Member, IEEE Nicolas Courty
email: nicolas.courty@irisa.fr

Rémi Flamary
email: remi.flamary@unice.fr

Senior Member, IEEE Devis Tuia
email: devis.tuia@geo.uzh.ch

Member, IEEE Alain Rakotomamonjy

Dt

Optimal Transport for Domain Adaptation

Keywords: Unsupervised Domain Adaptation, Optimal Transport, Transfer Learning, Visual Adaptation, Classification

Domain adaptation is one of the most challenging tasks of modern data analytics. If the adaptation is done correctly, models built on a specific data representation become more robust when confronted to data depicting the same classes, but described by another observation system. Among the many strategies proposed, finding domain-invariant representations has shown excellent properties, in particular since it allows to train a unique classifier effective in all domains. In this paper, we propose a regularized unsupervised optimal transportation model to perform the alignment of the representations in the source and target domains. We learn a transportation plan matching both PDFs, which constrains labeled samples of the same class in the source domain to remain close during transport. This way, we exploit at the same time the labeled samples in the source and the distributions observed in both domains. Experiments on toy and challenging real visual adaptation examples show the interest of the method, that consistently outperforms state of the art approaches. In addition, numerical experiments show that our approach

I. INTRODUCTION

Modern data analytics are based on the availability of large volumes of data, sensed by a variety of acquisition devices and at high temporal frequency. But this large amounts of heterogeneous data also make the task of learning semantic concepts more difficult, since the data used for learning a decision function and those used for inference tend not to follow the same distribution. Discrepancies (also known as drift) in data distribution are due to several reasons and are application-dependent. In computer vision, this problem is known as the visual adaptation domain problem, where domain drifts occur when changing lighting conditions, acquisition devices, or by considering the presence or absence of backgrounds. In speech processing, learning from one speaker and trying to deploy an application targeted to a wide public may also be hindered by the differences in background noise, tone or gender of the speaker. In remote sensing image analysis, one would like to leverage from labels defined over one city image to classify the land occupation of another city. The drifts October 6, 2016 DRAFT observed in the probability density function (PDF) of remote sensing images are caused by variety of factors:

different corrections for atmospheric scattering, daylight conditions at the hour of acquisition or even slight changes in the chemical composition of the materials.

For those reasons, several works have coped with these drift problems by developing learning methods able to transfer knowledge from a source domain to a target domain for which data have different PDFs.

Learning in this PDF discrepancy context is denoted as the domain adaptation problem [START_REF] Pan | A survey on transfer learning[END_REF]. In this work, we address the most difficult variant of this problem, denoted as unsupervised domain adaptation, where data labels are only available in the source domain.

We tackle this problem by assuming that the effects of the drifts can be reduced if data undergo a phase of adaptation (typically, a non-linear mapping) where both domains look more alike.

Several theoretical works [START_REF] Ben-David | Impossibility theorems for domain adaptation[END_REF], [START_REF] Mansour | Domain adaptation: Learning bounds and algorithms[END_REF], [START_REF] Germain | A PAC-Bayesian Approach for Domain Adaptation with Specialization to Linear Classifiers[END_REF] have emphasized the role played by the divergence between the data probability distribution functions of the domains.

These works have led to a principled way of solving the domain adaptation problem: transform data so as to make their distributions "closer", and use the label information available in the source domain to learn a classifier in the transformed domain, which can be applied to the target domain. Our work follows the same intuition and proposes a transformation of the source data that fits a least effort principle, i.e. an effect that is minimal with respect to a transformation cost or metric. In this sense, the adaptation problem boils down to: i) finding a transformation of the input data matching the source and target distributions and then ii) learning a new classifier from the transformed source samples.

This process is depicted in Figure 1. In this paper, we advocate a solution for finding this transformation based on optimal transport.

Optimal Transport (OT) problems have recently raised interest in several fields, in particular because OT theory can be used for computing distances between probability distributions. Those distances, known under several names in the literature (Wasserstein, Monge-Kantorovich or Earth Mover distances) have important properties: i)

They can be evaluated directly on empirical estimates of the distributions without having to smoothen them using non-parametric or semi-parametric approaches; ii)

By exploiting the geometry of the underlying metric space, they provide meaningful distances even when the supports of the distributions do not overlap. Leveraging from these properties, we introduce a novel framework for unsupervised domain adaptation, which consists in learning an optimal transportation based on empirical observations. In addition, we propose several regularization terms that favor learning of better transformations w.r.t.

the adaptation problem. They can either encode class information contained in the source domain or promote the preservation of neighborhood structures. An efficient algorithm is proposed for solving the resulting regularized optimal transport optimization problem. Finally, this framework can also easily be extended to the semisupervised case, where few labels are available in the target domain, by a simple and elegant modification in the optimal transport optimization problem.

The remainder of this Section presents related works, while Section II formalizes the problem of unsupervised domain adaptation and discusses the use of optimal transport for its resolution. Section III introduces optimal transport and its regularized version. Section IV presents the proposed regularization terms tailored to fit the domain adaptation constraints. Section V discusses algorithms for solving the regularized optimal transport problem efficiently. Section VI evaluates the relevance of October 6, 2016 DRAFT In the first family, methods which have been proposed include searching for projections that are discriminative in both domains by using inner products between source samples and transformed target samples [START_REF] Saenko | Adapting visual category models to new domains[END_REF], [START_REF] Kulis | What you saw is not what you get: domain adaptation using asymmetric kernel transforms[END_REF], [START_REF] Jhuo | Robust visual domain adaptation with low-rank reconstruction[END_REF].

Learning projections, for which labeled samples of the target domain fall on the correct side of a large margin classifier trained on the source data, have also been proposed [START_REF] Hoffman | Efficient learning of domain invariant image representations[END_REF]. Several works based on extraction of common features under pairwise constraints have also been introduced as domain adaptation strategies [START_REF] Ham | Semisupervised alignment of manifolds[END_REF], [START_REF]Heterogeneous domain adaptation using manifold alignment[END_REF], [START_REF] Tuia | Kernel manifold alignment for domain adaptation[END_REF].

The second family tackles the domain adaptation problem assuming, as in this paper, that no labels are available in the target domain. Besides works dealing with sample reweighting [START_REF] Sugiyama | Direct importance estimation with model selection and its application to covariate shift adaptation[END_REF], many works have considered finding a common feature representation for the two (or more) domains. Since the representation, or latent space, is common to all domains, projected labeled samples from the source domain can be used to train a classifier that is general [START_REF] Daumé | Frustratingly easy domain adaptation[END_REF], [START_REF]Domain adaptation via transfer component analysis[END_REF]. A common strategy is to propose methods that aim at finding representations in which domains match in some sense. For instance, adaptation can be performed by matching the means of the domains in the feature space [START_REF]Domain adaptation via transfer component analysis[END_REF], aligning the domains by their correlations [START_REF] Kumar | Generalized multiview analysis: A discriminative latent space[END_REF] or by using pairwise constraints [START_REF] Wang | Manifold alignment without correspondence[END_REF]. In most of these works, feature extraction is the key tool for finding a common latent space that embeds discriminative information shared by all domains.

Recently, the unsupervised domain adaptation problem has been revisited by considering strategies based on a gradual alignment of a feature representation. In [START_REF] Gopalan | Domain adaptation for object recognition: An unsupervised approach[END_REF],

authors start from the hypothesis that domain adaptation can be better estimated when comparing gradual dis-October 6, 2016 DRAFT tortions. Therefore, they use intermediary projections of both domains along the Grassmannian geodesic connecting the source and target eigenvectors. In [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF], [START_REF] Zheng | A Grassmann manifold-based domain adaptation approach[END_REF], all sets of transformed intermediary domains are obtained by using a geodesic-flow kernel. While these methods have the advantage of providing easily computable outof-sample extensions (by projecting unseen samples onto the latent space eigenvectors), the transformation defined remains global and is applied in the same way to the whole target domain. An approach combining sample reweighting logic with representation transfer is found in [START_REF] Zhang | Covariate shift in Hilbert space: A solution via surrogate kernels[END_REF], where authors extend the sample re-weighing to reproducing kernel Hilbert space through the use of surrogate kernels. The transformation achieved is again a global linear transformation that helps in aligning domains.

Our proposition strongly differs from those reviewed above, as it defines a local transformation for each sample in the source domain. In this sense, the domain adaptation problem can be seen as a graph matching problem [START_REF] Luo | Structural graph matching using the em algorithm and singular value decomposition[END_REF], [START_REF] Caetano | Grapihcal models and point pattern matching[END_REF], [START_REF] Caetano | Learning graph matching[END_REF] as each source sample has to be mapped on target samples under the constraint of marginal distribution preservation.

Optimal Transport and Machine Learning. The optimal transport problem has first been introduced by the French mathematician Gaspard Monge in the middle of the 19th century as a way to find a minimal effort solution to the transport of a given mass of dirt into a given hole. The problem reappeared in the middle of the 20th century in the work of Kantorovitch [START_REF] Kantorovich | On the translocation of masses[END_REF] and found recently surprising new developments as a polyvalent tool for several fundamental problems [START_REF] Villani | Optimal transport: old and new, ser. Grundlehren der mathematischen Wissenschaften[END_REF].

It was applied in a wide panel of fields, including computational fluid mechanics [START_REF] Benamou | A computational fluid mechanics solution to the monge-kantorovich mass transfer problem[END_REF], color transfer between multiple images or morphing in the context of image processing [START_REF] Rabin | Wasserstein barycenter and its application to texture mixing[END_REF], [START_REF] Ferradans | Regularized discrete optimal transport[END_REF], [START_REF] Bonneel | Sliced and radon Wasserstein barycenters of measures[END_REF], interpolation schemes in computer graphics [START_REF] Bonneel | Displacement interpolation using Lagrangian mass transport[END_REF], and economics, via matching and equilibriums problems [START_REF] Carlier | Numerical methods for matching for teams and Wasserstein barycenters[END_REF].

Despite the appealing properties and application success stories, the machine learning community has considered optimal transport only recently (see, for instance, works considering the computation of distances between histograms [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transportation[END_REF] or label propagation in graphs [START_REF] Solomon | Wasserstein propagation for semi-supervised learning[END_REF]); the main reason being the high computational cost induced by the computation of the optimal transportation plan.

However, new computing strategies have emerged [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transportation[END_REF], [START_REF] Cuturi | Fast computation of Wasserstein barycenters[END_REF], [START_REF] Bonneel | Sliced and radon Wasserstein barycenters of measures[END_REF] and made possible the application of OT distances in operational settings.

II. OPTIMAL TRANSPORT AND APPLICATION TO DOMAIN ADAPTATION

In this section, we present the general unsupervised domain adaptation problem and show how it can be addressed from an optimal transport perspective.

A. Problem and theoretical motivations

Let Ω ∈ R d be an input measurable space of dimension d and C the set of possible labels. P(Ω) denotes the set of all probability measures over Ω. The standard learning paradigm assumes the existence of a set of training data X s = {x s i } Ns i=1 associated with a set of class labels Y s = {y s i } Ns i=1 , with y s i ∈ C, and a testing set X t = {x t i } Nt i=1 with unknown labels. In order to infer the set of labels Y t associated with X t , one usually relies on an empirical estimate of the joint probability distribution P(x, y) ∈ P(Ω × C) from (X s , Y s ), and assumes that X s and X t are drawn from the same distribution P(x) ∈ P(Ω).

B. Domain adaptation as a transportation problem

In domain adaptation problems, one assumes the existence of two distinct joint probability distributions October 6, 2016 DRAFT P s (x s , y) and P t (x t , y), respectively related to a source and a target domains, noted as Ω s and Ω t . In the following, µ s and µ t are their respective marginal distributions over X. We also denote f s and f t the true labeling functions, i.e. the Bayes decision functions in each domain.

At least one of the two following assumptions is generally made by most domain adaptation methods:

• Class imbalance: Label distributions are different in the two domains (P s (y) = P t (y)), but the conditional distributions of the samples with respect to the labels are the same (P s (x s |y) = P t (x t |y));

• Covariate shift: Conditional distributions of the labels with respect to the data are equal (P s (y|x s ) = P t (y|x t ), or equivalently f s = f t = f ). However, data distributions in the two domains are supposed to be different (P s (x s ) = P t (x t )). For the adaptation techniques to be effective, this difference needs to be small [START_REF] Ben-David | Impossibility theorems for domain adaptation[END_REF].

In real world applications, the drift occurring between the source and the target domains generally implies a change in both marginal and conditional distributions.

In our work, we assume that the domain drift is due to an unknown, possibly nonlinear map of the input space T : Ω s → Ω t . This map may have a physical interpretation (e.g. change in the acquisition conditions, sensor drifts, thermal noise, etc.). It can also be directly caused by the unknown process that generates the data.

Additionnally, we also suppose that the transformation preserves the conditional distribution, i.e.

P s (y|x s ) = P t (y|T(x s )).

This means that the label information is preserved by the transformation, and the Bayes decision functions are tied through the equation f t (T(x)) = f s (x).

Another insight can be provided regarding the transformation T. From a probabilistic point of view, T transforms the measure µ in its image measure, noted T#µ, which is another probability measure over Ω t satisfying

T#µ(x) = µ(T -1 (x)), ∀x ⊂ Ω t (1)
T is said to be a transport map or push-forward from µ s to µ t if T#µ s = µ t (as illustrated in Figure 2.a).

Under this assumption, X t are drawn from the same PDF as T#µ s . This provides a principled way to solve the adaptation problem:

1) Estimate µ s and µ t from X s and X t (Equation ( 6))

2) Find a transport map T from µ s to µ t

3) Use T to transport labeled samples X s and train a classifier from them.

Searching for T in the space of all possible transformations is intractable, and some restrictions need to be imposed. Here, we propose that T should be chosen so as to minimize a transportation cost C(T) expressed as:

C(T) = Ωs c(x, T(x))dµ(x), (2)
where the cost function c : Ω s × Ω t → R + is a distance function over the metric space Ω. C(T) can be interpreted as the energy required to move a probability mass µ(x) from x to T(x).

The problem of finding such a transportation of minimal cost has already been investigated in the literature. For instance, the optimal transportation problem as defined by Monge is the solution of the following minimization problem:

T 0 = argmin T Ωs c(x, T(x))dµ(x), s.t. T#µ s = µ t (3)
The Kantorovitch formulation of the optimal transportation [START_REF] Kantorovich | On the translocation of masses[END_REF] is a convex relaxation of the above Monge

γ 0 = argmin γ∈Π Ωs×Ωt c(x s , x t )dγ(x s , x t ) (4)
In this formulation, γ can be understood as a joint probability measure with marginals µ s and µ t as depicted in Figure 2.b. γ 0 is also known as transportation plan [START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF]. It allows to define the Wasserstein distance of order p between µ s and µ t . This distance is formalized as

W p (µ s , µ t ) def = inf γ∈Π Ωs×Ωt d(x s , x t ) p dγ(x s , x t ) 1 p = inf γ∈Π E x s ∼µs,x t ∼µt d(x s , x t ) p 1 p (5)
where d is a distance and the corresponding cost function c(x s , x t ) = d(x s , x t ) p . The Wasserstein distance is also known as the Earth Mover Distance in the computer vision community [START_REF] Rubner | A metric for distributions with applications to image databases[END_REF] and it defines a metric over the space of integrable squared probability measures.

In the remainder, we consider the squared 2 Euclidean distance as a cost function, c(x, y) = xy 2 2 for computing optimal transportation. As a consequence, we evaluate distances between measures according to the squared Wasserstein distance W 2 2 associated with the Euclidean distance d(x, y) = xy 2 . The main rationale for this choice is that it experimentally provided the best result on average (as shown in the supplementary material). Nevertheless, other cost functions better suited to the nature of specific data can be considered, depending on the application at hand and the data representation, as discussed more in details in Section III-D.

III. REGULARIZED DISCRETE OPTIMAL TRANSPORT

This section discusses the problem of optimal transport for domain adaptation. In the first part, we introduce the OT optimization problem on discrete empirical distributions. Then, we discuss a regularized variant of this discrete optimal transport problem. Finally, we address the question of how the resulting probabilistic coupling can be used for mapping samples from source to target domain.

A. Discrete optimal transport

When µ s and µ t are only accessible through discrete samples, the corresponding empirical distributions can be written as

µ s = ns i=1 p s i δ x s i , µ t = nt i=1 p t i δ x t i (6)
where δ xi is the Dirac function at location

x i ∈ R d .
p s i and p t i are probability masses associated to the ith sample and belong to the probability simplex, i.e.

ns i=1 p s i = nt i=1 p t i = 1.
It is straightforward to adapt the Kantorovich formulation of optimal transport problem to the discrete case. We denote B the set of probabilistic couplings between the two empirical distributions defined as:

B = γ ∈ (R + ) ns×nt | γ1 nt = µ s , γ T 1 ns = µ t (7)
where 1 d is a d-dimensional vector of ones. The Kantorovitch formulation of the optimal transport [START_REF] Kantorovich | On the translocation of masses[END_REF] reads:

γ 0 = argmin γ∈B γ, C F (8)
where ., . F is the Frobenius dot product and C ≥ 0 is the cost function matrix, whose term C(i, j) = c(x s i , x t j ) denotes the cost to move a probability mass from x s i to x t j . As previously detailed, this cost was chosen as the squared Euclidean distance between the two locations,

i.e. C(i, j) = ||x s i -x t j || 2 2 .
October 6, 2016 DRAFT Note that when n s = n t = n and ∀i, j p s i = p t j = 1/n, γ 0 is simply a permutation matrix. In this case, the optimal transport problem boils down to an optimal assignment problem. In the general case, it can be shown that γ 0 is a sparse matrix with at most n s + n t -1 non zero entries, equating the rank of the constraint matrix expressing the two marginal constraints.

Problem ( 8) is a linear program and can be solved with combinatorial algorithms such as the simplex methods and its network variants (successive shortest path algorithms, Hungarian or relaxation algorithms). Yet, the computational complexity was shown to be O((n s + n t )n s n t log(n s +n t )) [1, p. 472, Th. 12.2] at best, which dampens the utility of the method when handling large datasets. However, the regularization scheme recently proposed by Cuturi [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transportation[END_REF] presented in the next section, allows a very fast computation of a transportation plan.

B. Regularized optimal transport

Regularization is a classical approach used for preventing overfitting when few samples are available for learning. It can also be used for inducing some properties on the solution. In the following, we discuss a regularization term recently introduced for optimal transport problem.

Cuturi [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transportation[END_REF] proposed to regularize the expression of the optimal transport problem by the entropy of the probabilistic coupling. The resulting informationtheoretic regularized version of the transport γ λ 0 is the solution of the minimization problem:

γ λ 0 = argmin γ∈B γ, C F + λΩ s (γ), (9)
where Ω s (γ) = i,j γ(i, j) log γ(i, j) computes the negentropy of γ. The intuition behind this form of regularization is the following: since most elements of γ 0 are 0 as the solution of a linear program, one can look for a smoother version of the transport, thus lowering its sparsity, by increasing its entropy. As a result, the optimal transport γ λ 0 will have a denser coupling between the distributions. Ω s (•) can also be interpreted as a Kullback-Leibler divergence KL(γ γ u ) between the joint probability γ and a uniform joint probability γ u (i, j) = 1 nsnt . Indeed, by expanding this KL divergence, we have KL(γ γ u ) = log n s n t + i,j γ(i, j) log γ(i, j). The first term is a constant w.r.t. γ, which means that we can equivalently use KL(γ γ u )

or Ω s (γ) = i,j γ(i, j) log γ(i, j) in Equation ( 9).

Hence, as the parameter λ weighting the entropybased regularization increases, the sparsity of γ λ 0 decreases and source points tend to distribute their probability masses toward more target points. When λ becomes very large (λ → ∞), the OT solution of Equation ( 9) converges toward γ λ 0 (i, j) → 1 nsnt , ∀i, j. Another appealing outcome of the regularized OT formulation given in Equation ( 9) is the derivation of a computationally efficient algorithm based on Sinkhorn-Knopp's scaling matrix approach [START_REF] Knight | The sinkhorn-knopp algorithm: Convergence and applications[END_REF]. This efficient algorithm will also be a key element in our methodology presented in Section IV.

C. OT-based mapping of the samples

In the context of domain adaptation, once the probabilistic coupling γ 0 has been computed, source samples have to be transported in the target domain. For this purpose, one can interpolate the two distributions µ s and µ t by following the geodesics of the Wasserstein metric [START_REF] Villani | Optimal transport: old and new, ser. Grundlehren der mathematischen Wissenschaften[END_REF]Chapter 7], parameterized by t ∈ [0, 1]. This defines a new distribution μ such that:

μ = argmin µ (1 -t)W 2 (µ s , µ) 2 + tW 2 (µ t , µ) 2 . (10)
Still following Villani's book, one can show that for a October 6, 2016 DRAFT squared 2 cost, this distribution boils down to:

μ = i,j γ 0 (i, j)δ (1-t)x s i +tx t j . (11)
Since our goal is to transport the source samples onto the target distribution, we are mainly interested in the case t = 1. For this value of t, the novel distribution μ is a distribution with the same support of µ t , since Equation ( 11) reduces to

μ = j pt j δ x t j . ( 12
)
with pt j = i γ 0 (i, j). The weights pt j can be seen as the sum of probability mass coming from all samples {x s i } that is transferred to sample x t j . Alternatively, γ 0 (i, j) also tells us how much probability mass of x s i is transferred to x t j . We can exploit this information to compute a transformation of the source samples. This transformation can be conveniently expressed with respect to the target samples as the following barycentric mapping:

x s i = argmin x∈R d j γ 0 (i, j)c(x, x t j ). (13)
where x s i is a given source sample and x s i is its corresponding image. When the cost function is the squared 2 distance, this barycenter corresponds to a weighted average and the sample is mapped into the convex hull of the target samples. For all source samples, this barycentric mapping can therefore be expressed as:

Xs = T γ 0 (X s ) = diag(γ 0 1 nt ) -1 γ 0 X t . ( 14
)
The inverse mapping from the target to the source domain can also be easily computed from γ T 0 . Interestingly, one can show [START_REF] Cuturi | Fast computation of Wasserstein barycenters[END_REF]Eq. 8] that this transformation is a first order approximation of the true n s Wasserstein barycenters of the target distributions. Also note that when marginals µ s and µ t are uniform, one can easily derive the barycentric mapping as a linear expression:

Xs = n s γ 0 X t and Xt = n t γ 0 X s (15)
for the source and target samples.

Finally, remark that if γ 0 (i, j) = 1 nsnt , ∀i, j, then each transported source point converges toward the center of mass of the target distribution that is 1 nt j x t j . This occurs when λ → ∞ in Equation ( 9).

D. Discussion

We discuss here the requirements and conditions of applicability of the proposed method.

Guarantees of recovery of the correct transformation.

Our goal for achieving domain adaptation is to uncover the transformation that occurred between source and target distributions. While the family of transformation that an OT formulation can recover is wide, we provide a proof that, for some simple affine transformations of discrete distributions, our OT solution is able to match source and target examples exactly. Theorem 3.1: Let µ s and µ t be two discrete distributions with n Diracs as defined in Equation [START_REF] Bonneel | Displacement interpolation using Lagrangian mass transport[END_REF]. If the following conditions hold 1) The source samples in µ s are x s i ∈ R d , ∀i ∈ 1, . . . , n such that x s i = x s j if i = j . 2) All weights in the source and target distributions are 1 n .

3) The target samples are defined as x t i = Ax s i + b i.e. an affine tranformation of the source samples. 4) b ∈ R d and A ∈ S + is a strictly positive definite matrix.

5) The cost function is

c(x s , x t ) = x s -x t 2 2 .
then the solution T 0 of the optimal transport problem ( 8) is so that

T 0 (x s i ) = Ax s i + b = x t i ∀i ∈ 1, . . . , n.
In this case, we retrieve the exact affine transformation on the discrete samples, which means that the label Part II], or learnt metrics [START_REF] Cuturi | Ground metric learning[END_REF]. Concave cost functions are also of particular use in real life problems [START_REF] Gangbo | The geometry of optimal transportation[END_REF].

Each different cost function will lead to a different OT plan γ 0 , but the cost itself does not impact the OT optimization problem, i.e. the solver is independent from the cost function. Nonetheless, since c(•, •) defines the Wasserstein geodesic, the interpolation between domains defined in Equation ( 10) leads to a different trajectory (potentially non-unique). Equation [START_REF] Caetano | Learning graph matching[END_REF], which corresponds to c(•, •), is a squared 2 distance, so it does not hold anymore. Nevertheless, the solution of [START_REF] Caetano | Grapihcal models and point pattern matching[END_REF] for t = 1 does not depend on the cost c and one can still use the proposed barycentric mapping [START_REF] Carreira-Perpinan | LASS: A simple assignment model with laplacian smoothing[END_REF].

For instance if the cost function is based on the 1 norm, the transported samples will be estimated using a component-wise weighted median. Unfortunately, for more complex cost functions, the barycentric mapping might be complex to estimate.

IV. CLASS-REGULARIZATION FOR DOMAIN

ADAPTATION

In this section we explore regularization terms that preserve label information and sample neighborhood during transportation. Finally, we discuss the semi-supervised case and show that label information in the target domain can be effectively included in he proposed model.

A. Regularizing the transport with class labels

Optimal transport, as it has been presented in the previous section, does not use any class information.

However, and even if our goal is unsupervised domain adaptation, class labels are available in the source domain. This information is typically used only during the decision function learning stage, which follows the adaptation step. Our proposition is to take advantage of the label information for estimating a better transport.

More precisely, we aim at penalizing couplings that match source samples with different labels to same target samples.

To this end, we propose to add a new term to the regularized optimal transport, leading to the following optimization problem: regularizer is that we would like each target sample to receive masses only from source samples that have the same label. As a consequence, we expect that a given target sample will be involved in the representation of transported source samples as defined in Equation ( 14), but only for samples from the source domain of the same class. This behaviour can be induced by means of a group-sparse penalty on the columns of γ.

min γ∈B γ, C F + λΩ s (γ) + ηΩ c (γ), (16)
This approach has been introduced in our preliminary work [START_REF] Courty | Domain adaptation with regularized optimal transport[END_REF]. In that paper, we proposed a p -1 regularization term with p < 1 (mainly for algorithmic reasons).

When applying a majoration-minimization technique on the p -1 norm, the problem can be cast as problem [START_REF] Bruzzone | Domain adaptation problems: A dasvm classification technique and a circular validation strategy[END_REF] and can be solved using the efficient Sinkhorn-Knopp algorithm at each iteration. However, this regularization term with p < 1 is non-convex and thus the proposed algorithm is guaranteed to converge only to local stationary points.

In this paper, we retain the convexity of the underlying problem and use the convex group-lasso regularizer 1 -2 instead. This regularizer is defined as Ideally, with this regularizer we expect that the masses corresponding to each group of labels are matching samples of the source and target domains exclusively. Hence, for the domain adaptation problem to have a relevant solution, the distributions of labels are expected to be preserved in both the source and target distributions. We thus need to have P s (y) = P t (y). This assumption, which is a classical assumption in the field of learning, is nevertheless a mild requirement since, in practice, small October 6, 2016 DRAFT deviations of proportions do not prevent the method from working (see reference [START_REF] Tuia | Multitemporal classification without new labels: a solution with optimal transport[END_REF] for experimental results on this particular issue).

Ω c (γ) = j cl ||γ(I cl , j)|| 2 , (17)
2) Laplacian regularization: This regularization term aims at preserving the data structure -approximated by a graph -during transport [START_REF] Ferradans | Regularized discrete optimal transport[END_REF], [START_REF] Carreira-Perpinan | LASS: A simple assignment model with laplacian smoothing[END_REF]. Intuitively, we would like similar samples in the source domain to also be similar after transportation. Hence, denote as xs i the transported source sample x s i , with xs i being linearly dependent on the transportation matrix γ through Equation [START_REF] Courty | Domain adaptation with regularized optimal transport[END_REF]. Now, given a positive symmetric similarity matrix S s of samples in the source domain, our regularization term is defined as

Ω c (γ) = 1 N 2 s i,j S s (i, j) xs i -xs j 2 2 , (18)
where S s (i, j) ≥ 0 are the coefficients of matrix S s ∈ R Ns×Ns that encodes similarity between pairs of source sample. In order to further preserve class structures, we can sparsify similarities for samples of different classes.

In practice, we thus impose S s (i, j) = 0 if y s i = y s j . The above equation can be simplified when the marginal distributions are uniform. In that case, transported source samples can be computed according to Equation [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transportation[END_REF]. Hence, Ω c (γ) boils down to

Ω c (γ) = Tr(X t γ L s γX t ), (19)
where L s = diag(S s 1)-S s is the Laplacian of the graph S s . The regularizer is therefore quadratic w.r.t. γ.

The regularization terms [START_REF] Daumé | Frustratingly easy domain adaptation[END_REF] or ( 19) are defined based on the transported source samples. When a similarity information is also available in the target samples, for instance, through a similarity matrix S t , we can take advantage of this knowledge and a symmetric Laplacian regularization of the form

Ω c (γ) = (1 -α)Tr(X t γ L s γX t ) + αTr(X s γL t γ X s ) (20)
can be used instead. In the above equation L t = diag(S t 1)-S t is the Laplacian of the graph in the target domain and 0 ≤ α ≤ 1 is a trade-off parameter that weights the importance of each part of the regularization term. Note that, unlike the matrix S s , the similarity matrix S t cannot be sparsified according to the class structure, since labels are generally not available for the target domain.

A regularization term similar to Ω c (γ) has been proposed in [START_REF] Ferradans | Regularized discrete optimal transport[END_REF] for histogram adaptation between images. However, the authors focused on displacements (x s i -x s i ) instead of on preserving the class structure of the transported samples.

B. Regularizing for semi-supervised domain adaptation

In semi-supervised domain adaptation, few labelled samples are available in the target domain [START_REF] Wang | Manifold alignment[END_REF]. Again, such an important information can be exploited by means of a novel regularization term to be integrated in the original optimal transport formulation. This regularization term is designed such that samples in the target domain should only be matched with samples in the source domain that have the same labels. It can be expressed as:

Ω semi (γ) = γ, M (21)
where M is a n s × n t cost matrix, with M(i, j) = 0 whenever y s i = y t j (or j is a sample with unknown label) and +∞ otherwise. This term has the benefit to be parameter free. It boils down to changing the original cost function C, defined in Equation [START_REF] Bredies | Equivalence of a generalized conditional gradient method and the method of surrogate functionals[END_REF], by adding an infinite cost to undesired matches. Smooth versions of this regularization can be devised, for instance, by using a probabilistic confidence of target sample x t j to belong to class y t j . Though appealing, we have not explored this latter option in this work. It is also noticeable that the October 6, 2016 DRAFT Laplacian strategy in Equation ( 20) can also leverage on these class labels in the target domain through the definition of matrix S t .

V. GENERALIZED CONDITIONAL GRADIENT FOR SOLVING REGULARIZED OT PROBLEMS

In this section, we discuss an efficient algorithm for solving optimization problem [START_REF] Cuturi | Ground metric learning[END_REF], that can be used with any of the proposed regularizers.

Firstly, we characterize the existence of a solution to the problem. We remark that regularizers given in Equations ( 17) and ( 18) are continuous, thus the objective function is continuous. Moreover, since the constraint set B is a convex, closed and bounded (hence compact) subset of R d , the objective function reaches its minimum on B. In addition, if the regularizer is strictly convex that minimum is unique. This occurs for instance, for the Laplacian regularization in Equation ( 18). Now, let us discuss algorithms for computing optimal transport solution of problem [START_REF] Cuturi | Ground metric learning[END_REF]. For solving a similar problem with a Laplacian regularization term, Ferradans et al. [START_REF] Ferradans | Regularized discrete optimal transport[END_REF] used a conditional gradient (CG) algorithm [START_REF] Bertsekas | Nonlinear programming[END_REF]. This approach is appealing and could be extended In this work, we aim for a more scalable algorithm. To this end, we consider an approach based on a generalization of the conditional gradient algorithm [START_REF] Bredies | A generalized conditional gradient method and its connection to an iterative shrinkage method[END_REF] denoted as generalized conditional gradient (GCG).

to
The framework of the GCG algorithm addresses the general case of constrained minimization of composite functions defined as function. This approach is justified when the resulting nonlinear optimization problem can be efficiently solved.

min γ∈B f (γ) + g(γ), (22)
The GCG algorithm has been shown by Bredies et al. [START_REF] Bredies | Equivalence of a generalized conditional gradient method and the method of surrogate functionals[END_REF] to converge towards a stationary point of Problem [START_REF] Germain | A PAC-Bayesian Approach for Domain Adaptation with Specialization to Linear Classifiers[END_REF]. In our case, since g(γ) is differentiable, stronger convergence results can be provided (see supplementary material for a discussion on convergence rate and duality gap monitoring).

More specifically, for problem [START_REF] Cuturi | Ground metric learning[END_REF] we can set

f (γ) = γ, C F + ηΩ c (γ) and g(γ) = λΩ s (γ). Supposing now that Ω c (γ) is differentiable, step 3 of Algorithm 1 boils down to γ = argmin γ∈B γ, C + η∇Ω c (γ k ) F + λΩ s (γ)
Interestingly, this problem is an entropy-regularized optimal transport problem similar to Problem (9) and can be efficiently solved using the Sinkhorn-Knopp scaling matrix approach.

In our optimal transport problem, Ω c (γ) is instantiated by the Laplacian or the group-lasso regularization term. Find the optimal step α k

α k = argmin 0≤α≤1 f (γ k + α∆γ) + g(γ k + α∆γ)
with ∆γ = γ * -γ k 5:

γ k+1 ← γ k + α k ∆γ, set k ← k + 1 6: until Convergence
The former is differentiable whereas the group-lasso is not when there exists a class cl and an index j for which γ(I cl , j) is a vector of 0. However, one can note that if the iterate γ k is so that γ k (I cl , j) = 0 ∀cl, ∀j, then the same property holds for γ k+1 . This is due to the exponentiation occurring in the Sinkhorn-Knopp algorithm used for the entropy-regularized optimal transport problem. This means that if we initialize γ 0 so that γ 0 (I cl , j) = 0, then Ω c (γ k ) is always differentiable. Hence, our GCG algorithm can also be applied to the group-lasso regularization, despite its nondifferentiability in 0.

VI. NUMERICAL EXPERIMENTS

In this section, we study the behavior of four different versions of optimal transport applied to DA problem. In the rest of the section, OT-exact is the original transport problem [START_REF] Bredies | Equivalence of a generalized conditional gradient method and the method of surrogate functionals[END_REF], OT-IT the Information theoretic regularized one [START_REF] Bruzzone | Domain adaptation problems: A dasvm classification technique and a circular validation strategy[END_REF], and the two proposed class-based regularized ones are denoted OT-GL and OT-Laplace, corresponding respectively to the group-lasso (Equation ( 17)) and Laplacian (Equation ( 18)) regularization terms. We also present some results with our previous class-label based regularizer built upon an p -1 norm: OT-LpL1 [START_REF] Courty | Domain adaptation with regularized optimal transport[END_REF].

A. Two moons: simulated problem with controllable complexity

In the first experiment, we consider the same toy example as in [START_REF] Germain | A PAC-Bayesian Approach for Domain Adaptation with Specialization to Linear Classifiers[END_REF]. The simulated dataset consists of two domains: for the source, the standard two entangled moons data, where each moon is associated to a specific class (See Figure 3(a)). The target domain is built by applying a rotation to the two moons, which allows to consider an adaptation problem with an increasing difficulty as a function of the rotation angle. This example is notably interesting because the corresponding problem is clearly non-linear, and because the input dimensionality is small, 2, which leads to poor performances when applying methods based on subspace alignment (e.g. [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF], [START_REF] Long | Transfer feature learning with joint distribution adaptation[END_REF]).

We follow the same experimental protocol as in [START_REF] Germain | A PAC-Bayesian Approach for Domain Adaptation with Specialization to Linear Classifiers[END_REF],

thus allowing for a direct comparison with the state-ofthe-art results presented therein. The source domain is composed of two moons of 150 samples each. The target domain is also sampled from these two shapes, with the same number of examples. Then, the generalization capability of our method is tested over a set of 1000 samples that follow the same distribution as the target domain. The experiments are conducted 10 times, and we consider the mean classification error as comparison criterion. As a classifier, we used a SVM with a Gaussian kernel, whose parameters were set by 5-fold crossvalidation. We compare the adaptation results with two state-of-the-art methods: the DA-SVM approach [START_REF] Bruzzone | Domain adaptation problems: A dasvm classification technique and a circular validation strategy[END_REF] and the more recent PBDA [START_REF] Germain | A PAC-Bayesian Approach for Domain Adaptation with Specialization to Linear Classifiers[END_REF], which has proved to provide competitive results over this dataset.

Results are reported in Table I. Our first observation is that all the methods based on optimal transport behave better than the state-of-the-art methods, in particular for low rotation angles, where results indicate that the geometrical structure is better preserved through the adaptation by optimal transport. Also, for large angle (e.g. 90 • ), the final score is also significantly better than other state-of-the-art method, but falls down to a 0.5

error rate, which is natural since in this configuration a transformation of -90 • , implying an inversion of labels, would have led to similar empirical distributions. This clearly shows the capacity of our method to handle large domain transformations. Adding the class-label information into the regularization also clearly helps for the midrange angle values, where the adaptation shows nearly optimal results up to angles < 40 • . For the strongest deformation (> 70 • rotation), no clear winner among the OT methods can be found. We think that, regardless of the amount and type of regularization chosen, the classification of test samples becomes too much tributary of the training samples. These ones mostly come from the denser part of µ s and as a consequence, the less dense parts of this PDF are not satisfactorily transported. This behavior can be seen in Figure 3d.

B. Visual adaptation datasets

We now evaluate our method on three challenging real world vision adaptation tasks, which have attracted a lot of interest in recent computer vision literature [START_REF] Patel | Visual domain adaptation: an overview of recent advances[END_REF]. We start by presenting the datasets, then the experimental protocol, and finish by providing and discussing the results obtained.

1) Datasets: Three types of image recognition problems are considered: digits, faces and miscellaneous objects recognition. This choice of datasets was already featured in [START_REF] Long | Transfer feature learning with joint distribution adaptation[END_REF]. A summary of the properties of each domain considered in the three problems is provided in Table II. An illustration of some examples of the different domains for a particular class is shown in Object recognition. We used the Caltech-Office dataset [START_REF] Saenko | Adapting visual category models to new domains[END_REF], [START_REF] Gopalan | Domain adaptation for object recognition: An unsupervised approach[END_REF], [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF], [START_REF] Zheng | A Grassmann manifold-based domain adaptation approach[END_REF], [START_REF] Patel | Visual domain adaptation: an overview of recent advances[END_REF]. The dataset contains images coming from four different domains: Amazon (online merchant), the Caltech-256 image collec-October 6, 2016 DRAFT tion [START_REF] Griffin | Caltech-256 Object Category Dataset[END_REF], Webcam (images taken from a webcam) and DSLR (images taken from a high resolution digital SLR camera). The variability of the different domains come from several factors: presence/absence of background, lightning conditions, noise, etc. We consider two feature sets:

• SURF descriptors as described in [START_REF] Saenko | Adapting visual category models to new domains[END_REF], used to transform each image into a 800 bins histogram.

These histograms are subsequently normalized and reduced to standard scores. • PCA, which consists in applying a projection on the first principal components of the joint source/target distribution (estimated from the concatenation of source and target samples);

• GFK, Geodesic Flow Kernel [START_REF] Gong | Geodesic flow kernel for unsupervised domain adaptation[END_REF];

• TSL, Transfer Subspace Learning [START_REF] Si | Bregman divergence-based regularization for transfer subspace learning[END_REF], which operates by minimizing the Bregman divergence between the domains embedded in lower dimensional spaces;

• JDA, Joint Distribution Adaptation [START_REF] Long | Transfer feature learning with joint distribution adaptation[END_REF], which extends the Transfer Component Analysis algorithm [START_REF]Domain adaptation via transfer component analysis[END_REF];

In unsupervised DA no target labels are available.

As a consequence, it is impossible to consider a crossvalidation step for the hyper-parameters of the different methods. However, and in order to compare the methods fairly, we follow the following protocol. For each source domain, a random selection of 20 samples per class (with the only exception of 8 for the DSLR dataset) is adopted.

Then the target domain is equivalently partitioned in a validation and test sets. The validation set is used to obtain the best accuracy in the range of the possible hyper-parameters. The accuracy, measured as the percent of correct classification over all the classes, is then evaluated on the testing set, with the best selected hyperparameters. This strategy normally prevents overfitting on the testing set. The experimentation is conducted 10 times, and the mean accuracy over all these realizations is reported.

We considered the following parameter range : for subspace learning methods (PCA,TSL, GFK, and JDA)

we considered reduced k-dimensional spaces with k ∈ {10, 20, . . . , 70}. A linear kernel was chosen for all the methods with a kernel formulation. For the all methods requiring a regularization parameter, the best value was searched in λ = {0.001, 0.01, 0. C→A ) where it achieves better scores. On both digits and objects recognition tasks, OT-GL significantly outperforms the baseline methods.

In the next experiment ( the proposed OT methods are still capable of improving significantly the performances of the final classification (up to more than 20 points in some case, e.g. D→A or A→W). This clearly shows how OT has the capacity to handle non-stationarity in the distributions that the deep architecture has difficulty handling. We also note that using the features from the 7th layer instead of the 6th does not bring a strong improvement in the classification accuracy, suggesting that part of the work of the 7th layer is already performed by the optimal transport. 4) Semi-supervised domain adaptation: In this last experiment, we assume that few labels are available in the target domain. We thus benchmark our semisupervised approach on SURF features extracted from the Office-Caltech dataset. We consider that only 3 labeled samples per class are at our disposal in the target domain. In order to disentangle the benefits of the labeled target samples brought by our optimal transport strategies from those brought by the classifier, we make a distinction between two cases: in the first one, denoted as "Unsupervised + labels", we consider that the label target samples are available only at the learning stage, after an unsupervised domain adaptation with optimal transport.

In the second case, denoted as "semi-supervised", labels in the target domain are used to compute a new transportation plan, through the use of the proposed semisupervised regularization term in Equation ( 21)).

Results are reported in Table V. They clearly show the benefits of the proposed semi-supervised regularization term in the definition of the transportation plan. A comparison with the state-of-the-art method of Hoffman and colleagues [START_REF]Efficient learning of domain-invariant image representations[END_REF] is also reported, and shows the competitiveness of our approach. This can be included with dedicated regularization terms.

We also plan to extend our optimal transport framework to the multi-domain adaptation problem, where the problem of matching several distributions can be cast as a multi-marginal optimal transport problem.

Fig. 1 :

 1 Fig. 1: Illustration of the proposed approach for domain adaptation. (left) dataset for training, i.e. source domain, and testing, i.e. target domain. Note that a classifier estimated on the training examples clearly does not fit the target data. (middle) a data dependent transportation map T γ0 is estimated and used to transport the training samples onto the target domain. Note that this transformation is usually not linear. (right) the transported labeled samples are used for estimating a classifier in the target domain.

October 6

 6 , 2016 DRAFT information are fully preserved during transportation. Therefore, one can train a classifier on the mapped samples with no generalization loss. We provide a simple demonstration in the supplementary material. Choosing the cost function. In this work, we have mainly considered a 2 -based cost function. Let us now discuss the implication of using a different cost function in our framework. A number of norm-based distances have been investigated by mathematicians [49, p 972].Other types of metrics can also be considered, such as Riemannian distances over a manifold[49,

where η ≥ 0 1 )Fig. 2 :

 012 Fig. 2: Illustration of the optimal transport problem. (a) Monge problem over 2D domains. T is a push-forward from Ω s to Ω t . (b) Kantorovich relaxation over 1D domains: γ can be seen as a joint probability distribution with marginals µ s and µ t . (c) Illustration of the solution of the Kantorovich relaxation computed between two ellipsoidal distributions in 2D. The grey line between two points indicate a non-zero coupling between them.

where || • || 2

 2 denotes the 2 norm and I cl contains the indices of rows in γ related to source domain samples of class cl. Hence, γ(I cl , j) is a vector containing coefficients of the jth column of γ associated to class cl. Since the jth column of γ is related to the jth target sample, this regularizer will induce the desired sparse representation in the target sample. Among other benefits, the convexity of the corresponding problem allows to use an efficient generic optimization scheme, presented in Section V.

  where f (•) is a differentiable and possibly non-convex function; g(•) is a convex, possibly non-differentiable function; B denotes any convex and compact subset of R n . As illustrated in Algorithm 1, all the steps of the GCG algorithm are exactly the same as those used for CG, except for the search direction part (Line 3). The difference is that GCG linearizes only part f (•) of the composite objective function, instead of the full objective

1 :: repeat 3 :

 13 Initialize k = 0 and γ 0 ∈ P 2With G ∈ ∇f (γ k ), solve γ = argmin γ∈B γ, G F + g(γ) 4:

Figure 4 .

 4 Figure 4.Digit recognition. As source and target domains, we use the two digits datasets USPS and MNIST, that share 10 classes of digits (single digits 0 -9). We randomly sampled 1, 800 and 2, 000 images from each original dataset. The MNIST images are resized to the same resolution as that of USPS(16 × 16). The grey levels of all images are then normalized to obtain a final common feature space for both domains. Face recognition. In the face recognition experiment, we use the PIE ("Pose, Illumination, Expression") dataset, which contains 32 × 32 images of 68 individuals taken under various pose, illumination and expressions conditions. The 4 experimental domains are constructed by selecting 4 distinct poses: PIE05 (C05, left pose), PIE07 (C07, upward pose), PIE09 (C09, downward pose) and PIE29 (C29, right pose). This allows to define 12 different adaptation problems with increasing difficulty (the most challenging being the adaptation from right to left poses). Let us note that each domain has a strong variability for each class due to illumination and expression variations.

Fig. 3 :

 3 Fig. 3: Illustration of the classification decision boundary produced by OT-Laplace over the two moons example for increasing rotation angles. The source domain is represented as coloured points. The target domain is depicted as points in grey (best viewed with colors).

• 2 )

 2 two DeCAF deep learning features sets [19]: these features are extracted as the sparse activation of the neurons from the fully connected 6th and 7th layers of a convolutional network trained on imageNet and then fine tuned on the visual recognition tasks considered here. As such, they form vectors with 4096 dimensions. Experimental setup: Following [23], the classification is conducted using a 1-Nearest Neighbor (1NN) classifier, which has the advantage of being parameter free. In all experiments, 1NN is trained with the adapted source data, and evaluated over the target data to provide a classification accuracy score. We compare our optimal

Fig. 4 :

 4 Fig. 4: Examples from the datasets used in the visual adaptation experiment. 5 random samples from one class are given for all the considered domains.

  schemes frequently outperform the competing state-ofthe-art methods.We believe that the framework presented in this paper will lead to a paradigm shift for the domain adaptation problem. Estimating a transport is much more general than finding a common subspace, but comes with the problem of finding a proper regularization term. The proposed class-based or Laplacian regularizers show very good performances, but we believe that other types of regularizer should be investigated. Indeed, whenever the transformation is induced by a physical process, one may want the transport map to enforce physical constraints.

  × Ω t ) with marginals µ s and µ t . The Kantorovitch problem seeks for a general coupling γ ∈ Π between Ω s and Ω t :

	problem. Indeed, let us define Π as the set of all
	probabilistic couplings ∈ P(Ω s
	October 6, 2016	DRAFT

  our problem. It is an iterative scheme that guaran-

	tees any iterate to belong to B, meaning that any of
	those iterates is a transportation plan. At each of these
	iterations, in order to find a feasible search direction,
	a CG algorithm looks for a minimizer of the objective
	function's linear approximation . Hence, at each iteration
	it solves a Linear Program (LP) that is presumably
	easier to handle than the original regularized optimal
	transport problem. Nevertheless, and despite existence
	of efficient LP solvers such as CPLEX or MOSEK, the

dimensionality of the LP problem makes this LP problem hardly tractable, since it involves n s × n t variables.

TABLE I :

 I Mean error rate over 10 realizations for the two moons simulated example.

TABLE II :

 II Summary of the domains used in the visual adaptation experiment

TABLE III :

 III Overall recognition accuracies in % obtained over all domains pairs using the SURF features. Maximum values for each pair is indicated in bold font.

	Domains	1NN	PCA	GFK	TSL	JDA	OT-exact OT-IT OT-Laplace OT-LpLq OT-GL
	U→M	39.00 37.83 44.16	40.66	54.52	50.67	53.66	57.42	60.15	57.85
	M→U	58.33 48.05 60.96	53.79	60.09	49.26	64.73	64.72	68.07	69.96
	mean	48.66 42.94 52.56	47.22	57.30	49.96	59.20	61.07	64.11	63.90
	P1→P2	23.79 32.61 22.83	34.29	67.15	52.27	57.73	58.92	59.28	59.41
	P1→P3	23.50 38.96 23.24	33.53	56.96	51.36	57.43	57.62	58.49	58.73
	P1→P4	15.69 30.82 16.73	26.85	40.44	40.53	47.21	47.54	47.29	48.36
	P2→P1	24.27 35.69 24.18	33.73	63.73	56.05	60.21	62.74	62.61	61.91
	P2→P3	44.45 40.87 44.03	38.35	68.42	59.15	63.24	64.29	62.71	64.36
	P2→P4	25.86 29.83 25.49	26.21	49.85	46.73	51.48	53.52	50.42	52.68
	P3→P1	20.95 32.01 20.79	39.79	60.88	54.24	57.50	57.87	58.96	57.91
	P3→P2	40.17 38.09 40.70	39.17	65.07	59.08	63.61	65.75	64.04	64.67
	P3→P4	26.16 36.65 25.91	36.88	52.44	48.25	52.33	54.02	52.81	52.83
	P4→P1	18.14 29.82 20.11	40.81	46.91	43.21	45.15	45.67	46.51	45.73
	P4→P2	24.37 29.47 23.34	37.50	55.12	46.76	50.71	52.50	50.90	51.31
	P4→P3	27.30 39.74 26.42	46.14	53.33	48.05	52.10	52.71	51.37	52.60
	mean	26.22 34.55 26.15	36.10	56.69	50.47	54.89	56.10	55.45	55.88
	C→A	20.54 35.17 35.29	45.25	40.73	30.54	37.75	38.96	48.21	44.17
	C→W	18.94 28.48 31.72	37.35	33.44	23.77	31.32	31.13	38.61	38.94
	C→D	19.62 33.75 35.62	39.25	39.75	26.62	34.50	36.88	39.62	44.50
	A→C	22.25 32.78 32.87 38.46	33.99	29.43	31.65	33.12	35.99	34.57
	A→W	23.51 29.34 32.05	35.70	36.03	25.56	30.40	30.33	35.63	37.02
	A→D	20.38 26.88 30.12	32.62	32.62	25.50	27.88	27.75	36.38	38.88
	W→C	19.29 26.95 27.75	29.02	31.81	25.87	31.63	31.37	33.44	35.98
	W→A	23.19 28.92 33.35	34.94	31.48	27.40	37.79	37.17	37.33	39.35
	W→D	53.62 79.75 79.25	80.50	84.25	76.50	80.00	80.62	81.38	84.00
	D→C	23.97 29.72 29.50	31.03	29.84	27.30	29.88	31.10	31.65	32.38
	D→A	27.10 30.67 32.98	36.67	32.85	29.08	32.77	33.06	37.06	37.17
	D→W	51.26 71.79 69.67	77.48	80.00	65.70	72.52	76.16	74.97	81.06
	mean	28.47 37.98 39.21	42.97	44.34	36.69	42.30	43.20	46.42	47.70
	a different optimization scheme, has globally inferior
	performances, except on some pairs of domains (e.g.

Table IV

 IV

		the DeCAF deep learning architecture [19]. We report
		the results of the experiment conducted on the Office-
		Caltech dataset, with the OT-IT and OT-GL regular-
		ization strategies. For comparison purposes, JDA is also
		considered for this adaptation task. The results show that,
		even though the deep learning features yield naturally a
	), we use the same	strong improvement over the classical SURF features,
	experimental protocol on different features produced by
	October 6, 2016	DRAFT

TABLE IV :

 IV Results of adaptation by optimal transport using DeCAF features.

			Layer 6			Layer 7
	Domains DeCAF	JDA	OT-IT OT-GL DeCAF	JDA	OT-IT	OT-GL
	C→A	79.25	88.04	88.69	92.08	85.27	89.63	91.56	92.15
	C→W	48.61	79.60	75.17	84.17	65.23	79.80	82.19	83.84
	C→D	62.75	84.12	83.38	87.25	75.38	85.00	85.00	85.38
	A→C	64.66	81.28	81.65	85.51	72.80	82.59	84.22	87.16
	A→W	51.39	80.33	78.94	83.05	63.64	83.05	81.52	84.50
	A→D	60.38	86.25	85.88	85.00	75.25	85.50	86.62	85.25
	W→C	58.17	81.97	74.80	81.45	69.17	79.84	81.74	83.71
	W→A	61.15	90.19	80.96	90.62	72.96	90.94	88.31	91.98
	W→D	97.50	98.88	95.62	96.25	98.50	98.88	98.38	91.38
	D→C	52.13	81.13	77.71	84.11	65.23	81.21	82.02	84.93
	D→A	60.71	91.31	87.15	92.31	75.46	91.92	92.15	92.92
	D→W	85.70	97.48	93.77	96.29	92.25	97.02	96.62	94.17
	mean	65.20	86.72	83.64	88.18	75.93	87.11	87.53	88.11

TABLE V :

 V Results of semi-supervised adaptation with optimal transport using the SURF features.

			Unsupervised + labels		Semi-supervised
		Domains	OT-IT	OT-GL	OT-IT	OT-GL	MMDT [28]
		C→A	37.0 ± 0.5	41.4 ± 0.5	46.9 ± 3.4	47.9 ± 3.1	49.4 ± 0.8
		C→W	28.5 ± 0.7	37.4 ± 1.1	64.8 ± 3.0	65.0 ± 3.1	63.8 ± 1.1
		C→D	35.1 ± 1.7	44.0 ± 1.9	59.3 ± 2.5	61.0 ± 2.1	56.5 ± 0.9
		A→C	32.3 ± 0.1	36.7 ± 0.2	36.0 ± 1.3	37.1 ± 1.1	36.4 ± 0.8
		A→W	29.5 ± 0.8	37.8 ± 1.1	63.7 ± 2.4	64.6 ± 1.9	64.6 ± 1.2
	s	A→D	36.9 ± 1.5	46.2 ± 2.0	57.6 ± 2.5	59.1 ± 2.3	56.7 ± 1.3
		W→C	35.8 ± 0.2	36.5 ± 0.2	38.4 ± 1.5	38.8 ± 1.2	32.2 ± 0.8
		W→A	39.6 ± 0.3	41.9 ± 0.4	47.2 ± 2.5	47.3 ± 2.5	47.7± 0.9
		W→D	77.1 ± 1.8 80.2 ± 1.6	79.0 ± 2.8	79.4 ± 2.8	67.0 ± 1.1
		D→C	32.7 ± 0.3	34.7 ± 0.3	35.5 ± 2.1	36.8 ± 1.5	34.1 ± 1.5
		D→A	34.7 ± 0.3	37.7 ± 0.3	45.8 ± 2.6	46.3 ± 2.5	46.9 ± 1.0
		D→W	81.9 ± 0.6	84.5 ± 0.4	83.9 ± 1.4	84.0 ± 1.5	74.1 ± 0.8
		mean	41.8	46.6	54.8	55.6	52.5

October 6, 2016 DRAFT

ACKNOWLEDGMENT

This work was partly funded by the Swiss National Science Foundation under the grant PP00P2-150593 and by the CNRS PEPS Fascido program under the Topase project.


Mélina Christodoulou

Guillain Mauviel
email: guillain.mauviel@univ-lorraine.fr

Jacques Lédé

Pascal Beaurain

Mathieu Weber

Hervé Legall

Francis Billaud

Novel vertical image furnace for fast pyrolysis studies

Keywords: image furnace, temperature, pyrometry, fluidized bed, modeling

Introduction

Ligno-cellulosic biomass is nowadays considered as a very promising renewable carbon resource. It represents a relevant alternative to reduce greenhouse gases emissions. During last decades, many scientists and industrials have pointed out its advantages for energy and chemicals production. However its use requires prior processing. Many options are possible such as the thermochemical routes. Pyrolysis is not only a bio-liquid/bio-char production process, but also the first chemical step inany thermal conversion processes like gasification or combustion. Theunderstanding and modelingof these processesrequirethe careful study of thisprimary pyrolysis step. In this context, this study included in the French GAYAproject, led by GDF Suez and funded by ADEME, which aims to produce Synthetic Natural Gas from ligno-cellulosic biomass. GAYA project will develop a demonstration platform for a new biomass gasification and methanation process. The gasification step produces the CO and H 2 mixture which isnecessary for the methanation step, which produces bio-methane. In order to up-scale and control this process, the gasification reactor needs to be modelled. An improvement of the thermochemical phenomena knowledge is necessary. These phenomena (pyrolysis, char gasification, tar cracking…) should be characterized through laboratory experiments close to the industrials conditions. In this context, our objective is to develop a setup which allows studying the primary solid pyrolysisclose to the conditions of the fluidized bed gasifier encountered in the GAYAproject. More specifically, the biomass particles should have the same spatial and temporal temperature evolutionsin the industrial process and in the laboratory experiment. In the case of an industrial reactor, it is not possible to characterize heat transfer conditions between the hot source and the biomass through asingle criteria, like hot source temperature or biomass heating rate. In fluidized bed, the heat flux transferred to the biomass by hot bed particles occurs by convection, radiation and conduction.Nevertheless the heat flux density could besimply defined by: Φ=hΔT

Since the global heat transfer coefficient h is very high in a fluidized bed (over 200 Wm -2 K -1 ), the local temperature of biomass increases sharply, stabilizes during pyrolysis [START_REF] Lédé | Biomass Pyrolysis: Comments on Some Sources of Confusions in the Definitions of Temperatures and Heating Rates[END_REF] and the resulting char finally tends to the bed temperature. The global heat transfer coefficient could be considered constant, but the heat flux density continuously lowersaccording to the decrease of the temperature difference.In order to reproduce this thermal behaviour, it has been imagined to develop an image furnace since it has the unique ability to provide the very high heat flux densities (more than 100 kWm -2 ) which areobserved in fluidized bed conditions (cf. table 1).

The image furnace is used since a long time for studying chemical reactions but also for simulating solar furnaces. It is based on the principle of focusing the light of a source onto the surface of a solid. First studies started in the 1950's [START_REF] Glaser | Thermal imaging techniques: Proceedings of a conference[END_REF]. The image furnaces presented an alternative to solar furnaces which have certain drawbackslike the variability of the available heat flux density according to the sun position and the weather conditions. It enables to heat rapidly and at high temperatures inert solids(T>2000K). Historically, the carbon arc has been the first type of lightsource. A great number of configurations were developed, depending for example on the methods for light beams concentration (ellipticor parabolic mirrors, etc). Xenon and mercury-xenon arc lampswere implemented in the 1960ssince they were easier to used andmore powerful sources [START_REF] Glaser | Thermal imaging techniques: Proceedings of a conference[END_REF]. A majority of studies concerned inorganic chemistry (measurements of oxides heat capacities, crystal growth, etc) [START_REF] Glaser | Thermal imaging techniques: Proceedings of a conference[END_REF][START_REF] Martin | Diffusion-controlled ignition of cellulosic materials by intense radiant energy[END_REF]. In the 1980s, many pioneering works report the use of the image furnace to study biomass thermal degradation (pyrolysis, gasification) [START_REF] Lede | Pyrolyse-flash de déchets ligno-cellulosiques en vue de leur valorisation par l'énergie solaire concentrée[END_REF][START_REF] Caubet | Thermochemical conversion of biomass: Gasification by flash pyrolysis study[END_REF][START_REF] Chan | Modelling and experimental verification of physical and chemical processes during pyrolysis of a large biomass particle[END_REF][START_REF] Antal | Design and operation of a solar fired biomass flash pyrolysis[END_REF][START_REF] Hopkins | The flash pyrolysis of cellulosic materials using concentrated visible light[END_REF][START_REF] Tabatabaie-Raissi Andm | Design and operation of a 30KWe/2KWth downward facing beam ARC image furnace[END_REF]. Later on, other works related to pyrolysis studiesby means of image furnace have been published [START_REF] Grønli | Mathematical Model for Wood Pyrolysis Comparison of Experimental Measurements with Model Predictions[END_REF]11,[START_REF] Boutin | Flash pyrolysis of cellulose pellets submitted to a concentrated radiation: Experiments and modelling[END_REF]13]. Generally image furnaces provide constant heat flux density [START_REF] Grønli | Mathematical Model for Wood Pyrolysis Comparison of Experimental Measurements with Model Predictions[END_REF]11,[START_REF] Authier | Wood Fast Pyrolysis: Comparison of Lagrangian and Eulerian Modeling Approaches with Experimental Measurements[END_REF]22] since the temperature of the light source is very high (over 4500K for xenon-arc lamp). This is not compatible with the decreasing heat flux density observed in industrial fluidized bed conditions. Consequently a new type of image furnace should be imagined. In order to evaluate if this new image furnace is able to reproduce the thermal conditions encountered in fluidized beds, the temperature evolution of the sample should be characterized.

In pyrolysis experiments, the temperature measurement is always difficult and questionable [START_REF] Lédé | Characterization of biomass fast pyrolysis[END_REF]. This difficulty is illustrated through a short comparison between different pyrolysis setups.

Ourcomparison is limited to five specific setups: a wire mesh reactor [START_REF] Hoekstra | Fast pyrolysis in a novel wiremesh reactor: initial decomposition products of pine wood[END_REF]16], a drop tube reactor [START_REF] Zanzi | Rapid high-temperature pyrolysis of biomass in a free-fall reactor[END_REF][START_REF] Zanzi | Rapid pyrolysis of agricultural residues at high temperature[END_REF], a radiant furnace [START_REF] Di Blasi | Pyrolytic behavior and products of some wood varieties[END_REF][START_REF] Di Blasi | Weight loss dynamics of wood chips under fast radiative heating[END_REF] and two image furnaces [START_REF] Grønli | Mathematical Model for Wood Pyrolysis Comparison of Experimental Measurements with Model Predictions[END_REF]11,[START_REF] Boutin | Flash pyrolysis of cellulose pellets submitted to a concentrated radiation: Experiments and modelling[END_REF][START_REF] Authier | Wood Fast Pyrolysis: Comparison of Lagrangian and Eulerian Modeling Approaches with Experimental Measurements[END_REF]22]. The table 1presentsthe main technical characteristics of the setups and the type of temperature measurement: thermocouple (TC) or pyrometry. Wire mesh reactor [START_REF] Hoekstra | Fast pyrolysis in a novel wiremesh reactor: initial decomposition products of pine wood[END_REF]16] Drop tube reactor [START_REF] Zanzi | Rapid high-temperature pyrolysis of biomass in a free-fall reactor[END_REF][START_REF] Zanzi | Rapid pyrolysis of agricultural residues at high temperature[END_REF] Radiant furnace [START_REF] Di Blasi | Pyrolytic behavior and products of some wood varieties[END_REF][START_REF] Di Blasi | Weight loss dynamics of wood chips under fast radiative heating[END_REF] Horizontal image furnace Vertical image furnace Present study [START_REF] Grønli | Mathematical Model for Wood Pyrolysis Comparison of Experimental Measurements with Model Predictions[END_REF]11] [ [START_REF] Authier | Wood Fast Pyrolysis: Comparison of Lagrangian and Eulerian Modeling Approaches with Experimental Measurements[END_REF]22]

Principle

Particles lie between two wire meshes.An electrical current pass through meshes which serve as resistance heaters.

Particles are transported at the top of the preheated reactor.They fall with the preheated inert gas.

The sample is suspended in a zone heated by infrared lamps.

Radiation of an arc lamp is concentrated at the surface of a biomass sample.

A lens concentrates the radiation An elliptical mirror concentrates the radiation. The parameters used, like the mass sample, the kind of heating, the shape and size of sample (solid or powder) are different according to the apparatus. The wire mesh and free fall use powders whereas the others use massive samples. In the case of pyrolysis, such a feature implies different internal heat transfer limitations: from chemical to heat transfer regime.

Kind of biomass samples

Mass in reactor

Regarding the temperature measurement, Di Blasi et al [START_REF] Di Blasi | Pyrolytic behavior and products of some wood varieties[END_REF][START_REF] Di Blasi | Weight loss dynamics of wood chips under fast radiative heating[END_REF] consider that the thermocouple placed at the centreline of the biomass sample is the reference for the pyrolysis temperature. Grønli & al [START_REF] Grønli | Mathematical Model for Wood Pyrolysis Comparison of Experimental Measurements with Model Predictions[END_REF] use thermocouples placed at different positions inside a big cylindrical sample. Hoekstra& al [START_REF] Hoekstra | Fast pyrolysis in a novel wiremesh reactor: initial decomposition products of pine wood[END_REF]16] determine the temperature by thermocouple and pyrometry. In their works they measure the wire mesh temperature whileit is impossible to determine the sample temperature. For Zanzi & al [START_REF] Zanzi | Rapid pyrolysis of agricultural residues at high temperature[END_REF][START_REF] Di Blasi | Pyrolytic behavior and products of some wood varieties[END_REF], the reference temperature is the temperature of the electrical heaters of the reactor as the gases introduced are preheatedat the same temperature. In the present study with the image furnace, the surface temperature of the sample is measured by pyrometry. In conclusion, it can be observed that the temperature reference varies significantly between the setups. Lédé [START_REF] Lédé | Biomass Pyrolysis: Comments on Some Sources of Confusions in the Definitions of Temperatures and Heating Rates[END_REF][START_REF] Lédé | Characterization of biomass fast pyrolysis[END_REF] and Narayan & Antal [START_REF] Narayan | Thermal Lag, Fusion, and the Compensation Effect during Biomass Pyrolysis[END_REF] showed that, in the case of biomass pyrolysis, there is a temperature stabilization phenomenon due to thermochemical reaction. The result is that a thermal lag exists between the heat source and the sample temperature. Hoekstra & al [START_REF] Hoekstra | Fast pyrolysis in a novel wiremesh reactor: initial decomposition products of pine wood[END_REF]16,[START_REF] Zanzi | Rapid high-temperature pyrolysis of biomass in a free-fall reactor[END_REF] take into account the lag in temperature between the mesh and the biomass sample. They calculated an indication of the reduction of average heating rate of the sample due to heat transfer resistance.

Finally, it appears that a direct measurement of the sample temperature is necessary but hard to achieve in fast pyrolysis conditions, since the spatial temperature gradients are high and the temperature evolution is rapid. Accurate measurements with thermocouples (TC) seemdifficult. Indeed good contact between pyrolysing biomass and the thermocouple itself should be achieved in order to be sure that the thermocouple temperature is the same as that of sample. However contact thermal resistances exist between the two interfaces. Besides the exact position of the thermocouple can be defined at +/-0,1 mm which may not be accurate enough in fast pyrolysis conditions. Conversely, pyrometry is non-invasive and sufficiently fast, but is restricted to surface. Attention should be paid to the light reflection over the sample which could blur the pyrometer.

The present paper reports first results obtained with a new image furnace operating with variable heat flux density. The pyrometry measurementsallow verifyingif the experimental thermal conditions are close to the fluidized bed thermal conditions.

Material and methods

Image furnace: the former horizontal setup

The new verticalimage furnace relies on the basis of the experimental setup used earlier in our laboratory, which has been described in details in previous papers [START_REF] Boutin | Flash pyrolysis of cellulose pellets submitted to a concentrated radiation: Experiments and modelling[END_REF]13,23,24,[START_REF] Agarwal | Transport phenomena in multi-particle systems-IV. Heat transfer to a large freely moving particle in gas fluidized bed of smaller particles[END_REF]. The basic principle of this former horizontal setup is to concentrate the radiation of a light source thanks to an elliptical mirror. The light beams are concentrated ona biomass sample, which is placed inside a cylindrical transparent quartzreactor swept by coolnitrogen. The advantages of this former setup are the followings:

-It is dedicated to selectively study primary solids reactions.

-Indeed, there are no secondary reactions in the gas phase because cool nitrogen flows through the quartz reactor. -The radiation is applied at a specific point of the sample.

-The nitrogen does notabsorb the lamp radiation conversely to the solid.

-The experiments are performed with thick samples (few millimetres thickness).

-The sample exposure time to the heat source is measured.

-The mass of all products is measured, which allows to achievemass balances. The drawbacksare:

-The samples must be thick in order to be self-supporting; experiment with fluid or powder samples are not possible. -The applied heat flux density is constant during one experiment.

-It is not possible to measure surface temperature and recover volatile products during the same experiment. -The optical properties of the biomass must be known to adjust a model to the experiments.

-The sample surfaceis small (the diameter equals 10 mm). This surface is limited by the space distribution of the radiation; it follows a Gaussian distribution [22]. At the focal point, the heat flux density can be considered uniformonly over ten millimetres.

The vertical image furnace apparatus

The novel vertical image furnace is based on the former horizontal setup. The light source is still an air cooled 5000 W xenon high pressure arc lamp (Osram XBO 5000 W/H CL OFR). This lamp is settled at the first focus of an elliptical mirror (cf. fig. 1). The light beams are concentrated at the second focus, where is placed the sample inside a cylindrical reactor.

Figure1. Scheme of the vertical image furnace.

The vertical image furnace underwent several modifications in order to overcome some of the disadvantages of the former horizontalsetup. It allowsworking with very thin layer since the sample is horizontal.It can be fine particles or a fluid.An originality of this verticalsetup is its ability to tune the heat flux density applied to the sample as afunction of time in order to reproduce the heat transfer conditions encountered in a thermo-chemical reactor. The light radiation is maximumwhen the sample is placed at the second focus of the elliptical mirror. When the sample is moved away from this second focus, the applied heat fluxdensity decreases. It has been chosen to displace the lamp instead of the sample. This feature would allow weighing directly the sample in future experiments. The lamp movement system is controlled by a computer, which allows imposing the lamp initial and final positions, as well as the motion speed of the lamp casing. The maximal travel distance of the lamp casing is 100 mm; the accuracy of the position is about 0.1mm. The maximum speed to move up is 18 mms -1 , and the maximum speed to move downis 65mms -1 .

A power sensor (Ophir thermal Head 1000W) is used to measure the heat fluxdensity according to the lamp position. The power sensor measures the available flux P. The window diameter D w used for the sensor equals 4 mm.Given this diameter and the measured power, it is possible to compute the heat flux density available by the equation 2.

Φ=4P/(πD w ²)

The flash times are controlled by an occultation shutter. Aceramic plate is fixed on a pneumatic track (Parker Origa, P210/20-32-500). In off position, the ceramic plate intercepts the light beams. During experiment, anexposure time is imposed by computer. The travelling time of the shutter is equal to 300 ms. Theshorter value of the flash time is 1 s with an accuracy of 10 ms.

Pyrolysis reactor

The reactor is made up of two parts, shown in figure 2. The first part is a stainless steel cylinder (inside diameter, 5.5×10 -2 m; height, 5×10 -2 m), including a Pyrex window (inside diameter, 3.5×10 -2 m; height, 3.3×10 -3 m) at its top. The second part is the air cooling jacket for the pyrometer, which is also swept by nitrogen and connected by a pipe to the top of the reaction chamber. There is an other sweep gas inlet under the Pyrex window. The flow rates of these sweep gases are controlled by mass flow meters (Brook Instruments 5850 S and 5850 TR, total flow 3x10 -5 m 3 s -1 [STP]). The sample is placed at the center of the first part. It is generally positioned at the second focus of the elliptical mirror through the use of two laser beams. The temperature of the sample surface is measured by pyrometry. The pyrometer Impac IN 5H plus cannot measure temperatures over 1183K.Its wavelengths range is between 8 and 14 µm. At this range,the char emissivity is 94% and the pyrometer measurement is not blurred by the image furnace light reflection (0.2 to 1.6 µm). The response time of the pyrometer is about 20 ms.

Figure2. Scheme of the reactor.

Sample

The samples are char particles,which are prepared in preliminaryexperiments i.e.wood samples pyrolysis in the image furnace at 560 kWm -2 . The heating time is 1 minto assure the total conversion of wood in char. The initial wood pelletsare 10 mm width and 3 mm thick.The obtained char pellets are 10 mm width and 2 mm thick (+/-0.1 mm).The experimental datapresented in this paper are measured on these char samples since the goal of this paper is to demonstrate the feasibility of heat flux density modulation with the image furnace. Since the study is limited to char samples, it allows avoiding taking pyrolytic reactions into account in the model. Furthermore, itwas simpler for a first approach to achieve measurements on non-reactive samples like char particles since there is no tar aerosols emissions in this case.These thick char samples are laid down on a ceramic support designed to minimize contact surface between sample and support. Thus the conductive heat losses are negligible.

Theoretical model

In fluidized bed, the heat flux transferred to the samples occurs by convection, radiation and conduction. A simple thermal modeling is used in order to determine the theoretical temperature evolution of a flat char particle immersed in a fluidized bed. The heat transfer is represented by a simple convection equation (Eq. 1) and uses global heat transfer coefficient h. Typically, this coefficientvaries between 200 and 1000 Wm -2 K -1 in fluidized beds [START_REF] Agarwal | Transport phenomena in multi-particle systems-IV. Heat transfer to a large freely moving particle in gas fluidized bed of smaller particles[END_REF][START_REF] Dyrness | Heat transfer in the splash zone of a bubbling fluidized bed[END_REF][START_REF] Di Blasi | Temperatures of Wood Particles in a Hot Sand Bed Fluidized by Nitrogen[END_REF][START_REF] Oka | Fluidized bed combustion Chapter3. Heat and mass transfer in fluidized beds[END_REF]. It is intended to compare the experimental surface temperature evolution of char with the theoretical evolution which would be encountered in a fluidized bed gasifier (bed temperature T r = 1123 K).The global heat transfer coefficient his assumed to be constant and equal to 500 Wm -2 K -1 .Other numerical data are reported in table 2. The cylindrical particle receives the heat flux density in the tangential direction of the wood fibres; the "x" axis is along this direction.

The energy conservation is represented in equation 3.

ρCp ∂T/∂t = λ (∂² T)/(∂x²)

(3) The initial conditionis the ambiant temperature uniformity (equation 4).

T (x,t=0) = T 0 (4) For the middle of the char particle (x = 0), the symmetry boundary condition is applied (equation 5).

(∂T/∂x) (x=0,t) = 0 (5) At the surface (x=L), the boundary condition is the heat flux density conservation (equation 6).

λ(∂T/∂x) (x=L, t) = -h(T-T r )

Thepartial differential equation ( 3) is solved by the methods of lines with the Adams-Bashforth-Moulton solver.

Results & Discussion

Heat flux density according to the lamp position

The figure3 represents the heat flux densitiesmeasured according to the lamp position. At zero, the heat flux density is the lowest (250 kWm -²). At 65 mm, the heat flux density is maximal (560 kWm -²): in this case, the sample is at the second focus of the elliptical mirror. In a fluidized bed reactor operatingat 1123 K, the initial heat flux density imposed on the sample surface ranges between 160 and 800 kWm -2 .It is in relatively good agreement with the heat flux densities obtained in the experimental setup. Besides, the vertical image furnace enables to vary the heat flux density applied to the sample. Thus it can be decreased over time in order to reproduce the heat flux density undergone by a particle inside a fluidized bed.

Results

Figure 4 shows the surface temperature vs. time evolution for different experimental conditions. At constant heat flux density, the lamp does not move and stay at a given position, 65mm for a heat flux density equals to 560kWm -² and 0mm for 250 kWm -2 .

The two experiments, performed at 560 kWm -2 are carried out after an interval of 2 months with char sample prepared in same conditions. Their surface temperature profiles are similar showing good repeatability of the measurements. The final temperature exceeds the pyrometer limit (T max = 1183K). The surface temperatureobtained at 560 kWm -2 increases less rapidly than the model temperature, but the final temperature largely exceeds the temperature of the fluidized bed reactor (T r =1123K). At 250 kWm -2 , the lowest available heat flux density, the surface temperature increases slowly in comparison withthe model temperature, but they tend to have similar final value. For short times, the experimental temperature is systematically lower than the model temperature: the difference is about 200K after 0.1 s, and 130 K near0.2 s. Above 0.7 s, this difference tends to reverse,butit has been shownthat itis possible to make these temperaturesconverge thanks to the heat flux density reduction. As a consequence, the maximum heat flux density of the image furnace should be raised to have a better agreement between the experiments and the modeling in the first 500milliseconds.

Discussion

The experiments are carried out with char samples in order to demonstrate that it is now possible to alter the heat flux density according to the exposure time. One of our main results is that the variation of the heat flux density enables to tend toward the modeling curve. This modification is relevant and judicious for reproducing fluidized bed reactor conditions. However, the surface temperature evolution measured shows that the heat flux density applied during the first 500ms is not high enough. The difference between the modeling and experimental surface temperatures is higher than 100K from 50 ms to 250 ms. Considering that for very thin sample (near the chemical regime) the pyrolysis lasts around 1s, it will be a critical point to increase the maximum heat flux density.

The preliminary temperature measurements have been made on char in order to avoid the potential problems due to tar aerosols producedby biomass pyrolysis. These aerosols could possibly mask the biomass surface and disturb the pyrometric measurements. In addition, the char has the advantage to undergo no chemical reactions which could infer a complex temperature stabilization phenomenon [START_REF] Lédé | Biomass Pyrolysis: Comments on Some Sources of Confusions in the Definitions of Temperatures and Heating Rates[END_REF][START_REF] Lédé | Characterization of biomass fast pyrolysis[END_REF][START_REF] Narayan | Thermal Lag, Fusion, and the Compensation Effect during Biomass Pyrolysis[END_REF]. The number of assumptions on physico-chemical parameters is also reduced. Further experiments will be made with biomass samples. In that case, and with the same heat flux densities profiles, the surface temperature evolution of biomass sample will be different because of the reflectivity differences and the chemical reactions. The aerosol masking effect will be managed thanks to the nitrogen sweep gas.

Conclusion

The new vertical image furnace has the main objective to reproduce the heat transfer conditions of fluidized bed gasifier. The comparison with other pyrolysis setups shows the difficulty of the sample temperature measurement in fast pyrolysis conditions. It is important to remind that it is not possible to consider that the temperature of the sample is the same as the temperature of the heat source (thermal lag effect). The pyrometric temperature measurement seems to be accurate if some limitations are taken into considerations : light reflection which could blur the pyrometer, emissivity determination, and restriction to surface temperature measurement.

Regarding the heat transfer conditions, the variation of the heat flux density enables to be in better agreement with the fluidized bed model than the configuration with a constant heat flux density. In this first approach, the vertical image furnace appears to be a good experimental setup to simulate fluidized bed conditions. Some modifications, like the raise of the maximum heat flux density, must be done to have a better accordance.

Our final objective is to improve the knowledge of the pyrolysis phenomenon in fluidized bed conditions. The next experiments will be performed on biomass samples. The study will also include the measurement of the temperature evolution of the exposed and unexposed surfaces of thick samples: this will bring interesting experimental data to be compared with theoretical model.

Nomenclature

Figure 3 .

 3 Figure 3. Heat flux density according to the lamp position.

Figure4.

  Figure4. Temperature profilesat the surface of wood char at constant heat fluxdensity andat variable heat flux density. Comparisonwith the model.

  Sample heat capacity (J kg -1 K -1 ) H Global heat transfer coefficient between the bed and the particle sample (Wm -2 K -1 ) L Characteristic length (m), i.e. thickness of the biomass sample in the image furnace experiments or half-thickness of the cylindrical pellet immerged in a fluidized bed gasifier. t Time (s) T Temperature (K) ΔT Temperature difference between the fluidized bed and the particle sample surface (K). P Power measured (W) x Axis in the tangential direction of wood fibres (m) Greek symbols Φ Available heat flux density (Wm -2 ) λ Sample thermal conductivity (Wm -1 K -1 ) ρ Sample density (kgm -3 ) σ Stefan-Boltzmann constant, 5.67 x 10 -8 (Wm -²K -4 )

Table 1 .

 1 Some operating conditions of different experimental setups.

Table 2 .

 2 Values of conditions and physicochemical parameters used to solve the model

	Parameters	Values	References
	Cp	420 + 2,09 T-6,85.10 -4 T²	[12]
	ρ	170	[21]
	λ	0.10	[12]
	L	2x10 -3	measured
	h	500	estimated
	T r	1123
	T 0	300	estimated

Acknowledgements

The research has been performed thanks to the French Environment and Energy Management Agency (ADEME) financial support through the GAYA Project led by GDF SUEZ.


R Walter

M Viret

S Singh

L Bellaiche

Revisiting Galvanomagnetic Effects in Conducting Ferromagnets Through Electromagnetic Field Angular Momentum Density and Magnetic Moment Coupling

, with an overview of the different physical phenomena readily explained by this coupling [1-4] and emphasis on various galvanomagnetic effects [2] -anisotropic magnetoresistance and the planar Hall effect.

It is known that reversing the vector resulting from the cross product of electric and magnetic fields reverses the sense of rotation of magnetic vortices, through the coupling of the magnetic toroidal moment with this cross product . By appropriately transforming this coupling energy, Ref. [1] demonstrated that the electromagnetic field angular momentum density can directly couple with magnetic moments to produce a physical energy. In that setting, this coupling reproduced the spin-current model in multiferroics and characterized a novel antiferroelectricity-driven magnetic anisotropy [1]. In conducting ferromagnets, it can explain the intrinsic anomalous Hall effect (AHE) and predict a novel Hall effect [3] as well as explain the anisotropic magnetoresistance (AMR) and the planar Hall effect (PHE) [2]. This energy also led to the prediction of novel spintronic effects [4] that bear resemblance to the so-called inverse Rashba-Edelstein effect. It may also be applicable to additional magneto-optical and spintronic phenomena.

Except for the phenomena discussed in Refs. [1] and [4], the proposed coupling has derived these effects according to a simple common method. The relevant electric and magnetic fields are first identified: for galvanomagnetic effects [2], an applied homogeneous electric field is used for the electric field and the Weiss field (proportional to magnetization) is used for the (homogeneous) magnetic field [START_REF] Nw Ashcroft | Solid State Physics[END_REF]. The coupling energy of a single conduction electron is taken as proportional to minus the scalar product of the field angular momentum and the magnetic moment of the electron, with some material dependent coupling constant as the proportionality coefficient. In this treatment of AMR and PHE, we are seeking anisotropy in longitudinal and transverse resistivity, so the electric field may adopt general Cartesian components along the longitudinal and transverse directions. Derivation of the appropriate expressions for AHE, AMR, and PHE then follows the standard derivation of the Hall effect in a Drude model. The equation of motion including the Lorentz force and the coupling force is considered component-wise in steady state; multiplying by the appropriate factor and summing over all electrons allows the component-wise equations to be expressed in terms of magnetization and current density. In steady state and with appropriate boundary conditions, the transverse applied electric field components cancel out the transverse current density, leaving only the longitudinal component of current density. Solving one of these resulting equations yields an expression for AHE. Solving for the longitudinal and transverse resistivities, the known expressions for AMR and PHE are recovered readily. This derivation of galvanomagnetic effects has several advantages. One, as far as the authors are aware, it provides the first unified explanation for AHE together with AMR and PHE. Moreover, provided the material dependent coupling constant and the proportionality coefficient between the Weiss field and magnetization are each linear in spin-orbit interaction, our model reproduces the respective first-and second-order dependencies of AHE and AMR on spin-orbit interaction. Second, since in our model the difference between the resistivities for magnetization parallel and perpendicular to the current is proportional to the material dependent coupling constant; if this constant is negative in some materials, the phenomenon of so-called negative AMR naturally results. Third, the derivation for AHE worked as well in semiclassical electron dynamics and this allowed the coupling constant there to be expressed in terms of Berry-phase curvature [3]; if this can be shown to work for AMR, then a novel Berry-phase curvature theory of AMR may be possible. Fourth, if materials can be engineered to achieve a particular coupling constant, this could lead to design of novel electronic devices.

We thank C Paillard for valuable discussions. R Walter acknowledges this material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-0957325. Furthermore he acknowledges support from a University of Arkansas Distinguished Doctoral Fellowship in Mathematics and Physics. This work is financially supported by ARO Grant W911NF-12-1-0085.