Basura Fernando

Amaury Habrard

Marc Sebban

Tinne Tuytelaars

Unsupervised Visual Domain Adaptation Using Subspace Alignment

In this paper, we introduce a new domain adaptation (DA) algorithm where the source and target domains are represented by subspaces described by eigenvectors. In this context, our method seeks a domain adaptation solution by learning a mapping function which aligns the source subspace with the target one. We show that the solution of the corresponding optimization problem can be obtained in a simple closed form, leading to an extremely fast algorithm. We use a theoretical result to tune the unique hyperparameter corresponding to the size of the subspaces. We run our method on various datasets and show that, despite its intrinsic simplicity, it outperforms state of the art DA methods.

Introduction

In classification, it is typically assumed that the labeled training data comes from the same distribution as that of the test data. However, many real world applications, especially in computer vision, challenge this assumption (see, e.g., the study on dataset bias in [START_REF] Torralba | Unbiased look at dataset bias[END_REF]). In this context, the learner must take special care during the learning process to infer models that adapt well to the test data they are deployed on. For example, images collected from a web camera are different from those taken with a DSLR camera. A classifier that would be trained on the former would likely fail to classify the latter correctly if applied without adaptation.

We refer to these different but related marginal distributions as domains. In order to build robust classifiers, it is necessary to take into account the shift between these two distributions. This issue is known as domain adaptation (DA). DA typically aims at making use of information coming from both source and target domains during the learning process to adapt automatically. We usually differentiate two different scenarios: [START_REF] Ben-David | Analysis of representations for adaptation[END_REF] the unsupervised setting where the training data consists of labeled source data and unlabeled target examples (see [START_REF] Margolis | A literature review of domain adaptation with unlabeled data[END_REF] for a survey); and (2) the semisupervised case where a large number of labels is available for the source domain and only a few labels are provided for the target domain. In this paper, we focus on the most difficult, unsupervised scenario.

As illustrated by recent results [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF][START_REF] Gopalan | Domain adaptation for object recognition: An unsupervised approach[END_REF], subspace based domain adaptation seems a promising approach to tackle unsupervised visual DA problems. In [START_REF] Gopalan | Domain adaptation for object recognition: An unsupervised approach[END_REF], Gopalan et al. generate intermediate representations in the form of subspaces along the geodesic path connecting the source subspace and the target subspace on the Grassmann manifold. Then, the source data are projected onto these subspaces and a classifier is learned. In [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF], Gong et al. propose a geodesic flow kernel which aims to model incremental changes between the source and target domains. In both papers, a set of intermediate subspaces is used to model the shift between the two distributions.

In this paper, we also make use of subspaces (composed of 𝑑 eigenvectors induced by a PCA), one for each domain. However, following the theoretical recommendations of [START_REF] Ben-David | Analysis of representations for adaptation[END_REF], we rather suggest to directly reduce the discrepancy between the two domains by moving closer the source and target subspaces. This is achieved by optimizing a mapping function that transforms the source subspace into the target one. From this simple idea, we design a new DA approach based on subspace alignment. The advantage of our method is two-fold: (1) by adapting the bases of the subspaces, our approach is global. This allows us to induce robust classifiers not subject to local perturbations; and (2) by aligning the source and target subspaces, our method is intrinsically regularized: we do not need to tune regularization parameters in the objective as imposed by a lot of optimizationbased DA methods.

Our subspace alignment is achieved by optimizing a mapping function which takes the form of a transformation matrix 𝑀 . We show that the optimal solution corresponds in fact to the covariance matrix between the source and target eigenvectors. From this transformation matrix, we derive a similarity function 𝑆𝑖𝑚(y S , y T ) to compare a source data y S with a target example y T . Thanks to a consistency theorem, we prove that 𝑆𝑖𝑚(y S , y T ), which captures the idiosyncrasies of the training data, converges uniformly to its true value. We show that we can make use of this theoretical result to tune the hyperparameter 𝑑, that tends to make our method parameter-free. The similarity function 𝑆𝑖𝑚(y S , y T ) can be used directly in a nearest neighbour classifier. Alternatively, we can also learn a global classifier such as support vector machines (SVM) on the source data after mapping them onto the target subspace.

As suggested by Ben-David et al. [START_REF] Ben-David | Analysis of representations for adaptation[END_REF], a reduction of the divergence between the two domains is required to adapt well. In other words, the ability of a DA algorithm to actually reduce that discrepancy is a good indication of its performance. A usual way to estimate the divergence consists in learning a linear classifier ℎ to discriminate between source and target instances, respectively pseudolabeled with 0 and 1. In this context, the higher the error of ℎ, the smaller the divergence. While such a strategy gives us some insight about the ability for a global learning algorithm (e.g. SVM) to be efficient on both domains, it does not seem to be suited to deal with local classifiers, such as the 𝑘-nearest neighbors. To overcome this limitation, we introduce a new empirical divergence specifically dedicated to local classifiers. We show through our experimental results that our DA method allows us to drastically reduce both empirical divergences.

The rest of the paper is organized as follows. We present the related work in section 2. Section 3 is devoted to the presentation of our DA method and the consistency theorem on the similarity measure deduced from the learned mapping function. In section 4, a comparative study is performed on various datasets. We conclude in section 5.

Related work

DA has been widely studied in the literature and is of great importance in many areas such as natural language processing [START_REF] Blitzer | Domain adaptation with structural correspondence learning[END_REF] or computer vision [START_REF] Torralba | Unbiased look at dataset bias[END_REF]. In this paper, we focus on the unsupervised domain adaptation setting that is well suited to vision problems since it does not require any labeling information from the target domain. This setting makes the problem very challenging and an important issue is to find out a relationship between the two domains. A common approach is to assume the existence of a domain invariant feature space and the objective of a large range of DA work is to approximate this space.

A classical strategy related to our work consists of learning a new domain-invariant feature representation by looking for a new projection space. PCA based DA methods have then been naturally investigated [START_REF] Chen | Extracting discriminative concepts for domain adaptation in text mining[END_REF][START_REF] Pan | Transfer learning via dimensionality reduction[END_REF][START_REF] Pan | Domain adaptation via transfer component analysis[END_REF] in order to find a common latent space where the difference between the marginal distributions of the two domains is minimized with respect to the Maximum Mean Discrepancy (MMD) divergence. Other strategies have been explored as well such as using metric learning approaches [START_REF] Kulis | What you saw is not what you get: Domain adaptation using asymmetric kernel transforms[END_REF][START_REF] Saenko | Adapting visual category models to new domains[END_REF] or canon-ical correlation analysis methods over different views of the data to find a coupled source-target subspace [START_REF] Blitzer | Domain adaptation with coupled subspaces[END_REF] where one assumes the existence of a performing linear classifier on the two domains.

In the structural correspondence learning method [START_REF] Blitzer | Domain adaptation with structural correspondence learning[END_REF], Blitzer et al. propose to create a new feature space by identifying correspondences among features from different domains by modeling their correlations with pivot features. Then, they concatenate source and target data using this feature representation and apply PCA to find a relevant common projection. In [START_REF] Chang | Robust visual domain adaptation with low-rank reconstruction[END_REF], Chang transforms the source data into an intermediate representation such that each transformed source sample can be linearly reconstructed by the target samples. This is however a local approach that may fail to capture the global structure information of the source domain. Moreover it is sensitive to noise and outliers of the source domain that have no correspondence in the target one.

Our method is also related to manifold alignment [START_REF] Wang | Manifold alignment without correspondence[END_REF][START_REF] Wang | Heterogeneous domain adaptation using manifold alignment[END_REF][START_REF] Zhai | Manifold alignment via corresponding projections[END_REF] whose main objective is to align two datasets from two different manifolds such that they can be projected to a common subspace. Most of these methods [START_REF] Wang | Heterogeneous domain adaptation using manifold alignment[END_REF][START_REF] Zhai | Manifold alignment via corresponding projections[END_REF] need correspondences from the manifolds and all of them exploit the local statistical structure of the data.

Recently, subspace based DA has demonstrated good performance in visual DA [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF][START_REF] Gopalan | Domain adaptation for object recognition: An unsupervised approach[END_REF]. These methods share the same principle: first they compute a domain specific d-dimensional subspace for the source data and another one for the target data, independently created by PCA. Then, they project source and target data into intermediate subspaces along the shortest geodesic path connecting the two d-dimensional subspaces on the Grassmann manifold. They actually model the distribution shift by looking for the best intermediate subspaces. These approaches are the closest to ours but, as mentioned in the introduction, it is more appropriate to align the two subspaces directly, instead of computing a large number of intermediate subspaces which can potentially be a costly tuning procedure. The effectiveness of our idea is supported by our experimental results.

As a summary, our approach has the following differences with existing methods:

We exploit the global covariance statistical structure of the two domains during the adaptation process in contrast to the manifold alignment methods that use local statistical structure of the data [START_REF] Wang | Manifold alignment without correspondence[END_REF][START_REF] Wang | Heterogeneous domain adaptation using manifold alignment[END_REF][START_REF] Zhai | Manifold alignment via corresponding projections[END_REF]. We project the source data onto the source subspace and the target data onto the target subspace in contrast to methods that project source data to the target subspace or target data to the source subspace such as [START_REF] Blitzer | Domain adaptation with coupled subspaces[END_REF]. Moreover, we do not project data to a large number of subspaces as in [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF][START_REF] Gopalan | Domain adaptation for object recognition: An unsupervised approach[END_REF]. Our method is totally unsupervised and does not require any target label information like constraints on cross-domain data [START_REF] Kulis | What you saw is not what you get: Domain adaptation using asymmetric kernel transforms[END_REF][START_REF] Saenko | Adapting visual category models to new domains[END_REF] or correspondences from across datasets [START_REF] Wang | Heterogeneous domain adaptation using manifold alignment[END_REF][START_REF] Zhai | Manifold alignment via corresponding projections[END_REF]. We do not apply PCA on cross-domain data like in [START_REF] Chen | Extracting discriminative concepts for domain adaptation in text mining[END_REF][START_REF] Pan | Transfer learning via dimensionality reduction[END_REF][START_REF] Pan | Domain adaptation via transfer component analysis[END_REF] as these approaches exploit only shared features in both domains. In contrast, we make use of the correlated features in both domains. Some of these features can be specific to one domain yet correlated to some other features in the other one allowing us to use both shared and domain specific features. As far as we know, this is the first attempt to use a subspace alignment method in the context of domain adaptation.

DA using unsupervised subspace alignment

In this section, we introduce our new subspace based DA method. We assume that we have a set 𝑆 of labeled data (resp. a set 𝑇 of unlabeled data) both lying in a given 𝐷dimensional space and drawn i.i.d. according to a fixed but unknown source (resp. target) distribution 𝒟 𝑆 (resp. 𝒟 𝑇 ). We denote the transpose operation by ′ .

In section 3.1, we explain how to generate the source and target subspaces of size 𝑑. Then, we present our DA method in section 3.2 which consists in learning a transformation matrix 𝑀 that maps the source subspace to the target one. From 𝑀 , we design a similarity function for which we derive a consistency theorem in section 3.3. This upper bound gives us some insight about how to tune the parameter 𝑑.

Subspace generation

Even though both the source and target data lie in the same 𝐷-dimensional space, they have been drawn according to different marginal distributions. Consequently, rather than working on the original data themselves, we suggest to handle more robust representations of the source and target domains and to learn the shift between these two domains. First, we transform every source and target data in the form of a 𝐷-dimensional z-normalized vector (i.e. of zero mean and unit standard deviation). Then, using PCA, we select for each domain 𝑑 eigenvectors corresponding to the 𝑑 largest eigenvalues. These eigenvectors are used as bases of the source and target subspaces, respectively denoted by 𝑋 𝑆 and 𝑋 𝑇 (𝑋 𝑆 , 𝑋 𝑇 ∈ ℝ 𝐷×𝑑 ). Note that 𝑋 ′ 𝑆 and 𝑋 ′ 𝑇 are orthonormal (thus, 𝑋 ′ 𝑆 𝑋 𝑆 = 𝐼 𝑑 and 𝑋 ′ 𝑇 𝑋 𝑇 = 𝐼 𝑑 where 𝐼 𝑑 is the identity matrix of size 𝑑). In the following, 𝑋 𝑆 and 𝑋 𝑇 are used to learn the shift between the two domains.

Domain adaptation with subspace alignment

As presented in section 2, two main strategies are used in subspace based DA methods. The first one consists in projecting both source and target data to a common shared subspace. However, since this only exploits shared features in both domains, it is not always optimal. The second one aims to build a (potentially large) set of intermediate representations. Beyond the fact that such a strategy can be costly, projecting the data to an intermediate common shared sub-space may lead to information loss in both source and target domains.

In our method, we suggest to project each source (y S ) and target (y T ) data (where y S , y T ∈ ℝ 1×𝐷 ) to its respective subspace 𝑋 𝑆 and 𝑋 𝑇 by the operations y S 𝑋 𝑆 and y T 𝑋 𝑇 , respectively. Then, we learn a linear transformation function that align the source subspace coordinate system to the target one. This step allows us to directly compare source and target samples in their respective subspaces without unnecessary data projections. To achieve this task, we use a subspace alignment approach. We align basis vectors by using a transformation matrix 𝑀 from 𝑋 𝑆 to 𝑋 𝑇 . 𝑀 is learned by minimizing the following Bregman matrix divergence:

𝐹 (𝑀 ) = ||𝑋 𝑆 𝑀 -𝑋 𝑇 || 2 𝐹 (1)
𝑀 * = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑀 (𝐹 (𝑀 )) (2)
where ||.|| 2 𝐹 is the Frobenius norm. Since 𝑋 𝑆 and 𝑋 𝑇 are generated from the first 𝑑 eigenvectors, it turns out that they tend to be intrinsically regularized. Therefore, we do not add a regularization term in the equation 1. It is thus possible to obtain a simple solution of equation 2 in closed form. Because the Frobenius norm is invariant to orthonormal operations, we can re-write equation 1 as follows:

𝐹 (𝑀 ) = ||𝑋 ′ 𝑆 𝑋 𝑆 𝑀 -𝑋 ′ 𝑆 𝑋 𝑇 || 2 𝐹 = ||𝑀 -𝑋 ′ 𝑆 𝑋 𝑇 || 2 𝐹 . ( 3
)
From this result, we can conclude that the optimal 𝑀 * is obtained as 𝑀 * = 𝑋 ′ 𝑆 𝑋 𝑇 . This implies that the new coordinate system is equivalent to 𝑋 𝑎 = 𝑋 𝑆 𝑋 ′ 𝑆 𝑋 𝑇 . We call 𝑋 𝑎 the target aligned source coordinate system. It is worth noting that if the source and target domains are the same, then 𝑋 𝑆 = 𝑋 𝑇 and 𝑀 * is the identity matrix.

Matrix 𝑀 transforms the source subspace coordinate system into the target subspace coordinate system by aligning the source basis vectors with the target ones. If a source basis vector is orthogonal to all target basis vectors, it is ignored. On the other hand, a high weight is given to a source basis vector that is well aligned with the target basis vectors.

In order to compare a source data y S with a target data y T , one needs a similarity function 𝑆𝑖𝑚(y S , y T ). Projecting y S and y T in their respective subspace 𝑋 𝑆 and 𝑋 𝑇 and applying the optimal transformation matrix 𝑀 * , we can define 𝑆𝑖𝑚(y S , y T ) as follows:

𝑆𝑖𝑚(y S , y T ) = (y S 𝑋 𝑆 𝑀 * )(y T 𝑋 𝑇 ) ′ = y S 𝑋 𝑆 𝑀 * 𝑋 ′ 𝑇 y T ′ = y S 𝐴y T ′ , ( 4
)
where 𝐴 = 𝑋 𝑆 𝑋 ′ 𝑆 𝑋 𝑇 𝑋 ′ 𝑇 . Note that Eq. 4 looks like a generalized dot product (even though 𝐴 is not necessarily We use 𝑆𝑖𝑚(y S , y T ) directly to perform a 𝑘-nearest neighbor classification task. On the other hand, since 𝑆𝑖𝑚(y S , y T ) is not PSD we can not make use of it to learn a SVM directly. As we will see in the experimental section, an alternative solution will consist in (i) projecting the source data via 𝑋 𝑎 into the target aligned source subspace and the target data into the target subspace (using 𝑋 𝑇 ), (ii) learn a SVM from this 𝑑-dimensional space. The pseudocode of our algorithm is presented in Algorithm 1.

Data

Consistency theorem on 𝑆𝑖𝑚(y S , y T )

The unique hyperparameter of our algorithm is the number 𝑑 of eigenvectors. In this section, inspired by concentration inequalities on eigenvectors [START_REF] Zwald | On the convergence of eigenspaces in kernel principal components analysis[END_REF], we derive an upper bound on the similarity function 𝑆𝑖𝑚(y S , y T ). Then, we show that we can make use of this theoretical result to efficiently tune 𝑑.

Let D𝑛 be the covariance matrix of a sample 𝐷 of size 𝑛 drawn i.i.d. from a given distribution and D its expected value over that distribution.

Theorem 1. We start by using a theorem from [START_REF] Zwald | On the convergence of eigenspaces in kernel principal components analysis[END_REF]. Let 𝐵 be s. )) 2

with probability at least 1 -𝛿 we have:

∥𝑋 𝑑 D -𝑋 𝑑 D𝑛 ∥ ≤ 4𝐵 √ 𝑛(𝜆 𝑑 -𝜆 𝑑+1 ) ( 1 + √ ln(1/𝛿) 2
)

.

From the previous theorem, we can derive the following lemma for the deviation between

𝑋 𝑑 D 𝑋 𝑑 D ′ and 𝑋 𝑑 D𝑛 𝑋 𝑑 D𝑛 ′ .
For the sake of simplification, we will use in the following the same notation 𝐷 (resp. 𝐷 𝑛 ) for defining either the sample 𝐷 (resp. 𝐷 𝑛 ) or its covariance matrix D (resp. D𝑛 ).

Lemma

( 1 + √ ln(1/𝛿) 2
)) 2

with probability at least 1 -𝛿 we have:

∥𝑋 𝑑 𝐷 𝑋 𝑑 𝐷 ′ -𝑋 𝑑 𝐷𝑛 𝑋 𝑑 𝐷𝑛 ′ ∥ ≤ 8 √ 𝑑 √ 𝑛 𝐵 (𝜆 𝑑 -𝜆 𝑑+1 ) ( 1 + √ ln(1/𝛿) 2
)
Proof.

∥𝑋 𝑑 𝐷 𝑋 𝑑 𝐷 ′ -𝑋 𝑑 𝐷𝑛 𝑋 𝑑 𝐷𝑛 ′ ∥ =∥𝑋 𝑑 𝐷 𝑋 𝑑 𝐷 ′ -𝑋 𝑑 𝐷 𝑋 𝑑 𝐷𝑛 ′ + 𝑋 𝑑 𝐷 𝑋 𝑑 𝐷𝑛 ′ -𝑋 𝑑 𝐷𝑛 𝑋 𝑑 𝐷𝑛 ′ ∥ ≤∥𝑋 𝑑 𝐷 ∥∥𝑋 𝑑 𝐷 ′ -𝑋 𝑑 𝐷𝑛 ′ ∥ + ∥𝑋 𝑑 𝐷 -𝑋 𝑑 𝐷𝑛 ∥∥𝑋 𝑑 𝐷𝑛 ′ ∥ ≤ 2 √ 𝑑 √ 𝑛 4𝐵 (𝜆 𝑑 -𝜆 𝑑+1 ) ( 1 + √ ln(1/𝛿) 2
)
The last inequality is obtained by the fact that the eigenvectors are normalized and thus ∥𝑋 𝐷 ∥ ≤ √ 𝑑 and application of Theorem 1 twice.

We now give a theorem for the projector of our DA method. ), then we have with probability at least 1 -𝛿

Theorem 2. Let

∥𝑋 𝑑 𝑆 𝑀𝑋 𝑑 𝑇 ′ -𝑋 𝑑 𝑆𝑛 𝑀 𝑛 𝑋 𝑑 𝑇𝑛 ′ ∥ ≤ 8𝑑 3/2 𝐵 ( 1 + √ ln(2/𝛿) 2 ) × ( 1 √ 𝑛 𝑆 (𝜆 𝑆 𝑑 -𝜆 𝑆 𝑑+1 ) + 1 √ 𝑛 𝑇 (𝜆 𝑇 𝑑 -𝜆 𝑇 𝑑+1 )
)

where 𝑀 𝑛 is the solution of the optimization problem of Eq 2 using source and target samples of sizes 𝑛 𝑆 and 𝑛 𝑇 respectively, and 𝑀 is its expected value.

Proof.

∥𝑋 𝑑 𝑆 𝑀𝑋 𝑑 𝑇 ′ -𝑋 𝑑 𝑆𝑛 𝑀𝑛𝑋 𝑑 𝑇𝑛 ′ ∥ = ∥𝑋 𝑑 𝑆 𝑋 𝑑 𝑆 ′ 𝑋 𝑑 𝑇 𝑋 𝑑 𝑇 ′ -𝑋 𝑑 𝑆𝑛 𝑋 𝑑 𝑆𝑛 ′ 𝑋 𝑑 𝑇𝑛 𝑋 𝑑 𝑇𝑛 ′ ∥ =∥𝑋 𝑑 𝑆 𝑋 𝑑 𝑆 ′ 𝑋 𝑑 𝑇 𝑋 𝑑 𝑇 ′ -𝑋 𝑑 𝑆 𝑋 𝑑 𝑆 ′ 𝑋 𝑑 𝑇𝑛 𝑋 𝑑 𝑇𝑛 ′ + 𝑋 𝑑 𝑆 𝑋 𝑑 𝑆 ′ 𝑋 𝑑 𝑇𝑛 𝑋 𝑑 𝑇𝑛 ′ -𝑋 𝑑 𝑆𝑛 𝑋 𝑑 𝑆𝑛 ′ 𝑋 𝑑 𝑇𝑛 𝑋 𝑑 𝑇𝑛 ′ ∥ ≤∥𝑋 𝑑 𝑆 𝑋 𝑑 𝑆 ′ 𝑋 𝑑 𝑇 𝑋 𝑑 𝑇 ′ -𝑋 𝑑 𝑆 𝑋 𝑑 𝑆 ′ 𝑋 𝑑 𝑇𝑛 𝑋 𝑑 𝑇𝑛 ′ ∥+ ∥𝑋 𝑑 𝑆 𝑋 𝑑 𝑆 ′ 𝑋 𝑑 𝑇𝑛 𝑋 𝑑 𝑇𝑛 ′ -𝑋 𝑑 𝑆𝑛 𝑋 𝑑 𝑆𝑛 ′ 𝑋 𝑑 𝑇𝑛 𝑋 𝑑 𝑇𝑛 ′ ∥ ≤∥𝑋 𝑑 𝑆 ∥∥𝑋 𝑑 𝑆 ′ ∥∥𝑋 𝑑 𝑇 𝑋 𝑑 𝑇 ′ -𝑋 𝑑 𝑇𝑛 𝑋 𝑑 𝑇𝑛 ′ ∥+ ∥𝑋 𝑑 𝑆 𝑋 𝑑 𝑆 ′ -𝑋 𝑑 𝑆𝑛 𝑋 𝑑 𝑆𝑛 ′ ∥∥𝑋 𝑑 𝑇𝑛 ∥∥𝑋 𝑑 𝑇𝑛 ′ ∥ ≤8𝑑 3/2 𝐵 ( 1 + √ ln(2/𝛿) 2 ) × ( 1 √ 𝑛𝑆(𝜆 𝑆 𝑑 -𝜆 𝑆 𝑑+1 ) + 1 √ 𝑛𝑇 (𝜆 𝑇 𝑑 -𝜆 𝑇 𝑑+1 )
) .

The first equality is obtained by replacing 𝑀 and 𝑀 𝑛 by their corresponding optimal solutions 𝑋 𝑑 𝑆 𝑋 𝑑 𝑇 ′ and 𝑋 𝑑 𝑆𝑛 𝑋 𝑑 𝑇𝑛 ′ from Eq 3. The last inequality is obtained by applying twice Lemma 1 and bounding the projection operators.

From Theorem 2, we can deduce a bound on the deviation between two successive eigenvalues. We can make use of this bound as a cutting rule for automatically determining the size of the subspaces. Let 𝑛 𝑚𝑖𝑛 = min(𝑛 𝑆 , 𝑛 𝑇 ) and (𝜆 𝑚𝑖𝑛 𝑑 -𝜆 𝑚𝑖𝑛 𝑑+1 ) = min((𝜆 𝑇 𝑑 -𝜆 𝑇 𝑑+1 ), (𝜆 𝑆 𝑑 -𝜆 𝑆 𝑑+1 )) and let 𝛾 > 0 be a given allowed deviation such that:

𝛾 ≥ ( 1 + √ ln 2/𝛿 2
) ( 16𝑑 3/2 𝐵 √ 𝑛 𝑚𝑖𝑛 (𝜆 𝑚𝑖𝑛 𝑑 -𝜆 𝑚𝑖𝑛 𝑑+1 )
)

.

Given a confidence 𝛿 > 0 and a fixed deviation 𝛾 > 0, we can select the maximum dimension 𝑑 𝑚𝑎𝑥 such that:

(𝜆 𝑚𝑖𝑛 𝑑𝑚𝑎𝑥 -𝜆 𝑚𝑖𝑛 𝑑𝑚𝑎𝑥+1 ) ≥ ( 1 + √ ln 2/𝛿 2
) ( 16𝑑 3/2 𝐵 𝛾 √ 𝑛 𝑚𝑖𝑛
) .

(5) For each 𝑑 ∈ {𝑑|1 . . . 𝑑 𝑚𝑎𝑥 }, we then have the guarantee that

∥𝑋 𝑑 𝑆 𝑀𝑋 𝑑 𝑇 ′ -𝑋 𝑑 𝑆𝑛 𝑀 𝑛 𝑋 𝑑 𝑇𝑛 ′ ∥ ≤ 𝛾.
In other words, as long as we select a subspace dimension d such that 𝑑 ≤ 𝑑 𝑚𝑎𝑥 , the solution 𝑀 * is stable and not over-fitting.

Divergence between source and target domains

The pioneer work of Ben-David et al. [START_REF] Ben-David | Analysis of representations for adaptation[END_REF] provides a generalization bound on the target error which depends on the source error and a measure of divergence, called the 𝐻Δ𝐻 divergence, between the source and target distributions 𝒟 𝑆 and 𝒟 𝑇 .

𝜖 𝑇 (ℎ) = 𝜖 𝑆 (ℎ) + 𝑑 𝐻Δ𝐻 (𝒟 𝑆 , 𝒟 𝑇 ) + 𝜆, ( 6
)
where ℎ is a learned hypothesis, 𝜖 𝑇 (ℎ) the generalization target error, 𝜖 𝑆 (ℎ) the generalization source error, and 𝜆 the error of the ideal joint hypothesis on 𝑆 and 𝑇 , which is supposed to be a negligible term if the adaptation is possible. Eq. 6 tells us that to adapt well, one has to learn a hypothesis which works well on 𝑆 while reducing the divergence between 𝒟 𝑆 and 𝒟 𝑇 . To estimate 𝑑 𝐻Δ𝐻 (𝒟 𝑆 , 𝒟 𝑇 ), a usual way consists in learning a linear classifier ℎ to discriminate between source and target instances, respectively pseudolabeled with 0 and 1. In this context, the higher the error of ℎ, the smaller the divergence. While such a strategy gives us some insight about the ability for a global learning algorithm (e.g. SVM) to be efficient on both domains, it does not seem to be suited to deal with local classifiers, such as the 𝑘-nearest neighbors. To overcome this limitation, we introduce a new empirical divergence specifically dedicated to local classifiers. Based on the recommendations of [START_REF] Ben-David | Domain adaptation-can quantity compensate for quality[END_REF], we propose a discrepancy measure to estimate the local density of a target point w.r.t. a given source point. This discrepancy, called Target density around source TDAS counts how many target points can be found on average within a 𝜖 neighborhood of a source point. More formally:

𝑇 𝐷𝐴𝑆 = 1 𝑛 𝑆 ∑ ∀y S |{y T |𝑆𝑖𝑚(y S , y T ) ≥ 𝜖}|. ( 7
)
Note that TDAS is associated with similarity measure 𝑆𝑖𝑚(y S , y T ) = y S 𝐴y T ′ where 𝐴 is the learned metric. As we will see in the next section, TDAS can be used to evaluate the effectiveness of a DA method under the covariate shift assumption and probabilistic Lipschitzness assumption [START_REF] Ben-David | Domain adaptation-can quantity compensate for quality[END_REF]. The larger the TDAS, the better the DA method.

Experiments

We evaluate our method in the context of object recognition using a standard dataset and protocol for evaluating visual domain adaptation methods as in [START_REF] Chang | Robust visual domain adaptation with low-rank reconstruction[END_REF][START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF][START_REF] Gopalan | Domain adaptation for object recognition: An unsupervised approach[END_REF][START_REF] Kulis | What you saw is not what you get: Domain adaptation using asymmetric kernel transforms[END_REF][START_REF] Saenko | Adapting visual category models to new domains[END_REF]. In addition, we also evaluate our method using various other image classification datasets.

DA datasets and data preparation

We provide three series of experiments on different datasets. In the first series, we use the Office dataset [START_REF] Saenko | Adapting visual category models to new domains[END_REF] and Caltech10 [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF] dataset that contain four domains altogether to evaluate all DA methods. The Office dataset consists of images from web-cam (denoted by W), DSLR images (denoted by D) and Amazon images (denoted by A). The Caltech10 images are denoted by C. We follow the same setup as in [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF]. We use each source of images as a domain, consequently we get four domains (A, C, D and W) leading to 12 DA problems. We denote a DA problem by the notation 𝑆 → 𝑇 . We use the image representations provided by [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF] for Office and Caltech10 datasets (SURF features encoded with a visual dictionary of 800 words). We follow the standard protocol of [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF][START_REF] Gopalan | Domain adaptation for object recognition: An unsupervised approach[END_REF][START_REF] Kulis | What you saw is not what you get: Domain adaptation using asymmetric kernel transforms[END_REF][START_REF] Saenko | Adapting visual category models to new domains[END_REF] for generating the source and target samples 1 .

In a second series, we evaluate the effectiveness of our DA method using other datasets, namely ImageNet (I), La-belMe (L) and Caltech-256 (C). In this setting we consider each dataset as a domain. We select five common objects (bird, car, chair, dog and person) for all three datasets leading to a total of 7719 images. We extract dense SIFT features and create a bag-of-words dictionary of 256 words using kmeans. Afterwards, we use LLC encoding and a spatial pyramid (2 × 2 quadrants + 3 × 1 horizontal + 1 full image) to obtain a 2048 dimensional image representation (similar data preparation as in [START_REF] Khosla | Undoing the damage of dataset bias[END_REF]).

In the last series, we evaluate the effectiveness of our DA method using larger datasets, namely PASCAL-VOC-2007 and ImageNet. We select all the classes of PASCAL-VOC-2007. The objective here is to classify PASCAL-VOC-2007 test images using classifiers that are built from the ImageNet dataset. To prepare the data, we extract dense SIFT features and create a bag-of-words dictionary of 256 using only Im-ageNet images. Afterwards, we use LLC encoding and spatial pyramids (2×2 + 3×1 + 1) to obtain a 2048 dimensional image representation.

Experimental setup

We compare our subspace DA approach with two other DA methods and three baselines. Each of these methods defines a new representation space and our goal is to compare the performance of a 1-Nearest-Neighbor (NN) classifier and a SVM classifier on DA problems in the subspace found.

We consider the DA methods Geodesic Flow Kernel (GFK [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF]) and Geodesic Flow Sampling (GFS [START_REF] Gopalan | Domain adaptation for object recognition: An unsupervised approach[END_REF]). They have indeed demonstrated state of the art performances achieving better results than metric learning methods [START_REF] Saenko | Adapting visual category models to new domains[END_REF] and better than those reported by Chang's method in [START_REF] Chang | Robust visual domain adaptation with low-rank reconstruction[END_REF]. Moreover, these methods are the closest to our approach. We also report results obtained by the following three baselines: Baseline 1: where we use the projection defined by the PCA subspace 𝑋 𝑆 built from the source domain to project both source and target data and work in the resulting representation. Baseline 2: where we use similarly the projection defined by the PCA subspace 𝑋 𝑇 built from the target domain. No adaptation NA: where no projection is made, we use the original input space without learning a new representation.

For each method, we compare the performance of a 1-Nearest-Neighbor (NN) classifier and of a SVM classifier 1 See supplementary material section 1.1 for the experimental details and additional results.

(with C parameter set to the mean similarity value obtained from the training set) in the subspace defined by each method. For each source-target DA problem in the first two series of experiments, we evaluate the accuracy of each method on the target domain over 20 random trials. For each trial, we consider an unsupervised DA setting where we randomly sample labeled data in the source domain as training data and unlabeled data in the target domain as testing examples. In the last series involving the PASCAL-VOC dataset, we rather evaluate the approaches by measuring the mean average precision over target data using SVM.

We have also compared the behavior of the approaches in a semi-supervised scenario by adding 3 labelled target examples to the training set for Office+Caltech10 series and 50 for the PASCAL-VOC series. This can be found in the supplementary material.

Selecting the optimal dimensionality

In this section, we present our procedure for selecting the space dimensionality d in the context of our method. The same dimensionality is used for Baseline1 and Baseline2. For GFK and GFS we follow the published procedures to obtain optimal results as presented in [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF]. First, we perform a PCA on the two domains and compute the deviation 𝜆 𝑚𝑖𝑛 𝑑 -𝜆 𝑚𝑖𝑛 𝑑+1 for all possible 𝑑 values. Then, using the theoretical bound of Eq: 5, we can estimate a 𝑑 𝑚𝑎𝑥 << 𝐷 that provides a stable solution with fixed deviation 𝛾 > 0 for a given confidence 𝛿 > 0. Afterwards, we consider the subspaces of dimensionality from 𝑑 = 1 to 𝑑 𝑚𝑎𝑥 and select the best 𝑑 * that minimizes the classification error using a 2 fold cross-validation over the labelled source data. This procedure is founded by the theoretical result of Ben-David et al. of Eq 6 where the idea is to try to move the domain distribution closer while maintaining a good accuracy on the source domain. As an illustration, the best dimensions for the Office dataset vary between 10 -50. For example, for the DA problem W → C, taking 𝛾 = 10 5 and 𝛿 = 0.1, we obtain 𝑑 𝑚𝑎𝑥 = 22 (see Figure 2) and by cross validation we found that the optimal dimension is 𝑑 * = 20.

Evaluating DA with divergence measures

Here, we propose to evaluate the capability of our method to move the domain distributions closer according to the measures presented in Section 3.4: the TDAS adapted to NN classification where a high value indicates a better distribution closeness and the 𝐻Δ𝐻 using a SVM where a value close to 50 indicates close distributions. We compute these discrepancy measures for the 12 DA problems coming from the Office and Caltech datasets and report the mean values over the 12 problems for each method in Table 1. We can remark that our approach reduces significantly the discrepancy between the source and target domains com- pared to the other baselines (highest TDAS value and lowest 𝐻Δ𝐻 measure). Both GFK and our method have lower 𝐻Δ𝐻 values meaning that these methods are more likely to perform well 2 .

Classification Results

Visual domain adaptation performance with Office/Caltech10 datasets: In this experiment we evaluate the different methods using Office [START_REF] Saenko | Adapting visual category models to new domains[END_REF]/Caltech10 [START_REF] Gopalan | Domain adaptation for object recognition: An unsupervised approach[END_REF] datasets which consist of four domains (A, C, D and W). The results for the 12 DA problems in the unsupervised setting using a NN classifier are shown in Table 2. In 9 out of the 12 DA problems our method outperforms the other ones. The results obtained in the semi-supervised DA setting (see supplementary material) confirm this behavior. Here our method outperforms the others in 10 DA problems.

The results obtained with a SVM classifier in the unsupervised DA case are shown in Table 3. Our method outperforms all the other methods in 11 DA problems. These results indicate that our method works better than other DA methods not only for NN-like local classifiers but also with more global SVM classifiers. Domain adaptation on ImageNet, LabelMe and Caltech-256 datasets : Results obtained for unsupervised DA using NN classifiers are shown in Table 4. First, we can remark that all the other DA methods achieve poor accuracy when LabelMe images are used as the source domain, while our method seems to adapt the source to the target reasonably well. On average, our method significantly outperforms all other DA methods.

A visual example where we classify ImageNet images using Caltech-256 images is shown in Figure 1. The nearest neighbor coming from Caltech-256 corresponds to the same class, even though the appearance of images are very different from the two datasets.

In Table 5 we report results using a SVM classifier for the unsupervised DA setting. In this case our method outperforms all other DA methods, confirming the good behavior of our approach.

Classifying PASCAL-VOC-2007 images using classifiers built on ImageNet : In this experiment, we compare the average precision obtained on PASCAL-VOC-2007 by a SVM classifier in both unsupervised and semi-supervised DA settings. We use ImageNet as the source domain and PASCAL-VOC-2007 as the target domain. The results are shown in Figure 3 for the unsupervised case and in the sup- plementary material for the semi-supervised one.

Our method achieves the best results for all the categories in both settings and outperforms all the methods on average. The semi-supervised DA seems to improve unsupervised DA by 10% (relative) in mAP. In the unsupervised DA setting, GFK improves by 7% in mAP over no adaptation while our method improves by 27% in mAP over GFK. In the semi-supervised setting our method improves by 13% in mAP over GFK and by 46% over no adaptation.

Conclusion

We present a new visual domain adaptation method using subspace alignment. In this method, we create subspaces for both source and target domains and learn a linear mapping that aligns the source subspace with the target subspace. This allows us to compare the source domain data directly with the target domain data and to build classifiers on source data and apply them on the target domain. We demonstrate excellent performance on several image classification datasets such as Office dataset, Caltech, ImageNet, LabelMe and Pascal-VOC. We show that our method outperforms state of the art domain adaptation methods using both SVM and nearest neighbour classifiers. We experimentally show that our method can be used on tasks such as labelling PASCAL-VOC images using ImageNet dataset for training. Due to its simplicity and theoretically founded stability, we believe that our method has the potential to be applied on large datasets consisting of millions of images. As future work we plan to extend our domain adaptation method to large image retrieval and on the fly learning of classifiers.

Figure 1 .

 1 Figure 1. Classifying ImageNet images using Caltech-256 images as the source domain. In the first row, we show an ImageNet query image. In the second row, the nearest neighbour image selected by our method is shown.

  t. for any vector x, ∥x∥ ≤ 𝐵, let 𝑋 𝑑 D and 𝑋 𝑑 D𝑛 be the orthogonal projectors of the subspaces spanned by the first d eigenvectors of D and D𝑛 . Let 𝜆 1 > 𝜆 2 > ... > 𝜆 𝑑 > 𝜆 𝑑+1 ≥ 0 be the first 𝑑 + 1 eigenvalues of D, then for any 𝑛 ≥ ( 4𝐵 (𝜆 𝑑 -𝜆 𝑑+1 )

Figure 2 .

 2 Figure 2. Finding a stable solution and a subspace dimensionality using the consistency theorem. Method NA Baseline 1 Baseline 2GFK OUR TDAS 1.25 3.34 2.74 2.84 4.26 HΔH 98.1 99.0 99.0 74.3 53.2 Table 1. Several distribution discrepancy measures averaged over 12 DA problems using Office dataset.

  Table 5. Recognition accuracy with unsupervised DA with SVM classifier (ImageNet (I), LabelMe (L) and Caltech-256 (C)).

Figure 3 .

 3 Figure 3. Train on ImageNet and classify PASCAL-VOC-2007 images using unsupervised DA with SVM.

:

  Source data 𝑆, Target data 𝑇 , Source labels 𝐿 𝑆 , Subspace dimension 𝑑 Result: Predicted target labels 𝐿 𝑇 𝑋 𝑆 ← 𝑃 𝐶𝐴(𝑆, 𝑑) ; 𝑋 𝑇 ← 𝑃 𝐶𝐴(𝑇, 𝑑) ; 𝑋 𝑎 ← 𝑋 𝑆 𝑋 ′ 𝑆 𝑋 𝑇 ; 𝑆 𝑎 = 𝑆𝑋 𝑎 ; 𝑇 𝑇 = 𝑇 𝑋 𝑇 ; 𝐿 𝑇 ← 𝐶𝑙𝑎𝑠𝑠𝑖𝑓 𝑖𝑒𝑟(𝑆 𝑎 , 𝑇 𝑇 , 𝐿

𝑆 ) ; Algorithm 1: Subspace alignment DA algorithm

Table 1 .

 1 Several distribution discrepancy measures averaged over 12 DA problems using Office dataset.

					GFK OUR
	TDAS	1.25	3.34	2.74	2.84	4.26
	HΔH	98.1	99.0	99.0	74.3	53.2

39.6 80.3 38.6 36.8 83.6Table 2 .

 2 .9 20.8 22.8 24.8 16.4 Baseline 1 38.0 29.8 35.5 30.9 29.6 31.3 Baseline 2 40.5 33.0 38.0 33.3 31.2 31.9 Recognition accuracy with unsupervised DA using a NN classifier (Office dataset + Caltech10).

	Method	C→A	D→A	W→A	A→C	D→C	W→C
	NA 21.5 26GFS [8] 36.9 32	27.5 35.3 29.4 21.7
	GFK [7]	36.9 32.5 31.1 35.6 29.8 27.2
	OUR	39.0 38.0 37.4 35.3 32.4 32.3
	Method	A→D	C→D	W→D	A→W	C→W	D→W
	NA	22.4 21.7 40.5 23.3 20.0 53.0
	Baseline 1 34.6 37.4 71.8 35.1 33.5 74.0
	Baseline 2 34.7 36.4 72.9 36.8 34.4 78.4
	GFS [8]	30.7 32.6 54.3 31.0 30.6 66.0
	GFK [7]	35.2 35.2 70.6 34.4 33.7 74.9
	OUR 37.6 Method C→A	D→A	W→A	A→C	D→C	W→C
	Baseline 1 44.3 36.8 32.9 36.8 29.6 24.9
	Baseline 2 44.5 38.6 34.2 37.3 31.6 28.4
	GFK	44.8 37.9 37.1 38.3 31.4 29.1
	OUR	46.1

2

See section 1.4 of supplementary material for more details.

42.0 39.3 39.9 35.0 31.8

	Method	A→D	C→D	W→D	A→W	C→W	D→W
	Baseline 1 36.1 38.9 73.6 42.5 34.6 75.4
	Baseline 2 32.5 35.3 73.6 37.3 34.2 80.5
	GFK	37.9 36.1 74.6 39.8 34.9 79.1
	OUR	38.8 39.4 77.9 39.6 38.9 82.3

Table 3 .

 3 Recognition accuracy with unsupervised DA using a SVM classifier(Office dataset + Caltech10).

	Method	L→C	L→I	C→L	C→I	I→L	I→C	AVG
	NA	46.0 38.4 29.5 31.3 36.9 45.5 37.9
	Baseline1 24.2 27.2 46.9 41.8 35.7 33.8 34.9
	Baseline2 24.6 27.4 47.0 42.0 35.6 33.8 35.0
	GFK	24.2 26.8 44.9 40.7 35.1 33.8 34.3
	OUR	49.1

41.2 47.0 39.1 39.4 54.5 45.0Table 4 .

 4 Recognition accuracy with unsupervised DA with NN classifier (ImageNet (I), LabelMe (L) and Caltech-256 (C)).

Acknowledgements :

The authors acknowledge the support of the FP7 ERC Starting Grant 240530 COGNIMUND, ANR LAMPADA 09-EMER-007-02 project and PASCAL 2 network of Excellence.


Benoit Coasne
email: coasne@mit.edu

Coralie Weigel

Alain Polian

Mathieu Kint

Jérome Rouquette

Julien Haines

Marie Foret

René Vacher

Benoit Rufflé
email: benoit.ruffle@univ-montp2.fr

Poroelastic Theory Applied to the Adsorption-Induced Deformation of Vitreous Silica

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Introduction

Gas transport in amorphous materials has been the subject of extensive studies over the past 40 years. Pioneering experiments on noble-gas solubility in vitreous silica were conducted up to 0.13 GPa by Shelby. [START_REF] Shelby | Helium Migration in Natural and Synthetic Vitreous Silica[END_REF][START_REF] Shelby | Pressure-Dependence of Helium and Neon Solubility in Vitreous Silica[END_REF] The pressure dependence of He and Ne solubilities suggested that the process could be explained by a statistical thermodynamical approach, i.e. Langmuir adsorption. In this picture, noble-gas solubility is described as a simple physical adsorption process with no chemical bonding to the rigid network. A maximum concentration of He and Ne atoms was then estimated to be 0.1 and 0.06 mole per mole of SiO 2 , respectively. It was argued that these maximal concentrations represent the tail of a log-normal probability distribution function of interstitial voids with a diameter around 0.2 nm and a width of about 0.1 nm, [START_REF] Shackelford | Interstitial Structure of Vitreous Silica[END_REF] a result confirmed by computer-generated models. [START_REF] Chan | Theoretical Study of the Interstice Statistics of the Oxygen Sublattice in Vitreous SiO 2[END_REF] Recent high pressure work is producing a new momentum to the field. [START_REF] Sato | Helium Penetrates into Silica Glass and Reduces its Compressibility[END_REF][START_REF] Shen | Effect of Helium on Structure and Compression Behavior of SiO 2 Glass[END_REF][START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF] Below 10 GPa, hydrostatic pressures P are obtained in diamond-anvil cells using a soft pressure transmitting medium such as a methanol-ethanol mixture (ME4:1). [START_REF] Klotz | Hydrostatic Limits of 11 Pressure Transmitting Media[END_REF] When such experiments are performed in the GPa pressure range for vitreous silica, the volume change of the sample agrees with that expected from its bulk modulus K 35-40 GPa. [START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF][START_REF] Kondo | Non-Linear Pressure-Dependence of the Elastic Moduli of Fused Quartz up to 3 GPa[END_REF][START_REF] Schroeder | Brillouin and Raman Scattering from Glasses under Pressure[END_REF][START_REF] Zha | Acoustic Velocities and Refractive Index of SiO 2 Glass to 57.5 GPa by Brillouin Scattering[END_REF][START_REF] Meade | Frequency Dependent Equation of State of Fused Silica up to 10 GPa[END_REF][START_REF] Tsiok | Logarithmic Kinetics of the Amorphous-Amorphous Transformations in SiO 2 and GeO 2 Glasses under High-Pressure[END_REF] The latter can be obtained by measuring the velocity of both longitudinal (v L ) and transverse ( v T ) acoustic waves, K = ρ(v 2 L -4 3 v 2 T ), where ρ is the mass density of the sample. On the other hand, the compressibility of vitreous silica inferred from the volumetric strain is strongly reduced when He is used as pressurizing medium, [START_REF] Sato | Helium Penetrates into Silica Glass and Reduces its Compressibility[END_REF][START_REF] Shen | Effect of Helium on Structure and Compression Behavior of SiO 2 Glass[END_REF] leading to the apparent bulk modulus K app 110-120 GPa. Using Brillouin light scattering, it is shown in [START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF] that K is only weakly affected by the presence of He, K 35-50 GPa, demonstrating that the much smaller apparent compressibility is not due to a stiffening of the network. The large difference between K app and K indicates that silica pressurized under He behaves as an open system. In that case, the static measurement underestimates the compressibility of the network by a term due to the gas charging the sample.

The complex interplay between fluid adsorption, mechanical response, and structural modifications in glasses remains fundamental, yet presently unresolved. It relates to the open and flexible structure of the glass which allows gas atoms to distend the network, in the same way as a porous material would deform upon fluid adsorption. [START_REF] Coussy | Mechanics and physics of porous solids[END_REF][START_REF] Serre | Role of Solvent-Host Interactions that Lead to Very Large Swelling of Hybrid Frameworks[END_REF][START_REF] Coudert | Thermodynamics of Guest-Induced Structural Transitions in Hybrid Organic-Inorganic Frameworks[END_REF][START_REF] Günther | Novel Insights into Nanopore Deformation Caused by Capillary Condensation[END_REF][START_REF] Neimark | Stress-Based Model for the Breathing of Metal-Organic Frameworks[END_REF][START_REF] Long | Pressure Enhancement in Carbon Nanopores: A Major Confinement Effect[END_REF][START_REF] Carmeliet | Nonlinear Poro-Elastic Model for Unsaturated Porous Solids[END_REF] Though glasses are not characterized by an open porosity it is expected that thermal motion opens the interstitial voids to gas atom diffusion. Hence, an adapted poromechanical description of adsorption-induced deformation of glasses makes sense. Such general poroelastic constitutive equations, which were revisited only recently, [START_REF] Brochard | Poromechanics of Microporous Media[END_REF] offer a very appealing theoretical framework. The present work reports a joint experimental and theoretical study providing a unifying and comprehensive picture of adsorption-induced deformation of bulk amorphous silica. We show that generalized poromechanics, a theory that was initially developed to describe the properties of porous elastic materials upon adsorption, can be extended to dense glass networks exhibiting only interstitial voids.

Methods

Experimental techniques

Using Ne as pressurizing medium, we measured the changes in volume of vitreous silica as a function of fluid pressure up to 9 GPa. The silica samples, Suprasil SF300 with less than 1 ppm OH content from Heraeus QuartzGlass, Germany, were loaded in a Chervin-type diamond-anvil cell using ruby chips for pressure calibration. [START_REF] Ayrinhac | Dynamical Origin of Anomalous Temperature Hardening of Elastic Modulus in Vitreous Silica[END_REF] The pressure dependence of the sample volume change was obtained by the measurement of the area of its image recorded with a CCD camera.

We further measured by Brillouin scattering the longitudinal and transverse sound velocities, of vitreous silica immersed in Ne up to 7 GPa. Brillouin spectra were obtained with the 514.5 nm line of a single-frequency argon-ion laser. A high resolution set-up [START_REF] Vacher | Finite Size Effects in Brillouin Scattering from Silica Glass[END_REF] was used in backscattering to precisely measure the longitudinal component. A Sandercock tandem interferometer was used in the platelet geometry to acquire simultaneously the longitudinal and transverse components. The Brillouin frequency shift ν B in an isotropic medium is given by

ν B = 2nv λ 0 sin θ 2 , (1)
where n is the refractive index, v the sound velocity of the mode responsible for the scattering, λ 0 the wavelength of the incident radiation in vacuum, and θ the scattering angle inside the sample. In backscattering this equation reduces to ν bs B = 2nv/λ 0 . In the symmetric platelet geometry, [START_REF] Whitfield | Elastic Moduli of NaCl by Brillouin Scattering at High-Pressure in a Diamond Anvil Cell[END_REF] one gets ν p B = 2v λ 0 sin θext 2 where θ ext is the angle between the incident and scattered beams outside the sample. ν p B does not depend on the refractive index of the sample. This gives a way to extract n from measurements of the same mode in both geometries,

n = ν bs B ν p B sin θ ext 2 . ( 2
)

Poromechanical Model

Adsorption-induced deformation (swelling or shrinkage) of porous materials can be described in the frame of classical poromechanics. This theory describes the volumetric strain and shear strain e ij induced by the adsorption of fluid molecules at a pressure P . [START_REF] Coussy | Mechanics and physics of porous solids[END_REF][START_REF] Biot | General Theory of Three-Dimensional Consolidation[END_REF] For an elastic solid having an intrinsic porosity Φ 0 and submitted to small strains, the constitutive equations read:

σ = K 0 -bP φ = Φ -Φ 0 = b + P/N s ij = 2G 0 e ij (3)
where K 0 and G 0 are the constant bulk and shear moduli while σ and s ij are the volume and shear stresses. φ is the porosity variation since Φ and Φ 0 are the porosities in the presence and in the absence of the adsorbed fluid, respectively. b and N , which are referred to as the Biot coefficient and modulus respectively, describe the coupling between the adsorption of molecules and the deformation of the material. A well-known drawback of Eqs. 3 is that the fluid confined in the porosity of the material is assumed to behave as the bulk fluid, i.e. no confinement effect. While this is a reasonable assumption for macroporous solids (pores larger than 50 nm), this does not hold for smaller pores such as mesopores (2-50 nm) and micropores (< 2 nm). Of course, this approximation is even less justified for the interstitial cavities (< 0.4 nm) existing in bulk glasses which arise from inherent density fluctuations. The theory has been recently revisited to include physical adsorption phenomena in microporous solids without any assumption on the state of the confined fluid and on the pore geometry. [START_REF] Brochard | Poromechanics of Microporous Media[END_REF] The approach is based on energy conservation and follows from first principles of thermodynamics. The constitutive equation, valid for an isotropic linear elastic porous solid under the hypothesis of small deformation, relates σ to and to the number of fluid atoms per unit volume of the undeformed solid n( , P ), the latter appearing naturally as a state variable of the system:

σ = K 0 - ∂ ∂ P 0 n( , p) V b (p) dp P (4)
Here V b (P ) is the molecular volume of the bulk fluid. Eq. 4 has been shown to be consistent with classical poromechanics, when the incorporated fluid is in its bulk state. [START_REF] Brochard | Poromechanics of Microporous Media[END_REF] Note that Eq.

(4) describes the thermodynamic equilibrium state of the system, after all memory effects have taken place.

In standard high pressure experiments in diamond anvil cells, the sample is immersed into the pressurizing fluid. The confining stress applied to the solid is thus equal to the opposite of the bulk fluid pressure, σ = -P and no wall or membrane separates the sample from the bulk fluid. The solid is free to shrink or to swell. This is the so-called unjacketed compression test in the field of poromechanics. In such experiments the pressure P of the bulk fluid drives both the amount of adsorbed atoms n u (P ) and the volumetric strain u (P ) of the solid, and Eq. 4 yields a relation that u (P ) must verify:

u (P ) = - P K 0 + 1 K 0 P 0 ∂n( , p) ∂ = u V b (p)dp (5)
Eq. 5 shows that the knowledge of the adsorption isotherms n( , P ) is sufficient to predict the pressure dependence of u (P ) and concomitantly n u (P ). The mechanical deformations can thus be predicted without having to simulate directly the material deformation induced upon adsorption at a given P . This approach is consistent with other thermodynamical treatments of adsorption-induced flexibility in Metal-Organic Framework and swelling of coals, [START_REF] Coudert | Thermodynamics of Guest-Induced Structural Transitions in Hybrid Organic-Inorganic Frameworks[END_REF][START_REF] Gor | Adsorption-Induced Deformation of Mesoporous Solids[END_REF] and has been recently extended to take into account the anisotropy of material properties. [START_REF] Espinoza | A Transverse Isotropic Model for Microporous Solids: Application to Coal Matrix Adsorption and Swelling[END_REF] In the specific case of coal swelling induced by CO 2 or CH 4 molecules, molecular simulations have shown that the adsorbed amount increases almost linearly with the volumetric strain, 21 n( , P ) n 0 (P )(1 + C(P ) ), where n 0 (P ) is the amount of adsorbed molecules at zero strain and C(P ) = ∂n( , P )/∂ . According to Eq. 5, the pressure dependence of the strain in unjacketed conditions u (P ) is then directly related to n 0 (P ) and C(P ) extracted from either measured or simulated adsorption isotherms,

u (P ) = - P K 0 + 1 K 0 P 0 n 0 (p)C(p)V b (p)dp (6)
which in turn gives the adsorbed amount of molecules in unjacketed conditions n u ( u , P ).

In our analysis of the adsorption-induced deformation of vitreous silica, we used a slightly more refined strategy to calculate u (P ). First, it is well documented that in the GPa pressure range, vitreous silica does not behave as a linear elastic solid. [START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF][START_REF] Kondo | Non-Linear Pressure-Dependence of the Elastic Moduli of Fused Quartz up to 3 GPa[END_REF][START_REF] Schroeder | Brillouin and Raman Scattering from Glasses under Pressure[END_REF][START_REF] Zha | Acoustic Velocities and Refractive Index of SiO 2 Glass to 57.5 GPa by Brillouin Scattering[END_REF][START_REF] Meade | Frequency Dependent Equation of State of Fused Silica up to 10 GPa[END_REF][START_REF] Tsiok | Logarithmic Kinetics of the Amorphous-Amorphous Transformations in SiO 2 and GeO 2 Glasses under High-Pressure[END_REF] The bulk modulus first decreases from K 0 36 GPa at ambient pressure to K 30 GPa around P = 2 GPa then increases rapidly with increasing P , reaching K 43 GPa at P = 6 GPa. A similar behavior [START_REF] Mantisi | Atomistic Response of a Model Silica Glass under Shear and Pressure[END_REF] is observed in the model silica we used which is described in the following section.

K decreases from K 0 56 GPa at ambient pressure to K 30 GPa around P = 8 GPa then increases at higher P . As an attempt to take into account the pressure dependence of K, one can use the secant bulk modulus K s (P ) = -P/ of the solid in the absence of any adsorbed fluid in Eq. 5. K s (P ) indeed relates the bulk fluid pressure P exerted on the sample to the volume deformation which would be observed without adsorption. K s (P ) was derived from. [START_REF] Mantisi | Atomistic Response of a Model Silica Glass under Shear and Pressure[END_REF] Such an attempt was made to test whether possible stiffening of the sample could change the predictions of the poromechanical model. In fact, rigorously, the model is only valid when K0 is used in Eq. ( 6). Using K 0 or K s (P ) leads to very similar results as K 0 K s (P ) in the pressure range considered in the present work as shown in Fig. 1.

Second, the simulated adsorption isotherms are not linear with strain but increase almost quadratically. An iterative procedure was then developed to calculate u (P ):

u i+1 (P ) = - P K s + 1 K s P 0 ∂n( , p) ∂ = u i V b (p)dp (7)
The unique solution for u (P ) is obtained after a couple of steps, using either 0 (P ) = -P/K 0 or 0 (P ) = 0 as initial values. The corresponding adsorbed amount n u ( u , P ) at the pressure P is then straightforwardly derived from u (P ) following the adsorption isotherms.

Molecular Simulations

Adsorption at room temperature of He and Ne in dense silica was simulated for different volumetric strains by means of Grand Canonical Monte Carlo simulations. In this tech- the dynamics of the system (even if the equivalence, i.e. ergodicity principle, between Molecular Dynamics and GCMC was demonstrated in the case of adsorption in porous media [START_REF] Sarkisov | Hysteresis in Monte Carlo and Molecular Dynamics Simulations of Adsorption in Porous Materials[END_REF] ).

While probing the dynamics of amorphous silica subjected to high pressure adsorption would provide interesting insights into the corresponding relaxation dynamics, it is out of the scope of the present paper.

The force field developed by [START_REF] Carré | New Fitting Scheme to Obtain Effective Potential from Car-Parrinello Molecular-Dynamics Simulations: Application to Silica[END_REF] was used to describe silica. In this model, the O and Si atoms, carrying partial charges, interact through a coulombic term and a repulsion/dispersion term described using a Buckingham potential, while Lennard-Jones (LJ) potentials were used for He and Ne. Unlike atom LJ parameters between the noble-gas atoms and silica were ob-tained from the like atom parameters using the Lorentz-Berthelot mixing rules (like atom LJ parameters for silica are taken from, 32 from 33 for He and [START_REF] Poling | The Properties of Gases and Liquids[END_REF] for Ne. The Lennard-Jones parameters, σ LJ and LJ , used to model the atomic interactions are given in Table 1. The electrostatic interaction was calculated using the Ewald sum to correct for the finite size of the simulation box.

Table 1: Lennard-Jones parameters used in the Grand Canonical Monte Carlo simulations. σ is in nm and in kJ/mol. Monte Carlo simulations as varying independently the deformation and the bulk fluid pressure P in the GPa range is not an easy task in laboratory experiments. The bulk modulus K of the sample at ambient pressure is estimated using Monte Carlo or Molecular Dynamics simulations in the isothermal-isobaric ensemble (NPT) from the pressure-induced volume variation of the system in the absence of any adsorbed fluid. The value found is similar to the value reported in [START_REF] Mantisi | Atomistic Response of a Model Silica Glass under Shear and Pressure[END_REF] where the same effective potential was used to simulate vitreous silica.

X σ X-X σ X-O σ X-Si X-X X-O X-

Results and Discussion

Revealing the adsorption-induced deformation

The measured volumetric strains of v-SiO 2 immersed in Ne fluid at pressures below 9 GPa are shown in Fig. 2. The results are compared with those obtained previously under He atmosphere at room temperature. [START_REF] Sato | Helium Penetrates into Silica Glass and Reduces its Compressibility[END_REF] In the latter case, when a pressure step is applied, the volumetric strain appears instantaneously on the time scale of the experiment. In contrast,

under Ne atmosphere at room temperature, the equilibrium is reached after a delay which can extend to hours or even days, depending on P and sample sizes. An example of Ne adsorption kinetics is illustrated in Fig. 3. When a pressure step from 0.2 to 2.1 GPa is applied at time t = 0, the volume first decreases to the value obtained with a non-penetrating pressurizing fluid ( -0.07), showing that Ne has not appreciably diffused into the network at this stage. At longer times, the volume increases and eventually reaches an equilibrium value.

This increase in volume shows that Ne penetration unexpectedly distends the silica network at constant external pressure. As shown in Fig. 2, the equilibrium values of the volumetric strain in Ne atmosphere are only slightly smaller than those found with He, [START_REF] Sato | Helium Penetrates into Silica Glass and Reduces its Compressibility[END_REF] indicating that the penetration of Ne produces almost the same effect as that of He. In both cases the change in volume is much smaller than the deformation obtained when vitreous silica is pressurized using the non-penetrating fluid ME4:1. [START_REF] Sato | Helium Penetrates into Silica Glass and Reduces its Compressibility[END_REF] and by the non-penetrating fluid ME4:1 (green triangles). [START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF] The solid and the dashed lines show the theoretical predictions as discussed in the text for He and Ne, respectively. The dotted line shows the deformation of the model silica in the absence of adsorption. [START_REF] Mantisi | Atomistic Response of a Model Silica Glass under Shear and Pressure[END_REF] Direct measurement of the amount of gas entering a solid upon high pressures is a very difficult task. We attempted to obtain this information in-situ from the refractive index of the He or Ne stuffed samples. We have used an effective medium approximation to estimate the mean polarisability of the composite. We apply the Clausius-Mossotti relation, 3 0 atomic density of the SiO 2 molecules, extracted from the volumetric strain measurements. α f is the polarisability of the fluid which hardly depends on P in the pressure range investigated in the present work. [START_REF] Letoullec | Refractive-Index Measurements of Dense Helium up to 16 GPa at T = 298 K -Analysis of its Thermodynamics and Electronic-Properties[END_REF][START_REF] Dewaele | Measurement of Refractive Index and Equation of State in Dense He, H 2 , H 2 O, and Ne under High Pressure in a Diamond Anvil Cell[END_REF] α SiO 2 is the strain-dependent polarisability of the silica network taken from the measurements at the same volumetric strain in a non-penetrating pressurizing medium. 11 Fig. 4 shows the strain dependence of the ratio (n 2 -1)/(n 2 + 2) for vitreous silica immersed in He and in ME4:1. In the former case, the error bars mainly reflect the uncertainty on the exact symmetric platelet geometry affecting the internal scattering angle θ in Brillouin scattering experiments. The difference between the symbols and the line at a

n 2 -1 n 2 +2 = N SiO 2 α SiO 2 +N f α f where N f is
given provides an estimate of N f ( ) in the stuffed sample. It must be stressed that N f can be estimated only when the difference between the two refractive indices is significant, i.e.

in the high pressure region. From the measured N f ( ) and (P ), N f (P ) can be calculated straightforwardly. In the upper pressure region investigated, we found N f 20 ± 4 at/nm 3 in He at a pressure P ∼ 6 GPa (for comparison the number density of silica at this He pressure is N SiO2 ∼ 23 SiO 2 /nm 3 ). Although a similar variation of the refractive index for silica immersed in Ne is observed, the higher Ne polarisability results in a lower adsorbed quantity of Ne atoms 7 ± 4 at/nm 3 in Ne. These values can be compared to those found for H 2 adsorbed in vitreous silica, 10.4 at/nm 3 at P = 6 GPa, using an ex-situ quenching technique. 37 Figure 4: Strain dependence of the ratio (n 2 -1)/(n 2 + 2) for vitreous silica immersed in He (symbols) and in ME4:1 (line [START_REF] Zha | Acoustic Velocities and Refractive Index of SiO 2 Glass to 57.5 GPa by Brillouin Scattering[END_REF][START_REF] Meade | Frequency Dependent Equation of State of Fused Silica up to 10 GPa[END_REF] ). The difference between the two data sets is related to the amount of He adsorbed in the silica network.

The equilibrium values of sound velocities of vitreous silica immersed in Ne up to 7 GPa are shown in Fig. 5 (blue dots) together with our previous results obtained in He (red squares). Sound velocities closely follow those found under He. As discussed in [START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF] for He, the anomalous minimum at 2 GPa in v L (P ) of vitreous silica (green triangles) [START_REF] Kondo | Non-Linear Pressure-Dependence of the Elastic Moduli of Fused Quartz up to 3 GPa[END_REF][START_REF] Schroeder | Brillouin and Raman Scattering from Glasses under Pressure[END_REF][START_REF] Zha | Acoustic Velocities and Refractive Index of SiO 2 Glass to 57.5 GPa by Brillouin Scattering[END_REF][START_REF] Tsiok | Logarithmic Kinetics of the Amorphous-Amorphous Transformations in SiO 2 and GeO 2 Glasses under High-Pressure[END_REF] is almost completely suppressed in Ne. This indicates that the gas-atoms penetrate into the silica network and first fill the largest interstitial voids, which are mainly responsible for the anomalous softening of the network. [START_REF] Shen | Effect of Helium on Structure and Compression Behavior of SiO 2 Glass[END_REF][START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF][START_REF] Huang | Structural Origin of Negative Thermal Expansion in High-Temperature Silica Polymorphs[END_REF][START_REF] Walker | The Origin of the Compressibility Anomaly in Amorphous Silica: A Molecular Dynamics Study[END_REF] Fig. 5 also shows that at a given P , the sound velocities in He or Ne are only slightly larger than those obtained in the non-penetrating pressurizing medium ME4:1. While a slight intrinsic stiffening cannot be ruled out (increase in the bulk modulus with increasing pressure), the fact the sound velocities in He or Ne are very close to those in the non-penetrating medium suggests that the much lower apparent compressibility in Fig. 2 is not due to a huge stiffening of the network but rather to an adsorption-induced deformation of the glass network as illustrated by Fig. 3. Such a strong effect of the pressurizing medium on the static compressibility, associated with adsorption, is usually observed in materials with open porosity, such as zeolites. [START_REF] Coasne | Enhanced Mechanical Strength of Zeolites by Adsorption of Guest Molecules[END_REF] Though glasses are We now turn to the predictions of the generalized poromechanical theory for u (P ) and n u (P ). Using Eq. 7 and the simulated adsorption isotherms of Fig. 6, the unique solution for u (P ) is found. The pressure dependence of u (P ) predicted by the poroelastic model is shown in Fig. 2 as solid and dashed lines for Ne and He, respectively. The calculated values reproduce rather nicely the experimental volumetric strains. The fact that the theoretical deformation u (P ) slightly overestimates the experimental data at low P is believed to be mostly due to the larger bulk modulus of the simulated silica. The volumetric strain of the model silica in the absence of adsorption indeed overestimates the experiments as shown in Fig. 2 by the dotted line and the triangles. The above results show that the poromechanical approach provides a framework for a quantitative description of the adsorption-induced deformation in vitreous silica. In particular, in agreement with the experimental data, our calculations show that the silica network expansion induced by He adsorption is larger than that induced by Ne adsorption.

The corresponding adsorbed amount n u ( u , P ) is shown by the bold red curves in Fig. 6 to enlighten the method and in Fig. in Fig. 7(a), illustrating the large amount of adsorbed gas-atoms. In this case, the He concentration is about 11 at/nm 3 , i.e. 0.53 mole of He per mole of SiO 2 , in qualitative agreement with the value of N f estimated from the experiments (20 ± 4 at/nm 3 for He and 7 ± 4 at/nm 3 for Ne). The amount of He adsorbed in silica is almost twice that of Ne at the same P . In both cases, the predicted n u (P ) varies non-monotonically with P . It first increases rapidly to a maximum around 4 and 6 GPa for Ne and He, respectively, and then decreases as the mechanical stress exerted by the fluid outside the sample induces significant volume contraction. We note that a similar pressure dependence has been recently obtained for the solubility of argon in liquid silica using very long molecular dynamics simulations of the complete system equilibrated at fixed T and P . [START_REF] Guillot | Noble Gases in High-Pressure Silicate Liquids: A Computer Simulation Study[END_REF]

Conclusion

The present study shows that the small apparent compressibility of silica under high He or Ne pressure is due to an adsorption-induced expansion. It further demonstrates that the elasticity of vitreous silica subjected to noble gas adsorption can be described by generalized poromechanics. This theory for the mechanical properties of microporous solids upon adsorption, is applied here to an amorphous solid exhibiting only interstitial voids, opening the way to a broad range of applications. We anticipate that similar conclusions will be reached for other glasses exhibiting a significant amount of interstitial free volume as in the case of GeO 2 or B 2 O 3 . The combined use of Monte Carlo simulations and poromechanics provides an efficient alternative to other theoretical approaches [START_REF] Coudert | Thermodynamics of Guest-Induced Structural Transitions in Hybrid Organic-Inorganic Frameworks[END_REF][START_REF] Gor | Adsorption-Induced Deformation of Mesoporous Solids[END_REF][START_REF] Guillot | Noble Gases in High-Pressure Silicate Liquids: A Computer Simulation Study[END_REF][START_REF] Sarda | Breaking of Henry's Law for Noble Gas and CO 2 Solubility in Silicate Melt under Pressure[END_REF] describing the complete system equilibrated in isothermal-isobaric or osmotic ensembles but inherently limited to short time scales and by possible equilibration issues. Poromechanics associated to Monte Carlo simulations thus might give the opportunity to extend this study to dense materials, where the equilibration time is long, such as glasses and melts of geological interest, actinide oxides fuels or irradiated metallic alloys.

Figure 1 :

 1 Figure1: Pressure dependence of the bulk modulus of model silica (red triangles).[START_REF] Mantisi | Atomistic Response of a Model Silica Glass under Shear and Pressure[END_REF] Blue dots are the secant bulk modulus used in the poromechanical model and the green dashed line shows the P = 0 value K 0 .

1 }Figure 2 :

 12 Figure2: Volumetric strain of v-SiO 2 as a function of the pressure exerted by an adsorbing fluid, Ne (blue dots), He (red squares),[START_REF] Sato | Helium Penetrates into Silica Glass and Reduces its Compressibility[END_REF] and by the non-penetrating fluid ME4:1 (green triangles).[START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF] The solid and the dashed lines show the theoretical predictions as discussed in the text for He and Ne, respectively. The dotted line shows the deformation of the model silica in the absence of adsorption.[START_REF] Mantisi | Atomistic Response of a Model Silica Glass under Shear and Pressure[END_REF]

Figure 3 :

 3 Figure 3: Volumetric strain as a function of time after a pressure step from 0.2 to 2.1 GPa in Ne. The dotted line is a guide for the eye. The equilibrium value is shown by a full symbol. The dashed line marks the expected deformation of silica immersed in the non-penetrating fluid ME4:1.

  u m e t r i c s t r a i n ε v -S i O 2

Figure 5 :

 5 Figure5: Sound velocities of v-SiO 2 as a function of P immersed in Ne (blue dots), He (red squares),[START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF] and ME4:1 (green triangles).[START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF] LA and TA denote the longitudinal and transverse acoustic velocities, respectively.

3. 2 Figure 6 :

 26 Figure 6: Room temperature adsorption isotherms in v-SiO 2 from GCMC simulations as a function of the bulk fluid pressure P : Ne (a), He (b). The symbols represent the calculated values whereas the thin lines are guides for the eye. The vertical dashed lines mark the unstrained volume. The bold red lines show the amount of gas-atoms adsorbed in silica predicted by the poroelastic constitutive equations in unjacketed conditions n u ( u , P ).

Figure 7 :

 7 Figure 7: (a) Typical atomic configuration for He adsorbed at room temperature in silica glass at a bulk fluid pressure P = 6 GPa and a fixed volumetric strain = -0.03. To emphasize the large amount of adsorbed fluid atoms, about 11 at/nm 3 , only He atoms are shown (blue spheres). The silica network is shown by sticks that are orange on the Si side and red on the O side. The simulation cube is projected along its [x, y, z] diagonal. (b) Pressure dependence of the amount of Ne and He atoms adsorbed by v-SiO 2 predicted by the poroelastic model in unjacketed conditions.

= 1 -

 1 Alternatively, the adsorption capacity can be quantified by the tangent Biot coefficient, b tan ( , P ) = V b (P ) ∂n( ,P ) ∂ K/K app in the unjacketed case. The tangent Biot coefficients derived from the experiments are b tan 0.7 for He and b tan 0.6 for Ne, in qualitative agreement with the values given by molecular simulations (b tan = 0.83 and b tan = 0.74 for He and Ne at 2 GPa). Interestingly, such high Biot coefficients are usually encountered in porous materials such as limestone or sandstone. 41,42

Acknowledgement

The authors thank Eric Courtens for his critical reading of the manuscript, Sébastien Clément and Rémy Vialla for their help during the experiments and Christelle Dupas for the sample preparation. The work was partially supported by Région Languedoc-Roussillon (Omega Platform).

Supporting Information Available

Additional data showing the He and Ne adsorbed amounts as a function of pressure for different strains. This material is available free of charge via the Internet at http://pubs. acs.org/.

Graphical TOC Entry


Leonardo S Cardoso
email: leonardo.cardoso@insa-lyon.fr

Abdelbassat Massouri

Benjamin Guillon

Paul Ferrand

Florin Hutu

Guillaume Villemaud

Tanguy Risset

Jean-Marie Gorce

CorteXlab: A Facility for Testing Cognitive Radio Networks in a Reproducible Environment

While many theoretical and simulation works have highlighted the potential gains of cognitive radio, several technical issues still need to be evaluated from an experimental point of view. Deploying complex heterogeneous system scenarios is tedious, time consuming and hardly reproducible. To address this problem, we have developed a new experimental facility, called CorteXlab, that allows complex multi-node cognitive radio scenarios to be easily deployed and tested by anyone in the world. Our objective is not to design new software defined radio (SDR) nodes, but rather to provide a comprehensive access to a large set of high performance SDR nodes. The CorteXlab facility offers a 167 m 2 electromagnetically (EM) shielded room and integrates a set of 24 universal software radio peripherals (USRPs) from National Instruments, 18 PicoSDR nodes from Nutaq and 42 IoT-Lab wireless sensor nodes from Hikob. CorteXlab is built upon the foundations of the SensLAB testbed and is based the free and open-source toolkit GNU Radio. Automation in scenario deployment, experiment start, stop and results collection is performed by an experiment controller, called Minus. CorteXlab is in its final stages of development and is already capable of running test scenarios. In this contribution, we show that CorteXlab is able to easily cope with the usual issues faced by other testbeds providing a reproducible experiment environment for CR experimentation.

I. INTRODUCTION

It is widely accepted that cognitive radio (CR) has the potential to promote high spectral efficiencies, through a better re-use of radio resources. However, to be able to realise the complex CR scenarios envisioned for the future, correct usage of software defined radio (SDR) technologies is needed. While many works have shown the potentials of CR, there are still many technical difficulties standing in the way between the idealised and a practical CR usage. Several of such technical issues are easier to be dealt experimentally, rather than from an analytical or simulation point of view. This is specifically the case when large scale systems are employed such as distributed/cooperative spectrum sensing [START_REF] Cardoso | Spectrum Sensing in Cognitive Radio Networks[END_REF] and massive MIMO [START_REF] Larsson | Massive mimo for next generation wireless systems[END_REF].

In the last ten years, experimenting with SDRs has been made possible by the commoditization of radio platforms and the availability software toolkits, such as the universal software radio peripheral (USRP) and GNU Radio, respectively. They have popularised -and even democratised -the experimentation on radio, allowing researchers, engineers and amateurs to build and test radio transceiver technologies, ranging from the most simple (e.g. [START_REF] Katz | Using software defined radio (sdr) to demonstrate concepts in communications and signal processing courses[END_REF]) to the state-of-the-art in radio communications (e.g. [START_REF] Massey | Implementation of a real-time wireless interference alignment network[END_REF]). However, while the deployment of an experimental scenario over a couple of radio nodes is manageable, it becomes a daunting task when many nodes are involved. This becomes especially hard when multiple types of radio nodes, with different capabilities and characteristics integrate a heterogeneous testbed. Indeed, the testbed operator must sequentially connect to each one of the host computers to deploy the code that implements the radio processing, start the experiment and, when finished, collect the results possibly spread over several computers. If any kind of synchronised or timed start procedure is needed, then it becomes impossible for a single operator. Another common issue of traditional testbeds is the lack of reproducibility of experiments. The scientific value of testing and comparing cutting edge radio techniques depends on the ability to reproduce experiments. Unmanaged environments offer unpredictable propagation and interference fluctuations, that are usually disregarded in experimental results. Due to the random nature of both of these impairments, it is very unlikely that the same conditions will be offered when re-executing the experiment, which will probably affect the assessment of the results obtained.

To address all of these issues we introduce CorteXlab [START_REF]Cortexlab: Cognitive radio testbed[END_REF], a testbed composed of 84 heterogeneous and high performance radio nodes deployed in an electromagnetically isolated room, for cutting edge radio experimentation. With CorteXlab, our main objective is to provide a unified and comprehensive access to a large set of heterogeneous nodes in a reproducible environment, in the aim to foster experimental development of future radio techniques. CorteXlab offers 24 USRPs 2932 from National Instrument, 18 PicoSDRs from Nutaq and 42 IoT-Lab wireless sensor nodes from Hikob. It makes available the full potential of SDR, through the widely accepted GNU Radio toolkit as well as high performance real-time field programable gate array (FPGA) development. The SDR hardware available in CorteXlab ranges from simplistic wireless sensor network (WSN) nodes to full blown 4x4 MIMO SDR nodes with agile radio capabilities. Through a carefully designed backbone network, those radio nodes are capable of cooperating to emulate complex radio technologies such as network multiple input -multiple output (MIMO) [START_REF] Venkatesan | Network mimo: Overcoming intercell interference in indoor wireless systems[END_REF], interference alignment (IA) [START_REF] Cadambe | Interference alignment and the degrees of freedom for the k user interference channel[END_REF] and physical layer based relay networks [START_REF] Louie | Practical physical layer network coding for two-way relay channels: performance analysis and comparison[END_REF].

CorteXlab [START_REF]Cortexlab: Cognitive radio testbed[END_REF] is developed, along with 8 other testbeds, under the framework of a nationwide French program Future Internet of Things (FIT) [START_REF]Equipex fit: Future internet of things[END_REF]. FIT aims to develop an experimental facility, a federated and competitive infrastructure with international visibility and a broad panel of users. It will provide this facility with a set of complementary components that enable experimentation on innovative services for academic and industrial users. The project will give a means to experiment on mobile wireless communications at the network and application layers thereby accelerating the design of advanced networking technologies for the future internet. In this work, CorteXlab is introduced, detailing its structure, inner workings and main differences with respect to the currently available testbeds.

The remainder of this paper is organized as follows. In the next section a brief review of the state of the art in experimental testbed for CR is presented. Then, in sections III through VI, a full description of CorteXlab is given. Finally, conclusions and perspectives are drawn in section VIII.

II. STATE OF THE ART OF EXPERIMENTAL TESTBEDS

Large-scale CR testbeds are mandatory to develop and evaluate the performance of upcoming physical (PHY) and medium access control (MAC) layers and future CR techniques. Unfortunately, testing new algorithms on real testbeds is complex and time-consuming. Several research groups already obtained interesting results from their own testbeds, but comparing all results and methods fairly and with a good level of reproducibility.

Whereas numerous testbeds are available for specific wireless network standards (sensor or 802.11-oriented), only a few large-scale testbeds have been developed having full SDR and CR capabilities. Appart from on-going projects such as CREW [START_REF]Cognitive radio experimentation world[END_REF] or TRIAL [START_REF]Trial environment for cognitive radio and networks[END_REF] and some other small testbeds involving less than 10 nodes, we found only two testbeds developed respectively at Rutgers University, Orbit [START_REF]Orbit -wireless network testbed[END_REF] and at Virginia Tech., Cornet [START_REF]Virginia tech cognitive radio network testbed[END_REF], where USRPs have been adopted.

The Orbit testbed [START_REF] Raychaudhuri | Overview of the orbit radio grid testbed for evaluation of next-generation wireless network protocols[END_REF] counts with an impressive number of nodes, a total of 400, dispatched on the ceilings of an experiment room. It also uses a clever scheduling and deployment system, called ORBIT Management Framework (OMF) [START_REF] Rakotoarivelo | Omf: a control and management framework for networking testbeds[END_REF], to manage experiments on those 400 nodes. For CR experiments, however, only 28 nodes are of the USRP kind, since the main focus of Orbit is on higher layer experiments. The remaining nodes implement a fixed 802.11 PHY/MAC layer. A portal website is used to control all user interaction to the platform.

The Cornet testbed, unlike Orbit, was created from the ground up for PHY/MAC layer research and supports CR experimentation. It counts with a total of 48 nodes, distributed over 4 levels of an academic building in the Virginia Tech campus. The USRP radios use a custom developed radio frequency (RF) board which supports several non ISM frequencies. Those frequencies were made available by the federal communications commission for Cornet's exclusive use. The USRPs are connected to a host computer that performs the signal processing in software, and which are accessible to the user via secure shell (SSH) connection. In spite of the impressive node count and the possibility to transmit freely on non ISM bands, users of Cornet are still faced with a frugal calendar based reservation system and one-by-one access to the nodes, which hinders large scale deployments. Furthermore, since the nodes are on different floors, they have limited radio accessibility to nodes on different floors. Last but not least, after creating a user account anyone can have access to any node at any time, and consequently to the code installed by other users. This intrinsic insecurity can scare security sensitive users away from the platform.

On both cases, the registered users can remotely program and run experiments on the USRPs. Still, the USRPs adopted therein are not the bleeding edge hardware available. In both cases, the cognitive radio nodes are based on an old USRP technology which present some limitations. We remark that both facilities have their nodes deployed in a conventional environment, which means that the system may suffer from external interference while itself may produce interference on any external system.

III. THE CORTEXLAB FACILITY

Using as our starting point the current state of these front line testbeds, we have developed CorteXlab, a new facility for large scale radio experimentation. The main objective of CorteXlab is to enable users to run real-time communications with customized application (APP) (with traffic generation), transport control (TCP), network (NET), MAC and PHY layers, implementing current (WIFI, Zigbee, LTE, LTE-A) and upcoming (5G) standards. Users all over the world will be able to schedule experiments and access CorteXlab through a custom built web portal and a standard secure shell (SSH) remote connection. CorteXlab deploys in a electromagnetically (EM) shielded room a heterogeneous set of nodes, including WSN and SDR nodes. By providing access to different nodes technologies, we broaden the scenario possibilities, hopefully appealing to a larger audience of research and development engineers.

Our testbed is complementary to Cornet and Orbit in the following aspects:

• A shielded experimentation room reinforces the reproducibility and the control of the interference environment; • Heterogeneous high performance nodes are adopted, fostering agility, flexibility and real-time performance; • An experimentation control plane, called Minus is introduced to control the execution of the experiment. Each of these aspects are further detailed in the following.

IV. EXPERIMENTATION ROOM

CorteXlab counts with a completely shielded room to ensure EM isolation with the outside world. This allows the use of any (restricted or open) frequency between 300 MHz and 5 GHz, which might appeal to researchers looking to test specific frequency or standard related techniques. It also improves reproducibility since all sources of interference and channel impairments are confined to the interior of the room. To avoid excessive reflections, EM wave absorbing foams cover all walls and roof, leaving room from some reflections to occur. A glimpse of the experimentation room, before the installation of the radio platforms can be seen in Figure 1. A measurement campaign has shown an attenuation of more than 80 dB is ensured in the band [1000 MHz-5 GHz] as seen in Figure 2, with at least 60 dB throughout the whole target band. We are well aware that by shielding the room, a risk is taken on producing an unrealistic propagation environment. To properly assess the radio environment with the EM shielding, two approaches are currently under study. In the first approach, an extensive ray-tracing simulation campaign is under way to provide an initial understanding of the expected kind of propagation environment. This campaign will allow (if needed) the subsequent placement of absorbers or reflection surfaces to improve the radio characteristics of the room. The second approach consists on running a campaign of channel measurements to properly assess the effectiveness of the changes made in the first step, as well as to provide real channel measurements to our future users. The experimentation room will also count with several network-connected sensors, such as humidity and temperature for external monitoring of the conditions and traceability of any relevant changes to the environment. Finally, cameras will also be installed in the room to diminish human intervention to a minimum.

V. RADIO PLATFORMS

As previously stated, we chose a mix of three types of nodes in the CorteXlab: low power WSN nodes, general-purpose SDR nodes and real-time high performance SDR nodes.

A. WSN platforms

The WSN nodes are powered by a Cortex A8 processor and count with an off-the-shelf CC2420 802.15.4 radio interface operating at 2.4 GHz, built by a start-up called Hikob. These nodes embed a complete Linux environment and can be programmed to implement different MAC, NET and TCP layers. They will be used to test techniques such as low consumption routing for internet of things and distributed calculation for wireless sensor networks. They can also be used as medium sensing nodes (in the ISM band) that cooperate with SDR nodes to form an intelligent distributed spectrum sensing tier.

B. General purpose SDR platforms

Represented by the National Instruments USRP 2932, the general-purpose SDR nodes will use (but are not limited to) the GNU Radio toolkit for rapid prototyping of transmission techniques mostly reliant on the general purpose processor (GPP) of the host PC. The USRP 2932 is a high end radio platform, counting with a 400 MHz -4.4 GHz RF board, bandwidth of up to 40 MHz (with reduced dynamic range, nominal band of 20 MHz), a precise OCXO clock source and a 1 gigabit ethernet (GigE) connection to the host PC. The host PC is based on a Linux environment and will allow users to test not only PHY layer techniques, but also MAC, NET, TRA and APP, by deploying custom kernel modules that can coexist with the stock implementations easily.

Both the use of Linux and GNU Radio will facilitate the development and test at the user's own computer before bringing the experiment over to CorteXlab. Furthermore, the GNU Radio and Linux communities have progressed a lot in the last decade, offering good and free support to users as well as a multitude of open and accessible examples, that can be downloaded off the internet and used without additional charges.

C. High performance SDR platforms

The high end SDR nodes, are composed by the Nutaq PicoSDR radio platforms. In CorteXlab they will come in two flavors: 2x2 and 4x4 MIMO. The 2x2 PicoSDRs possess two Radio420x mezzanine cards on top of a Perseus board, that counts with a powerful Virtex6 FPGA. It will connect to the host PC through a GigE. The 4x4 MIMO PicoSDRs possess four Radio420x mezzanine cards and two Virtex6 FPGAs. It can connect to the host PC through either a GigE or a 4x PCI express (PCIe), the latter offering high bandwidth for full MIMO operation. All Radio420x can tune in the 300 MHz to 3.8 GHz range and can step up to 28 MHz in bandwidth. As with the USRP case, the host PCs are Linux based and will allow users to test PHY and above layers via custom kernel modules.

Initially, two programming modes will be supported by the PicoSDRs:

• GNU Radio: this method will work more or less the same way as it does for the USRPs, with the added possibility of 2x2 and 4x4 MIMO. In this mode the user will be able to choose either a passthrough FPGA design, giving direct access to the radio interface, or any one of our custom PHY layer intellectual properties (IP)s, including OFDM and 802.14.5. GNU Radio blocks interfacing to either of these FPGA IPs will be available. • Board software development kit (BSDK): this is the typical Nutaq method for programming the PicoSDRs and consists of part VHDL FPGA and part C code. It brings full flexibility, allowing a user to bring his own IP to the PicoSDR FPGAs, through ready made bitstreams that will be flashed to the FPGA at the beginning of the experiment. The C code, in the form of a Linux executable, will interface to the FPGA inputs and outputs through a readily available driver that uses either the GigE or PCIe interfaces, and can contain not only PHY layer signal processing but also MAC and above implementations. We know that this method is more challenging than the simple GNU Radio one, and thus we will offer the possibility to use one of our custom PHY IPs (OFDM and 802.14.5) as a starting point. It should be noted that, to use the BSDK method, users will have to own a license of the Xilinx development suite. Finally, every SDR node is co-located with a single WSN node. They are laid out in the experimentation room in an uniform pattern, as shown in Figure 3, where the node positions (marked with a small "+" symbol) are shown in the floor plan of the room. The nodes are attached to the ceiling and are given a 1.8 m clearance is between them.

VI. EXPERIMENTATION CONTROL PLANE

In order to promote the automated control of all nodes during an experiment, an experimentation control plane was created, called Minus. It, allows for the scheduling, deployment, start, finish and results collection of user experiments. Each experiment is organized as a Minus task, and contains:

• Scenario description: a textual file with the list of nodes, their roles and their parameters; • Scenario roles: the actual firmware, GNU Radio Python scripts, pre-compiled libraries, pre-compiled C code and FPGA bitstreams that define the node behavior during the experiment; • Role parameters: the configuration values that refine the behavior of each role. To better understand the task organization, lets consider the following spectrum sensing example. Suppose the user wants to create a cooperative spectrum sensing system with two tiers. He chooses a set of USRPs to be small cell base stations (SCBSs), which he further divides into those operating at frequency f 1 with band b 1 (first tier) and those operating at frequency f 2 with band b 2 (second tier). Then he selects another set of USRPs to be user equipments (UEs) that switch automatically between tiers, (f 1 , b 1 ) and (f 2 , b 2 ), according to the estimated probability of missed detection. In this case, there are only two roles: SCBSs and UEs which can be defined by two GNU Radio scripts, say "scbs.py" and "ue.py". There are also only two parameter sets: "-f1 2.412G -W1 1M" and "-f2 2.490G -W2 2M" for each tier involved.

Minus is divided into a server and a client side. At the server side, based on the scenario description file, Minus is able to turn on the related nodes, transfer the task files, start the experiment simultaneously on all nodes (or with a pre-determined delay between them), capture the end of the task, turn nodes off and collect results. The results are later on copied to a predefined disk space accessible only by the concerned user. Along with the results, standard output and standard error files are generated to aid the user in finding problems in the task execution if any.

The Minus client side is composed of a set of command line executables that enable the user to create a task file, submit a task, query the server status of the server and abort a task. The user can access to the client side of Minus through a special machine, called "Airlock" responsible for hosting all userside services. Finally, through Minus, no direct user access is allowed on the nodes themselves, guaranteeing the security and longevity of the testbed as a whole. Even in the case of node issues, Minus provides a remotely operated reinstall, to restore the node normal operation.

VII. SYSTEM ARCHITECTURE

To support Minus, a complex network infrastructure was built. To enforce security, all networking is divided into virtual LANs and filtered through firewalls. A services server hosts all virtualized core services, including Minus. A storage server automatically backs up all important data to avoid loss in case of failure.

The CorteXlab network architecture is shown in Figure 4. All SDR nodes possess two network interfaces. The management interface (black connections) is hidden from the user and allows Minus to control the nodes, whereas the data interface is open to the user, and allows communication between nodes. The latter interface can be used to implement cooperative schemes, where nodes can exchange information. These interfaces can also be used to allow one node to control several others, for example in a distributed MIMO approach.

CONCLUSION

Cognitive radio is a paradigm that refers to dynamic radio resource sharing among heterogeneous wireless systems. It is clearly expected to play a fundamental role in a near future, allowing coexistence of multiple radios in a unique frequency band. Cognitive radio will only unlock the high spectral efficiencies foreseen if software defined radio technologies are correctly exploited. In this paper we have introduced CorteXlab, a facility for experimentation with cognitive radio at the PHY layer. At the best of our knowledge, CorteXlab is the first facility offered to the research community allowing to remotely test cognitive radio scenarios in a completely reproducible environment. Compared to existing facilities, CorteXlab provides three important new features: coexistence of heterogeneous technologies, a shielded room and the automatic scenario deployment, through Minus. Last but not least, the co-existence of simple low power wireless sensor networks nodes together with complex software radio nodes will open the door to new original and complex scenarios. In the final stages of its development, CorteXlab is currently open only to restricted partners.

Fig. 1 .

 1 Fig. 1. The experimentation room.

Fig. 2 .

 2 Fig. 2. Signal attenuation in the experimentation room with respect to outside.

Fig. 3 .

 3 Fig. 3. Experimentation room floor plan. Nodes are positioned at the "+" symbols.

Fig. 4 .

 4 Fig. 4. CorteXlab network architecture.

ACKNOWLEDGMENT

This work is funded by the project FIT (Future Internet of Things) from the program 'equipement d'excellence' of the French government.