESZSL [20]	55.1	53.9	34.9 ± 0.0 34.9 ± 0.0
	Sync [19]	54.1	55.6	56.4 ± 0.9 54.8 ± 0.6
	SAE [21]	33.4	33.3	56.2 ± 0.0 53.3 ± 0.0
			AwA2 [7]
	Linear V → S	n/a	n/a	68.2 ± 0.0 49.7 ± 0.0
	Linear S → V	n/a	n/a	85.5 ± 0.0 68.9 ± 0.0
	ALE [16]	80.3	62.5	80.3 ± 2.2 62.9 ± 2.3
	DeViSE [17] 68.6	59.7	76.6 ± 1.6 62.1 ± 1.6
	SJE [18]	69.5	61.9	80.4 ± 2.9 62.2 ± 1.2
	ESZSL [20]	75.6	58.6	70.5 ± 0.0 50.8 ± 0.0
	Sync [19]	71.2	46.6	65.6 ± 0.8 58.1 ± 0.8
	SAE [21]	80.7	54.1	81.1 ± 0.0 62.8 ± 0.0
			SUN [25]
	Linear V → S	n/a	n/a	50.8 ± 0.0 46.7 ± 0.0
	Linear S → V	n/a	n/a	62.8 ± 0.0 61.5 ± 0.0
	ALE [16]	59.1	58.1	63.5 ± 0.5 59.4 ± 0.3
	DeViSE [17] 57.5	56.5	61.6 ± 0.3 58.4 ± 0.3
	SJE [18]	57.1	53.7	59.9 ± 0.6 55.3 ± 1.1
	ESZSL [20]	57.3	54.5	21.0 ± 0.0 16.0 ± 0.0
	Sync [19]	59.1	56.3	58.5 ± 0.5 56.6 ± 1.8
	SAE [21]	42.4	40.3	63.7 ± 0.0 61.0 ± 0.0
			aPY [26]
	Linear V → S	n/a	n/a	30.0 ± 0.0 31.6 ± 0.0
	Linear S → V	n/a	n/a	42.2 ± 0.0 40.0 ± 0.0
	ALE [16]	30.9	39.7	16.0 ± 5.7 30.4 ± 0.9
	DeViSE [17] 35.4	39.8	30.3 ± 3.2 29.0 ± 1.1
	SJE [18]	32.0	32.9	18.9 ± 2.7 33.5 ± 1.9
	ESZSL [20]	34.4	38.3	31.6 ± 0.0 16.3 ± 0.0
	Sync [19]	39.7	23.9	34.1 ± 0.5 35.5 ± 0.1
	SAE [21]	8.3	8.3	31.7 ± 0.0 29.0 ± 0.0

Table 7 :

 7 GZSL score (harmonic mean of A U →C and A S→C , per class accuracy)

	with and without calibration and GZSL regularization. On Proposed Split [4],
	averaged over 5 runs.
	Method	Reported in [4]		Ours
	with calibration	-	-
	with λ * GZSL	-	-	-
		CUB [6]
	Linear V → S	n/a	18.2 ± 0.0 34.3 ± 0.0 35.5 ± 0.0
	Linear S → V	n/a	32.5 ± 0.0 41.9 ± 0.0 43.5 ± 0.0
	ALE [16]	34.4	35.6 ± 1.3 45.1 ± 1.3 46.2 ± 1.0
	DeViSE [17]	32.8	35.1 ± 0.6 43.6 ± 1.1 43.4 ± 0.7
	SJE [18]	33.6	29.7 ± 1.7 41.2 ± 0.6 44.2 ± 1.0
	ESZSL [20]	21.0	17.9 ± 0.0 33.7 ± 0.0 33.9 ± 0.0
	Sync [19]	19.8	33.2 ± 1.0 46.2 ± 0.8 47.6 ± 0.2
	SAE [21]	13.6	25.7 ± 0.0 43.1 ± 0.0 43.1 ± 0.0
	Average (CUB)	25.9	28.5	41.1	42.2
		AwA2 [7]
	Linear V → S	n/a	8.3 ± 0.0	47.3 ± 0.0 48.1 ± 0.0
	Linear S → V	n/a	44.3 ± 0.0 62.7 ± 0.0 64.0 ± 0.0
	ALE [16]	23.9	26.9 ± 1.5 55.8 ± 1.7 55.8 ± 1.7
	DeViSE [17]	27.8	17.4 ± 4.1 54.6 ± 1.2 54.6 ± 1.2
	SJE [18]

Table 8 :

 8 ZSL and GZSL scores with 10-crop features, evaluated with per class (p.c.) and per sample (p.s.) accuracies. With calibration and λ * GZSL . On P.S.[START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF], averaged over 5 runs.

	7 ± 0.0

AwA2 was recently proposed in[START_REF] Xian | Zero-shot learning -a comprehensive evaluation of the good, the bad and the ugly[END_REF] as a replacement for the Animals with Attributes (AwA) dataset[START_REF] Lampert | Attribute-based classification for zero-shot visual object categorization[END_REF] whose images are not publicly available.