Michaël Perrot
email: michael.perrot@univ-st-etienne.fr

Amaury Habrard
email: amaury.habrard@univ-st-etienne.fr

Damien Muselet
email: damien.muselet@univ-st-etienne.fr

Marc Sebban
email: marc.sebban@univ-st-etienne.fr

Modeling Perceptual Color Differences by Local Metric Learning

Keywords: Color difference, Metric learning, Uniform color space

Having perceptual differences between scene colors is key in many computer vision applications such as image segmentation or visual salient region detection. Nevertheless, most of the times, we only have access to the rendered image colors, without any means to go back to the true scene colors. The main existing approaches propose either to compute a perceptual distance between the rendered image colors, or to estimate the scene colors from the rendered image colors and then to evaluate perceptual distances. However the first approach provides distances that can be far from the scene color differences while the second requires the knowledge of the acquisition conditions that are unavailable for most of the applications. In this paper, we design a new local Mahalanobis-like metric learning algorithm that aims at approximating a perceptual scene color difference that is invariant to the acquisition conditions and computed only from rendered image colors. Using the theoretical framework of uniform stability, we provide consistency guarantees on the learned model. Moreover, our experimental evaluation shows its great ability (i) to generalize to new colors and devices and (ii) to deal with segmentation tasks.

Introduction

In computer vision, the evaluation of color differences is required for many applications. For example, in image segmentation, the basic idea is to merge two neighbor pixels in the same region if the difference between their colors is "small" and to split them into different regions otherwise [START_REF] Bitsakos | An experimental study of color-based segmentation algorithms based on the mean-shift concept[END_REF]. Likewise, for visual salient region detection, the color difference between one pixel and its neighborhood is also the main used information [1], as well as for edge and corner detection [START_REF] Van De Weijer | Boosting color saliency in image feature detection[END_REF][START_REF] Van De Weijer | Edge and corner detection by photometric quasi-invariants[END_REF]. On the other hand, in order to evaluate the quality of color images, Xue et al. have shown that the pixel-wise mean square difference between the original and distorted image provides very good results [START_REF] Xue | Perceptual fidelity aware mean squared error[END_REF]. As a last example, the orientation of gradient which is the most widely used feature for image description (SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF], HOG [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]) is evaluated as the ratio between vertical and horizontal differences.

Depending on the application requirement, the used color difference may have different properties. For material edge detection, it has to be robust to local photometric variations such as highlights or shadows [START_REF] Van De Weijer | Edge and corner detection by photometric quasi-invariants[END_REF]. For gradient-based color descriptors, it has to be robust to acquisition condition variations [START_REF] Burghouts | Performance evaluation of local colour invariants[END_REF][START_REF] Van De Sande | Evaluating color descriptors for object and scene recognition[END_REF] or discriminative [START_REF] Van De Weijer | Boosting color saliency in image feature detection[END_REF]. For most applications and especially for visual saliency detection [1], image segmentation [START_REF] Bitsakos | An experimental study of color-based segmentation algorithms based on the mean-shift concept[END_REF] or image quality assessment [START_REF] Xue | Perceptual fidelity aware mean squared error[END_REF], the color difference has to be above all perceptual, i.e. proportional to the color difference perceived by human observers. In the computer vision community, some color spaces such as CIELAB or CIELUV are known to be closer to the human perception of colors than RGB. It means that distances evaluated in these spaces are more perceptual than distances in the classical RGB spaces (which are known to be non uniform). Thus, by moving from RGB to one of these spaces with a default transformation [START_REF] Stokes | A standard default color space for the internet: sRGB[END_REF][START_REF] Union | Parameter values for the hdtv standards for production and international programme exchange[END_REF], the results of many applications have improved [1,2,[START_REF] Bitsakos | An experimental study of color-based segmentation algorithms based on the mean-shift concept[END_REF][START_REF] Khan | Discriminative color descriptor[END_REF][START_REF] Mojsilovic | A computational model for color naming and describing color composition of images[END_REF]. Nevertheless, it is important to know that this default approach provides a perceptual distance between the colors in the rendered image (called image-wise color distance) and not between the colors as they appear to a human observer looking at the real scene (called scene-wise color distance). The transformation from the scene colors to the image rendered colors is a succession of non-linear transformations which are device specific (white balance, gamma correction, demosaicing, compression, . . . ). For some applications such as image quality assessment, it is required to use the image-wise color distances since only the rendered image colors need to be compared, whatever the scene colors. But for a lot of other applications such as image segmentation, saliency detection, . . . , we claim that a scene-wise perceptual color distance should be used. Indeed, in these cases, the aim is to be able to evaluate distances as they would have been perceived by a human observing the scene and not after the camera transformations. Some solutions exist [START_REF] Kim | A new in-camera imaging model for color computer vision and its application[END_REF] to get back to scene colors from RGB camera outputs but they require calibrated acquisition conditions (known illumination, known sensor sensitivities, RAW data available,. . . ).

In this paper we propose a method to estimate scene-wise color distances from non calibrated rendered image colors. Furthermore, we go a step further towards an invariant color distance. This invariance property means that, considering one image representing two color patches, the distance is predicting how much difference would have perceived a human observer looking at the two real patches under standard fixed viewing conditions, such as the ones recommended by the CIE (Commission Internationale de l'Eclairage) in the context of color difference assessment [START_REF] Sharma | The ciede2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations[END_REF]. In other words, whatever the acquisition device or the illuminant, an invariant scene-wise distance should return stable values.

Since the acquisition condition variability is huge, rather than using models of invariance [START_REF] Burghouts | Performance evaluation of local colour invariants[END_REF][START_REF] Van De Sande | Evaluating color descriptors for object and scene recognition[END_REF] and models of acquisition devices [START_REF] Kim | A new in-camera imaging model for color computer vision and its application[END_REF][START_REF] Xiong | From pixels to physics: Probabilistic color de-rendering[END_REF], we propose to automatically learn an invariant perceptual distance from training data. In this context, our objective is three-fold and takes the form of algorithmic, theoretical and practical contributions:

-First, we design a new metric learning algorithm [37] dedicated to approximate reference perceptual distances from the image rendered RGB space. It aims at learning local Mahalanobis-like distances in order to capture the non linearity required to get a scene-wise perceptual color distance.

-Second, modeling the regions as a multinomial distribution and making use of the theoretical framework of uniform stability, we derive consistency guarantees on our algorithm that show how fast the empirical loss of our learned metric converges to its true generalization value.

-Lastly, to learn generalizable distances, we create a dataset of color patches that are acquired under a large range of acquisition conditions (different cameras, illuminations, viewpoints). We claim that this dataset [37] may play the role of benchmark for the computer vision community.

The rest of this paper is organized as follows: Section 2 is devoted to the presentation of the related work in color distances and metric learning. In Section 3, we present the experimental setup used to generate our dataset of images. Then, we introduce our new metric learning algorithm and perform a theoretical analysis. Finally, Section 4 is dedicated to the empirical evaluation of our algorithm. To tackle this task, we perform two kinds of experiments: first, we assess the capability of the learned metrics to generalize to new colors and devices; second, we evaluate their relevance in a segmentation application. We show that in both settings, our learned metrics outperform the state of the art.

Related Work

Perceptually uniform color distance

A large amount of work has been done by color scientists around perceptual color differences [START_REF] Wyszecki | Color Science: Concepts and Methods, Quantitative Data and Formulas[END_REF][START_REF] Huang | Evaluation of threshold color differences using printed samples[END_REF][START_REF] Sharma | The ciede2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations[END_REF], where the required inputs of the proposed distances are either reflectance spectra or the device-independent color components CIE XYZ [START_REF] Wyszecki | Color Science: Concepts and Methods, Quantitative Data and Formulas[END_REF]. These features are obtained with particular devices such as spectrophotometer or photoelectric colorimeter [START_REF] Wyszecki | Color Science: Concepts and Methods, Quantitative Data and Formulas[END_REF]. It is known that neither the euclidean distance between reflectance spectra nor the euclidean distance between XYZ vectors are perceptual, i.e. these distances can be higher for two colors that look similar than for two colors that look different. Consequently, some color spaces such as CIELAB or CIELUV have been designed to be more perceptually uniform. In those spaces, specific color difference equations have been proposed to improve perceptual uniformity over the simple euclidean distance [START_REF] Huang | Evaluation of threshold color differences using printed samples[END_REF]. The ∆E 00 [START_REF] Sharma | The ciede2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations[END_REF] distance is one nice example of such a distance. It corresponds to the difference perceived by a human looking at the two considered colors under standard viewing conditions recommended by the CIE (illuminant D65, illuminance of 1000 lx, etc.).

However, it is worth noting that in most of the computer vision applications, the available information does not take the form of a reflectance spectra or some device-independent components, as assumed above. Indeed, the classical acquisition devices are cameras that use iterative complex transforms from the irradiance (amount of light) collected by each CCD sensor cell to the pixel intensity of the output image [START_REF] Kim | A new in-camera imaging model for color computer vision and its application[END_REF]. These device-dependent transforms are color filtering, white-balancing, gamma correction, demosaicing, compression, etc. [START_REF] Xiong | From pixels to physics: Probabilistic color de-rendering[END_REF] which are designed to provide pleasant images and not to accurately measure colors. Consequently, the available RGB components in color images do not allow us to get back to the original spectra or XYZ components. To overcome this limitation, two main strategies have been suggested in the literature: either by applying a default transformation from RGB components to L * a * b * (CIELAB space) or L * u * v * (CIELUV space) assuming a given configuration, or by learning a coordinate transform to actual L * a * b * components under particular conditions.

Using default transformations A classical strategy consists in using a default transformation from the available RGB components to XYZ and then to L * a * b * or L * u * v * [1,[START_REF] Bitsakos | An experimental study of color-based segmentation algorithms based on the mean-shift concept[END_REF][START_REF] Khan | Discriminative color descriptor[END_REF][START_REF] Mojsilovic | A computational model for color naming and describing color composition of images[END_REF]. This default transformation assumes an average gamma correction of 2.2 [START_REF] Stokes | A standard default color space for the internet: sRGB[END_REF], color primaries close to ITU-R BT.709 [START_REF] Union | Parameter values for the hdtv standards for production and international programme exchange[END_REF] and D65 illuminant (Daylight). Finally, from the estimated L * a * b * or L * u * v * (denoted L * a * b * and L * u * v * respectively) of two pixels, one can make use of the euclidean distance. In the case of L * a * b * , one can use L * a * b * to estimate more complex and accurate distances such as ∆E 00 via its estimate ∆E 00 ( [START_REF] Sharma | The ciede2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations[END_REF]), that will be used in our experimental study as a baseline. As discussed in the introduction, when using this approach, the provided color distance characterizes the difference between the colors in the rendered image after the camera transformations and is not related to the colors of the scene.

Learning coordinate transforms to L * a * b * For applications requiring the distances between the colors in the scene, the acquisition conditions are calibrated first and then the images are acquired under these particular conditions [START_REF] Larraín | Use of digital images to estimate {CIE} color coordinates of beef[END_REF][START_REF] Len | Color measurement in l*a*b* units from rgb digital images[END_REF]. Therefore, the camera position and the light color, intensity and positions are fixed and a set of images of different color patches are acquired. Meanwhile, under the same exact conditions, a colorimeter measures the actual L * a * b * components (in the scene) for each of these patches. In [START_REF] Len | Color measurement in l*a*b* units from rgb digital images[END_REF], they learn then the best transform from camera RGB to actual L * a * b * components with a neural network. In [START_REF] Larraín | Use of digital images to estimate {CIE} color coordinates of beef[END_REF], they first apply the default transform presented before from camera RGB to L * a * b * and then learn a polynomial regression (until quadratic term) from the L * a * b * to the true L * a * b * . However, it is worth mentioning that in both cases the learned transforms are accurate only under these acquisition conditions. Thus, these approaches can not be applied on most of the computer vision applications where such an information is unavailable.

From our knowledge, no previous work has both underlined and answered the problem of the approximations that are made during the estimation of the L * a * b * components in the very frequent case of uncalibrated acquisitions. The standard principle consisting in applying a default transform leads to distances that are only coarsely perceptual with respect to the scene colors. We will see in the rest of this paper that rather than sequentially moving from space to space with inaccurate transforms, a better way consists in learning a perceptual metric directly in the image rendered RGB space. This is a matter of metric learning for which we present a short survey in the next section.

Metric learning

Metric learning (see [3] for a survey) arises from the necessity for a lot of applications to accurately compare examples. The underlying idea is to define application dependent metrics which are able to capture the idiosyncrasies of the data at hand. Most of the existing work in metric learning is focused on learning a Mahalanobis-like distance of the form d M (x, x ′ ) = (xx ′ ) T M(xx ′ ), where M is a positive semi-definite (PSD) matrix to optimize. Note that using a Cholesky decomposition of M, the Malahanobis distance can be seen as a Euclidean distance computed after applying a learned data linear projection.

The work of [START_REF] Xing | Distance metric learning, with application to clustering with side-information[END_REF] where the authors maximize the distance between dissimilar points while maintaining a small distance between similar points has been pioneering in this field. Following this idea, Weinberger and Saul [START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF] propose to learn a PSD matrix dedicated to improve the k-nearest neighbors algorithm. To do so, they force their metric to respect local constraints. Given triplets (z i , z j , z k ) where z j and z k belong to the neighborhood of z i , z i and z j being of the same class, and z k being of opposite class, the constraints impose that z i should be closer to z j than to z k with a margin ε. To overcome the PSD constraint, which requires a costly projection of M onto the cone of PSD matrices, Davis et al. [START_REF] Davis | Information-theoretic metric learning[END_REF] optimize a Bregman divergence under some proximity constraints between pairs of points. The underlying idea is to learn M such that it remains close to a matrix M 0 defined a-priori. If the Bregman divergence is finite, the authors show that M is guaranteed to be PSD.

An important limitation of learning a unique global metric such as a Mahalanobis distance comes from the fact that no information about the structure of the input space is taken into account. Moreover, since a Mahalanobis distance boils down to projecting the data into a new space via a linear transformation, it does not allow us to capture non linearity. Learning local metrics is one possible way to deal with these two issues 1 . In [START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF], the authors propose a local version of [START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF], where a clustering is performed as a preprocess and then a metric is learned for each cluster. In [START_REF] Wang | Parametric local metric learning for nearest neighbor classification[END_REF], Wang et al. optimize a combination of metric bases that are learned for some anchor points defined as the means of clusters constructed, for example, by the K-Means algorithm. Other local metric learning algorithms have been recently proposed, only in a classification setting, such as [START_REF] Xiong | Random forests for metric learning with implicit pairwise position dependence[END_REF] which makes use of random forests and absolute position of points to compute a local metric; in [START_REF] Huang | Reduced-rank local distance metric learning[END_REF], a local metric is learned based on a conical combination of Mahalanobis metrics and pair-wise similarities between the data; a last example of this non exhaustive list comes from [START_REF] Semerci | Mixtures of large margin nearest neighbor classifiers[END_REF], where the authors learn a mixture of local Mahalanobis distances.

Learning a perceptual color distance

In this section, we present a way to learn a perceptual distance that is invariant across acquisition conditions. First, we explain how we have created an image dataset designed for this purpose. Then, making use of the advantages of learning local metrics, we introduce our new algorithm that aims at accurately approximating a perceptual color distance in different parts of the RGB space. We end this section by a theoretical analysis of our algorithm.

Creating the dataset

Given two color patches, we want to design a perceptual distance not disturbed by the acquisition conditions. So we propose to use pairs of patches for which we can measure the true perceptual distance under standard viewing conditions and to image them under different other conditions.

The choice of the patches is key in this work since all the distances will be learned from these pairs. Consequently, the colors of the patches have to be well distributed in the RGB cube in order to be able to well approximate the color distance between two new pairs that have not been seen in the training set. Moreover, as we would like to learn a local perceptual distance, we need pairs of patches whose colors are close from each other. According to [START_REF] Sharma | The ciede2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations[END_REF], ∆E 00 seems to be a good candidate for that because it is designed to compare similar colors. Finally, since hue, chroma and luminance differences impact the perceptual color difference [START_REF] Sharma | The ciede2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations[END_REF], the patches have to be chosen so that all these three variations are represented among the pairs. Given these three requirements, we propose to use two different well-known sets of patches, namely the Farnsworth-Munsell 100 hue test and the Munsell atlas (see Fig. 1). The Farnsworth-Munsell 100 hue test is one of the most famous color vision tests which consists in ordering 84 patches in the correct order and any misplacement can point to some sort of color vision deficiency. Since these 84 patches are well distributed on the hue wheel, their colors will cover a large area of the RGB cube when imaging them under an important range of acquisition conditions. Furthermore, consecutive patches are known to have very small color differences and then, learning perceptual distances from such pairs is a good purpose. This set is constituting the main part of our dataset. Nevertheless, the colors of these patches first, are not highly saturated and second, they mostly exhibit hue variations and relatively small luminance and chroma differences. In order to cope with these weaknesses, we add to this dataset the 238 patches constituting the Munsell Student Color Set [START_REF] Munsell | A pigment color system and notation[END_REF]. These patches are characterized by more saturated colors and the pairs of similar patches mostly exhibit luminance and chroma variations (since only the 5 principal and 5 intermediate hues are provided in this student set).

To build the dataset, we first use a spectroradiometer (Minolta CS 1000) in order to measure the spectra of each color patch of the Farnsworth set, the spectra of the Munsell atlas patches being available online2 . Five measurements have been done in our light cabinet and the final spectra are the average of each measurement. From these spectra, we evaluate the L * a * b * coordinates of each patch under D65 illuminant. Then, we evaluate the distance ∆E 00 between all Note that the available ∆E 00 have been evaluated in the standard viewing conditions recommended by the CIE for color difference assessment and we would like to obtain these reference distances whatever the acquisition conditions. Consequently, we propose to use 4 different cameras, namely Kodak DCS Pro 14n, Konica Minolta Dimage Z3, Nikon Coolpix S6150 and Sony DCR-SR32 and a large variety of lights, viewpoints and backgrounds (since background also perturbs the colors of the patches). For each camera, we acquire 50 images of each Farnsworth pair and 15 of each Munsell pair (overall, 41, 800 imaged pairs). Finally, after all these measurements and acquisitions, we have for each image of a pair, two image rendered RGB vectors and one reference distance ∆E 00 .

Local metric learning algorithm

In this section, our objective is to approximate the reference distance ∆E 00 by a metric learning approach in the RGB space which aims at optimizing K local metrics plus one global metric. For this task, we perform a preprocess by dividing the RGB space into K local parts thanks to a clustering step. From this, we deduce K+1 regions defining a partition C 0 , C 1 , . . . , C K over the possible pairs of patches. A pair p = (x, x ′ ) belongs to a region C j , 1 ≤ j ≤ K if both x and x ′ belong to the same cluster j, otherwise p is assigned to region C 0 . In other words, each region C j corresponds to pairs related to cluster j, while C 0 contains the remaining pairs whose points do not belong to the same cluster. Then, we approximate ∆E 00 by learning a Mahalanobis-like distance in every C j (j = 0, 1, . . . , K), represented by its associated PSD 3 × 3 matrix M j .

Each metric learning step is done from a finite-size training sample of n j triplets

T j = {(x i , x ′ i , ∆E 00 )} n j
i=1 where x i and x ′ i represent color patches belonging to the same region C j and ∆E 00 (x i , x ′ i ) (∆E 00 for the sake of simplicity) their associated perceptual distance value. We define a loss function l on any pair of patches (x, x ′ ):

l(M j , (x, x ′ , ∆E 00 )) = ∆ 2 T j -∆E 00 (x, x ′ )
(j = 0, 1 . . . , K) of triplets T j = {(x i , x ′ i , ∆E 00 )} n j i=1 (where x i , x ′ i ∈ C j and ∆E ab (x i , x ′ i ) < 5)
for j = 0 → K do Learn M j by solving the convex optimization Problem (1) using T j (xx ′ ) T M j (xx ′ ), l measures the error made by a learned distance M j . We denote the empirical error over T j by εT j (M j ) = 1 n j (x,x ′ ,∆E 00 )∈T j l(M j , (x, x ′ , ∆E 00 )). We suggest to learn the matrix M j that minimizes εT j via the following regularized problem:

arg min M j 0 εT j (M j ) + λ j M j 2 F , (1)
where λ j > 0 is a regularization parameter and • F denotes the Frobenius norm. To obtain a proper distance, M j must be PSD (denoted by M j 0) and thus has to be projected onto the PSD cone as previously explained. Due to the simplicity of M j (3 × 3 matrix), this operation is not costly3 . It is worth noting that our optimization problem takes the form of a simple regularized least absolute deviation formulation. The interest of using the least absolute deviation, rather than a regularized least square, comes from the fact that it enables accurate estimates of small ∆E 00 values.

The pseudo-code of our metric learning algorithm is presented in Alg. 1. Note that to solve the convex problem 1, we use a classical interior points approach. Moreover, parameter λ j is tuned by cross-validation.

Discussion about Local versus Global Metric Note that in our approach, the metrics learned in the K regions C 1 , . . . , C K are local metrics while the one learned for region C 0 is rather a global metric considering pairs that do not fall in the same region. Beyond the fact that such a setting will allow us to derive generalization guarantees on our algorithm, it constitutes a straightforward solution to deal with patches at test time that would not be concerned by the same local metric in the color space. In this case, we make use of the matrix M 0 associated to partition C 0 . Another possible solution may consist in resorting to a Gaussian embedding of the local metrics. However, because this solution would imply learning additional parameters, we suggest in this paper to make use of this simple and efficient (parameters-wise) strategy. In the segmentation experiments of this paper, we will notice that M 0 is used in only ∼20% of the cases. Finally, note that if K = 1, this boils down to learning only one global metric over the whole training sample. In the next section, we justify the consistency of this approach.

Theoretical study

In this part, we provide a generalization bound justifying the consistency of our method. It is derived by considering (i) a multinomial distribution over the regions, and (ii) per region generalization guarantees that are obtained with the uniform stability framework [START_REF] Bousquet | Stability and generalization[END_REF].

We assume that the training sample T = ∪ K j=0 T j is drawn from an unknown distribution P such that for any (x, x ′ , ∆E 00 ) ∼ P , ∆E 00 (x, x ′ ) ≤ ∆ max , with ∆ max the maximum distance value used in our context. We assume any input instance x to be normalized such that x ≤ 1, where • is the L2-norm 4 .

The K + 1 regions C 0 , . . . , C K define a partition of the support of P . In partition C j , let D j = max (x,x ′ ,∆E 00 )∼P (C j ) ( xx ′ ) be the maximum distance between two elements and P (C j ) be the marginal distribution.

Let M = {M 0 , M 1 , . . . , M K } be the K+1 matrices learned by our Alg. 1. We define the true error associated to M by ε(M) = K j=0 ε P (C j ) (M j )P (C j ) where

ε P (C j ) (M j ) = E (x,x ′ ,∆E 00 )∼P (C j ) l (M j , (x, x ′ , ∆E 00 )) is the local true risk for C j . The empirical error over T of size n is defined as εT (M) = 1 n K j=0 n j εT j (M j ) where εT j (M j ) = 1 n j (x,x ′ ,∆E 00 )∈T j l (M j , (x, x ′ , ∆E 00 )) is the empirical risk of T j .
Generalization bound per region C j To begin with, for any learned local matrix M j , we provide a bound on its associated local true risk ε P (C j ) (M j ) in function of the empirical risk εT j (M j ) over T j .

Lemma 1 (Generalization bound per region).

With probability 1δ, for any matrix M j related to a region C j , 0 ≤ j ≤ K, learned with Alg. 1, we have:

|ε P (C j ) (M j ) -εT j (M j )| ≤ 2D 4 j λ j n j + 4D 4 j λ j +∆ max ( 2D 2 j λ j +2∆ max ) ln( 2 δ ) 2n j .
The proof of this lemma is provided in the supplementary material and is based on the uniform stability framework. It shows that the consistency is achieved in each region with a convergence rate in O(1/ √ n). When the region is compact, the quantity D j is rather small making the bound tighter.

Generalization bound for Alg. 1 The generalization bound of our algorithm is based on the fact that the different marginals P (C j ) can be interpreted as the parameters of a multinomial distribution. Thus, (n 0 , n 1 , . . . , n K ) is then a IID multinomial random variable with parameters n = n j=0 n j and (P (C 0 ), P (C 1 ), . . . , P (C K )). Our result makes use of the Bretagnolle-Huber-Carol concentration inequality for multinomial distributions [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF] which is recalled in the supplementary material for the sake of completeness (this result has also been used in [START_REF] Xu | Robustness and Generalization[END_REF] in another context).

We are now ready to introduce the main theorem of the paper.

Theorem 1 Let C 0 , C 1 , . . . , C k be the regions considered, then for any set of metrics M = {M 0 , . . . , M K } learned by Alg. 1 from a data sample T of n pairs, we have with probability at least 1δ that

ε(M) ≤ εT (M) + L B 2(K + 1) ln 2 + 2 ln 2/δ n + 2(KD 4 + 1) λn + 4(KD 4 + 1) λ + ∆ max ( 2(KD 2 + 1) √ λ +2(K + 1)∆ max ) ln( 4(K+1) δ ) 2n
,

where

D = max 1≤j≤K D j , L B = max{ ∆ max √ λ , ∆ 2
max } is the bound on the loss function and λ = min 0≤j≤K λ j is the minimum regularization parameter among the K + 1 learning problems used in Alg. 1.

The proof of this theorem is provided in the supplementary material. The first term after the empirical risk comes from the application of the Bretagnolle-Huber-Carol inequality with a confidence parameter 1δ/2. The last terms are derived by applying the per region consistency Lemma 1 to all the regions with a confidence parameter 1δ/2(K + 1) and the final result is derived thanks to the union bound.

This result justifies the global consistency of our approach with a standard convergence rate in O(1/ √ n). We can remark that if the local regions C 1 , . . . , C n are rather small (i.e. D is significantly smaller than 1), then the last part of the bound will not suffer too much on the number of regions. On the other hand, there is also a trade-off between the number/size of regions considered and the number of instances falling in each region. It is important to have enough examples to learn good models.

Experiments

Evaluating the contribution of a metric learning algorithm can be done in two ways: (1) assessing the quality of the metric itself, and (2) measuring its impact once plugged in an application. In the following, we first evaluate the generalization ability of the learned metrics on our dataset. Then, we measure their contribution in a color segmentation application.

Evaluation on our dataset

To evaluate the generalization ability of the metrics, we conduct two experiments: We assess the behavior of our approach when it is applied (i) on new unseen colors and (ii) on new patches coming from a different unseen camera. In these experiments, we consider all the pairs of patches (x, x ′ ) of our dataset characterized by a ∆E ab < 5, resulting in 41, 800 pairs. Due to the large amount of data, combined with the relative simplicity of the 3×3 local metrics, we notice that the algorithm is rather insensible to the choice of λ. Therefore, we use λ = 1 in all our experiments. The displayed results are the average over 5 runs. To estimate the performance of our metric we use two criteria we want to make as small as possible. The first one is the mean absolute difference, computed over a test set T S, between the learned metric ∆ T -i.e. the metric learned with Alg. 1 -w.r.t. a training set of pairs T and the reference ∆E 00 . As a second criterion, we use the STRESS5 measure [START_REF] Melgosa | Performance of recent advanced colordifference formulas using the standardized residual sum of squares index[END_REF]. Roughly speaking, it evaluates quadratic differences between the learned metric ∆ T and the reference ∆E 00 . We compare our approach to the state of the art where ∆ T is replaced by ∆E 00 [START_REF] Sharma | The ciede2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations[END_REF] in both criteria, i.e. transforming from rendered image RGB to L * a * b * and computing the ∆E 00 distance.

Generalization to unseen colors

In this experiment, we perform a 6-fold cross validation procedure over the set of patches. Thus we obtain, on average, 27927 training pairs and 13873 testing pairs. The results are shown on Fig. 2(a) according to an increasing number of clusters (from 1 to 70). We can see that using our learned metric ∆ T instead of the state of the art estimate ∆E 00 [START_REF] Sharma | The ciede2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations[END_REF] enables significant improvements according to both criteria (where the baselines are 1.70 for the mean and 48.05 for the STRESS). Note that from 50 clusters, the quality of the learned metric declines slightly while remaining much better than ∆E 00 . Figure 2(a) shows that K = 20 seems to be a good compromise between a high algorithmic complexity (the higher K, the larger the number of learned metrics) and good performances of the models. When K = 20, using a Student's t test over the mean absolute differences and a Fisher test over the STRESS, our method is significantly better than the state of the art with a p-value < 1 -10 . Figure 2(a) also emphasizes the interest of learning several local metrics. Indeed, optimizing 20 local metrics rather than only one is significantly better with a p-value smaller than 0.001 for both criteria.

Generalization to unseen cameras In this experiment, our model is learned according to a 4-fold cross validation procedure such that each fold corresponds to the pairs coming from a given camera. Thus we learn the metric on a set of 31350 pairs and test it on a set of 10450 pairs. Therefore, this task is more complicated than before. The results are presented in Fig. 2(b). We can note that our approach always outperforms the state of the art for the mean criterion (of baseline 1.70). Regarding the STRESS, we are on average better when using between 5 to 60 clusters. Beyond 65 clusters, the performances decrease significantly. This behavior likely describes an overfitting phenomenon due to the fact that a lot of local metrics have been learned that are more and more specialized for 3 out of 4 cameras, and unable to generalize well to the fourth one. For this series of experiments, K = 20 is still a good value to deal with the trade-off between complexity and efficiency. Using a Student's t test over the mean absolute differences and a Fisher test over the STRESS, our method is significantly better with p-values respectively < 1 -10 and < 0.006. The interest of learning several local metrics rather than only one is still confirmed. Applying statistical comparison tests between K = 20 and K = 1 leads to small p-values < 0.001.

Thus for both series of experiments, K = 20 appears to be a good number of clusters and allows significant improvements. Therefore, we suggest to take this value in the next section to tackle a segmentation problem. Before that, let us finish this section by geometrically showing the interest of learning local metrics. Figure 3(a) shows ellipsoids uniformly distributed in the RGB space whose surface corresponds to the RGB colors lying at the corresponding learned local distance of 1 from the center of the ellipsoid. It is worth noting that the variability of the shapes and orientations of the ellipsoids is high, meaning that each local metric could capture local specificities of the color space. The experimental results presented in the next section will prove this claim.

Application to image segmentation

In this experiment, we evaluate the performance of our approach in a color based image segmentation application. We propose to use the approach from [START_REF] Bitsakos | An experimental study of color-based segmentation algorithms based on the mean-shift concept[END_REF] that suggests a nice extension of the classical mean-shift algorithm by accounting color information. Furthermore, the authors show that the more perceptual the used distance, the better the results. Especially, by using the default transform from the available camera RGB to the L * u * v * , they significantly improve the segmentation results over the simple RGB coordinates. Our aim is not to propose a new segmentation algorithm but to use the exact algorithm proposed in [START_REF] Bitsakos | An experimental study of color-based segmentation algorithms based on the mean-shift concept[END_REF] working in the RGB space and to replace in their code (publicly available) the distance between two colors with our learned color distance ∆ T . By this way, we can compare the perceptual property of our distance with this of the recommended default approach (euclidean distance in the L * u * v * space).

Therefore, we take exactly the same protocol as [START_REF] Bitsakos | An experimental study of color-based segmentation algorithms based on the mean-shift concept[END_REF]. We use the same 200 images taken from the well-known Berkeley dataset and the associated groundtruth that is constituted by 1087 segmented images provided by humans. In order to assess the quality of the segmentation, as recommended by [START_REF] Bitsakos | An experimental study of color-based segmentation algorithms based on the mean-shift concept[END_REF], we use the average Boundary Displacement Error (BDE) and the Probabilistic Rand Index (PRI). Note that the better the quality of the segmentation, the lower the BDE and the higher the PRI. The segmentation algorithm proposed in [START_REF] Bitsakos | An experimental study of color-based segmentation algorithms based on the mean-shift concept[END_REF] has one main parameter which is the color distance threshold under which two neighbor pixels (or sets of pixels) have to be merged in the same segment. As in [START_REF] Bitsakos | An experimental study of color-based segmentation algorithms based on the mean-shift concept[END_REF], we plot the evolution of the quality criteria versus the average segment size (see Figs. 3(b) and3(c)). For comparison, we have run the code from [START_REF] Bitsakos | An experimental study of color-based segmentation algorithms based on the mean-shift concept[END_REF] for the parameters providing the best results in their paper, namely "CMS Luv/N.", corresponding to their color mean-shift (CMS) applied in the L * u * v * color space. The results of CMS applied in the RGB color space with the classical euclidean distance are plotted as "CMS RGB/N." and those of CMS applied with our color distance in the RGB color space are plotted as "CMS Local Metric/N.".

For both criteria, we can see that our learned color distance significantly improves the quality of the results over the two other approaches, i.e. it provides a segmentation that is closer to the one computed by humans. This is truer when the segment size is increasing (right part of the plots). It is important to understand that increasing the average segment size (moving to the right on the plots) is like merging neighbor segments in the images. So by analyzing the curves, we can see that for the classical approaches ("CMS Luv/N." and "CMS RGB/N."), it seems that the segments that are merged together when moving to the right on the plot are not the ones that would be merged by humans. That is why both criteria are worst (BDE increases and PRI decreases) on the right for these methods. On the other hand, it seems that our distance is more accurate when merging neighbor segments since for high average segment sizes, our results are much better. This point can be observed in Fig. 4, where the segment size is high, i.e. when the number of clusters is low (50), the segmentation provided by RGB or L * u * v * are far from the ground truth, unlike our approach which provides nice results. To get the same perceptual result, both methods require about 500 clusters. We provide more segmentation comparisons in the supplementary material.

Conclusion

In this paper, we presented a new local metric learning approach for approximating perceptual distances directly in the rendered image RGB space. Our method outperforms the state of the art for generalizing to unseen colors and to unseen camera distortions and also in a color image segmentation task. The model is both efficient -for each pair one only needs to find the two clusters of the patches and to apply a 3 × 3 matrix -and expressive thanks to the local aspect allowing us to model different distortions in the RGB space. Moreover, we derived a generalization bound ensuring the consistency of the learning approach. Finally, we designed a dataset of color patches which can play the role of a benchmark for the computer vision community.

Future work will include the use of metric combination approaches together with more complex regularizers on the set of models (mixed and nuclear norms for example). Another perspective concerns the spatial continuity of the learned metrics. Even though Fig. 3(a) shows ellipsoids that tend to be locally regular leading to a certain spatial continuity, our model does not explicitely deal with this issue. One solution may consist in resorting to a Gaussian embedding of the local metrics. From a practical side, the development of transfer learning methods for improving the generalization to unknown devices could be an interesting direction. Another different perspective would be to learn photometric invariant distances.

Modeling Perceptual Color Differences by Local Metric Learning Supplementary Material 1

Michaël Perrot, Amaury Habrard, Damien Muselet, and Marc Sebban LaHC, UMR CNRS 5516, Université Jean-Monnet, F-42000, Saint-Étienne, France {michael.perrot,amaury.habrard,damien.muselet,marc.sebban}@univ-st-etienne.fr

Overview of the supplementary material

This supplementary material is organised into two parts. In Section 2 we provide the proofs of the lemma and the theorem presented in Section 3.3 of the paper, while Section 3 presents some examples of image segmentation.

Theoretical analysis

This section presents the proofs of Lemma 1 and Theorem 1 from Section 3.3 of the paper. Lemma 1 is proved in Section 2.1 and Theorem 1 is proved in Section 2.2.

Generalization bound per region C j

First, we recall our optimization problem considered in each region C j : arg min

M j 0 F T j (M j ) (1)
where

F T j (M j ) = εT j (M j ) + λ j M j 2 F , εT j (M j ) = 1 n j (x,x ,∆E00)∈T j l(M j , (x, x , ∆E 00 )),
and

l(M j , (x, x , ∆E 00 )) = x -x ) T M j (x -x -∆E 00 (x, x ) 2 .
Here εT j (M j ) stands for the empirical risk of a matrix M j over a training set T j , of size n j , drawn from an unknown distribution P (C j ). The true risk ε P (Cj ) (M j ) is defined as follows:

ε P (Cj ) (M j ) = E (x,x ,∆E00)∼P (Cj ) l(M j , (x, x , ∆E 00 )) .
In this section, T i j denotes the training set obtained from T j by replacing the i th example of T j by a new independent one. Moreover, we have ∆ max = max 0≤j≤K max (x,x ,∆E00)∼P (Cj ) {∆E 00 (x, x )} and

D j = max (x,x ,∆E00)∼P (Cj ) ( x -x ) ≤ 1 1 .
To derive such a generalization bound, we need to consider loss functions that fulfill two properties: k-lipschitz continuity (Definition A) and (σ, m)-admissibility (Definition B).

Definition A (k-lipschitz continuity) A loss function l(M j , (x, x , ∆E 00 )) is k-lipschitz w.r.t. its first argument if, for any matrices M j , M j and any example (x, x , ∆E 00 ), there exists k ≥ 0 such that:

l(M j , (x, x , ∆E 00 )) -l(M j , (x, x , ∆E 00 )) ≤ k M j -M j F .
This k-lipschitz property ensures that the loss deviation does not exceed the deviation between matrices M j and M j with respect to a positive constant k.

Definition B ((σ, m)-admissibility) A loss function l(M j , (x, x , ∆E 00 )) is (σ, m)-admissible, w.r.t. M j , if it is convex w.r.t. its first argument and for two examples (x, x , ∆E 00 (x, x )) and (t, t , ∆E 00 (t, t )), we have:

l(M j , (x, x , ∆E 00 (x, x ))) -l(M j , (t, t , ∆E 00 (t, t ))) ≤ σ |∆E 00 (x, x ) -∆E 00 (t, t )| + m.
Definition B bounds the difference between the losses of two examples by a value only related to the ∆E 00 values plus a constant independent from M j . Let us introduce a last concept which is required to derive a generalization bound.

Definition C (Uniform stability) In a region C j , a learning algorithm has a uniform stability in K nj , with

K ≥ 0 a constant, if ∀i, sup (x,x ,∆E00)∼P (Cj ) l(M j , (x, x , ∆E 00 )) -l(M i j , (x, x , ∆E 00 )) ≤ K n j ,
where M j is the matrix learned on the training set T j and M i j is the matrix learned on the training set T i j . The uniform stability guarantees that the solutions learned with two close training sets are not significantly different and that the variation converges in O(1/n j ).

To prove Lemma 1 of the paper, we need several additional lemmas and one more theorem which are not presented in the paper. First we show that our loss is k-lipschitz continuous, (σ, m)-admissible and that our algorithm respects the property of uniform stability. For the sake of readability, we number these lemmas and this theorem with capital letters.

Lemma A (k-lipschitz continuity) Let M j and M j be two matrices for a region C j and (x, x , ∆E 00 ) be an example. Our loss l(M j , (x, x , ∆E 00 )) is k-lipschitz with k = D 2 j . Proof. We now provide a lemma that will help to prove Lemma C on the (σ, m)-admissibility of our loss function.

l(M j , (x, x , ∆E 00 )) -l(M j , (x, x , ∆E 00 )) = x -x ) T M j (x -x -∆E 00 (x, x ) 2 -x -x ) T M j (x -x -∆E 00 (x, x ) 2 ≤ x -x ) T M j (x -x -x -x ) T M j (x -x (2.1) = x -x ) T M j -M j (x -x ≤ x -x M j -M j F x -x (2.2) ≤ D 2 j M j -M j F (2.3) Inequality (2.
Lemma B Let M j be an optimal solution of Problem (1), we have

M j ≤ ∆ max λ j .
Proof. Since M j is an optimal solution of Problem (1), we have then:

F T j (M j ) ≤ F T j (0) ⇔ 1 n j (x,x ,∆E00)∈T j l(M j , (x, x , ∆E 00 )) + λ j M j 2 F ≤ 1 n j (x,x ,∆E00)∈T j l(0, (x, x , ∆E 00 )) + λ j 0 2 F ⇒ λ j M j 2 F ≤ 1 n j (x,x ,∆E00)∈T j l(0, (x, x , ∆E 00 )) (3.1) ⇒ λ j M j 2 F ≤ ∆ 2 max (3.2) ⇒ M j F ≤ ∆ max λ j .
Inequality (3.1) comes from the fact that our loss is always positive and that 0 F = 0. (3.2) is obtained by noting that l(0, (x, x , ∆E 00 )) ≤ ∆ 2 max .

Lemma C ((σ, m)-admissibility) Let (x, x , ∆E 00 (x, x )) and (t, t , ∆E 00 (t, t )) be two examples and M j be the optimal solution of Problem (1). The loss l(M j , (x, x , ∆E 00 )) is (σ, m)-admissible with σ = 2∆ max and m =

2D 2 j ∆max √ λj . Proof. l(M j , (x, x , ∆E 00 (x, x ))) -l(M j , (t, t , ∆E 00 (t, t ))) = x -x ) T M j (x -x -∆E 00 (x, x ) 2 -t -t ) T M j (t -t -∆E 00 (t, t ) 2 ≤ x -x ) T M j (x -x -t -t ) T M j (t -t + ∆E 00 (t, t ) 2 -∆E 00 (x, x ) 2 (4.1) ≤ x -x ) T M j (x -x + t -t ) T M j (t -t + ∆E 00 (t, t ) 2 -∆E 00 (x, x ) 2 (4.2) ≤ 2 max (x,x ) x -x ) T M j (x -x + ∆E 00 (t, t ) 2 -∆E 00 (x, x ) 2 ≤ 2D 2 j ∆ max λ j + ∆E 00 (t, t ) 2 -∆E 00 (x, x ) 2 (4.3) ≤ 2D 2 j ∆ max λ j + |∆E 00 (t, t ) + ∆E 00 (x, x )| |∆E 00 (t, t ) -∆E 00 (x, x )| ≤ 2D 2 j ∆ max λ j + 2∆ max |∆E 00 (t, t ) -∆E 00 (x, x )| .
Inequalities (4.1) and (4. We will now prove the uniform stability of our algorithm but before to present this proof, we need the following Lemma.

Lemma D Let F T j () and F T i j () be the functions to optimize, M j and M i j their corresponding minimizers, and λ j the regularization parameter used in our algorithm. Let ∆M j = M j -M i j , then, we have, for any t ∈ [0, 1],

M j 2 F -M j -t∆M j 2 F + M i j 2 F -M i j + t∆M j 2 F ≤ 2kt λ j n j ∆M j F . (5)
Proof. This proof is similar to the proof of Lemma 20 in [1] which we recall for the sake of completeness. εT i j () is a convex function, thus, for any t ∈ [0, 1], we can write:

εT i j (M j -t∆M j ) -εT i j (M j ) ≤ t εT i j (M i j ) -εT i j (M j ) , ( 6
)
εT i j (M i j + t∆M j ) -εT i j (M i j ) ≤ t εT i j (M j ) -εT i j (M i j ) . (7)
By summing inequalities ( 6) and ( 7) we obtain

εT i j (M j -t∆M j ) -εT i j (M j ) + εT i j (M i j + t∆M j ) -εT i j (M i j ) ≤ 0. (8)
Since M j and M i j are minimizers of F T j () and F T i j (), we can write:

F T j (M j ) -F T j (M j -t∆M j ) ≤ 0, (9)
F T i j (M i j ) -F T i j (M i j + t∆M j ) ≤ 0. (10)
By summing inequalities ( 9) and ( 10), we obtain

εT j (M j ) -εT j (M j -t∆M j ) + λ j M j 2 F -λ j M j -t∆M j 2 F + εT i j (M i j ) -εT i j (M i j + t∆M j ) + λ j M i j 2 F -λ j M i j + t∆M j 2 F ≤ 0. (11)
We can now sum inequalities ( 8) and ( 11) to obtain

εT j (M j ) -εT i j (M j ) -εT j (M j -t∆M j ) + εT i j (M j -t∆M j )+ λ j M j 2 F -λ j M j -t∆M j 2 F + λ j M i j 2 F -λ j M i j + t∆M j 2 F ≤ 0. (12)
From ( 12), we can write:

λ j M j 2 F -λ j M j -t∆M j 2 F + λ j M i j 2 F -λ j M i j + t∆M j 2 F ≤ B (13)
with B = εT i j (M j ) -εT j (M j ) + εT j (M j -t∆M j ) -εT i j (M j -t∆M j ). We are now looking for a bound on B:

B ≤ εT j (M j -t∆M j ) -εT i j (M j -t∆M j ) + εT i j (M j ) -εT j (M j ) ≤ 1 n j (x,x ,∆E00)∈T j l(M j -t∆M j , (x, x , ∆E 00 )) - (t,t ,∆E00)∈T i j l(M j -t∆M j , (t, t , ∆E 00 )) + (t,t ,∆E00)∈T i j l(M j , (t, t , ∆E 00 )) - (x,x ,∆E00)∈T j l(M j , (x, x , ∆E 00 )) = 1 n j l(M j -t∆M j , (x i , x i , ∆E 00
))l(M j -t∆M j , (t i , t i , ∆E 00 )) + l(M j , (t i , t i , ∆E 00 ))l(M j , (x i , x i , ∆E 00 )) (14.1) ≤ 1 n j l(M j -t∆M j , (x i , x i , ∆E 00 ))l(M j , (x i , x i , ∆E 00 )) + l(M j , (t i , t i , ∆E 00 ))l(M j -t∆M j , (t i , t i , ∆E 00 )) (14.2)

≤ 1 n j k M j -t∆M j -M j F + k M j -M j + t∆M j F (14.3) ≤ 2kt n j ∆M j F .
Equality (14.1) comes from the fact that T j and T i j only differ by their i th example, inequality (14.2) is due to the triangle inequality and (14.3) is obtained thanks to the k-lipschitz property of our loss (Lemma A).

Then combining the bound on B with equation ( 13) and dividing both sides by λ j gives the Lemma.

We can now show the uniform stability property of the approach.

Lemma E (Uniform stability) Given a training sample T j of n j examples drawn i.i.d. from P (C j ), our algorithm has a uniform stability in K nj with K = 2D 4 j λj .

Proof. By setting t = 1 2 in Lemma D, one can obtain for the left hand side:

M j 2 F -M j - 1 2 ∆M j 2 F + M i j 2 F -M i j + 1 2 ∆M j 2 F = 1 2 ∆M j 2 F
and thus:

1 2 ∆M j 2 F ≤ 2k 1 2 λ j n j ∆M j F , which implies ∆M j F ≤ 2k λ j n j .
Since our loss is k-lipschitz (Lemma A) we have:

l(M j , (x, x , ∆E 00 )) -l(M i j , (x, x , ∆E 00 )) ≤ k ∆M j F ≤ 2k 2 λ j n j .
In particular, sup (x,x ,∆E00) l(M j , (x, x , ∆E 00 ))l(M j , (x, x , ∆E 00 )) ≤ 2k 2 λ j n j .

By recalling that k = D 2 j (Lemma A) and setting K = 2k 2 λj , we get the lemma. We now recall the McDiarmid inequality [2], used to prove our main theorem.

Theorem A (McDiarmid inequality) Let X 1 , ..., X n be n independent random variables taking values in X and let

Z = f (X 1 , ..., X n ). If for each 1 ≤ i ≤ n, there exists a constant c i such that sup x1,...,xn,x i ∈X |f (x 1 , ..., x n ) -f (x 1 , ..., x i , ..., x n )| ≤ c i , ∀1 ≤ i ≤ n, then for any > 0, Pr [|Z -E [Z]| ≥ ] ≤ 2 exp -2 2 n i=1 c 2 i .
Using Lemma E about the stability of our algorithm and the McDiarmid inequality we can derive our generalization bound. For this purpose, we replace Z by R T j = ε P (Cj ) (M j ) -εT j (M j ) in Theorem A and we need to bound E T j R T j and R T j -R T i j , which is done in the following two lemmas.

Lemma F For any learning method of estimation error R T j and satisfying a uniform stability in K nj , we have

E T j R T j ≤ K n j .
Proof.

E T j R T j ≤ E T j E (x,x ,∆E00) l(M j , (x, x , ∆E 00 )) -εTj (M j ) ≤ E Tj ,(x,x ,∆E00)    l(M j , (x, x , ∆E 00 )) - 1 n j (xk,x k ,∆E00)∈T j l(M j , (x k , x k , ∆E 00 ))    ≤ E Tj ,(x,x ,∆E00)    1 n j (xk,x k ,∆E00)∈T j l(M j , (x, x , ∆E 00 )) -l(M j , (x k , x k , ∆E 00 ))    ≤ E Tj ,(x,x ,∆E00)    1 n j (xk,x k ,∆E00)∈T j l(M k j , (x k , x k , ∆E 00 )) -l(M j , (x k , x k , ∆E 00 ))    (15.1) ≤ K n j . ( 15.2)
Inequality (15.1) comes from the fact that T j and (x, x , ∆E 00 ) are drawn i.i.d. from the distribution P (C j ) and thus we do not change the expected value by replacing one example with another, (15.2) is obtained by applying triangle inequality followed by the property of uniform stability (Lemma E).

Lemma G For any matrix M j learned by our algorithm using n j training examples, and any loss function l satisfying the (σ, m)-admissibility, we have 16.5) comes from the fact that T j and T i j only differ by their i th example. (16.6) comes from the (σ, m)-admissibility of our loss (Lemma C). Noting that |∆E 00 (t i , t i ) -∆E 00 (x i , x i )| ≤ ∆ max gives inequality (16.7).

R T j -R T k j ≤ 2K + (∆ max σ + m) n j . Proof. R T j -R T i j = ε P (Cj ) (M j ) -εT j (M j ) -ε P (Cj ) (M i j ) -εT i j (M i j ) = ε P (Cj ) (M j ) -εT j (M j ) -ε P (Cj ) (M i j ) + εT i j (M i j ) -εT j (M i j ) + εT j (M i j ) ≤ ε P (Cj ) (M j ) -ε P (Cj ) (M i j ) + εT j (M i j ) -εT j (M j ) + εT i j (M i j ) -εT j (M i j ) (16.1) ≤ E (x,x ,∆E00) l(M j , (x, x , ∆E 00 )) -l(M i j , (x, x , ∆E 00 )) + εT j (M i j ) -εT j (M j ) + εT i j (M i j ) -εT j (M i j ) (16.2) ≤ K n j + εT j (M i j ) -εT j (M j ) + εT i j (M i j ) -εT j (M i j ) (16.3) ≤ K n j + 1 n j (x,x ,∆E00)∈T j l(M i j , (x, x , ∆E 00 )) -l(M j , (x, x , ∆E 00 )) + εT i j (M i j ) -εT j (M i j ) ≤ K n j + K n j + εT i j (M i j ) -εT j (M i j ) (16.4) = 2K n j + 1 n j l(M i j , (t i , t i , ∆E 00 )) -l(M i j , (x i , x i , ∆E 00 )) (16.5) ≤ 2K n j + 1 n j (σ |∆E 00 (t i , t i ) -∆E 00 (x i , x i )| + m) (16.6) ≤ 2K + (∆ max σ + m) n j . ( 16
Lemma 1 (Generalization bound) With probability 1δ, for any matrix M j related to a region C j , 0 ≤ j ≤ K, learned with Algorithm 1, we have:

ε P (Cj ) (M j ) ≤ εT j (M j ) + 2D 4 j λ j n j + 4D 4 j λ j +∆ max ( 2D 2 j λ j +2∆ max ) ln( 2 δ ) 2n j .
Proof. Using the McDiarmid inequality (Theorem A) and Lemma G we can write:

Pr R T j -E T j R T j ≥ ≤ 2 exp   - 2 2 n j=1 2K+(5σ+m) nj 2    ≤ 2 exp - 2 2 1 nj (2K + (5σ + m)) 2 .
Then, by setting:

δ = 2 exp - 2 2 1 nj (2K + (5σ + m)) 2
we obtain:

= (2K + (∆ max σ + m)) ln 2 δ 2n j
and thus:

Pr R T j -E T j R T j < > 1 -δ.
Then, with probability 1δ:

R T j < E T j R T j + ⇔ ε P (Cj ) (M j ) -εT j (M j ) < E T j R T j + ⇔ ε P (Cj ) (M j ) < εT j (M j ) + K n j + (2K + (∆ max σ + m)) ln 2 δ 2n j .
The last equation is obtained by using Lemma F and replacing K, σ and m by their respective values gives the lemma.

We showed that our approach is locally consistent. In the next section, we show that our algorithm globally converges in O(1/ √ n).

Generalization bound for Algorithm 1

We consider the partition C 0 , C 1 , . . . , C K over pairs of examples considered by Algorithm 1. We first recall the concentration inequality that will help us to derive the bound.

We recall the true and empirical risks. Let M = {M 0 , M 1 , . . . , M K } be the K +1 matrices learned by our algorithm. The true error associated to M is defined as ε(M) = K j=0 ε P (Cj ) (M j )P (C j ) where ε P (Cj ) (M j ) is the local true risk for C j . The empirical error over T of size n is defined as εT (M) = 1 n K j=0 n j εT j (M j ) where εT j (M j ) is the empirical risk of T j . Before proving the main theorem of the paper we introduce an additional lemma showing a bound on the loss function.

Lemma H Let M = {M 0 , M 1 , . . . , M K } be any set of metrics learned by Algorithm 1 from a data sample T of n pairs, for any 0 ≤ j ≤ K, we have that for any example (x, x , ∆E 00 ) ∼ P (C j ): l(M j , (x, x , ∆E 00 )) ≤ L B , with L B = max{ ∆max √ λ , ∆ 2 max }.

Proof.

l(M j , (x, x , ∆E 00 )) = xx ) T M j (xx -∆E 00 (x, x ) We can now prove the main theorem of the paper. where D = max 1≤j≤K D j , L B is the bound on the loss function and λ = min 0≤j≤K λ j is the minimum regularization parameter among the K + 1 learning problems used in Algorithm 1.

Proof. Let n j be the number points of T that fall into the partition C j . (n 0 , n 1 , . . . , n K ) is a IID multinomial random variable with parameters n and (P (C 0 ), P (C 1 ), . . . , P (C K )).

Fig. 1 .

 1 Fig. 1. Some images from our dataset showing (first row) the 84 used Farnsworth-Munsell patches or (second row) the 238 Munsell patches under different conditions.

  Generalization to new cameras.

Fig. 2 .

 2 Fig. 2. (a): Generalization of the learned metrics to new colors; (b) Generalization of the learned metrics to new cameras. For (a) and (b), we plotted the Mean and STRESS values as a function of the number of clusters. The horizontal dashed line represents the STRESS baseline of ∆E 00 . For the sake of readability, we have not plotted the mean baseline of ∆E 00 at 1.70.

Fig. 3 .

 3 Fig. 3. (a) Interest of learning local metrics. We took 27 points uniformly distributed on the RGB cube. Around each point we plotted an ellipsoid where the surface corresponds to the RGB colors lying at a learned distance of 1. In this case we used the metric learned by our algorithm using K = 20. (b) Boundary Displacement Error (lower is better) versus the average segment size. (c) Probabilistic Rand Index (higher is better) versus the average segment size.

Fig. 4 .

 4 Fig. 4. Segmentation illustration. When the number of clusters is low (around 50), the segmentation provided by RGB or L * u * v * are far from the ground truth, unlike our approach which provides nice results. To get the same perceptual result, both methods require about 500 clusters.

1 )

 1 is due to the triangle inequality, (2.2) is obtained by application of the Cauchy-Schwarz inequality and some classical norm properties. (2.3) comes from the definition of D j . Setting k = D 2 j gives the Lemma.

2 )

 2 are obtained by applying the triangle inequality respectively twice and once, (4.3) comes from the fact that M j F ≤ ∆max √ λj (Lemma B) and that xx ≤ D j . Setting σ = 2∆ max

Proposition 1 ( 2 ,

 12 [3]). Let (n 0 , n 1 , . . . , n K ) an IID multinomial random variable with parameters n = K j=0 n j and (P (C 0 ), P (C 1 ), . . . , P (C K )). By the Breteganolle-Huber-Carol inequality we have:P r K j=0 nj n -P (C j ) ≥ η ≤ 2 K exp -nη2 hence with probability at least 1δ, K j=0 n j n -P (C j ) ≤ 2K ln 2 + 2 ln(1/δ) n .

2 ≤ 2 ( 18

 2218 max xx ) T M j (xx , ∆E 00 (x, x ) .1) comes from the fact that any matrix M j is positive semi definite and thus we are taking the absolute difference of two positive values. Inequality (18.2) is obtained by using the Cauchy-Schwarz inequality, the Lemma B with λ = min 0≤j≤K λ j and the inequality xx ≤ 1. Inequality (18.3) is due to the definition of ∆ max .

Theorem 1

 1 Let C 0 , C 1 , . . . , C K be the regions considered and M = {M 0 , M 1 , . . . , M K } any set of metrics learned by Algorithm 1 from a data sample T of n pairs, we have with probability at least 1δ that ε(M) ≤ε T (M) + L B 2(K + 1) ln 2 + 2 ln(2/δ)

ModelingEEEEEE 4 j

 4 Perceptual Color Differences by Local Metric Learning -SupplementaryMaterial 1 9 |ε(M) -εT (M)| = E (x,x ,∆E00)∼P [l(M, (x, x , ∆E 00 ))] -εT (M) = K j=0 E (x,x ,∆E00)∼P |(x,x ,∆E00)∈Cj l(M j , (x, x , ∆E 00 )) P (C j ) -εT (M) = K j=0 E (x,x ,∆E00)∼P |(x,x ,∆E00)∈Cj l(M j , (x, x , ∆E 00 )) P (C j ) -K j=0 E (x,x ,∆E00)∼P |(x,x ,∆E00)∈Cj l(M j , (x, x , ∆E 00 )) (x,x ,∆E00)∼P |(x,x,∆E00)∈Cj l(M j , (x, x , ∆E 00 )) (x,x ,∆E00)∼P |(x,x ,∆E00)∈Cj l(M j , (x, x , ∆E 00 )) P (C j ) (x,x ,∆E00)∼P |(x,x ,∆E00)∈Cj l(M j , (x, x , ∆E 00 )) (x,x ,∆E00)∼P |(x,x ,∆E00)∈Cj l(M j , (x, x , ∆E 00 )) (x,x ,∆E00)∼P |(x,x ,∆E00)∈Cj l(M j , (x, x , ∆E 00 )) P (C j ) -(x,x ,∆E00)∼P |(x,x ,∆E00)∈Cj l(M j , (x, x , ∆E 00 )) x ,∆E00)∼P |(x,x ,∆E00)∈Cj l(M j , (x, x , ∆E 00 )) -εT j (M j ) x ,∆E00)∼P |(x,x ,∆E00)∈Cj l(M j , (x, x , ∆E 00 )) -εT j (M j ) λ j n j + 2D

Fig. 1 .

 1 Fig.1. Illustration of segmentation provided by the color mean-shift algorithm applied in the RGB components (third column), on L * u * v * components (fourth column) and by using our learned distance directly in the RGB components (fifth column). First column represents the original image and the second one the ground truth.

  .7) Inequalities (16.1) and (16.2) are due to the triangle inequality. (16.3) and (16.4) come from the uniform stability (Lemma E). (

Note that kernel learning is another solution to consider non linearity in the data.

https://www.uef.fi/spectral/spectral-database

We noticed during our experiments that M j is, most of the time, PSD without requiring any projection on the cone.

Since we work in the RGB cube, any patch belongs to [0; 255] 3 and it is easy to normalize each coordinate by 255 √ 3.

STandardized REsidual Sum of Squares.

We assume the examples to be normalized such that x ≤ 1.

Acknowledgments

The final publication is available at http://link.springer.com/.

Inequalities (19.1) and (19.2) are due to the triangle inequality. (19.3) comes from the application of Lemma H. Inequality (19.4) is obtained by applying Proposition 1 with probability 1δ/2. (19.5) is due to the application of Lemma 1 with probability 1δ/(2(K + 1)) for each of the (K + 1) learning problems. Inequality (19.6) is obtained by cancelling out the n j , noting that √ n j ≤ √ n and taking D = max 1≤i≤n D j . Note that D 0 = 1 corresponds to the partition used by the global metric.

Eventually by the union bound we obtained the final result with probability 1δ.

Image Segmentation

In this section, we illustrate the application of the color mean-shift algorithm presented in our paper. We apply color mean-shift on RGB components, on L * u * v * components and by using our learned distance directly in the RGB components. The overall quantitative results for the Berkeley dataset are provided in the paper and we propose to show some qualitative results on this dataset in Figure 1. As explained in the paper, the number of segments in the resulting images is not a parameter of the algorithm, as a consequence it is not easy to obtain images with the same number of segments for the three algorithms (RGB, L * u * v * and Metric learning). Thus, given an image, by playing with the color distance threshold, we have tried to obtain the same segment numbers as the corresponding ground truth for the three algorithms. However, the color mean-shift algorithm provides some very small segments, specially for the RGB and L * u * v * color spaces. Consequently, for each test, in Figure 1, we have mentioned between brackets, first, the number of segments, and second, the number of segments whose size is more than 150 pixels. For a fair comparison, we use this last number as reference for each image, i.e. this number is almost constant and close to the ground truth for each row.

It is worth mentioning that the ground truth segmentation has always very few segments. Thus, starting from a large number of small segments, the used algorithm is grouping them by considering their color differences. Consequently, the used color distance is crucial when we want to obtain small number of segments as provided by the ground truth. We can see in Figure 1, that when working in the RGB or L * u * v * color spaces, some segments that are perceptually different are merged while some other similar segments are not. Most of the time, the color mean-shift is working well when using our distance. This point was already checked quantitatively on the whole Berkeley dataset in the paper.


Peter Imkeller
email: imkeller@math.hu-berlin.de

Thibaut Mastrolia
email: mastrolia@ceremade.dauphine.fr

Dylan Possamaï
email: possamai@ceremade.dauphine.fr

Anthony Réveillac
email: anthony.reveillac@insa-toulouse.fr

A note on the Malliavin-Sobolev spaces

Keywords: Malliavin calculus, Wiener space, functional analysis, stochastic Gâteaux differentiability AMS 2010 subject classification: Primary: 60H07; Secondary: 46B09

In this paper, we provide a strong formulation of the stochastic Gâteaux differentiability in order to study the sharpness of a new characterization, introduced in [6], of the Malliavin-Sobolev spaces. We also give a new internal structure of these spaces in the sense of sets inclusion.

Introduction

Consider the classical W 1,2 (R d ) Sobolev space on R d (d ≥ 1) consisting of mappings f which are square integrable and which admit a square integrable weak derivative. This space is known to be the completion of continuously differentiable functions with respect to the Sobolev seminorm. As proved by Nikodym, f is an element of W 1,2 (R d ) if and only if there exist versions of f along particular directions (see [8, (C) page 717] for a precise statement) whose Radon-Nikodym derivatives are square integrable. In an infinitedimensional framework, similar characterizations can also be obtained. For instance, consider the classical Wiener space (Ω, F ) defined as the space of continuous functions Ω, on [0, T ] (with T a fixed positive real number) vanishing at 0, and where F is the σ-algebra of Borel sets on Ω, for the topology induced by the uniform norm. Let P 0 be the Wiener measure on Ω, that is the unique measure on (Ω, F ) which makes the coordinate process W a standard Brownian motion. Let in addition H be the Cameron-Martin space, that is the space of absolutely continuous functions on [0, T ] with square integrable (with respect to Lebesgue's measure on [0, T ]) Radon-Nikodym derivative. In that context, one can define the two following spaces:

(S) the closure of so-called cylindrical random variables on Ω with respect to the Sobolev seminorm • 1,2 (where the weak derivative is the Malliavin derivative).

(K-S) the set of random variables Z on Ω which are ray absolutely continuous (RAC) and which satisfies the stochastic Gâteaux differentiability property (SGD) i.e. which admit a Gâteaux derivative in any direction of the Cameron-Martin space H.

More precisely, a random variable Z on (Ω, F , P 0 ) satisfies Property (SGD), if for any element h in H,

lim ε→0 Z • τ εh -Z ε P0 = DZ, h H ,
where the limit is taken in probability, DZ denotes the Gâteaux derivative of Z and where τ εh denotes the shift operator on the Wiener space. The space (S) has been introduced by Shigekawa in [START_REF] Shigekawa | Derivatives of Wiener functionals and absolute continuity of induced measures[END_REF], as a natural candidate to be a Sobolev space whose derivative is given by the Malliavin derivative (whose definition will be recalled in the next section), whereas the space (K-S) has been introduced by Kusuoka and Stroock in [START_REF] Kusuoka | Applications of the Malliavin calculus[END_REF]. In comparison to the finite-dimensional case, Sugita proved in [START_REF] Sugita | On a characterization of the Sobolev spaces over an abstract Wiener space[END_REF] that these two spaces coincide. We will refer to them later on as D 1,2 . To be more precise, Characterization (K-S) (essentially) means that Z ∈ D 1,2 ⇐⇒ Z satisfies (RAC) and (SGD).

In practice Property (RAC) may not be easy to handle. Recently, it was shown in [6, Theorem 4.1] that the characterization (RAC) + (SGD) can actually be replaced, equivalently, by a stronger version of (SGD) that we name strong stochastically Gâteaux differentiability, or (SSGD) for short, which can be roughly stated as (a precise statement will be given in the next section): for any element h in H

lim ε→0 Z • τ εh -Z ε L q = DZ, h H ,
where the limit is taken in L q spaces. According to [START_REF] Mastrolia | On the Malliavin differentiability of BSDEs[END_REF], we obtain

Z ∈ D 1,2 ⇐⇒ ∃q ∈ (1, 2) s.t. Z satisfies (SSGD 2 (q)),
where a random variable Z on (Ω, F , P 0 ) is said to satisfy (SSGD 2 (q)) if (Z and ∇Z are square integrable and) for any element h in H

lim ε→0 Z • τ εh -Z ε L q = ∇Z, h H ,
where ∇Z denotes the Malliavin derivative of Z. In other words, the (RAC) property from the characterization of D 1,2 can be dropped by strengthening property (SGD). One can also prove the following (see [START_REF] Mastrolia | On the Malliavin differentiability of BSDEs[END_REF]Lemma 4.2]):

Z ∈ D 1,2 ⇐⇒ ∀q ∈ (1, 2), Z satisfies (SSGD 2 (q)). (1.1)
In particular, one can wonder if

Z ∈ D 1,2 ⇐⇒ Z satisfies (SSGD 2 (2))? (1.2) In view of (1.1), it is clear that Z satisfies SSGD 2 (2) implies that Z is in D 1,2 .
The question is then is it also a necessary condition? The aim of this article is to answer this question by the negative. Indeed we provide in Theorem 3.1 a random variable Z such that Z ∈ D 1,2 and Z does not satisfies (SSGD 2 (2)).

Incidentally, this question is related to proving that

D 1,2+ := ε>0 D 1,2+ε D 1,2 ,
which we address at the end of the paper (as Theorem 3.3).

The rest of the paper is organized as follows. In Section 2 we fix all the notations, give a rigorous definition of the strong stochastic Gâteaux differentiability, and we highlight the wellposedness of this definition by giving some classical properties concerning the space of random variables which satisfy this (strong) differentiability property. Then, in Section 3 we answer the question given in (1.2) and we study more precisely the structure of the Malliavin-Sobolev spaces.

Preliminaries

Notations

We fix throughout the paper a time horizon T > 0.

Let Ω := C 0 ([0, T ], R) be the canonical Wiener space of continuous function ω from [0, T ] to R such that ω(0) = 0. F denotes the Borel σ-algebra on Ω, for the uniform topology. Let W := (W t ) t∈[0,T ] be the canonical Wiener process, that is, for any t in [0, T ], W t denotes the evaluation mapping: W t (ω) := ω t for any element ω in Ω. We set F o := (F o t ) t∈[0,T ] the natural filtration of W , and remind the reader that F = F o T . Under the Wiener measure P 0 , the process W is a standard Brownian motion and we denote by F := (F t ) t∈[0,T ] the usual augmentation (which is right-continuous and complete) of F o under P 0 . Unless otherwise stated, all the expectations considered in this paper will have to be understood as expectations under P 0 , and all notions of measurability for elements of Ω will be with respect to the filtration F or the σ-field F T . Unless otherwise stated, topological spaces are endowed with their Borel σ-algebra.

We set

H := f : [0, T ] -→ R, Borel-measurable, s.t. T 0 |f (s)| 2 ds < +∞ ,
and we denote by H the Cameron-Martin space defined as:

H := h : [0, T ] -→ R, ∃ ḣ ∈ H, h(t) = t 0 ḣ(x)dx, ∀t ∈ [0, T ] ,
which is an Hilbert space equipped with the inner product h 1 , h 2 H := T 0 ḣ1 (t) ḣ2 (t)dt, for any (h 1 , h 2 ) ∈ H × H, and with associated norm h 2 H := h, h H . For any h in H, we will always denote by ḣ a version of its Radon-Nykodym density with respect to the Lebesgue measure. Define next for any Hilbert space K and for any p ≥ 1, L p (K) as the set of all F T -measurable random variables Z which are valued in an Hilbert space K, and such that

Z p L p (K) < +∞, where Z L p (K) := (E [ Z p K ]) 1/p .
Let S be the set of polynomial cylindrical functionals, that is the set of random variables Z of the form

Z = f (W (h 1 ), . . . , W (h n )), (h 1 , . . . , h n ) ∈ H n , f ∈ R n [X], for some n ≥ 1, (2.1)
where W (h) := T 0 ḣs dW s for any h in H. For any Z in S of the form (2.1), the Malliavin derivative ∇Z of Z is defined as the following H-valued random variable:

∇Z := n i=1 f xi (W (h 1 ), . . . , W (h n ))h i (•), (2.2)
where f xi := df dxi . Denote then by D 1,p the closure of S with respect to the Malliavin-Sobolev semi-norm • 1,p , defined as:

Z 1,p := (E [|Z| p ] + E [ ∇Z p H ]) 1/p .
One of the main tools that we will use throughout this paper is the shift operator along directions in the Cameron-Martin space. More precisely, for any h ∈ H, we define the following shift operator τ h : Ω -→ Ω by

τ h (ω) := ω + h.
Note that the fact that h belongs to H ensures that τ h is a measurable shift on the Wiener space. One of the main techniques when working with shifts on the path space is the famous Cameron-Martin formula.

Proposition 2.1. (Cameron-Martin Formula, see e.g. [9, Appendix B.1]) Let Z be a F Tmeasurable random variable and let h be in H. Then, when both sides are well-defined

E[Z • τ h ] = E Z exp T 0 ḣ(s)dW s - 1 2 T 0 | ḣ(s)| 2 ds .
Fix ε > 0 and h ∈ H, we defined the difference quotient of Z along h denoted by X ε as

X ε := Z • τ εh -Z ε .
Our analysis will be based on notions of Gâteaux Differentiability on the Wiener space. Before introducing a new definition in this context we recall the classical formulation due to Kusuoka and Stroock in [START_REF] Kusuoka | Applications of the Malliavin calculus[END_REF].

Definition 2.2 (Stochastically Gâteaux Differentiability, [START_REF] Kusuoka | Applications of the Malliavin calculus[END_REF]). Let p > 1 and Z be in L p (R).

Z is said to be Stochastically Gâteaux Differentiable if there exists DZ ∈ L p (H) such that for any h in H

lim ε→0 Z • τ εh -Z ε = DZ, h H , in probability under P 0 .
We refer this property as (SGD).

We now introduce a stronger formulation of (SGD) that we name Strong Stochastically Gâteaux Differentiability.

Definition 2.3 (Strong Stochastically Gâteaux Differentiability).

Let p > 1 and Z be in L p (R). Z is said to be Strongly Stochastically Gâteaux Differentiable of order (p, q) if there exist q ∈ (1, p) and DZ ∈ L p (H) such that for any h in H

lim ε→0 E Z • τ εh -Z ε -DZ, h H q = 0.
We denote this property by (SSGD p (q)) and we define G p (q) the space of random variables Z which satisfy (SSGD p (q)) with Gâteaux derivative DZ in L p (H).

We name this property Strong Stochastically Gâteaux Differentiability as it strengthens the Stochastic Gâteaux Differentiability property introduced in [START_REF] Kusuoka | Applications of the Malliavin calculus[END_REF] where the L q -convergence above is replaced with convergence in probability.

Remark 2.4. According to [6, Theorem 4.1 and Lemma 4.2], Z enjoys Property (SSGD p (q)) for an element q in (1, p) if and only if there exists DZ ∈ L p (H) such that for any q ′ ∈ (1, p) and for any h in H, it holds that

lim ε→0 E Z • τ εh -Z ε -DZ, h H q ′ = 0.
As a consequence, if there exists q in (1, p) such that Z satisfies Property (SSGD p (q)), then it also satisfies Property (SSGD p (q ′ )) for any q ′ ∈ (1, p).

In view of Remark 2.4, we from now write (SSGD p ) for (SSGD p (q)) with q in (1, p). Similarly, we write G p for G p (q). We now recall the main result of [START_REF] Mastrolia | On the Malliavin differentiability of BSDEs[END_REF].

Theorem 2.5 ([6], Theorem 4.1). Let Z be a real random variable. Then

Z ∈ D 1,p ⇐⇒ Z satisfies (SSGD p ).
In other words, D 1,p = G p .

At this stage, we would like point out that from the definition of Property (SSGD p ) we do not consider the case where the difference quotient associated to an element in G p converges in L p . This induces a stronger property than (SSGD p ) defined below.

Definition 2.6. Let p > 1 and Z be in L p (R). We say that Z satisfies (SSGD p (p)) if there exists DZ ∈ L p (H) such that for any h in H

lim ε→0 E Z • τ εh -Z ε -DZ, h H p = 0.
We define G p (p) the space of random variables Z which satisfy (SSGD p (p)) with Gâteaux derivative DZ in L p (H).

Obviously, in view of Theorem 2.5, Property (SSGD p (p)) is sufficient for a given random variable to belong to D 1,p . This fact was somehow known in the literature ( [START_REF] Bogachev | Differentiable measures and the Malliavin calculus[END_REF] or [START_REF] Janson | Gaussian Hilbert spaces[END_REF]) although the characterization of Theorem 2.5 was not. As a consequence, it is a natural question to wonder whether or not Condition (SSGD 2 (2)) is also necessary for an element Z to belong to D 1,2 . In other words Do the spaces D 1,2 and G 2 (2) coincide ?

Addressing this question is the main goal of the next section.

Main results

Throughout this section, we fix T = 1 and we set φ(x) := e -x 2 /2 √ 2π for any x ∈ R. Let a be in R and consider the following random variable

Z := f (W 1 ), (3.1) with f (x) := e x 2 4
x -a (2π)

1 4 1 x≥ √ 2a + g(x)1 x< √ 2a
, where g is a bounded continuously differentiable map with bounded derivative, such that f is continuously differentiable on R. Notice that f is a nonnegative and nondecreasing mapping when x ≥ √ 2a.

Theorem 3.1. Let p > 1. There exists Z in D 1,p such that Z does not satisfy (SSGD p (p)).

Proof. For the sake of simplicity we prove the result for p = 2.

Let Z be the random variable defined by (3.1) with a > 3 2 . We prove that Z ∈ D 1,2 and that Z does not satisfy (SSGD 2 (2)). By definition of f ,

f ′ (x) = 1 x≥ √ 2a e x 2 4 (-ax -a-1 + 1 2 x 1-a )(2π) 1 4 + g ′ (x)1 x< √ 2a .
The proof is divided in two steps.

Step 1. Z ∈ D 1,2 . There exists C > 0 such that

E |Z| 2 ≤ 2 +∞ √ 2a x -2a dx + √ 2a -∞ |g(x)| 2 φ(x)dx ≤ C +∞ √ 2a x -2a dx + 1 < +∞, since a > 1 2 .
Besides, there exists C ′ > 0 such that

E |f ′ (W 1 )| 2 ≤ 2 +∞ √ 2a (-ax -a-1 + 1 2 x 1-a ) 2 dx + √ 2a -∞ |g ′ (x)| 2 φ(x)dx = C ′ +∞ √ 2a (x -2a-2 + x -2a+2 + x -2a )dx + 1 < +∞, since a > 3 2 .
We hence deduce that Z ∈ D 1,2 .

Step 2. Z does not satisfy (SSGD 2 (2)). Since Z is in D 1,2 , we can compute its Malliavin derivative:

∇Z, h H = 1 0 f ′ (W 1 ) ḣr dr = f ′ (W 1 )h 1 .
In addition, from [8, Theorem 3], for any h ∈ H, it holds that

X ε := Z • τ εh -Z ε P0 -→ ε→0 ∇Z, h H .
Thus, Z belongs to G 2 (2) if and only if (|X ε | 2 ) ε∈(0,1) is uniformly integrable for every h in H. In fact, in our example, for any ε ∈ (0, 1), |X ε | 2 is even not integrable. Indeed, let h ≡ 1 in H and fix ε ∈ (0, 1). Since f is nondecreasing and nonnegative on [ √ 2a, +∞), we obtain

I := E |X ε | 2 = R f (x + ε) -f (x) ε 2 φ(x)dx ≥ +∞ √ 2a f (x + ε) -f (x) ε 2 φ(x)dx ≥ +∞ √ 2a f (x + ε) -f (x + ε/2) ε 2 φ(x)dx = +∞ √ 2a 1 ε ε ε/2 f ′ (x + s)ds 2 φ(x)dx.
According to the mean value theorem, for any x ∈ [ √ 2a, +∞), there exists s x ε ∈ (ε/2, ε) such that

Z x ε := ε -1 ε ε/2 f ′ (x + s)ds = f ′ (x + s x ε ).
Thus,

I ≥ +∞ √ 2a e (x+s x ε ) 2 4 ψ a (s x ε , x) 2 e -x 2 2 dx = +∞ √ 2a Γ ε,x |ψ a (s x ε , x)| 2 dx,
where ψ a (s, x) := -a(x + s) -a-1 + 1 2 (x + s) 1-a and Γ ε,x := e (s x ε ) 2 2 +s x ε x . Since s x ε ∈ (ε/2, ε), we deduce that I = +∞. Thus, for any ε > 0, |X ε | 2 is not integrable, so the family (|X ε | 2 ) ε∈(0,1) is not uniformly integrable, and from the (converse part) of the dominated convergence Theorem, Z / ∈ G 2 (2).

Remark 3.2. In the previous example we can recover the result of Theorem 2.5 since Z ∈ D 1,2 , that is (|X ε | q ) ε is indeed (uniformly) integrable when q < 2, and so Z ∈ G 2 (q). Assume that 1 < q < q ′ < 2, one could show that for any ε ∈ (0, 1) and x ∈ [ √ 2a, +∞) there exists

s x ε ∈ (0, ε) such that sup ε∈(0,1) E |X ε | q ′ = sup ε∈(0,1) +∞ √ 2a |f ′ (x + s x ε )| q ′ φ(x)dx + sup ε∈(0,1) √ 2a -∞ |f ′ (x + s x ε )| q ′ φ(x)dx = sup ε∈(0,1) +∞ √ 2a e (x+s x ε ) 2 4 ψ a (s x ε , x) q ′ e -x 2 2 (2π) 1 2 -q ′ 4 dx + sup ε∈(0,1) √ 2a -∞ |g ′ (x + s x ε )| q ′ φ(x)dx = sup ε∈(0,1) +∞ √ 2a Γ ε,x |ψ a (s x ε , x)| q ′ (2π) q ′ 4 -1 2 e x 2 ( q ′ 2 -1) 2 dx + sup ε∈(0,1) √ 2a -∞ |g ′ (x + s x ε )| q ′ φ(x)dx,
where ψ a (s, x) := -a(x + s) -a-1 + 1 2 (x + s) 1-a and Γ ε,x := e q ′ (s x ε ) 2 +2q ′ s x ε x 4 .

Since the second term on the right-hand side above is uniformly bounded in ε and since q ′ ∈ (1, 2) we deduce that sup ε∈(0,1)

E |X ε | q ′ <
+∞. Thus, from the de la Vallée-Poussin Criterion and the dominated convergence Theorem, Z ∈ G 2 (q).

We have thus constructed a random variable Z in D 1,2 which does not satisfy (SSGD 2 (2)). Hence, according to Remark 2.4, for any ε > 0, Z / ∈ D 1,2+ε . Theorem 3.1 is linked to another question, namely the right-continuity of the Malliavin-Sobolev spaces.

Theorem 3.3. Let p > 1. Define D 1,p+ := ε>0 D 1,p+ε . We have D 1,p+ G p (p) D 1,p . (3.2)
Proof. Theorem 3.1 gives immediately the last inclusion G p (p) D 1,p . Now we turn to the first inclusion. Let Z in D 1,p+ , then there exists ε > 0 such that Z ∈ D 1,p+ε . From Theorem 2.5 together with Remark 2.4, we obtain that Z ∈ D 1,p+ε = G p+ε ⊂ G p (p). Hence,

D 1,p+ ⊂ G p (p).
However, G p (p) ⊂ D 1,p+ . For the sake of simplicity, assume that p = 2. We aim at constructing a random variable Z which is in G 2 (2) but not in D 1,2+ . Fix some η, µ such that 0 < η < µ < e -1 and such that

• x -→ 1/(x| log(x)| 8 ) is decreasing on (0, µ + η). (3.3) • x -→ x log(x) 6 log x log(x) 6 = - x log(x) 6 log x log(x) 6 is increasing on (0, η]. (3.4) • x -→ x(log(x)) -6 is increasing on (0, η]. (3.5)
Notice that (3.3) implies that the map x → 1/(x| log(x)| i ) is decreasing on (0, µ + η) for i ∈ {5, 6, 7, 8}. Set Z := f (W 1 ) where f : R -→ R is defined by

f (x) := F (x)1 x∈(0,µ] + G(x)1 x>µ , (3.6)
where G is a smooth function on R * + with compact support such that f is continuously differentiable on R * + , and where F : R * + -→ R is continuous and continuously differentiable on (0, µ] and defined by

F (x) = √ x log(x) 3 , F ′ (x) = 1 2 √ x log(x) 3 - 1 √ x log(x) 4 .
Then, using Bertrand's integrals Theory, one can easily prove that

Z ∈ D 1,2 but Z / ∈ D 1,2+ . Nevertheless, one can show that Z ∈ G 2 (2). Indeed, from [8, Theorem 3], for any h ∈ H X ε := Z • τ εh -Z ε P0 -→ ε→0 ∇Z, h H . Hence, Z belongs to G 2 (2) if and only if (|X ε | 2 ) ε∈(0,η/µ
) is uniformly integrable for every h in H. According to the de la Vallée-Poussin Criterion (see e.g. the remark immediately after Theorem 1.1 in [START_REF] Hu | A note on the de la Vallée Poussin criterion for uniform integrability[END_REF]) if there exists a non-negative measurable map ψ from R + into R + such that for any

h 1 = 0 lim x→+∞ ψ(x) x = +∞, sup ε∈(0,η/|h1|) E ψ(|X ε | 2 ) < +∞, then (|X ε | 2 ) ε∈(0,η/|h1|) is uniformly integrable for all h ∈ H. Take ψ(x) := x |log(x)| 1 x>0 defined on R + . Notice that if h 1 = 0 then X ε = 0 so, (|X ε | 2 ) ε∈(0,η/|h1|
) is uniformly integrable. We now distinguish two cases.

Case h 1 > 0. We have

I := sup ε∈(0,η/h1) E ψ(|X ε | 2 ) = I +∞ µ + I µ 0 + I 0 -∞ , where
I +∞ µ := sup ε∈(0,η/h1) +∞ µ ε -1 εh1 0 f ′ (x + s)ds 2 log   ε -1 εh1 0 f ′ (x + s)ds 2   φ(x)dx I µ 0 := sup ε∈(0,η/h1) µ 0 ε -1 εh1 0 f ′ (x + s)ds 2 log   ε -1 εh1 0 f ′ (x + s)ds 2   φ(x)dx I 0 -∞ := sup ε∈(0,η/h1) 0 -εh1 f (x + εh 1 ) -f (x) ε 2 log f (x + εh 1 ) -f (x) ε 2 φ(x)dx.
From the definition (3.6) of f , we obtain

I +∞ µ = sup ε∈(0,η/h1) +∞ µ ε -1 εh1 0 G ′ (x + s)ds 2 log   ε -1 εh1 0 G ′ (x + s)ds 2   φ(x)dx I µ 0 = sup ε∈(0,η/h1) µ 0 ε -1 εh1 0 f ′ (x + t)dt 2 log   ε -1 εh1 0 f ′ (x + t)dt 2   φ(x)dx I 0 -∞ = sup ε∈(0,η/h1) 0 -εh1 f (x + εh 1 ) ε 2 log f (x + εh 1 ) ε 2 φ(x)dx.
According to the mean value theorem, for any ε ∈ (0, η/h 1 ) and for any x ∈ R + , there exist s x ε ∈ (0, εh 1 ) and t x ε ∈ (0, εh 1 ) such that

ε -1 εh1 0 G ′ (x + s)ds = G ′ (x + s x ε ) and ε -1 εh1 0 f ′ (x + t)dt = f ′ (x + t x ε ).
Hence, from the definition of G (we remind the reader that G has compact support), we deduce that I +∞ µ < +∞. Now, we denote by i(ε, µ) the subset of (0, µ) defined by i(ε, µ) := {x ∈ (0, µ)|x + t x ε ≤ µ}. There exists a constant C > 0 which may vary from line to line such that

I µ 0 = sup ε∈(0,η/h1) µ 0 |f ′ (x + t x ε )| 2 log |f ′ (x + t x ε )| 2 φ(x)dx ≤ C sup ε∈(0,η/h1) i(ε,µ) 1 (x + t x ε )| log(x + t x ε )| 6 log log(x + t x ε ) -2 2 √ x + t x ε log(x + t x ε ) 4 2 φ(x)dx + sup ε∈(0,η/h1) i(ε,µ) 1 (x + t x ε )| log(x + t x ε )| 8 log log(x + t x ε ) -2 2 √ x + t x ε log(x + t x ε ) 4 2 φ(x)dx + sup ε∈(0,η/h1) (0,µ)\i(ε,µ) |G ′ (x + t x ε )| 2 log |G ′ (x + t x ε )| 2 φ(x)dx = C sup ε∈(0,η/h1) i(ε,µ) 1 (x + t x ε )| log(x + t x ε )| 6 log | log(x + t x ε ) -2| 2 4(x + t x ε ) log(x + t x ε ) 8 φ(x)dx + sup ε∈(0,η/h1) i(ε,µ) 1 (x + t x ε )| log(x + t x ε )| 8 log | log(x + t x ε ) -2| 2 4(x + t x ε ) log(x + t x ε ) 8 φ(x)dx + 1 .
Since µ is smaller than e -1 , we can ensure that for any ε ∈ (0, η/h 1 ) and x ∈ i(ε, µ), we have

|log(|log(x + t x ε )|)| ≤ | log(x + t x ε )|.
Hence, there exists C > 0 such that

I 1 ≤ C sup ε∈(0,η/h1) 8 i=5 i(ε,µ) 1 (x + t x ε )| log(x + t x ε )| i dx.
It is then clear, from the properties of Bertrand's integrals together with the choice of η, µ according to (3.3), that I µ 0 < +∞. Besides,

I 0 -∞ = sup ε∈(0,η/h1) 0 -εh1 f (x + εh 1 ) ε 2 log f (x + εh 1 ) ε 2 φ(x)dx = sup ε∈(0,η/h1) 1 ε 2 εh1 0 x log(x) 6 log x ε 2 log(x) 6 φ(x -εh 1 )dx = I 1 + I 2 ,
where

I 1 = sup ε∈(0,η/h1) 1 ε 2 εh1 0 x log(x) 6 log(ε -2 ) φ(x -εh 1 )dx I 2 = sup ε∈(0,η/h1) 1 ε 2 εh1 0 x log(x) 6 log x log(x) 6 φ(x -εh 1 )dx.
We first show that I 1 is finite. From (3.5) and since η < e -1 < 1 we obtain

I 1 ≤ sup ε∈(0,η/h1) |h 1 | 2 | log(ε -2 )| log(εh 1 ) 6 φ(0) ≤ |h 1 | 2 φ(0) - 2 log(η) 5 + | log(h 1 )| log(η) 6 < +∞.
We now turn to I 2 and we obtain from (3.4) together with (3.3) that

I 2 ≤ sup ε∈(0,η/h1) |h 1 | 2 log(εh 1 ) 6 log εh 1 log(εh 1 ) 6 φ(0) ≤ - 7|h 1 | 2 log(η) 5 φ(0) < +∞, which implies in turn I 0 -∞ < +∞.
Case h 1 < 0. Similarly to the previous case, we have

J := sup ε∈(0,η/|h1|) E ψ(|X ε | 2 ) = J +∞ + J µ + J 0 ,
where From the mean value Theorem, one shows that J +∞ < +∞. Besides, by making the substitution y = x -ε|h 1 | in J 0 we deduce that J 0 ≤ e |η| 2 2 I 0 -∞ < +∞. Now, turn to J µ . According to the mean value theorem, for any ε ∈ (0, η/|h 1 |) and for any x ∈ (ε|h 1 |, µ + ε|h 1 |], there exists t x ε ∈ (0, ε|h 1 |) such that

J +∞ := sup ε∈(0,η/|h1|) +∞ µ+ε|h1| ε -1 ε|h1| 0 f ′ (x -s)ds 2 log   ε -1 ε|h1| 0 f ′ (x -s)ds 2   φ(x)dx J µ := sup ε∈(0,η/|h1|) µ+ε|h1| ε|h1| ε -1 ε|h1| 0 f ′ (x -s)ds 2 log   ε -1 ε|h1| 0 f ′ (x -s)ds 2   φ(x)dx J 0 := sup ε∈(0,η/ε|h1|) ε|h1| 0 f (x -ε|h 1 |) -f (x) ε 2 log f (x -ε|h 1 |) -f (x) ε 2 φ(x)dx.
ε -1 ε|h1| 0 f ′ (x -t)dt = f ′ (x -t x ε ).
So, following the same lines as in the proof of I µ 0 < +∞, we obtain J µ < +∞. Indeed,

J µ = sup ε∈(0,η/|h1|) µ+ε|h1| ε|h1| |f ′ (x -t x ε )| 2 log |f ′ (x -t x ε )| 2 φ(x)dx ≤ sup ε∈(0,η/|h1|) j(ε,µ) |f ′ (x -t x ε )| 2 log |f ′ (x -t x ε )|
1 (x -t x ε )| log(x -t x ε )| i dx .
In addition, (3.3) implies that This concludes the proof.

From the definition ( 3 . 6 )

 36 of f , we obtain

| 2 φ

 2 η/|h1|) (ε|h1|,µ+ε|h1|)\j(ε,µ) |f ′ (x -t x ε )| 2 log |f ′ (x -t x ε )(x)dx, with j(ε, µ) := {x ∈ (ε|h 1 |, µ + ε|h 1 |), x -t x ε ≤ µ}. Hence, J µ ≤ C sup ε∈(0,η/|h1|) j(ε,µ)

J µ ≤ C 1 1

 11 ε|h 1 |)| log(x -ε|h 1 |)| i dx ≤ C 1 + sup ε∈(0,η/h1) ε|h 1 |)| log(x -ε|h 1 |)| i dx = y| log(y)| i dy < +∞.

Acknowledgments

Thibaut Mastrolia is grateful to Région Île-de-France for financial support and acknowledges INSA de Toulouse for its warm hospitality.


Jo Guldi et David Armitage, The History Manifesto

Frédéric Clavert Docteur en histoire contemporaine, Frédéric Clavert a étudié les sciences politiques et l'histoire contemporaine à Strasbourg et à Leeds. Ses recherches s'orientent aujourd'hui vers les relations entre les banquiers centraux et la construction européenne d'une part, sur les sources de l'historien.ne à l'ère numérique d'autre part. Il a été titulaire d'une bourse du DAAD et a obtenu l'aide à la publication du Prix Pierre Grappin. Après cinq ans comme chercheur (Histoire de l'intégration européenne et Humanités numériques) au Centre virtuel de la connaissance sur l'Europe (Luxembourg), il est désormais ingénieur de recherche pour le LabEx "Ecrire une Histoire Nouvelle de l'Europe"

23/12/2014

Face à de grandes questions qui se posent à l'ère de la mondialisation -qu'elles soient climatiques et environnementales, sociales ou politiquesles historiens Jo Guldi (Brown university) et David Armitage (Harvard) publient un appel aux armes pour tous ceux qui croient au rôle des sciences historiques dans la société :"The History Manifesto is a call to arms to historians and everyone interested in the role of history in contemporary society." ( p. i). Ce livre peut également être compris comme une histoire de la notion de longue durée, de l'article fondateur de Fernand Braudel en 1958 aux signes récents de son retour, en passant par son reflux, des années 1970 aux années 1990/2000. Dans cette histoire de la longue durée, les auteurs tentent de montrer que son reflux s'explique pour des raisons intellectuelles (la micro-histoire) mais aussi sociologiques (la manière dont s'est structuré le champs universitaire). Son retour est, pour les auteurs, lié à la technologie (la big data et les humanités numériques) et au politique. Cette histoire est dialectique: la longue durée d'aujourd'hui n'est pas exactement celle de Fernand Braudel -son contexte intellectuel et technique est très différent, lui donnant une souplesse que son ancêtre n'avait pas, notamment grâce aux différents niveaux de granularité que nous offrent les données historiques aujourd'hui disponibles. Elle offre également une très grande pertinence pour un public plus large, y compris dans le domaine politique.

Les auteurs tentent notamment de rappeler que l'histoire offre à ceux qui veulent penser le futur trois approches: « a sense of destiny and free will, counterfactual thinking, and thinking about utopias. » (p. 10). Ces trois approches historiques éloignent l'histoire des modèles de lois naturelles d'autres sciences sociales (notamment l'anthropologie) ou économiques. La thèse centrale des auteurs est ainsi que ces approches sont un remède crucial pour une société paralysée par la pensée de court-terme, car l'histoire permet d'élargir la manière dont nous pensons nos futurs possibles, en se reposant sur l'analyse des phénomènes climatiques, sociaux, politiques (et autres) dans une perspective de très long terme.

Pour des raisons intellectuelles, mais aussi -et peut-être surtout -sociologiques (la professionnalisation des sciences historiques: la course à la publication, la nécessité de montrer que l'on maîtrise des compétences professionnelles très précises), la longue durée à laissé la place à une pensée de court-terme et, parallèlement, les historiens se sont retirés du débat public, laissant place à d'autres scientifiques, plus intéressés par les analyses de courtterme. Mais les évolutions technologiques récentes favorisent un retour à la longue durée. La constitution de volumes massifs de données (big data) forceront de plus en plus ceux qui veulent analyser à s'interroger sur la provenance de leurs données, leur histoire et leur pertinence pour le futur. Ainsi, affirment les auteurs: "This change in the life of data may determine a major shift for the university of the future, where historical thinkers will have an increasingly important role to play as the arbiters of big data." (p.12).

Après cette introduction de grande ampleur, qui donne un aperçu global de l'argumentaire des auteurs, le livre est structuré en quatre chapitres développant leur analyse. Le premier chapitre décrit l'émergence de la longue durée -concept-clé du livre et utilisé en français dans ce texte anglais -, sa retraite face au court-termisme (chapitre 2) avant de s'interroger sur l'interaction entre ces deux temporalités (chapitre 3) puis d'argumenter que les avancées techniques et la collecte de volumes massifs de données permettent, aujourd'hui, d'avoir une vision de longue durée tout en disposant d'une granularité suffisamment fine pour la combiner avec des approches plus proches des gens (chapitre 4).

Sans rentrer dans le détail de ces chapitres, nous donneront ici un exemple éclairant. Les auteurs détaillent, dans le chapitre 3 notamment, l'émergence de l'inquiétude climatique. La question centrale qu'ils posent aux climatologues, finalement, n'est pas celle de l'origine humaine du réchauffement climatique -les travaux scientifiques la prouve -mais celle de leur discours qui, depuis les années 1970, reste catastrophiste. Or, argumentent Guldi et Armitage, au vu de l'évolution des recherches sur le climat, le discours qui les entourent aurait dû aussi changer. C'est dans ce genre de cas que les historiens, selon les auteurs, devraient avoir un rôle: insérer une problématique actuelle dans une vision historique de long terme, dépassant les exemples très connus et médiatiques de civilisations -notamment les Mayas -qui ont disparu ou été affaiblies par leur incapacité à gérer une crise environnementale et rappelant aussi que certaines civilisations ont su survivre et s'adapter à ce type de crise, y compris l'Occident médiéval.

The History Manifesto recoupe, finalement, des arguments d'autres publications récentes -nous pensons notamment au Big Data in history de Patrick Manning 1 dont nous avons publié une critique 2 -en les réinsérant dans l'historiographie du second XXe siècle et dans un contexte où l'historien.ne a un rôle très important à jouer, par sa démarche critique, 1 Manning Patrick, Big data in history, New York, Palgrave Pivot, 2013.

2 Clavert Frédéric, « Patrick Manning, Big Data in History », Lectures, 28 février 2014. qu'il.elle peut appliquer d'une part aux discours court-termistes, d'autre part aux données massives, à leur analyse et aux conclusions que l'on peut en donner.

Depuis sa publication, The History Manifesto s'est attiré une certaine critique, y compris dans sa tentative de voir les historien.ne.s comme pouvant parler au pouvoir politique. Nous nous cantonnerons ici à mentionner certaines de ces critiques.

Tim Hitchcock (University of Sussex)3 estime que ne voir l'histoire que comme une science sociale, comme un macroscope, porte le risque d'oublier que les sciences humaines peuvent appréhender la complexité des phénomènes humains par la lecture proche des sources: "The real power of work in this tradition, lay in its ability to deploy emotive and powerful detail in the context of the largest of political and economic stories". Claire Lemercier (Centre de Sociologie des Organisations, IEP de Paris) a, entre autres, contesté, avec des arguments solides, l'utilisation des chiffres par les deux auteurs de ce manifeste, montrant que l'on ne pouvait pas parler de retraite de la longue durée après 1970 [START_REF] Lemercier | La longue durée : une histoire sans histoire ?[END_REF] . Elle a également pointé les faiblesses de l'ouvrage dans le domaine des archives ouvertes et de la diffusion des recherches historiques [START_REF] Lemercier | L'histoire et ses publics : une question d'historiographie ou de modes de diffusion ?[END_REF] . Commentant l'un des deux billets de Claire Lemercier, Serge Noiret (Institut Européen de Florence) 6 s'est étonné de l'impasse faite par les deux auteurs sur la notion de public history. Paul Bertrand (Université catholique de Louvain), dans trois publications sur son carnet de recherche7, n'est pas moins critique notamment sur la manière dont les auteurs envisagent les archives (négativement connotées dans le manifeste) et les données (considérées comme plus positives). Gabriel Galvez-Behar 8 (Lille

Hitchcock Tim,« Big Data, Small

Data and Meaning », Historionycs, http://historyonics.blogspot.fr/2014/11/big-data-small-data-and-meaning_9.html, 11 septembre 2014 (consulté