Léo Gautheron
email: leo.gautheron@univ-st-etienne.fr

Amaury Habrard

Emilie Morvant

Marc Sebban

Metric Learning from Imbalanced Data with Generalization Guarantees

Since many machine learning algorithms require a distance metric to capture dis/similarities between data points, metric learning has received much attention during the past decade. Surprisingly, very few methods have focused on learning a metric in an imbalanced scenario where the number of positive examples is much smaller than the negatives, and even fewer derived theoretical guarantees in this setting. Here, we address this difficult task and design a new Mahalanobis metric learning algorithm (IML) which deals with class imbalance. We further prove a generalization bound involving the proportion of positive examples using the uniform stability framework. The empirical study performed on a wide range of datasets shows the efficiency of IML.

Introduction

Metric learning [START_REF] Bellet | Metric learning[END_REF][START_REF] Kulis | Metric learning: A survey[END_REF], subfield of representation learning, consists of designing a pairwise function able to measure the dis/similarity between two data points. This issue is key in machine learning where such metrics are at the core of many algorithms, like k-nearest neighbors (kNN), SVMs, k-Means, etc. To construct a dis/similarity measure suitable for a given task, most metric learning algorithms optimize a loss function which aims at bringing closer examples of the same label while pushing apart examples of different labels. In practice, metric learning is usually performed with cannot link/must link constraints-x and x should be dis/similar [START_REF] Davis | Informationtheoretic metric learning[END_REF][START_REF] Lu | Neighborhood repulsed metric learning for kinship verification[END_REF][START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF][START_REF] Xiang | Learning a mahalanobis distance metric for data clustering and classification[END_REF][START_REF] Xing | Distance metric learning with application to clustering with side-information[END_REF][START_REF] Zadeh | Geometric mean metric learning[END_REF]-or relative constraints-x should be more similar to x than to x [START_REF] Lee | Rank-based distance metric learning: An application to image retrieval[END_REF][START_REF] Schultz | Learning a distance metric from relative comparisons[END_REF][START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF][START_REF] Zheng | Person re-identification by probabilistic relative distance comparison[END_REF].

In this paper, we focus on the family of metric learning algorithms that construct a Mahalanobis distance d M (x, x ) = √ (xx ) M(xx ) parameterized by a positive semidefinite matrix M. Learning a Mahalanobis distance leads to several nice properties: (i) d M is a generalization of the Euclidean distance; (ii) it induces a projection such that the distance between two points is equivalent to their Euclidean distance after a linear projection; (iii) M can be low rank implying a projection in a lower dimensional latent space; (iv) it involves optimization problems that are often convex and thus easy to solve. Two famous representatives of Mahalanobis distance learning are LMNN (Large Margin Nearest Neighbor [START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF]) and ITML (Information-Theoretic Metric Learning [START_REF] Davis | Informationtheoretic metric learning[END_REF]), which are both designed to improve the accuracy of the kNN classification rule in the latent space. The principle of LMNN is the following: for each training example, its k nearest neighbors of the same class (the target neighbors) should be closer than examples of other classes (the impostors). ITML uses a LogDet regularization and minimizes (resp. maximizes) the distance between examples of the same (resp. different) class. Another recent Mahalanobis distance learning algorithm is GMML (Geometric Mean Metric Learning) [START_REF] Zadeh | Geometric mean metric learning[END_REF] where the metric is computed using a closed form solution of an unconstrained optimization problem involving similar and dissimilar pairs. In light of these learning procedures, it is worth noticing that the loss functions optimized in LMNN, ITML and GMML (and in most pairwise metric learning methods) tend to favor the majority class as there is no distinction between the constraints involving examples of the majority class and the constraints on the minority class. This strategy is thus not well suited when dealing with imbalanced datasets. An illustration of this phenomenon on the spectfheart dataset from the UCI repository is shown in Figure 1. We observe that decreasing the proportion of minority examples tends to generate a metric which classifies (with a kNN rule) all the examples as the majority class, thus leading to an accuracy close to 1. On the other hand, the F1measure, which is much more adapted to imbalanced scenarios (it involves both the false positive and false negative rates), de- creases with the proportion of positives, showing that the classifier missed many positives, usually considered as the examples of interest. This problem of learning from imbalanced data has been widely tackled in the literature [START_REF] Branco | A survey of predictive modeling on imbalanced domains[END_REF][START_REF] He | Learning from imbalanced data[END_REF]. Classic methods typically make use of over/under-sampling techniques [START_REF] Drummond | C4. 5, class imbalance, and cost sensitivity: why under-sampling beats over-sampling[END_REF][START_REF] Estabrooks | A multiple resampling method for learning from imbalanced data sets[END_REF][START_REF] Liu | Exploratory undersampling for classimbalance learning[END_REF][START_REF] Aggarwal | Outlier Analysis[END_REF] or create synthetic samples in the neighborhood of the minority class-e.g., using SMOTE-like strategies [START_REF] Chawla | Smote: Synthetic minority over-sampling technique[END_REF][START_REF] Chawla | Smoteboost: Improving prediction of the minority class in boosting[END_REF][START_REF] Han | Borderline-smote: a new oversampling method in imbalanced data sets learning[END_REF] or resorting to adversarial techniques [START_REF] Douzas | Effective data generation for imbalanced learning using conditional generative adversarial networks[END_REF]. However, these methods may lead to over or under-fitting and are often subject to an inability to generate enough diversity, especially in a highly imbalanced scenario. Other strategies aim at addressing imbalanced situations directly during the learning process. They include cost-sensitive methods [START_REF] Elkan | The foundations of cost-sensitive learning[END_REF][START_REF] Zadrozny | Cost-sensitive learning by cost-proportionate example weighting[END_REF] which require prior knowledge on the miss-classification costs, the optimization of imbalanceaware criteria [START_REF] Frery | Efficient top rank optimization with gradient boosting for supervised anomaly detection[END_REF][START_REF] Mcfee | Metric learning to rank[END_REF][START_REF] Vogel | A probabilistic theory of supervised similarity learning for pointwise roc curve optimization[END_REF] which are often non convex, or ensemble methods based on bagging and boosting strategies [START_REF] Galar | A review on ensembles for the class imbalance problem: bagging-, boosting-, and hybrid-based approaches[END_REF] that can be computationally expensive.

Unlike the state of the art, we suggest in this paper to address the problem of learning from imbalanced data by optimizing a metric suited to scenarios where the positive data are very scarce. As far as we know, very few methods were designed in this setting. Feng et al. [START_REF] Feng | Learning a distance metric by balancing kl-divergence for imbalanced datasets[END_REF] propose to regularize a standard metric learning problem by using the KL-divergence between the classes. Wang et al. [START_REF] Wang | Iterative metric learning for imbalance data classification[END_REF] proposed IMLS that learns a classic metric and then performs a sampling on the training data to account the imbalance. However, as we will see in our experimental study, better performances can be achieved by resorting to a metric dedicated specifically to deal with the imbalance of the application at hand. Note that deep metric learning methods have also received attention by the community to address the problem of imbalanced data [START_REF] Liu | Defect detection on el images based on deep feature optimized by metric learning for imbalanced data[END_REF][START_REF] Wang | Dynamic curriculum learning for imbalanced data classification[END_REF]. However these methods often require large training datasets, like in visual tasks. However, this requirement is not always fulfilled by the application at hand. Moreover, it is worth noticing that none of the previous approaches come with theoretical guarantees, a gap we will fill in this paper. In order to implicitly control the rates of false positives and false negatives, we propose a new algorithm, called IML for Imbalanced Metric Learning, which accounts carefully the nature of the pairwise constraints (by decompos-ing them with respect to the labels involved in the pairs) and weights their impact in the loss function so as to maximize the F1-measure. Beyond this algorithmic contribution, we further provide a theoretical analysis of IML using the uniform stability framework [START_REF] Bousquet | Stability and generalization[END_REF]. We derive a generalization bound which has the advantage to involve the proportion of minority examples. This bound provides some insight into the way to tune the weighting parameters to counterbalance the negative impact of imbalanced datasets.

The paper is organized as follows. Section 2 describes our algorithm IML which takes the form of a simple regularized convex problem. Section 3 is dedicated to the theoretical analysis. We perform an experimental study of our approach in Section 4 before concluding in Section 5.

IML: Imbalanced Metric Learning

Notations and Setting

In this paper, we deal with binary classification tasks where X ⊆ R d is a d-dimensional input space and Y = {-1, +1} is the binary output space. We further define Z = X × Y as the joint space where z = (x, y) ∈ Z is a labeled example. In supervised classification, an algorithm is provided with a learning sample S = {z i =(x i , y i )} n i=1 of n observations. These observations are drawn i.i.d. from a fixed yet unknown distribution D over Z. We denote this set of n i.i.d. observations as S ∼ D n . We assume that the learning sample is defined as S = S + ∪ S -, with S + the set of positive examples and S -the set of negative examples such that the number of positives n + = |S + | is smaller than the number of negatives n -= |S -| (we say that +1 is the minority class and -1 the majority one). The final objective of the learner is to construct a classifier f : X → Y (from a hypothesis space F ) which behaves well on unseen data drawn from D. Here, we aim at constructing a Mahalanobis distance which induces a new space in which a kNN will work well on both classes. The Mahalanobis distance is a type of metric parameterized by a positive semidefinite (PSD) matrix M ∈ R d×d that can be decomposed as M = L T L, where L ⊆ R r×d is a projection induced by M (where r is the rank of M). A nice property is that the Mahalanobis distance between two points x and x is equivalent to the Euclidean distance after having projected x and x in the r-dimensional space, i.e.,

d 2 M (x, x ) = (x -x ) M(x -x ) = (Lx -Lx ) (Lx -Lx ) .

Problem Formulation

Classic Mahalanobis metric learning algorithms [START_REF] Bellet | Metric learning[END_REF][START_REF] Cao | Generalization bounds for metric and similarity learning[END_REF][START_REF] Jin | Regularized distance metric learning: Theory and algorithm[END_REF] are usually expressed as follows:

min M 0 F(M) = 1 n 2 (z,z )∈S 2 (M, z, z ) + λReg(M), (1)
where one wants to minimize the trade-off between a convex loss over all pairs of examples and a regularization Reg under the PSD constraint M 0. The major drawback of this formulation is that the loss gives the same importance to any pair of examples (z, z ) whatever the labels y and y . Intuitively, this is not well suited to imbalanced data where the minority class is the set of examples of interest. Some metric learning algorithms [START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF][START_REF] Zadeh | Geometric mean metric learning[END_REF] allow to weight the role played by the must-link and cannot-link constraints. However, the problem still holds because the labels of the examples are not directly taken into account.

Our simple idea is to decompose further the sets of pairs of examples based on their labels. Each set can then weighted differently during the optimization to reduce the negative effect of the imbalance.

Let us define the loss function of our IML algorithm as follows:

(M, z, z ) =                    a 1 (M, z, z ) if y= + 1 and y = +1, (1-a) 1 (M, z, z ) if y= -1 and y = -1, b 2 (M, z, z ) if y= + 1 and y = -1, (1-b) 2 (M, z, z ) if y= -1 and y = +1, (2)
with the two functions 1 and 2 defined as F where I is the identity matrix and . F is the Frobenius norm. It aims at avoiding overfitting by enforcing M to be close to the identity matrix I. Said differently, we aim at being close to the Euclidean distance while satisfying the best the semantic constraints.

1 (M, z, z ) = [d 2 M (x, x ) -1] + and 2 (M, z, z ) = [1+m-d 2 M (x, x )] +
All things considered, our IML algorithm takes the form of the following convex problem:

min M 0 F(M) = 1 n 2 (z,z )∈Sim + a 1 (M, z, z ) + (z,z )∈Sim - (1-a) 1 (M, z, z ) + (z,z )∈Dis + b 2 (M, z, z ) + (z,z )∈Dis - (1-b) 2 (M, z, z ) + λ M -I 2 F , (3)
where the four sets Sim + , Dis + , Dis -and Sim -are defined as subsets of S × S respectively as: Sim + ⊆ S + × S + , Dis + ⊆ S + × S -, Dis -⊆ S -× S + and Sim -⊆ S -× S -. If we look more closely at the proposed Equation ( 3), when all pairs from S × S are involved, Sim + and Sim -contain respectively n + n + and n -n -pairs while Dis + and Dis -contain respectively n + n -and n -n + pairs. This means that the pairs in Dis + and Dis -are symmetric and these two sets might be merged. However, metric learning methods rarely consider all the possible pairs as it becomes quite inefficient in the presence of a large number of examples. Possible strategies to select the pairs include a random subsampling [START_REF] Davis | Informationtheoretic metric learning[END_REF][START_REF] Xiang | Learning a mahalanobis distance metric for data clustering and classification[END_REF][START_REF] Xing | Distance metric learning with application to clustering with side-information[END_REF][START_REF] Zadeh | Geometric mean metric learning[END_REF] or a selection based on the nearest neighbors rule [START_REF] Lu | Neighborhood repulsed metric learning for kinship verification[END_REF][START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF]. For this reason, it might make sense to separate the two sets Dis + and Dis -and allows us to weight them differently as (i) they may not consider the same subsets of pairs, and (ii) may not capture the same geometric information. Another interpretation of such a decomposition in an imbalanced learning setting is the following: if z is selected as belonging to the neighborhood of z, the minimization of the four terms of Equation ( 3) can be seen as a nice way to implicitly optimize with a kNN rule the true positive, false negative, false positive and true negative rates respectively.

Generalization bound for IML

In this section, we provide a theoretical analysis of our algorithm using the uniform stability framework [START_REF] Bousquet | Stability and generalization[END_REF]. This framework can be adapted to any metric learning algorithm A [START_REF] Bellet | Metric learning[END_REF][START_REF] Jin | Regularized distance metric learning: Theory and algorithm[END_REF] taking the following form:

min M 0 G(M) = (z,z )∈S 2 A (M, z, z ) R(M) +λReg(M) , (4)
where R(M) is the empirical loss of M on S, and A is any loss function that is q-Lipschitz1 and (σ, p)-admissible2 . To prove a uniform stability-based generalization bound the algorithm A has to be stable-meaning that its output does not change significantly under a small modification of Saccording to the following definition.

Definition 1 ([18] Eq. ( 5)). A metric learning algorithm A has uniform stability in β ≥ 0 w.r.t. the loss function A if ∀i ∈ {1, . . . , n} the following holds

∀S ∈ Z n , sup z,z A (M, z, z ) -A (M i , z, z ) ≤ β ,
where M is learned from S, and M i is learned from S i , the set obtained by replacing the i th example in S by another also i.i.d. from D.

If an algorithm has uniform stability, then it is possible to derive an upper bound on its generalization error using the Mc-Diarmid inequality [START_REF] Bousquet | Stability and generalization[END_REF] recalled below.

Theorem 1 (McDiarmid Inequality, [3] Th.
2). Let G : Z n → R be any function for which there exists constants c i , ∀i ∈ {1, . . . , n} s.t.

sup S∈Z n ,z i ∈Z |G(S) -G(S i )| ≤ c i , then ∀ >0, P S G(S) -E S G(S) ≥ ≤ 2 exp       -2 2 n i=1 c 2 i       .
where P S denotes the probability with respect to the random draw of the sample S from D n . Then, one can derive the following theorem.

Theorem 2 ([2] Th. 8.11). Let S be a sample of n randomly selected training examples and M be the PSD matrix learned from an algorithm A with stability β. Assuming that the loss A is q-Lipschitz and (σ, p)-admissible, with probability at least 1 -δ over the random choice of S ∼ D n , we have the following bound on the true risk R(M)

R(M) ≤ R(M) + 2β + 2nβ + 2(2σ + p) ln 2/δ 2n .
Unlike standard PAC results [START_REF] Valiant | A theory of the learnable[END_REF], this kind of generalization bound has two advantages: (i) it takes into consideration properties of the algorithm, and (ii) it offers tools to deal with the fact that the pairs of examples are usually not drawn i.i.d. from D × D [START_REF] Bellet | Metric learning[END_REF]. In the rest of this section, we first show that our loss is q-Lipschitz; then, we prove that our algorithm is stable, and finally, we derive a generalization bound on its true risk using the McDiarmid inequality. Note that in the following, we assume that the norm of any example is upper-bounded by a constant, i.e., ∀x ∈ R d , x ≤ B. Lemma 1. Let M, M be any matrices and (z, z ) any pair of labeled examples, then the loss , as defined in Equation (2), is q-Lipschitz w.r.t. its first argument, i.e., we have

(M, z, z ) -(M , z, z ) ≤ q M -M F , with q = 4B 2 .
Proof. See Appendix A.

Let M be the matrix learned from S and Sim + , Dis + , Dis -and Sim -be the subsets of pairs coming from S × S as described in Section 2. The true and empirical losses are respectively defined as:

R(M) = E z∼D,z ∼D (M, z, z ) and R(M) = 1 n 2 (z,z )∈Sim + a 1 (M, z, z ) + (z,z )∈Sim - (1-a) 1 (M, z, z ) + (z,z )∈Dis + b 2 (M, z, z ) + (z,z )∈Dis - (1-b) 2 (M, z, z ) .
where E z∼D,z ∼D denotes the expectation with respect to the random draw of z and z according to D. Thus Problem (3) can be reformulated as:

min M 0 F(M) = R(M) + λ M -I 2 F .
Uniform Stability of IML. We now proceed to show that our algorithm satisfies Definition 1. For that purpose, we introduce a lemma similar to Lemma 20 in [START_REF] Bousquet | Stability and generalization[END_REF] and Lemma 8.6 in [START_REF] Bellet | Metric learning[END_REF].

Lemma 2. Let matrices M * and M * i be the minimizers of F on S and S i respectively. Let ∆M * = M * i -M * and ρ = n + n the proportion of minority examples. Then for any t ∈ [0, 1] we have

M * -I 2 F -M * +t∆M * -I 2 F + M * i -I 2 F -M * i -t∆M * -I 2 F ≤ a(2ρ -1) + 2(1 -ρ) λn 2qt M * i -M * F .
Proof. See Appendix B.

Note that the parameter b of Problem (3) does not appear in this Lemma. While in the experiments, in order to scale to large datasets, the pairs will be generated by using the kneighborhood of the examples, we derived the proof in Appendix B by using all the pairs, that allowed us to get rid of the parameter b. This enables us to provide a more general result that does not depend on additional parameters (here the k of the k-nearest-neighbor rule).

Along with the q-Lipschitz property of Lemma 1, Lemma 2 allows us to prove that IML is stable.

Lemma 3. IML has uniform stability with

β = 2q 2 a(2ρ -1) + 2(1 -ρ) λn .
Proof. See Appendix C.

Derivation of the main result. To prove our bound, we follow the derivation of Theorem 8.11 in [START_REF] Bellet | Metric learning[END_REF]. We first provide two lemmas, and then we use them in conjunction with the McDiarmid inequality to derive a generalization bound.

First, we introduce a lemma that bounds the difference on the empirical risk over S and S i . Lemma 4. Let M * be the optimal solution of Problem (3). We have

R(M * ) -R i (M * ) ≤ 2 a(2ρ -1) + 1 -ρ 4B 2 M * F + 2(1 -ρ)(1+m) n .
where R and R i denote respectively the empirical risk over S and S i .

Proof. See Appendix D.

Using Lemma 4 and the stability of Lemma 3, we introduce a lemma similar to Lemma 8.10 in [START_REF] Bellet | Metric learning[END_REF]. Lemma 5. Let matrices M * and M * i be the minimizers of F on S and S i respectively. As IML has stability β, we have

R(M * ) -R(M * ) -R(M * i ) -R i (M * i ) ≤ 2nβ + D n = c i , with D = 2 a(2ρ -1) + 1 -ρ 4B 2 M * F + 2(1 -ρ)(1+m).
Proof. See Appendix E.

We are now able to state our main result.

+ 1 -ρ 4B 2 M * F + 2(1 -ρ)(1+m) .
Proof. See Appendix F.

Discussion. The difference between our Theorem 3 and classic bounds of the form of Theorem 2 is that proportion of minority examples ρ = n + n and the weight of the similar minority pairs a appear in the two terms β and D. Classic metric learning bounds are derived in a balanced setting where ρ = 0.5 (i.e. positives and negatives are balanced) and where the parameters a and b are equal to 0.5. It is worth noting that plugging these values in our bound allows us to retrieve the constant β = 2q 2 λn as derived in Theorem 8.7 in [START_REF] Bellet | Metric learning[END_REF]. This means that our β formulation is a generalization of the standard stability constants in Metric Learning. Regarding the term D, the decomposition into four terms allows us to get a tighter bound by a factor 4 with

D = 4B 2 M F + (1 + m) while D = 16B 2 M F + 4(1 + m) in [2].
Another interesting interpretation of our bound is that when ρ tends to 0, i.e., the dataset is more and more imbalanced, a classic metric learning method (with a = b = 0.5) will converge slower. Indeed, in such a situation, we get a stability constant β which would tend to 3q 2 λn > 2q 2 λn while by parameterizing by a, we have 2q 2 (-a+2) λn . In this case, a value of a close to 1 allows us to reduce the negative effect of the imbalance.

Experiments

Datasets. In this section, we provide an empirical study of IML on 22 datasets. They come mainly from the UCI 3 and Keel4 repositories. The 'splice' dataset comes from LIBSVM 5 . All datasets are normalized such that each feature has a mean of 0 and a variance of 1.

For the sake of simplicity, we have chosen binary datasets, described in Table 1 where the minority class is given by the columns "Label". Note that IML can easily be generalized to multiclass problems by learning one metric per class in a standard "one-versus-all" strategy, and then applying a majority vote.

Optimization Details. Like most Mahalanobis metric learning algorithms, IML requires that the learned matrix M is PSD. There exist different methods to enforce the PSD constraint [START_REF] Kulis | Metric learning: A survey[END_REF]. A classic solution consists in performing a Projected Gradient Descent where one alternates a gradient descent step and a (costly) projection onto the cone of PSD matrices. The advantage is that the problem remains convex w.r.t. M [START_REF] Xing | Distance metric learning with application to clustering with side-information[END_REF], ensuring that one will attain the optimal solution of the problem by correctly setting the projection step in the gradient descent. Another solution [START_REF] Weinberger | Fast solvers and efficient implementations for distance metric learning[END_REF] is based on the fact that if M is PSD, it can be rewritten as M = L L. Therefore, instead of learning M, one can enforce M to be PSD in a cheaper way by directly learning the projection matrix L ∈ R r×d (where r is the rank of M). This can be done thanks to a gradient descent by computing the gradient of the problem w.r.t. L (instead of M). The implementation 6 we propose is based on this latter approach [START_REF] Weinberger | Fast solvers and efficient implementations for distance metric learning[END_REF] where we make use of the L-BFGS-B algorithm [START_REF] Zhu | Algorithm 778: L-bfgs-b: Fortran subroutines for large-scale bound-constrained optimization[END_REF] from the SciPy Python library to optimize our problem: it takes as input our initial point (the identity matrix), the optimization problem of Equation ( 3), and its gradient, then it performs a gradient descent that returns the projection matrix L minimizing Problem (3). To prevent us from tuning r and finding the best rdimensional projection space, we set r = d in the experiments.

As discussed at the end of Section 2, the pairs of examples considered by IML in its four terms are chosen using the nearest neighbors rule. Indeed, we noted experimentally that the algorithms using this strategy (LMNN [START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF], IMLS [START_REF] Wang | Iterative metric learning for imbalance data classification[END_REF] and IML) perform better than the ones using a random selection strategy (ITML [START_REF] Davis | Informationtheoretic metric learning[END_REF] and GMML [START_REF] Zadeh | Geometric mean metric learning[END_REF]).

Experimental setup. All along our experiments, we use a 3NN classifier after projection of the training and test data using the metric learned. The metrics considered in the comparative study are the Euclidean distance and the ones learned by LMNN [START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF], ITML [START_REF] Davis | Informationtheoretic metric learning[END_REF], GMML [START_REF] Zadeh | Geometric mean metric learning[END_REF], IMLS [START_REF] Wang | Iterative metric learning for imbalance data classification[END_REF] and IML.

For each dataset, we generate randomly 20 stratified splits of 70% training examples and 30% test data (same class proportions in training and test) and report the mean results over the 20 splits. The parameters are tuned by 5-fold cross-validation on the training set through a grid search using the following parameter ranges: for LMNN and IMLS, µ ∈ {0, 0.05, . . . , 1} (k is fixed to 3); for ITML, γ ∈ {2 -10:10 }; for GMML t ∈ {0, 0.05, . . . , 1}; and for IML we fix a = b = n - n and we tune m ∈ {1, 10, 100, 1000, 10000} and λ ∈ {0, 0.01, 0.1, 1, 10} (k is also fixed to 3). Note that to study the impact of the hyperparameters a and b and the impact of how the pairs of examples were selected, we performed an experiment where we artificially decreased the proportion of minority examples in the datasets. The results of this experiment are given in Appendix G.

The results of the first experiment are reported in Table 2. On average, the F1-measure of 72.3% obtained by IML is the best in comparison to 70.8% for LMNN and IMLS, 70.1% for ITML, 69.3% for GMML and 67.3% for the Euclidean distance. Overall, IML has also the best average rank of 1.52. We note that IML generally gives better performances on the datasets considered no matter how much they are balanced or not. This means that our reweighting scheme of the pairs can not only improve the performances in an imbalanced setting but is also competitive in more classic scenarios.

Second experiment. To address imbalanced data issues, classic machine learning algorithms typically resort to over/undersampling techniques [START_REF] Aggarwal | Outlier Analysis[END_REF] or create synthetic samples in the neighborhood of the minority class-e.g., using SMOTE-like strategies [START_REF] Chawla | Smote: Synthetic minority over-sampling technique[END_REF]. We now aim at studying the behavior of those methods when used as a pre-process of the metric learning procedures. We consider the results of Table 2 as baselines. We compare them to the performances obtained after performing prior to metric learning an over-sampling using SMOTE and a Random Under Sampling (RUS) strategy of the negative data. We use the implementations of these methods from the Python library imbalanced-learn [START_REF] Lemaître | Imbalanced-learn: A python toolbox to tackle the curse of imbalanced datasets in machine learning[END_REF].

The results obtained are reported in Table 3 for SMOTE and in Table 4 for RUS. Note that the results from Tables 2, 3 and 4 where computed using the same training/test splits and the same validation folds and are thus comparable. In each of the three settings considered, IML obtains the best results showing that it is more appropriate for improving the F1-measure. We also note that SMOTE allows one to increase significantly the performances of all methods, while there is no gain with RUS in comparison with an approach without sampling. This increase of performance tells us that SMOTE and IML are more complementary than competitors with different objectives (re-balancing for the former and representation learning for the latter).

Conclusion and perspectives

In this paper, we propose to revisit the classic formulation of metric learning algorithms that learn a Mahalanobis metric in the light of imbalanced data issues. Unlike the state of the art methods that do not make any distinction between the labels of similar pairs, we propose to decompose the usual loss with respect to the different possible labels involved in the pairs. This decomposition allows us to give specific weights to each type of pairs in order to improve the performance on the minority class. We derive a generalization bound specific to the imbalanced setting showing a convergence term depending on the class imbalance and illustrating the hardness of learning from imbalanced data. Our experimental evaluation shows that we are able to outperform state of the art metric learning algorithms in terms of F1-measure over balanced and imbalanced datasets. We believe that our work gives rise to exciting perspectives when facing imbalanced data. Among them, we want to study how our algorithm could be adapted to learn non-linear metrics. From an algorithmic point of view, we would like to extend our method by deriving a closed form solution in a similar way as done by Zadeh et al. [START_REF] Zadeh | Geometric mean metric learning[END_REF] to drastically reduce the computation time while maintaining good performances.

Fig. 1 .

 1 Fig.1. Illustration on the spectfheart dataset of the negative impact of classic metric learning algorithms when facing an increasing imbalance in the dataset. On the left, as the proportion of minority examples decreases, the metric learning algorithms tend to classify all the examples as the majority class, with an accuracy close to 1. On the right, using the F1-measure, we note that the learned metrics plugged in a kNN actually miss many positives, usually considered as the examples of interest.

  and where [value] + = max(0, value) is the Hinge loss and m ≥ 0 a margin parameter. The idea of 1 is to bring examples of the same class at a distance less than 1 while 2 aims to push far away examples of different classes at a distance larger than 1 plus a given margin m. Both parameters a and b take values in [0, 1]. The parameter a controls the trade-off between bringing closer the minority examples and bringing closer the majority examples. While the second parameter b controls the trade-off between keeping far away the majority examples from the neighborhood of minority ones, and keeping far away minority examples from the neighborhood of majority ones. Moreover, we set the regularization term of Equation (1) as Reg(M) = M-I 2

Theorem 3 (

 3 Generalization bound for IML). Let S be a sample of n = n + + n -randomly selected training examples with ρ = n + n the proportion of minority examples and let M * be the optimal solution learned from problem (3) having stability β. With probability at least 1-δ over the random choice of S ∼ D n , we have R(M * ) ≤ R(M * ) + 2β + (2nβ + D) ln 2/δ 2n with β = 2q 2 a(2ρ -1) + 2(1 -ρ) λn and D = 2 a(2ρ -1)

Table 1 .

 1 Description of the datasets (n: number of examples, d: number of features, c: number of classes) and the class chosen as positive (Label), its cardinality (n + ) and its percentage (%).

	Name	n	d c Label	n +	% Name	n	d	c	Label	n +	%
	splice	3175 60 2	-1 1527 48.10% glass	214 11	6	1	70 32.71%
	sonar	208 60 2	R	97 46.64% newthyroid	215	5	3	2, 3	65 30.23%
	balance	625	4 3	L	288 46.08% german	1000 23	2	2 300 30.00%
	australian	690 14 2	1	307 44.49% vehicle	846 18	4	van 199 23.52%
	heart	270 13 2	2	120 44.44% spectfheart	267 44	2	0	55 20.60%
	bupa	345	6 2	1	145 42.03% hayes	160	4	3	3	31 19.38%
	spambase 4597 57 2	1 1812 39.42% segmentation 2310 19	7 window 330 14.29%
	wdbc	569 30 2	M	212 37.26% abalone	4177 10 28	8 568 13.60%
	iono	351 34 2	b	126 35.90% yeast	1484	8 10	ME3 163 10.98%
	pima	768	8 2	1	268 34.90% libras	360 90 15	1	24	6.66%
	wine	178 13 3	1	59 33.15% pageblocks	5473 10	5	3, 4, 5 231	4.22%

Table 2 .

 2 Average F1-measure ± standard deviation over 20 splits using different metric learning algorithms.

	Dataset	Euclidean	LMNN	ITML	GMML	IMLS	IML
	hayes	44.9 ± 13.2 57.2 ± 12.5 55.4 ± 8.7 52.7 ± 10.8 57.2 ± 12.5 54.9 ± 9.2
	wine	94.9 ± 2.2	96.0 ± 2.9	96.3 ± 3.3	95.3 ± 3.1	96.0 ± 2.9	96.6 ± 2.1
	sonar	69.2 ± 5.3	70.6 ± 6.5	70.6 ± 5.9	69.1 ± 5.0	71.1 ± 6.7	74.6 ± 3.7
	glass	66.0 ± 3.4	63.6 ± 5.2	62.6 ± 5.2	67.2 ± 3.6	63.6 ± 5.2	66.6 ± 4.3
	newthyroid	83.4 ± 4.2	88.1 ± 5.2	89.8 ± 5.2	91.1 ± 2.5	88.1 ± 5.2	91.3 ± 2.6
	spectfheart	34.8 ± 12.3 39.1 ± 8.4	34.4 ± 7.9 29.1 ± 11.4 38.6 ± 8.7	42.4 ± 8.7
	heart	76.8 ± 2.1	74.8 ± 3.2	76.8 ± 2.9	76.9 ± 3.6	74.6 ± 3.1	77.1 ± 3.1
	bupa	49.8 ± 4.4	50.1 ± 5.0	51.3 ± 4.8	52.0 ± 5.3	50.1 ± 5.0	52.5 ± 5.1
	iono	67.8 ± 6.7	70.8 ± 3.9	73.4 ± 5.4	72.0 ± 5.4	71.0 ± 4.0	76.1 ± 2.9
	libras	48.4 ± 15.1 68.3 ± 12.2 65.5 ± 15.3 56.1 ± 16.3 66.6 ± 10.3 67.9 ± 12.1
	wdbc	94.2 ± 1.3	93.5 ± 1.7	94.3 ± 1.1	94.4 ± 1.3	93.4 ± 2.2	95.2 ± 1.1
	balance	87.4 ± 1.8	89.8 ± 1.3	93.0 ± 1.4	90.3 ± 1.3	89.8 ± 1.3	90.6 ± 1.2
	australian	79.9 ± 1.7	81.7 ± 2.0	82.0 ± 1.9	81.0 ± 2.6	81.4 ± 2.0	81.9 ± 1.8
	pima	56.2 ± 1.9	55.9 ± 3.3	57.5 ± 3.0	56.7 ± 3.0	55.9 ± 3.3	57.2 ± 2.7
	vehicle	80.5 ± 2.4	92.6 ± 1.0	90.2 ± 2.4	90.1 ± 1.7	92.5 ± 1.2	91.8 ± 1.9
	german	35.3 ± 2.8	37.3 ± 3.9	37.4 ± 3.3	37.1 ± 3.3	37.8 ± 3.7	38.4 ± 3.5
	yeast	73.2 ± 2.3	74.9 ± 2.8	74.2 ± 3.1	73.5 ± 2.6	74.5 ± 2.7	75.4 ± 2.4
	segmentation	81.8 ± 2.4	85.3 ± 2.1	79.6 ± 3.0	80.8 ± 3.1	85.6 ± 2.3	86.0 ± 2.5
	splice	76.3 ± 0.7	86.5 ± 0.8	79.7 ± 1.4	76.3 ± 1.3	88.0 ± 0.9	87.4 ± 0.6
	abalone	22.6 ± 2.1	22.1 ± 2.1	21.2 ± 3.0	21.6 ± 1.7	22.1 ± 2.1	23.0 ± 1.9
	spambase	85.3 ± 0.9	88.4 ± 0.8	87.8 ± 1.0	86.8 ± 0.8	88.7 ± 0.5	89.3 ± 0.8
	pageblocks	71.9 ± 3.0	71.8 ± 3.2	69.7 ± 5.1	73.7 ± 2.9	71.8 ± 3.1	73.4 ± 2.6
	Mean	67.3 ± 4.2	70.8 ± 4.1	70.1 ± 4.3	69.3 ± 4.2	70.8 ± 4.0	72.3 ± 3.5
	Average Rank	5.00	3.57	3.57	4.00	3.35	1.52

Table 3 .

 3 Same experiment as in Table2after having applied the SMOTE algorithm[START_REF] Chawla | Smote: Synthetic minority over-sampling technique[END_REF] until n + = n -.

	Dataset	Euclidean	LMNN	ITML	GMML	IMLS	IML
	hayes	68.0 ± 6.8 64.4 ± 7.6 67.8 ± 7.8	69.5 ± 7.2	64.2 ± 7.6 68.6 ± 7.2
	wine	92.7 ± 2.8 95.3 ± 3.0 96.3 ± 2.6	94.4 ± 3.2	95.4 ± 2.9 96.7 ± 2.2
	sonar	72.6 ± 4.2 73.0 ± 6.4 72.6 ± 4.5	71.2 ± 4.2	72.7 ± 6.0 75.2 ± 4.8
	glass	66.6 ± 2.9 66.1 ± 4.0 64.6 ± 3.2	67.4 ± 3.8	66.1 ± 4.2 66.2 ± 3.5
	newthyroid	87.6 ± 3.5 88.7 ± 4.0 91.6 ± 3.2	89.9 ± 4.3	88.7 ± 4.0 90.7 ± 2.2
	spectfheart	47.4 ± 2.3 41.9 ± 8.5 46.7 ± 6.9	49.1 ± 4.4	40.9 ± 7.3 46.1 ± 7.8
	heart	77.3 ± 2.0 75.0 ± 2.6 75.7 ± 4.1	77.2 ± 3.9	74.4 ± 3.0 76.9 ± 2.3
	bupa	54.1 ± 3.1 55.4 ± 3.4 53.9 ± 3.7	55.9 ± 4.1	55.4 ± 3.4 54.8 ± 3.5
	iono	78.4 ± 2.6 77.7 ± 3.7 77.5 ± 3.8	78.2 ± 3.9	76.9 ± 3.9 79.9 ± 3.9
	libras	68.3 ± 8.1 76.7 ± 8.5 69.7 ± 13.8 69.2 ± 11.0 69.0 ± 14.7 78.1 ± 9.4
	wdbc	93.4 ± 1.3 93.5 ± 2.2 94.0 ± 1.4	93.6 ± 1.5	93.0 ± 2.2 94.8 ± 1.3
	balance	87.4 ± 1.9 89.6 ± 1.5 92.1 ± 1.4	89.8 ± 1.9	89.5 ± 1.5 90.5 ± 1.4
	australian	80.3 ± 1.6 80.7 ± 3.2 82.4 ± 1.5	81.1 ± 1.8	81.1 ± 3.2 82.1 ± 1.6
	pima	60.1 ± 2.6 60.1 ± 2.1 60.8 ± 2.2	60.3 ± 2.8	60.1 ± 2.1 61.2 ± 2.0
	vehicle	80.6 ± 2.1 92.0 ± 1.6 89.9 ± 3.1	89.5 ± 2.1	92.3 ± 1.5 91.1 ± 1.4
	german	46.3 ± 2.2 45.4 ± 3.5 46.4 ± 1.8	46.0 ± 2.3	45.1 ± 3.4 47.1 ± 2.0
	yeast	65.9 ± 2.9 67.1 ± 3.7 70.4 ± 2.8	68.4 ± 2.3	68.2 ± 3.7 70.4 ± 3.0
	segmentation	82.0 ± 1.9 83.8 ± 2.9 81.6 ± 2.2	81.5 ± 1.8	84.8 ± 3.2 84.8 ± 2.2
	splice	74.9 ± 0.9 86.3 ± 0.8 79.6 ± 1.2	76.4 ± 1.3	87.9 ± 1.1 87.2 ± 0.6
	abalone	32.3 ± 0.7 31.4 ± 1.7 31.9 ± 0.8	31.7 ± 1.1	31.5 ± 1.2 32.3 ± 1.0
	spambase	85.9 ± 0.7 88.5 ± 0.6 87.4 ± 0.9	87.2 ± 0.8	88.8 ± 0.8 89.4 ± 0.8
	pageblocks	62.0 ± 2.9 61.5 ± 4.1 55.5 ± 4.0	61.5 ± 3.5	61.0 ± 4.1 62.5 ± 3.5
	Mean	71.1 ± 2.7 72.5 ± 3.6 72.2 ± 3.5	72.2 ± 3.3	72.1 ± 3.9 73.9 ± 3.1
	Average Rank	4.26	3.91	3.39	3.39	4.17	1.87

Table 4 .

 4 Same experiment as inTable 2 after having applied a Random Under Sampling of the negative examples until n -= n + . ± 10.6 35.6 ± 10.9 38.2 ± 12.2 36.4 ± 12.6 35.6 ± 10.9 39.3 ± 13.

	Dataset	Euclidean	LMNN	ITML	GMML	IMLS	IML
	hayes	63.4 ± 9.0	67.7 ± 7.4	64.7 ± 6.7	66.4 ± 8.1	67.7 ± 7.4	66.0 ± 7.6
	wine	91.2 ± 2.6	94.1 ± 2.8	94.2 ± 3.8	92.7 ± 3.7	94.1 ± 2.9	93.9 ± 3.2
	sonar	70.4 ± 5.2	73.1 ± 6.3	70.2 ± 5.7	69.9 ± 5.5	71.1 ± 9.2	74.4 ± 4.8
	glass	64.6 ± 3.5	63.2 ± 4.5	61.1 ± 4.9	64.6 ± 3.1	62.7 ± 5.0	64.5 ± 4.7
	newthyroid	86.6 ± 4.6	91.4 ± 5.0	91.1 ± 4.9	90.6 ± 3.3	91.4 ± 5.0	92.3 ± 2.6
	spectfheart	44.2 ± 3.9	42.6 ± 8.0	46.7 ± 4.6	45.9 ± 5.6	42.6 ± 8.0	48.8 ± 6.3
	heart	77.4 ± 2.0	75.8 ± 3.3	76.6 ± 2.5	77.3 ± 1.9	75.5 ± 3.3	77.1 ± 2.1
	bupa	53.8 ± 4.1	54.1 ± 4.8	54.6 ± 4.1	55.7 ± 3.6	54.1 ± 4.8	55.0 ± 3.2
	iono	73.1 ± 5.2	73.3 ± 4.1	75.4 ± 3.4	74.7 ± 3.2	73.3 ± 4.1	77.3 ± 3.0
	libras	34.3 8
	wdbc	93.7 ± 1.2	92.9 ± 1.6	93.6 ± 1.8	93.2 ± 2.1	92.3 ± 2.5	94.7 ± 1.4
	balance	87.5 ± 1.5	90.1 ± 1.3	92.8 ± 1.5	90.1 ± 1.7	90.2 ± 1.3	90.7 ± 1.4
	australian	80.4 ± 1.7	81.7 ± 2.2	82.2 ± 1.6	81.5 ± 2.3	81.7 ± 2.2	82.5 ± 2.2
	pima	60.8 ± 2.7	60.5 ± 2.1	62.2 ± 1.7	60.8 ± 2.4	60.4 ± 2.1	61.2 ± 2.5
	vehicle	74.0 ± 3.1	89.7 ± 1.6	87.7 ± 2.6	85.5 ± 3.1	89.7 ± 2.0	89.3 ± 1.9
	german	46.7 ± 1.6	46.9 ± 2.5	47.3 ± 2.3	47.5 ± 1.6	46.2 ± 2.0	48.0 ± 1.8
	yeast	57.2 ± 4.5	60.8 ± 4.6	60.9 ± 3.8	59.7 ± 3.8	61.2 ± 4.9	61.9 ± 3.9
	segmentation	64.6 ± 3.1	70.4 ± 2.4	65.7 ± 3.4	64.3 ± 2.9	72.4 ± 2.9	74.2 ± 1.8
	splice	75.9 ± 0.7	86.5 ± 0.6	79.5 ± 1.6	76.2 ± 1.2	87.9 ± 0.9	87.2 ± 0.6
	abalone	32.8 ± 1.1	32.5 ± 1.4	32.5 ± 1.3	31.6 ± 1.0	32.2 ± 1.3	32.6 ± 1.4
	spambase	85.0 ± 1.0	88.1 ± 1.1	86.8 ± 1.2	86.2 ± 1.2	88.4 ± 0.7	88.7 ± 0.8
	pageblocks	46.8 ± 3.7	50.2 ± 4.8	43.0 ± 5.2	48.3 ± 4.5	49.6 ± 5.6	49.1 ± 4.0
	Mean	66.6 ± 3.5	69.1 ± 3.8	68.5 ± 3.7	68.1 ± 3.6	69.1 ± 4.0	70.4 ± 3.4
	Average Rank	4.83	3.65	3.30	3.96	3.43	1.83

A function f is q-Lipschitz w.r.t. its first argument if for any u, v, | f (u)f (v)| ≤ q|u -v|.

[START_REF] Bellet | Metric learning[END_REF] A loss A is (σ, p)-admissible w.r.t. its first argument M if it is convex in M and if ∀z 1 , z 2 ,

z 3

, z 4 , | A (M, z 1 , z 2 )-A (M, z 3 , z 4 )|≤σ|y[START_REF] Estabrooks | A multiple resampling method for learning from imbalanced data sets[END_REF] -y 34 |+p with y i j = + 1 if y i =y j and y i j = -1 otherwise.

https://archive.ics.uci.edu/ml/datasets.html

http://sci2s.ugr.es/keel/datasets.php

https://www.csie.ntu.edu.tw/ ~cjlin/libsvmtools/ datasets/binary.html#splice

We will make the source code available online in case of acceptance.


Sara Bellali

Jacques Bou Khalil

Anthony Fontanini

Didier Raoult

Jean-Christophe Lagier
email: jclagier@yahoo.fr

Prof Jean-Christophe

A new protectant medium preserving bacterial viability after freeze drying

Keywords: freeze-drying, protectant medium, preservation, bacterial viability, survival, gut microbiota

Freeze-drying technology has been widely considered for decades as a suitable technique to preserve microorganisms. However, protective agents must be added prior to freeze drying to improve the survival and storage stability of the bacteria.

The objective of our study was to evaluate the effect of a new protectant medium containing sucrose (10%), trehalose (10%), skimmed milk (10%) and antioxidants on the viability of gut bacteria under different storage conditions. Two strains were tested, Escherichia coli and Akkermansia muciniphila, as examples of facultative aerobic and anaerobic bacteria, respectively. We studied the cell viability and bacterial morphology in 5 fecal samples in the presence and absence of this protectant medium using plating technique, flow cytometry and scanning electron microscopy.

The results of bacterial viability assessed by plating method showed that the protectant medium yielded higher survival rates for both strains whatever the storage conditions (85-93%) compared to normal saline solution (0.36-37.50%). It also showed its effectiveness on fecal samples, where bacterial viability after freeze-drying was 89.47±7.63% and 84.01±7.44%, as evidenced by flow cytometry analysis and plating method. However unprotected samples showed the lowest cell viability at 19.01±12.88% and 13.23±9.56%, as measured by flow cytometry and plating method. In addition, bacterial size and shape were conserved in the protectant medium. In contrast, storage without protectant medium severely damaged bacterial morphology.

In conclusion, our study is the first to use morphological features as well as culture-dependant and culture-independent tests to evaluate the effectiveness of a new protectant medium.

Introduction

Freeze-drying has been the most commonly used technique to enhance the storage stability of probiotics for decades. Probiotics are defined as living microorganisms that, once administered in adequate amounts, confer a health benefit for the host [START_REF] Who | Evaluation of health and nutritional properties of powder milk and live lactic acid bacteria[END_REF]. A variety of probiotics such as yeast, Lactobacillus, and Bifidobacterium species have been successfully preserved by freeze-drying [START_REF] Biavati | Bifidobacteria: history, ecology, physiology and applications[END_REF][START_REF] Gomes | Bifidobacterium spp. and Lactobacillus acidophilus: biological, biochemical, technological and therapeutical properties relevant for use as probiotics[END_REF].

Recently, a novel group of obligate anaerobic bacteria was considered as the next generation of probiotics in the treatment of inflammatory bowel disease [START_REF] Everard | Cross-talk between Akkermansia muciniphila and intestinal epithelium controls diet-induced obesity[END_REF][START_REF] Van Immerseel | Butyric acid-producing anaerobic bacteria as a novel probiotic treatment approach for inflammatory bowel disease[END_REF]. Among them, Faecalibacterium prausnitzii is a strict anaerobic bacterium that constitutes 3-5% of all fecal bacteria [START_REF] Miquel | Faecalibacterium prausnitzii and human intestinal health[END_REF] and is considered a highly abundant butyrate producer [START_REF] Breyner | Microbial Anti-Inflammatory Molecule (MAM) from Faecalibacterium prausnitzii shows a protective effect on DNBS and DSS-Induced colitis model in mice through Inhibition of NF-κB pathway[END_REF]. Akkermansia muciniphila, a strict anaerobic mucin-degrading bacterium, is also considered an abundant candidate of the healthy human microbiota (1-5%) [START_REF] Derrien | Akkermansia muciniphila gen. nov., sp. nov., a human intestinal mucin-degrading bacterium[END_REF]. To our knowledge, only one recent study reported the preservation of Akkermansia muciniphila by freeze-drying [START_REF] Marcial-Coba | Viability of microencapsulated Akkermansia muciniphila and Lactobacillus plantarum during freeze-drying, storage and in vitro simulated upper gastrointestinal tract passage[END_REF].

In addition, the human microbiota is known to play an important role in health and disease [START_REF] Wang | The Human Microbiota in Health and Disease[END_REF]. Several studies have already reported that fecal microbiota transplantation (FMT) provided highly effective treatment of Clostridium difficile infection [START_REF] Brandt | Long-term follow-up of colonoscopic fecal microbiota transplant for recurrent Clostridium difficile infection[END_REF][START_REF] Hocquart | Early Fecal Microbiota Transplantation Improves Survival in Severe Clostridium difficile Infections[END_REF][START_REF] Kassam | Fecal microbiota transplantation for Clostridium difficile infection: systematic review and meta-analysis[END_REF][START_REF] Surawicz | Guidelines for diagnosis, treatment, and prevention of Clostridium difficile infections[END_REF][START_REF] Van Nood | Duodenal infusion of donor feces for recurrent Clostridium difficile[END_REF]. In recent years, FMT has emerged, evolving from the use of fresh fecal microbiota to the cryopreservation of fecal microbiota [START_REF] Hamilton | Standardized frozen preparation for transplantation of fecal microbiota for recurrent Clostridium difficile infection[END_REF]. However, colonoscopic administration remains invasive to patients and is therefore also complicated for the healthcare units at the technical and organizational level. Other FMT modalities are reported to be easier and less complicated. In fact, the orally-administered FMT via encapsulated fresh microbiota [START_REF] Louie | Fecal microbiome transplantation (FMT) via oral fecal microbial capsules for recurrent Clostridium difficile infection (rCDI)[END_REF] and the frozen/thawed capsules seem to be the best choice [START_REF] Youngster | Oral, capsulized, frozen fecal microbiota transplantation for relapsing Clostridium difficile infection[END_REF]. In addition, oral FMT showed the same clinical efficiencies as fresh FMT in the treatment of recurrent Clostridium difficile infection [START_REF] Lee | Frozen vs Fresh Fecal Microbiota Transplantation and Clinical Resolution of Diarrhea in Patients With Recurrent Clostridium difficile Infection: A Randomized Clinical Trial[END_REF]. However, these FMT administrations are limited by storage and transport conditions.

In order to overcome these complications and simplify these procedures, we suggested oral freeze-dried FMT, a far gentler and esthetically pleasing format that could be selfadministrated [START_REF] Staley | Successful Resolution of Recurrent Clostridium difficile Infection using Freeze-Dried, Encapsulated Fecal Microbiota; Pragmatic Cohort Study[END_REF]. It would also be less expensive and time-consuming. To date, few studies have addressed the issue of freeze-dried fecal samples ready to be used for fecal transplantation [START_REF] Hecker | Fecal Microbiota Transplantation by Freeze-Dried Oral Capsules for Recurrent Clostridium difficile Infection[END_REF][START_REF] Hirsch | Effectiveness of fecal-derived microbiota transfer using orally administered capsules for recurrent Clostridium difficile infection[END_REF][START_REF] Jiang | Randomised clinical trial: faecal microbiota transplantation for recurrent Clostridum difficile infection -fresh, or frozen, or lyophilised microbiota from a small pool of healthy donors delivered by colonoscopy[END_REF][START_REF] Staley | Successful Resolution of Recurrent Clostridium difficile Infection using Freeze-Dried, Encapsulated Fecal Microbiota; Pragmatic Cohort Study[END_REF].

In fact, freeze-drying combines freezing and drying stresses that were known to be more detrimental to sensitive bacteria than cryopreservation [START_REF] Heylen | Safeguarding bacterial resources promotes biotechnological innovation[END_REF]. Before the freeze-drying or lyophilization process, bacterial suspensions need to be frozen first and subsequently dried under vacuum. However, without suitable protectant medium, freezedrying severely damages cell membranes and proteins [START_REF] Carpenter | Stabilization of phosphofructokinase with sugars during freeze-drying: characterization of enhanced protection in the presence of divalent cations[END_REF][START_REF] Carpenter | Modes of stabilization of a protein by organic solutes during desiccation[END_REF][START_REF] Crowe | Are freezing and dehydration similar stress vectors? A comparison of modes of interaction of stabilizing solutes with biomolecules[END_REF][START_REF] Leslie | Trehalose and sucrose protect both membranes and proteins in intact bacteria during drying[END_REF], causing a decrease in viability [START_REF] Panoff | Cold stress responses in mesophilic bacteria[END_REF][START_REF] Wolfe | Freezing, drying, and/or vitrification of membrane-solute-water systems[END_REF]. Furthermore, the importance of adding protectants such as disaccharides (e.g. trehalose, sucrose) [START_REF] Leslie | Trehalose and sucrose protect both membranes and proteins in intact bacteria during drying[END_REF], polyols (e.g. mannitol, sorbitol) (Ana S. Carvalho et al., 2003;[START_REF] Efiuvwevwere | Mannitol-enhanced survival of Lactococcus lactis subjected to drying[END_REF] and proteins (e.g. skimmed milk) [START_REF] Castro | Evidence of membrane damage in Lactobacillus bulgaricus following freeze drying[END_REF] prior to freezing or drying, played a major role in preserving cells and improving their storage viability. Meanwhile, their protective mechanisms remained unclear and not fully understood. Three major protective mechanisms were reported: (i) preventing the intra and extracellular ice formation [START_REF] Baumann | Preservation of lactic cultures[END_REF][START_REF] Fowler | Cryo-Injury and Biopreservation[END_REF] (ii) water replacement hypothesis by hydrogen bonds formation (Leslie et al., 1995) (ii) or glassy matrix formation [START_REF] Crowe | The role of vitrification in anhydrobiosis[END_REF].

Each protectant medium impacts differently microorganisms and no universal protectant medium has yet been developed [START_REF] Champagne | The Freeze-Drying of Lactic Acid Bacteria[END_REF][START_REF] Sanders | Environmental stress responses in Lactococcus lactis[END_REF]. The glycerol is one of the most commonly cryoprotectant used in preparation of frozen liquid suspensions of microbiota [START_REF] Costello | Establishing a Fecal Microbiota Transplant Service for the Treatment of Clostridium difficile Infection[END_REF][START_REF] Hamilton | Standardized frozen preparation for transplantation of fecal microbiota for recurrent Clostridium difficile infection[END_REF][START_REF] Kaito | Fecal microbiota transplantation with frozen capsules for a patient with refractory acute gut graft-versus-host disease[END_REF][START_REF] Satokari | Simple faecal preparation and efficacy of frozen inoculum in faecal microbiota transplantation for recurrent Clostridium difficile infection -an observational cohort study[END_REF]. However, this cryoprotectant is not recommended for freeze drying and is not useful due to its viscosity which can lead to a sticky and an insufficient dried product not amenable to encapsulation. For this, having a suitable protectant medium capable of preserving anaerobic and aerobic bacteria during freeze drying process can be challenging [START_REF] Font De Valdez | Comparative study of the efficiency of some additives in protecting lactic acid bacteria against freeze-drying[END_REF]. Here, we investigate the effect of a protectant medium containing sucrose (10%), trehalose (10%), skimmed milk (10%) and antioxidants to preserve Akkermansia muciniphila and Escherichia coli during freezing, freeze-drying, and subsequent storage for 30 days at different conditions. This protectant medium was also used in a specific process to preserve fecal microbiota used in FMT. Thus, we evaluated the effectiveness and preservation rate of this protectant medium by studying cell viability and bacterial morphology in samples. To this end, we used plating technique, flow cytometry and scanning electron microscopy.

Materials & methods

Preservation of individual bacteria

1.1.Protectant medium composition

In this work, we used a protectant medium selected on the basis of our previous study patented in 2017 under the following number (N° WO/2018/234645). This medium contains per g/L in phosphate buffered saline (PBS) (Life Technologies, Paiseley, United Kingdom) the following elements: sucrose (10g); skimmed milk (10g); trehalose (5g); CaCl2 (0.1g); MgCl2 (0.1g); KOH (0.3/0.6g) and tree antioxidants being: Ascorbic acid (1g); Uric acid (0.4g) and Glutathione (0.1g). The pH of the medium was about 7.3 ± 0.2. Skimmed milk was sterilized at 121°C for 15 min and the rest of the solution (antioxidants and sugar) was sterilized by filtration through 0.22 μm filters.

1.2.Bacterial strains and growth conditions

In order to prove the effectiveness of the protectant medium, facultative aerobic and strict anaerobic strains were tested. Escherichia coli (CSUR P1966) was grown in Columbia sheep blood agar plates (BioMérieux, Marcy l'Etoile, France) at 37°C for 24 hours under aerobic conditions. Akkermansia muciniphila strain (CSUR P6566) was grown in Columbia sheep blood agar plates at 37°C for 48 hours under anaerobic conditions using a GasPak generator (Becton Dickinson Microbiology Systems, Sparks, MD, USA). Both strains were isolated from fecal materials of two healthy donors in our laboratory.

1.3.Freeze-drying procedures

Firstly, we chose the best time to harvest bacterial colonies with higher viable rates and less dead bacteria based on flow cytometry measurements and microscopic observations at different incubation times (12, 24 and 75 hours). Fresh Escherichia coli and Akkermansia muciniphila were harvested directly from agar plates after 24 hours and 48 hours, respectively, and mixed with two solutions: (a) normal saline solution (NaCl 0.9%) (Fresenius Kabi, Sevres, France) as control, and (b) the protectant medium (milk 10% + sucrose 10% + trehalose 5%+ antioxidants), with a final concentration of 2-3x10 10 CFU/mL and 5-4x10 10 CFU/mL, respectively, for Akkermansia muciniphila and Escherichia coli.

Bacterial suspensions in (a) normal saline solution and in (b) the protectant medium were placed into three 2 mL, type I (Wheaton, Millville, NJ, USA) serum vials and frozen at -80°C for 5 hours. Suspensions were desiccated in a DELTA 1-24 LSC-CHRIST freeze-dryer at a condenser temperature of -80°C and at a chamber pressure of 0.63 mbar for 12 hours at 0°C, followed by 3 hours at + 30°C. After freeze-drying, vials were sealed manually, and stored at +4°C (Figure 1).

1.4.Storage conditions

Bacterial suspensions in presence of protectant medium or saline solution were stored for 48 hours and for 30 days under different conditions (Figure 1):

(i) Frozen at -80°C, (ii) Frozen at -196°C by dipping the vials into liquid nitrogen (LN2), (iii) Frozen at 80°C for 5 hours then freeze-dried, (iiii) At +4°C.

After freeze-drying, samples were rehydrated and homogenized with phosphate buffered saline (PBS) to return to their original volume (500 µL) at 25°C and incubated at room temperature.

Preservation of fecal samples

2.1.Stool sample collection and preparation

Fresh fecal samples were obtained from 5 healthy donors (2 women, 3 men) from France, with a normal body mass index. These volunteers were not subjected to any feeding trial, specific diet, or antibiotic treatment for the last six months prior to sampling. Stool samples were collected in a sterile stool container under anaerobic conditions by using a GasPak generator (Becton, Dickinson and Sparks, USA) and were then immediately transported to the laboratory. Donors were informed of the study and signed informed consents. The project received the IHU Méditerranée Infection ethics committee agreement under number 2016-011.

2.2.Stool sample preparation

Sample preparations were carried out under a sterile hood. Briefly, 50g of fecal matter was blenderized (BOSCH Ultracompact 400W device) with 250 mL of normal saline solution (NaCl 0.9%) (Fresenius Kabi, Sevres, France) for 5 min, then sieved using coffee strainers to remove food debris. The slurry was centrifuged for 15 min at 6,000 x g. The pellet was suspended in one half of the initial volume in 2 media: (a) the normal saline solution used as control, and (b) the protectant medium (skimmed milk 10%+sucrose10% + trehalose 5% + antioxidants). Then, bacterial suspensions were frozen at -80°C for 5 hours and directly freeze-dried under the same conditions as described above, and finally stored at +4°C (Figure 1).

Cell viability determination

Bacterial viability of each storage condition, (a) in normal saline solution and (b) in protectant medium was calculated before and after freeze-drying using viable counts and flow cytometry method.

Enumeration of bacteria by plate count

Decimal dilutions were prepared with PBS. One hundred µL of the sample suspension was mixed with 900 µL of PBS, vortexed for 10 seconds and serially diluted with PBS. The Columbia agar plates were divided up into three or four lines into which 10µL of each dilution were speared onto it. Each dilution was plated in triplicates. The plates were incubated at 37°C for 48 h under aerobic and anaerobic (GasPak generator) conditions.

The number of colony-forming units per milliliter (CFU/mL) was determined and the viability was defined as the percentage ratio of viable cells after freeze-drying and viable cells before freeze-drying using the following equation:

Survival % = ⁄ ⁄
x 100

LIVE/DEAD enumeration

The membrane integrity was determined using the LIVE/DEAD BacLight Bacterial Viability Kit (Molecular Probes, Invitrogen, USA) as described previously [START_REF] Bellali | Among live and dead bacteria, the optimization of sample collection and processing remains essential in recovering gut microbiota components[END_REF].

Scanning electron microscopy for bacterial morphology evolution

Fresh, frozen and freeze-dried samples were either directly smeared onto microscopy slides or cyto-centrifuged on cytospin slides. Slides were then processed to image acquisition, after staining with PTA (phosphotungstic acid 1 %) in order to check morphological appearance changes and cell integrity.

We used a table top scanning electron microscope SEM (Hitachi TM4000 Plus) to evaluate bacterial structures. The SEM has a capability of observing specimen in low vacuum pressure (100 Pa to 101 Pa) to reduce charge-up on the specimen's surface by the irradiated electrons.

Evacuation time after the loading of specimens into the SEM Chamber is shorter than 2 minutes, which is much quicker than conventional SEMs with high vacuum condition. The imaging process for all samples in the presence and absence of protectant medium was acquired at the same acquisition settings regarding magnification, intensity and voltage mode.

All settings are displayed on micrographs.

Statistical analysis

All experiments were carried out in triplicate. All viability data are expressed as the means standard deviations (SDs). The difference between two means before and after freeze-drying was calculated using the Mann-Whitney t-test (Prism v5.0, GraphPad). Means were considered significantly different when P-value was less than 0.05. Differences between fresh viable cells and all storage conditions were analyzed by one-way analysis of variance followed by a Bonferroni's multiple comparisons test (Prism v5.0, GraphPad).

Results

Evolution of the protectant medium efficacy by reporting the viability of individual bacteria after freezing and freeze-drying 1. Viability of Escherichia coli and Akkermansia muciniphila after freezing and freeze drying

The effect of protectant medium and saline water solution on the viability of Escherichia coli and Akkermansia muciniphila under the different storage conditions (for 48 hours and 30 days of storage) are summarized in Figure 2 and compared to fresh cultures suspended in protectant medium and saline water solution.

(a) Freezing at -80°C

The viability of E. coli with protectant medium (85.00%) did not vary after 48 hours of freezing and remained stable after 30 days of storage (84.00%). Similarly, the viability of A. muciniphila did not greatly decreased after 48 hours of freezing (86.67%) and even after 30 days of storage (75.00%). However, the freezing in presence of normal saline had significantly reduced the viability of E. coli and A. muciniphila to 25% and 0.67% respectively. Thus, after 30 days of storage the viability remained reduced to 24.17% and 0.04% respectively.

(b) Freezing in liquid nitrogen (-196°C)

Similar results were obtained after fast freezing in liquid nitrogen, where frozen E. coli and A. muciniphila with protectant medium exhibited high survival rates of 93.00% and 90.00% respectively. The viability of both strains remained unchanged, even after 30 days of storage (90.00% and 86.67%, respectively). However, unprotected E. coli and A. muciniphila showed low viability of 37.50% and 25.56% respectively, compared to that observed in the presence of a protectant medium. After 30 days of storage, the viability of E. coli and A. muciniphila was 25.00% and 17.00% respectively.

(c) Freeze-drying

Results showed that freeze-dried E. coli and A. muciniphila in the presence of protectant medium had the highest survival rates of 85.00% and 91.67% respectively after 48 hours. It remained stable after 30 days of storage, 80.00% and 83.33% respectively. In the presence of normal saline solution, the freeze-drying process severely damaged cell viability of E. coli (2.67%) and A. muciniphila (0.36%) compared to freezing. The viability of E. coli and A. muciniphila significantly decreased to 0.29% and 1x10 8 CFU/mL; 0.02%, respectively.

(d) Storage at +4°C

Bacterial suspensions of E. coli and A. muciniphila in the presence of protectant medium, exhibited high survival rates of 80.00% and 90.00% respectively after 48 hours of storage at 4°C. Meanwhile, the viability of A. muciniphila and E. coli was declined to 60.00% and 41.65% respectively after 30 days of storage. However, both A. muciniphila and E. coli, suspended in normal saline water, were affected within the 48 hours of storage, with a large drop of viability (29.17% and 33.33%, respectively) after 48 hours. After 30 days of storage, cell viability of E. coli and A. muciniphila (25.00%) and (16.67%), respectively were higher than freezing or freeze drying.

Evaluation of bacterial morphology and shape integrity by scanning electron microscopy after freezing and freeze-drying

Bacterial morphology and integrity of both Escherichia coli and Akkermansia muciniphila treated with (a) normal saline solution or with (b) protectant medium in fresh, frozen and freeze-dried state were evaluated for each conservation condition.

Micrographs of samples in the presence of normal saline solution showed damaged

Escherichia coli cells under all storage conditions tested (freezing or freeze-drying). The degenerative aspect of the bacteria was clearly observed (Figure 3b1,3b2,3c1,3c2,3d1,3d2).

Similar results were obtained in Akkermansia muciniphila with an irregularity in bacterial shapes and an increase in cell size due to osmotic stress (2.59±1.45µm versus 0.78±0.24µm) (Figure 4b1,4b2,4b3,4b4). Nevertheless, morphology of both bacteria was preserved in the presence of the protectant medium, which shape and size seemed to be better conserved.

Frozen or freeze-dried Akkermansia muciniphila were predominantly spherical, although some were found to be irregular or elongated in the same way as fresh cells (Figure 4c1,4c3,4d1,4d3). Interestingly, we detected small shapes of Escherichia coli that were osmoticallydehydrated due to sucrose and trehalose contained in the protectant medium (Figures 3e1,3e2,3f1,3f2,3g1,3h1).

Validation of the proof of concept by evaluating the effect of the protectant medium on the viability of freeze-dried fecal samples

Enumeration of fecal material by plate method

Total bacterial counts (anaerobic and aerobic counts), and bacterial viability were presented in Table 1. Total bacterial counts ranged from 3.42x10 9 CFU/mL to 6.92x10 9 CFU/mL, and from 4.10x10 9 CFU/mL to 8.75x10 9 CFU/mL, respectively, for samples resuspended in normal saline solution and in the protectant medium.

Before freeze-drying, we noticed differences between total cell counts in all fresh fecal samples, whatever protected or not, so that the number of anaerobic bacteria was ten times higher than that of aerobic bacteria in five samples. Moreover, anaerobic bacteria were much more numerous for samples suspended in protectant medium (5.90x10 9 ±1.92 x10 9 ) than in normal saline solution (4.25 x10 9 ±1.47 x10 9 ) (Table 1).

After freeze-drying, all five freeze-dried fecal samples with normal saline solution decreased in viability and had a lower average survival rate of 13.23±9.56%, ranging from 1.69% to 26.27%, compared to samples dried in the presence of protectant medium, producing an average survival rate of 84.01±7.44% ranging from 75.60% to 94.12% (Table 1).

As shown in Figure 5A, we found significant differences before and after freeze-drying when the fecal sample was dried with normal saline solution (P=0.01). On the other hand, no significant differences were found for samples freeze-dried in the presence of protectant medium as compared to fresh fecal samples (Figure 5B).

2.1.Enumeration of fecal samples by flow cytometry

In addition to plate count methods, we used the flow cytometry method to assess the viability of freeze-dried fecal samples. IP and SYTO9 were used simultaneously for viability assessment before and after freeze-drying. Three bacterial populations were observed; live (SYTO9-stained), dead (IP-stained) and injured, which were double stained (IP/SYTO9stained).

The relative percentages of live, dead and injured bacterial populations obtained before and after freeze-drying for the five fecal samples are presented in Figure 6. Before freeze-drying, the relative percentage of viable fecal bacteria in the presence of normal saline solution were: 84.01% (sample 1), 66.65% (sample 2), 61.93% (sample 3), 68.90% (sample 4), and 65.98%(sample 5).Therefore, after freeze-drying it decreased to 16.67% (sample 1), 23.48% (sample 2), 5.85% (sample 3), 18.67% (sample 4) and 2.27% (sample 5) (Figure 6).

Simultaneously, dead and injured bacterial populations increased, whereas, in the presence of the protectant medium, we did not observe any significant changes before (79.87%, 64.52%, 68.38%, 73.13%, and 70.73%) and after (73.10%, 56.77%, 55.54%, 74.12%, 60.31%) freezedrying (Figure 6). Overall, the presence of protectant medium during freeze-drying showed a higher protection of bacteria (89.47±7.63%), while normal saline solution damaged most of them (19.01±12.88%). These results were similar to those found by plate counting method. Furthermore, using statistical analysis, we found a significant difference (P= 0.0079) between live bacterial counts before and after freeze-drying of samples dried with saline solution (Figure 5C). Meanwhile, viable count of samples dried with protectant medium were not statistically significant (Figure 5D).

Electron microscopy of fecal material before and after freeze drying

The processed fecal samples showed significant differences of their bacterial components at the level of shape and size. Before freeze-drying, stool samples in the presence of normal saline solution showed a bigger bacterial size (1.06±0.15µm) (Figure 7a1-a2) compared to those suspended in protectant medium (0.70±0.14µm) (Figure 7b1-b2). Furthermore, after freeze-drying, the protectant medium conferred a more stable state to the bacteria, where size and shape remained conserved and smaller (0.61±0.17µm) (Figure 7d1-d2) in contrast to the normal saline solution (1.05±0.16µm) (Figure 7c1-c2).

Discussion

Freeze-drying in the presence of a protective medium to ensure optimal bacterial viability plays a key role in microbiology nowadays. This quality of preservation is extremely needed in FMT and probiotics production.

In this study, we evaluated the impact of freeze drying on the bacterial viability using the normal saline solution as a control typically used in the majority (62%) of clinical studies of FMT [START_REF] Gough | Systematic review of intestinal microbiota transplantation (fecal bacteriotherapy) for recurrent Clostridium difficile infection[END_REF][START_REF] Van Nood | Duodenal infusion of donor feces for recurrent Clostridium difficile[END_REF], and a new protectant medium containing sucrose (10%), trehalose (10%), skimmed milk (10%) and three antioxidants (uric acid, ascorbic acid and glutathione). Those compounds were found in several studies, in combination or individually, to be effective in protecting bacteria against the injuries during freezing or freeze drying and improved storage stability. Several investigators have reported the positive effect of trehalose on bacterial survival during freeze drying [START_REF] Jain | Trehalose and protein stability[END_REF][START_REF] Mensink | How sugars protect proteins in the solid state and during drying (review): Mechanisms of stabilization in relation to stress conditions[END_REF]. Trehalose is also known as an antioxidant protecting membranes against oxidative stress [START_REF] Herdeiro | Trehalose protects Saccharomyces cerevisiae from lipid peroxidation during oxidative stress[END_REF]. In addition, trehalose and sucrose were the most disaccharides commonly used as protectants during the freezing-drying process [START_REF] Broeckx | Drying techniques of probiotic bacteria as an important step towards the development of novel pharmabiotics[END_REF][START_REF] Carpenter | Modes of stabilization of a protein by organic solutes during desiccation[END_REF]A. S. Carvalho et al., 2003A. S. Carvalho et al., , 2002;;[START_REF] Crowe | The role of vitrification in anhydrobiosis[END_REF][START_REF] Crowe | Interactions of sugars with membranes[END_REF][START_REF] Leslie | Trehalose and sucrose protect both membranes and proteins in intact bacteria during drying[END_REF][START_REF] Linders | Effect of Added Carbohydrates on Membrane Phase Behavior and Survival of DriedLactobacillus plantarum[END_REF][START_REF] Paiva | Biotechnological Applications of the Disaccharide Trehalose[END_REF][START_REF] Zayed | Influence of trehalose and moisture content on survival of Lactobacillus salivarius subjected to freeze-drying and storage[END_REF]. These two sugars were capable of protecting cell membranes by water replacement involved hydrogen bonding between sugars and polar-head groups of the phospholipids [START_REF] Crowe | Are freezing and dehydration similar stress vectors? A comparison of modes of interaction of stabilizing solutes with biomolecules[END_REF][START_REF] Leslie | Trehalose and sucrose protect both membranes and proteins in intact bacteria during drying[END_REF][START_REF] Rudolph | Membrane stabilization during freezing: The role of two natural cryoprotectants, trehalose and proline[END_REF]. They also were capable of reducing the ice formation by increasing the shrinkage of cells before freezing [START_REF] Fowler | Cryo-Injury and Biopreservation[END_REF]. Remarkably, this dehydration mechanism was observed in our results carried out by scanning electron microscopy, where cells of E. coli were found dehydrated before and after freezing and drying due to the presence of sugars that induced osmosis-derived dehydration. In contrast, in the absence of sugars, cells of E. coli and Akkermansia muciniphila were found injured and severally damaged after freezing and drying due to ice crystal formation, as described previously in several studies [START_REF] Champagne | The Freeze-Drying of Lactic Acid Bacteria[END_REF][START_REF] Sanders | Environmental stress responses in Lactococcus lactis[END_REF]. Such a mechanism still remains unclear.

Proteins also provided an additional protective effect by covering the cells and balancing the cell membranes during freeze-drying and storage [START_REF] Buitink | High Critical Temperature above Tg May Contribute to the Stability of Biological Systems[END_REF]. Indeed, skimmed milk was selected as an efficient drying medium [START_REF] Bevilacqua | Bifidobacteria as Potential Functional Starter Cultures: A Case Study by MSc Students in Food Science and Technology (University of Foggia, Southern Italy)[END_REF][START_REF] Hubálek | Protectants used in the cryopreservation of microorganisms[END_REF], it contains proteins providing an additional protective layer for the cells (Abadias et al., 2001;[START_REF] Carvalho | Relevant factors for the preparation of freeze-dried lactic acid bacteria[END_REF] and stabilizing membrane components [START_REF] Castro | Changes in the cell membrane of Lactobacillus bulgaricus during storage following freeze-drying[END_REF][START_REF] Selmer-Olsen | Effect of protective solutes on leakage from and survival of immobilized Lactobacillus subjected to drying, storage and rehydration[END_REF]. Another feature of skimmed milk is its ability to dry easier and provide a higher yield of dry matter. According to the study reported by [START_REF] Zayed | Influence of trehalose and moisture content on survival of Lactobacillus salivarius subjected to freeze-drying and storage[END_REF], the addition of skimmed milk to the mixture of trehalose and sucrose gave a higher survival rate during subsequent storage (Abadias et al., 2001).

Furthermore, previous studies reported that the mixture of many components of protectant medium (e.g. sucrose + trehalose, sucrose + trehalose + skimmed milk…etc.) could result in a better protection of microorganisms than single component due to additive or synergic protective effects [START_REF] Berner | Effect of protective agents on the viability of Lactococcus lactis subjected to freeze-thawing and freeze-drying[END_REF][START_REF] Celik | Factors influencing the stability of freeze-dried stressresilient and stress-sensitive strains of bifidobacteria[END_REF][START_REF] Hubálek | Protectants used in the cryopreservation of microorganisms[END_REF][START_REF] Jalali | Stability evaluation of freeze-dried Lactobacillus paracasei subsp. tolerance and Lactobacillus delbrueckii subsp. bulgaricus in oral capsules[END_REF][START_REF] Keivani Nahr | Optimization of the nanocellulose based cryoprotective medium to enhance the viability of freeze dried Lactobacillus plantarum using response surface methodology[END_REF][START_REF] Ming | Formulation of Protective Agents for Improvement of Lactobacillus salivarius I 24 Survival Rate Subjected to Freeze Drying for Production of Live Cells in Powderized Form[END_REF][START_REF] Sharma | Standardization of lyophilization medium for Streptococcus thermophilus subjected to viability escalation on freeze drying[END_REF][START_REF] Yang | Freeze-drying of live attenuated Vibrio anguillarum mutant for vaccine preparation[END_REF][START_REF] Yu | Optimization of protectant, salinity and freezing condition for freeze-drying preservation of Edwardsiella tarda[END_REF][START_REF] Zhang | Orthogonal array deciphering MRS medium requirements for isolated Lactobacillus rhamnosus ZY with cell properties characterization[END_REF].

Regarding total anaerobic bacteria in fecal samples prior to freeze drying, we found that samples suspended with our protectant medium had more anaerobes than samples without protectant. The higher number of anaerobes was explained in our previous study (N° WO/2018/234645) by the presence of antioxidants in our protectant medium preserving sensitive anaerobic bacteria from oxygen. In addition to Akkermansia muciniphila, our protectant medium showed superior efficacy to preserve other fastidious anaerobes such as Treponema denticola and Treponema pectinovorum (Data unpublished) and also extremely oxygen-sensitive (EOS) bacteria such as Methanobrevibacter smithii and Faecalibacterium prausnitzii (Data unpublished).

Creating a medium protecting the majority of the gut bacteria was challenging. We chose Escherichia coli as an example of aerobic bacteria and Akkermansia muciniphila as an example of anaerobic bacteria in addition to its beneficial effects. We recommend the freeze drying of anaerobic and aerobic bacteria using the protectant medium comprising sucrose (10%), trehalose (10%), skimmed milk (10%) and antioxidants. We believe that this protectant medium holds great promises for therapeutic purposes such as fecal microbiota transplant and probiotics.

The idea of creating this protectant medium comes from many conservation strategies and for the first time we used morphological, culture-dependent and culture-independent tests to evaluate its effectiveness. Further studies will be conducted to test freeze-dried microbiota transplantation in oral capsules using this protectant medium. IHU Méditerranée Infection, and for the installation of a TM4000 microscope at IHU Méditerranée Infection facility. The authors also thank Magdalen Lardière for English reviewing. hours and 30 days of storage. Freeze-dried cells were stored at 4°C. Viability was measured before and after all storage conditions using the plate counting method. Statistically significant differences between fresh cells and all storage conditions were analyzed by oneway analysis of variance followed by Bonferroni's multiple comparisons test. * P < 0.01 * * P < 0.001 * * * P < 0.0001, ns: not significant.

Samples

Figure 1 :

 1 Figure 1: Schematic figure illustrating the design of the protocols established to evaluate the

Figure 2 :

 2 Figure 2: Effect of protectant medium and saline water solution on the viability of (A)

Figure 3 :

 3 Figure 3: Scanning electron micrographs of Escherichia coli conserved in normal saline

Figure 4 :

 4 Figure 4: Scanning electron micrographs of Akkermansia muciniphila conserved in normal

Figure 5 :

 5 Figure 5: Statistical differences of viable count of 5 fecal samples before and after freeze

Figure 6 :

 6 Figure 6: Relative abundance and flow cytometry measurements (Cells/mL) of Live/Dead

Figure 7 :

 7 Figure 7: Scanning electron micrographs of stool sample conserved in normal saline

Acknowledgments

We sincerely thank Takashi Irie, Kyoko Imai, Shigeki Matsubara, Taku Sakazume, Yusuke Ominami, Hishada Akiko and the Hitachi team of Japan (Hitachi High-Technologies Corporation, Science & Medical Systems Business Group 24-14, Nishi-shimbashi 1-chome,

Funding source

This study was supported by the Institut Hospitalo-Universitaire (IHU) Méditerranée Infection, the National Research Agency under the program « Investissements d'avenir », reference ANR-10-IAHU-03. SB, JCL and DR are co-inventors of a patent (N° WO/2018/234645).

Competing Interests

The authors declare no competing interests.