Rémi Bertrand

H Robe

Damien Texier

Y Zedan

E Feulvarch

P Bocher

Heterogeneous mechanical behavior of dissimilar aluminum alloys FSWed joints assessed by OHR-DIC characterization

Keywords: FSWed dissimilar aluminum joint, Hybrid lightweight design, Digital image correlation, Tensile behavior

come

Introduction

Currently, reducing the weight of transport structure is an important challenge. The aluminum alloys (AA) 7020 comply with this constraint. AA7020 is a medium strength Al-Zn-Mg alloy well known for its good mechanical performance [1]. However, Al-Cu-Mg alloys, such as AA2139, will progressively replace 7XXX alloys because of the conjunction of higher mechanical strength and lower density [2]. Unfortunately, the intensive use of AA2139 is limited due to the production cost. Building hybrid structure by assembling medium-performance and highperformance aluminum alloys as a function of the local stresses in the structure is an attractive solution for cost-effective design strategies. In order to assemble these dissimilar alloys, Friction Stir Welding (FSW) is particularly suitable. Several process parameters, such as speeds [3], axial force [4], plates position [5], were reported to influence both the material flow of the weld and the mechanical behavior. In the present study, various dissimilar joints AA2139-T8/AA7020-T651 were produced with a large range of industrial welding parameters in order to correlate these parameters with the local mechanical behavior.

Materials and procedures

Two 5 mm-thick sheets made of AA2139-T8 on the advancing side (AS) and AA7020-T651 on the retreating side (RS) were friction stir welded with a Triflat TM tool. The rolling direction of October 11-13, 2017 the sheets was always perpendicular of the welding direction. Five process parameters were used, with a fixed tilt angle of 1.5°, a rotation speed ranging from 600 to 1000 rev.min -1 and an advance speed ranging from 250 to 550 mm.min -1 . Welds were studied after a 5 months natural aging time. FSWed samples were extracted and prepared in order to perform cross-section macrographs and Vickers microhardness maps. Macro-and microtensile tests were carried out. For each condition, local strain fields of the deformed joint were assessed by OHR-DIC [6] in the cross-welding direction/normal direction plane for various loadings.

Results and discussion

Defect-free joints were produced with a rather similar macrostructure in this range of welding parameters.

For all welding conditions, macrographs revealed typical features of FSWed joints in terms of grain size evolution in the various areas (nugget, TMAZ, HAZ and base materials). An example is given Fig. 1a. Hardness maps (Fig. 1b) showed that nugget have globally a lower hardness than both base material. The two materials in the nugget exhibited very comparable hardness values, making them hard to distinguish. It worth mentioning that the minimum hardness region (MHR) was systematically found on the AA7020 side (RS) at a distance between 7 and 13 mm far from the center line (HAZ). This distance decrease when the rotation speed increase. The MHR is due to a thermal effect leading to precipitate coarsening. The macroscopic tensile performances of the whole joints were similar in this range of welding parameters. Joint necking and fracture always occurred on the MHR. However, a step-by-step local plastic strain measurements campaign (Fig. 1c) revealed that strain localization first occurred in the nugget at 283 MPa. Strain localization happens at the MHR for a stress level about 20 MPa higher. While the hardness map and the fracture surface features have suggested coarse precipitates and high ductile material in the MHR, the early plasticity in the nugget region was not expected from microhardness measurements. This early strain localization could play an important feature in fatigue. Differences in strain hardening of the nugget and the MHR together with some stress triaxiality related to the thickness of the related soft regions lead to intense strain localization in the MHR at high deformations.

Conclusions

OHR-DIC allowed to highlight the heterogeneous behavior of the joint for several stages of the tensile test. In particular, it has shown that the area where the deformation begins, i.e. the nugget, is not the fracture area, i.e. the MHR. This is explained by the disparities of locale tensile behavior of these areas.

Figure 1 :

 1 Figure 1: (a) microstructure, (b) microhardness and (c) local plastics strain in loading direction for various stresses in cross-section (250 mm.min -1 , 600 rev.min -1 )


T Bui

J-M Loubes

L Risser

& P Balaresque

Distribution regression model with a Reproducing Kernel Hilbert Space approach

Keywords: Regression, Reproducing kernel Hilbert space, Wasserstein distance, Transient evoked otoscoutic emission

In this paper, we introduce a new distribution regression model for probability distributions. This model is based on a Reproducing Kernel Hilbert Space (RKHS) regression framework, where universal kernels are built using Wasserstein distances for distributions belonging to W 2 (Ω) and Ω is a compact subspace of R. We prove the universal kernel property of such kernels and use this setting to perform regressions on functions. Different regression models are first compared with the proposed one on simulated functional data. We then apply our regression model to transient evoked otoascoutic emission (TEOAE) distribution responses and real predictors of the age.

Introduction

Regression analysis is a predictive modeling technique that has been widely studied over the last decades with the goal to investigate relationships between predictors and responses (inputs and outputs) in regression models, see for instance [START_REF] Azaïs | Le modèle linéaire par l'exemple[END_REF][START_REF] Kutner | Applied linear regression models[END_REF] and references therein. When the inputs belong to functional spaces, different strategies have been investigated and used in several application domains about functional data analysis [START_REF] Neter | Applied linear statistical models[END_REF][START_REF] Ramsay | Applied functional data analysis: methods and case studies[END_REF]. Extensions of the Reproducing Kernel Hilbert Space (RKHS) framework became recently popular to extend the results of the statistical learning theory in the context of regression of functional data as well as to develop estimation procedures of functional valued functions f [START_REF] Preda | Regression models for functional data by reproducing kernel hilbert spaces methods[END_REF][START_REF] Kadri | Nonlinear functional regression: a functional rkhs approach[END_REF]. This framework is particularly important in the field of statistical learning theory because of the so-called Representer theorem, which states that every function can be written as a linear combination of the kernel function evaluated at training points [START_REF] Berlinet | Reproducing kernel Hilbert spaces in probability and statistics[END_REF].

In our framework, we aim to solve the regression problem with inputs belonging to probability distribution spaces, whose responses are probability distributions and whose predictors are real values. Specially, we consider the model

y i = f (µ i ) + i , (1)
where {µ i } n i=1 are probability distributions on R, {y i } n i=1 are real numbers and the i represent an independent and identically distributed Gaussian noise. As in classical regression models, this setting estimates an unknown function f from the observations {(µ i , y i )} n i=1 .

The framework of [START_REF] Smola | A hilbert space embedding for distributions[END_REF] recently became popular to embed probability distributions into RKHS. It solves the learning problem of distribution regression in a two-stage sampled setting and use the analytical solution of a kernel ridge regression problem to regress from probability measures to real-valued observations. Specifically, the authors embed a distribution to an RKHS H(k) induced by a kernel k which is defined on set of distribution inputs. The regression function is composed of an unknown function f and an element of H(k), where H(K) is the RKHS induced by kernel K defined on the set of mean embeddings of distributions to RKHS H(k). Whereas the relation between the random distribution and the real number response can be learnt by using directly Representer theorem for the regularized empirical risk over RKHS . In what follows, we will consider kernels built using the Wasserstein distance. Details on Wasserstein distances and their links with optimal transport problems can be found in [START_REF] Villani | Optimal transport: old and new[END_REF]. Some kernels with this metric have been developed in [START_REF] Kolouri | Sliced Wasserstein kernels for probability distributions[END_REF][START_REF] Peyré | Gromov-Wasserstein averaging of kernel and distance matrices[END_REF]. We focus here in the work in [START_REF] Bachoc | A gaussian process regression model for distribution inputs[END_REF], in which the authors built a family of positive definite kernel. Within the setting of this paper, we will construct a RKHS corresponding to this kind of kernels to apply the theory of RKHS. More specifically, for an input belonging to Wasserstein spaces, the authors of [START_REF] Bachoc | A gaussian process regression model for distribution inputs[END_REF] built a class of positive definite kernels that are functions of Wasserstein distances. More interestingly, the framework of [START_REF] Christmann | Universal kernels on non-standard input spaces[END_REF], the authors provided a kind of universal kernel tittled the Gaussian-type RBF-kernel. This result is really useful for this paper because from [START_REF] Sriperumbudur | Universality, characteristic kernels and rkhs embedding of measures[END_REF][START_REF] Micchelli | Universal kernels[END_REF] we can build easily a RKHS from a universal kernel. Hence by using the good universal properties, that will be mentioned in Section 3, we will define a new method to construct the RKHS from a universal kernel which is supported by the family of positive definite kernels. Then, we will get a particular estimation from the Representer theorem for an unknown function f in the regression model with distribution inputs.

The paper is structured as follows: In Section 2, we first recall important concepts about kernels on Wasserstein spaces W 2 (R). We then give a brief introduction to Wasserstein spaces on R and explain how positive definite kernel done are constructed in [START_REF] Bachoc | A gaussian process regression model for distribution inputs[END_REF]. Section 3 deals with the proposed setting of distribution regression models. We motivate there the use of universal kernels and build an estimation function for the learning problem. We then assess the numerical performance of this method in Section 4. The tests are first performed on simulated generated data to compare our model with state-of-the-art ones. Then, we study the relationship between the age and hearing sensitivity by using TEOAEs recording that are acquired by stimulating with a very short but strong broadband stimulus. These recordings are then the ear responses by emitting a sound track on a given frequency. More precisely, we predict the age of the subjects, on which they were acquired using the proposed distribution regression model, from TEOAE data. Discussions are finally drawn Section 5.

2 Kernel on Wasserstein space W 2 (R)

The Wasserstein space on R

Let us consider the set W 2 (R) of probability measures on R with a finite moment of order two. For two µ, ν probability distributions in W 2 (R), we denote Π(µ, ν) the set of all probability measures π over the product set R × R with first (resp. second) marginal µ (resp. ν). The transportation cost with quadratic cost function, which we denote quadratic transportation cost, between two measures µ and ν is defined as:

T 2 (µ, ν) = inf π∈Π(µ,ν)
|x -y| 2 dπ(x, y).

(

) 2
This transportation cost allows to endow the set W 2 (R) with a metric by defining the quadratic Monge-Kantorovich (or quadratic Wasserstein) distance between µ and ν as

W 2 (µ, ν) = T 2 (µ, ν) 1/2 .
A probability measure π in Π(µ, ν) performing the infimum in ( 2) is called an optimal coupling. This vocabulary transfers to a random vector (X 1 , X 2 ) with distribution π. We will call W 2 (R) endowed with the distance W 2 the Wasserstein space. More details on Wasserstein spaces and their links with optimal transport problems can be found in [START_REF] Villani | Optimal transport: old and new[END_REF].

For distributions in R, the Wasserstein distance can be written in a simpler way as follows: For any µ ∈ W 2 (R), we denote by F -1 µ the quantile function associated to µ. Given a uniform random variable U on [0, 1], F -1 µ (U ) is the random variable with law µ. Then, for every µ and ν the random vector (F -1 µ (U ), F -1 ν (U )) is the optimal coupling (see [START_REF] Villani | Optimal transport: old and new[END_REF]), where F -1 is defined as

F -1 µ (t) = inf{u, F µ (u) ≥ t}. (3)
In this case, the simplest expression for the Wasserstein distance is given in [START_REF] Whitt | Bivariate distributions with given marginals[END_REF]:

W 2 (µ, ν) = E(F -1 µ (U ) -F -1 ν (U )) 2 . ( 4
)
Topological properties of Wasserstein spaces are reviewed in [START_REF] Villani | Optimal transport: old and new[END_REF]. Hereafter, compacity will be required and will be obtained as follows: let Ω ⊂ R be a compact subset, then the Wasserstein space W 2 (Ω) is also compact. In this paper, we consider Wasserstein spaces W 2 (Ω), where Ω is a compact subset on R endowed with the Wasserstein distance W 2 . Hence for any µ ∈ W 2 (Ω), we denote

F µ | Ω : Ω → [a, b] with [a, b] ⊂ [0, 1]
the distribution function restricted on a compact subset Ω. We also define F -1 µ | Ω as:

F -1 µ | Ω (t) = inf {u ∈ Ω, F µ | Ω (u) ≥ t}, ∀t ∈ [a, b]. (5)
Given a uniform random variable V on [a, b], F -1 µ | Ω is a random variable with law µ. By inheriting properties from W 2 (R) for every µ and ν in W 2 (Ω), the random vector

F -1 µ | Ω (V ), F -1 ν | Ω (V
) is an optimal coupling. In this case, we consider in this paper the simplest expression for the Wasserstein distance between µ and ν in W 2 (Ω):

W 2 (µ, ν) = E(F -1 µ | Ω (V ) -F -1 ν | Ω (V )) 2 . ( 6
)

Kernel

Constructing a positive definite kernel defined on the Wasserstein space is not obvious and was recently done in [START_REF] Bachoc | A gaussian process regression model for distribution inputs[END_REF]. For sake of completeness, we recall here briefly this construction.

Theorem 2.1. Let k Θ : W 2 (Ω) × W 2 (Ω) → R
with the parameter Θ := (γ, H, l) such that γ = 0 and l > 0 defined as

k Θ (µ, ν) := γ 2 exp - W 2H 2 (µ, ν) l . ( 7
)
Then for 0 < H ≤ 1, k Θ is a positive definite kernel.
The proof of this Theorem directly follows Theorem 2.2 and Propositions 2.3. In this paper we use Theorem 2.1 to study the properties of such kernel in the RKHS regression framework.

The following theorem that can be found in [START_REF] Cowling | Harmonic analysis on semigroups[END_REF] or referred to Theorem III.1 in [START_REF] Bachoc | A gaussian process regression model for distribution inputs[END_REF], also provides a generic way to construct kernel using completely monotone functions.

Theorem 2.2. (Schoenberg) Let F : R + → R + be a completely monotone function, and K a negative definite kernel. Then (x, y) → F (K(x, y)) is a positive definite kernel.

The following proposition which can be found in [START_REF] Bachoc | A gaussian process regression model for distribution inputs[END_REF], finally gives conditions on the exponent H to achieve negative definite kernel using exponents of the Wasserstein distance.

Proposition 2.3. The function W 2H 2 is a negative definite kernel if and only if 0 < H ≤ 1.
Proof. The proof of Theorem 2.1 follows immediately below from Theorem 2.2 and Proposition 2.3. Applying Proposition 2.3, we deduce that W 2H 2 (µ, ν) with H = 1 is a negative definite kernel for all µ, ν in W 2 (Ω). We can easily see that e -λx with λ positive is a completely monotone function. Let us then consider a mapping as follows:

F : R + → R + x → γ 2 e -λx , where γ 2 > 0, x = W 2 2 (µ, ν) with λ = 1 l , l > 0.
Then F is also a completely monotone function. From Theorem 2.2, k Θ is a positive definite kernel.

Regression

Setting

In this section, we aim to define a regression function with distribution inputs. The problem of distribution regression consists in estimating an unknown function f :

W 2 (Ω) → R by using observations (µ i , y i ) in W 2 (Ω) × R for all i = 1, • • • , n .
We recall observes in (1) as follows

y i = f (µ i ) + i . (8)
To provide a general form for functions defined on distributions, we will use the RKHS framework. Let k Θ : W 2 (Ω) × W 2 (Ω) → R be defined in Theorem 2.1. For a fixed valid Θ, we define a space F 0 as follows:

F 0 := span {k Θ (•, µ) : µ ∈ W 2 (Ω)} .
And F 0 is endowed with the inner product

f n , g m F 0 = n i=1 m j=1 α i β j k Θ (µ i , ν j ),
where

f n (•) = n i=1 α i k Θ (•, µ i ) and g m (•) = m j=1 β j k Θ (•, ν j ). The norm in F 0 is corresponding to the inner product, f n 2 F 0 = n i=1 α 2 i k Θ (µ i , µ i ). ( 9
)
Let F be the space of all continuous real-valued functions from W 2 (Ω) to R. The set F 0 consists of all functions in F which are uniform limits of functions of form f n . We want to approximate F 0 as well as possible F. Following universal approximating property that is a universal kernel k Θ has a property that F 0 = F. Hence we will consider in the following section that a universal kernel of k Θ to prove that F 0 is dense in F and that F 0 = F.

From that for all f, g belong in F, the inner product is well defined as following formula

f, g F := lim n→∞ f n , g n F 0 . (10)
Coming back to our problem, we want to estimate the unknown function f by an estimation function f obtained by minimizing the regularized empirical risk over the RKHS F. For this consider, we solve the solution of the minimization problem

f = argmin f ∈F n i=1 |y i -f (µ i )| 2 + λ f 2 F , (11)
where λ ∈ R + is the regularization parameter. Using the Representer theorem, this leads to the following expression for f ,

f : µ → f (µ) := n j=1 αj k Θ (µ, µ j ), (12)
where {α j } n j=1 are parameters typically obtained from training data.

Universal kernel

First, we recall the definition of a universal kernel and the main theorem to ensure universal properties of positive definite kernels in Theorem 2.1.

Definition 3.1. Let C(X) be the space of continuous bounded functions on compact domain X.

A continuous kernel k on domain X is called universal if the space of all functions induced by k is dense in C(X), i.e, for all f ∈ C(X) and every > 0 there exists a function g induced by k with f -g ∞ ≤ .

For more information on universal kernel and RKHS, we refer to Chapter 4 in [START_REF] Villani | Optimal transport: old and new[END_REF] and [START_REF] Sriperumbudur | Universality, characteristic kernels and rkhs embedding of measures[END_REF], [START_REF] Micchelli | Universal kernels[END_REF]. Theorem 3.2. Let choose the parameter Θ in [START_REF] Berlinet | Reproducing kernel Hilbert spaces in probability and statistics[END_REF] such that γ = 0, l > 0 and

H = 1. The kernel k Θ : W 2 (Ω) × W 2 (Ω) → R defined in (7) is universal.
The proof of this Theorem relies on the two following Proposition 3.3 and Proposition 3.4.

Proposition 3.3. Let F µ | Ω : Ω → [a, b] with [a, b] ⊂ [0, 1] be the distribution function re- stricted on a compact subset Ω of R, F -1 µ | Ω be defined by F -1 µ | Ω (t) = inf {u ∈ Ω, F µ | Ω (u) ≥ t} for all t ∈ [a, b]. Then F µ | Ω is continuous if and only if F -1 µ | Ω is strictly increasing on [inf ranF µ | Ω , sup ranF µ | Ω ]. F µ | Ω is strictly increasing if and only if F -1 µ | Ω is continuous on ranF µ | Ω , where ranF µ | Ω := {F µ | Ω (x) : x ∈ Ω}, the range of F µ | Ω .
See e.g [START_REF] Embrechts | A note on generalized inverses[END_REF] for a proof of Proposition 3.3. Proposition 3.4. Let X be a compact metric space and H be a separable Hilbert space such that there exist a continuous and injective map: ρ : X → H. For σ > 0, the Gaussian-type RBF-kernel k σ : X × X → R is the universal kernel, where

k σ (x, x ) := exp(-σ 2 ρ(x) -ρ(x ) 2 H ), x, x ∈ X.
See a part of Theorem 2.2 in [START_REF] Christmann | Universal kernels on non-standard input spaces[END_REF] for proof of Proposition 3.4.

Proof. Proof of Theorem 3.2. From Proposition 3.3 with the conditions including the distribution restricted on Ω, F µ | Ω be continuous and F µ | Ω be strictly increasing on Ω, then there exists a continuous and injective map

ρ : W 2 (Ω) → L 2 [a, b] µ → ρ(µ) := F -1 µ | Ω . F -1 µ | Ω is continuous on [a, b] and strictly increasing on [inf ranF µ | Ω , sup ranF µ | Ω ].
We consider a Wasserstein space W 2 (Ω) metrized by the Wasserstein distance W 2 with Ω be a compact subset on R and L 2 [a, b] be the usual space of square integrable functions on [a, b]. For σ in Proposition 3.4 is exactly defined by 1/ √ l for all l > 0. We have

k Θ (µ, ν) = γ 2 exp - F -1 µ | Ω -F -1 ν | Ω 2 L 2 [a,b]
l is the universal kernel from Proposition 3.4. We complete the proof of Theorem 3.2.

The minimization program in [START_REF] Peyré | Gromov-Wasserstein averaging of kernel and distance matrices[END_REF] can be solved explicitely using the representer theorem of [START_REF] Kimeldorf | Some results on tchebycheffian spline functions[END_REF]. Note that Schölkopf and Smola [START_REF] Smola | Learning with kernels[END_REF] give a simple proof of a more general version of the theorem. Define c ij as follows

c ij = γ 2 exp - W 2 2 (µ i , µ j ) l and α = (α 1 , • • • , α n ) T , Y = (y 1 , • • • , y n ) T .
Now we take the matrix formulation of ( 11) we obtain

min α trace((Y -Cα)(Y -Cα) T ) + λtrace(Cαα T ), (13)
where the operation trace is defined as

trace(A) = n i=1 a ii with A = (a ii ) n i=1 .
Taking the derivative of (13) with respect to vector α, we find that α satisfies the system of linear equations

(C + λI)α = Y. ( 14
) Hence f (µ) = n j=1 αj k Θ (µ, µ j ), (15)
with α = (C + λI) -1 Y. (16)
4 Numerical Simulations and Real data application 4.1 Simulation

Overview of the simulation procedure

In this section, we investigate the regression model for predicting the regression function from distributions. Particularly, we want to estimate the unknown function f in model ( 8) by using the proposed estimation f in [START_REF] Micchelli | Universal kernels[END_REF], so we need to present how we can optimize the parameters in this formula. We then compare the regression model based on RKHS induced by our universal kernel function to more classical kernel functions operating on projections of the probability measures on finite dimensional spaces. We address the input-output map given by

f (ν) = m ν 0.05 + σ ν , ( 17
)
where ν is a Gaussian distribution of mean m ν and variance σ 2 ν . We consider the ground truth function f that we compare with a predicted function f , such as:

f (ν) = γ 2 n j=1 αj exp - W 2 2 (ν, µ j ) l , (18)
where the Wasserstein distance between two Gaussian distribution is calculated using:

W 2 2 (µ, ν) = (m µ -m ν ) 2 + (σ µ -σ ν ) 2 ,
where µ ∼ N (m µ , σ 2 µ ) and ν ∼ N (m ν , σ 2 ν ). Each value αj is estimated using Eq. ( 16) which depends on parameter λ > 0. Thus our proposed estimation function f depends totally on the three parameters λ > 0, γ = 0 and l > 0. To understand the effects of these parameters on f , we define reference values of λ, γ and l. We then generate a training set including the normal distributions ν i ∼ N (m ν i , σ ν i ) such that cor(ν i , ν j ) = 0, ∀i, j = 1, • • • n, with n be a size of training set. In this simulation, we take n = 200. From the training set {(ν i , f (ν i ))} n=200 i=1 , we fit two regression models which we call "Wasserstein" and "Legendre", for which we provide more details below. Then we evaluate the quality of the two regression models on a test set of size n t of the form {(ν t,i , f (ν t,i ))} nt i , where ν t,i is generated in the same way as ν i above. We consider the following quality criteria, that is the root mean square error (RMSE) to see the qualify of our regression model

RM SE 2 ( f , f ) = 1 n t nt i=1 f (ν t,i ) -f (ν t,i ) 2 .

Detail on the regression models

We refer our model by Wasserstein and introduce briefly Legendre regression models. Wasserstein model first propose the estimated function as follows

f (ν t,i ) = γ 2 n j=1 αj exp - (m ν j -m ν t,i ) 2 + (σ ν j -σ ν t,i ) 2 l , (19)
where ν t,i , i = 1, • • • , n t belong to testing set with size n t , and ν j , j = 1, • • • , n belong to training set with size n. The estimated function f depends on three parameters γ = 0, λ > 0 and l > 0.

The Legendre model is based on kernel functions operating on finite dimensional linear projections of the distributions. For a Gaussian distribution µ ∼ N (m, σ 2 ) with density f µ (t) =

1 √ 2πσ exp -(t-m) 2 2σ 2
and support [0, 1], we compute for i = 0, • • • , θ -1:

a i (µ) = 1 0 1 √ 2πσ exp - (t -m) 2 2σ 2 p i (t)dt,
where p i is the i-th normalized Legendre polynomial, with

1 0 p 2 i (t)dt = 1.
The integer θ is called the order of the decomposition. Then k L operators on the vector (a 0 (ν), • • • , a θ-1 (ν) and is of the form

k L (ν 1 , ν 2 ) = γ 2 exp - θ-1 i=0 |a i (ν 1 ) -a i (ν 2 )| l i . ( 20
)
Thus the estimated regression function f in this case is calculated by following function

f (ν t,i ) = γ 2 exp - θ-1 i=0 |a i (ν t,i ) -a i (ν j )| l i .
We just consider the orders of the decomposition 5 and 10. We fix l i = l for all i = 1, • • • , θ -1, this estimated function depends also on three parameters γ = 0, λ > 0 and l > 0. We see the two graphs, one follows values of λ in the left side and l in the right side. RMSE will be minimized, in which it is lower than 0.08, with 0 < λ < 15 and l big enough. We note that when 0 < l < 1 RMSE is quite a big value for all λ > 0, so we avoid to chose these values of l.

Result

In simulation, we will see the effects of parameters λ > 0, γ = 0 and l > 0 on RMSE between predicted function f (ν i,t ) and exact function f (ν i,t ) through the testing set {(ν i,t )} nt i=1 . We also take two sizes of testing set n t = 500, n t = 700 to see the changes of RMSE. We just show the detailed presentation about choosing the optimal parameters on the "Wasserstein" model.

Case of testing set size n t = 500 : Now we consider RMSE in the case of n t = 500 under the different fixed values γ = 1/2, 1, 10 and running λ > 0 separated by 30 values from 0.005 to 30, l > 0 separated by 25 values from 0.005 to 20. Let us see here the values of RMSE with the different cases of γ in following Figure 1,2,3.

We realize through three choices of γ = 1/2, 1, 10 that the values γ give the same impact of RMSE variations, but the smallest RMSE in the case γ = 1. In following this stimulation, we fix the value of γ = 1 and run the values of λ and l to see the changes of RMSE in the case of bigger size of testing set. Figure 2: In the case of n t = 500, fixing a value γ = 1, we run λ > 0 separated by 30 values from 0.005 to 30, l > 0 separated by 25 values from 0.005 to 20. We see the two graphs, one follows values of λ in the left side and l in the right side. The variations of RMSE in this case is not change significantly with the case of γ = 1/2, however, it looks smaller than the case γ = 1/2. We also see that RMSE will be minimized by two case: first 0 < λ < 1 and l big enough; second λ > 1 for all l > 1 and RMSE is quite big at 0 < l < 1 all λ > 0. Case of testing set size n t = 700 : Now we consider RMSE in the case of n t = 700 in Figure 4 for a fixed value γ = 1 and running λ > 0 separated by 30 values from 0.005 to 30, l > 0 separated by 25 values from 0.005 to 20. We want to see the affects of testing set size on RMSE. Then we consider directly about the estimated regression function effects under parameters λ and l. As far as we known, there exists oversmoothing and undersmoothing issues which happen sometimes in the learning problem when the error component is small, but the estimated function is oversmooth or undesmooth. See the Figure 5, to more clearly about our regression model with the exact function defined in [START_REF] Cowling | Harmonic analysis on semigroups[END_REF].

And finally, we consider the different RMSE's between "Wasserstein" and "Legendre" model by choosing the values γ = 1, λ = 5 and l = 10 under considering n t = 700. In Table 1, we show the values of RMSE quality criteria for the "Wasserstein" and "Legendre" distribution regression models. From the values of the RMSE criterion, the "Wasserstein" model clearly outperforms the other models. The RMSE of the "Legendre" models slightly decreases when the order increases, and stay well above the RMSE of the "Wasserstein" model. Hence from the Figure 5 and Table 1, we can see that by choosing the optimal parameters for γ, l and λ we can obtain a very well estimation function f without the under-smoothing and over-smoothing issues in the learning problem. Our regression model stay well above the RMSE criterion.

Figure 4: In the case of n t = 700, fixing a value γ = 1, we run λ > 0 separated by 30 values from 0.005 to 30, l > 0 separated by 25 values from 0.005 to 20. RMSE is almost lower than 0.06 when λ > 1 and l > 1. However for 0 < λ < 1, we can also obtain the small RMSE when l big enough. This figure provides a view about size of testing set, in which for the big enough of testing set size we will obtain the smaller RMSE under of the optimal parameters γ, λ and l. The green line presents an exact function, which is many more variations, we desire find the optimal parameters to obtain a more smooth curve. From Figure 4, we can chose these parameters following RMSE, however, in some cases it happens over-smoothing and under-smoothing. See in this Figure, the blue line looks like have some desirable properties when we chose the big enough values of λ and l. model RMSE "Wasserstein" 0.04 "Legendre" order 5 0.15 "Legendre" order 10 0.11

Table 1: RMSE values of different quality criteria for the "Wasserstein" and "Legendre" distribution regression models. The "Wasserstein" is based on universal kernel function operating directly on the input Gaussian distributions, while "Legendre" is based on linear projections of the Gaussian distribution inputs on finite-dimensional spaces. For "Legendre", the order value is the dimension of the projection space. The quality criteria is the root mean square error (RMSE) should be minimal. The "Wasserstein" distribution regression model clearly outperforms the "Legendre" with two orders 5 and 10.

Our interpretation for these results is that, because of the nature of the simulated data (µ i , f (µ i )) working directly on distributions and with the Wasserstein distance, is more appropriate than using linear projections. Indeed, in particular, two distributions with similar means and small variances are close to each other with respect to both Wasserstein distance and the value of the output function f . However,the probability density functions of the two distributions are very different from each other with respect to the L 2 distance in the case that the ratio between the two variances is large. Hence linear projections based on probability density functions is inappropriate in the setting considered here.

Application on evolution of hearing sensitivity

An otoacoustic emission (OAE) is a sound which is generated from within the inner ear. OAEs can be measured with a sensitive microphone in the ear canal and provide a noninvasive measure of cochlear amplification (see Chapter: Hearing basics in [START_REF] Eggermont | Hearing Loss: Causes, Prevention, and Treatment[END_REF]). Recording of OAEs has become the main method for newborn and infant hearing screening (see Chapter: Early Diagnosis and Prevention of Hearing Loss in [START_REF] Eggermont | Hearing Loss: Causes, Prevention, and Treatment[END_REF]). There are two types of OAEs: spontaneous otoacoustic emissions (SOAEs), which can occur without external stimulation, and evoked otoacoustic emissions (EOAEs), which require an evoking stimulus. In this paper, we consider a type of EOAEs that is Transient-EOAE (TEOAE) (see for instance in [START_REF] Joris | Frequency selectivity in old-world monkeys corroborates sharp cochlear tuning in humans[END_REF]), in which the evoked response from a click covers the frequency range up to around 4kHz. More precisely, each TEOAE models the ability of the cochlea to response to some frequencies in order to transform a sound into an information that will be processed by the brain. So to each observation is associated a curve (the Oto-Emission curve) which describes the response of the cochlea at several frequencies to a sound. The level of response depends on each individual and each stimulus should be normalized, but the way each individual reacts is characteristic of its physiological characteristic. Hence to each individual is associated a curve, which after normalization, it is considered as a distribution µ describing the repartition of the responses for different frequencies ranging from 0 to 10 kHz. These distributions are shown in Figure 6 and Table 2.

Name

Age 0(Hz) 39.06 78. The relationship between age and hearing sensitivity is investigated in [START_REF] O-Uchi | Age-related changes in evoked otoacoustic emission in normal-hearing ears[END_REF][START_REF] Collet | Age-related changes in evoked otoacoustic emissions[END_REF] The results show that when age increases, the presence of EOAEs by age group and the frequency peak in spectral analysis decreases and EOAE threshold increases. The differences in EOAE have been also reported between age classes in humans. These results convey the idea that the response evolves with age and that the effect of ages in hearing issues is deeply related to the changes of the cochlear properties. Hence our model uses as input these distributions µ and try to build

Discussion

In this paper, we have introduced a new estimated function for regression model with distribution inputs. More precisely, we effectively used class of positive definite kernel produced by Wasserstein distance, built in [START_REF] Bachoc | A gaussian process regression model for distribution inputs[END_REF] by proving that it is a kind of universal kernel. Researching the universal kernel theories, we detected a very good property of our universal kernel to build a RKHS. Then we obtained a particular estimation from Representer theorem for our distribution regression problem, these works showed that the relation between the random distribution and the real number response can be learnt by using directly the regularized empirical risk over RKHS. Our proposed estimation is clearly better than state-of-the-art-ones in simulated data. More interestingly, we researched successfully TEOAE curve of each individual in human population as a distribution which after normalization. We then investigated the relationship between age and its TEOAE that the response involves with age and the effect of age in hearing issues is deeply related to the change of cochlear. This is a new interesting approach in the field of Biostatistics, in which we indicated the evolution of hearing capacity under statistical domain -distribution regression model. We believe that our paper tackles an important issue for data science experts willing to predict problems in regression with probability distributions as input. The extension of this work on distributions for general dimensions should be addressed in a further work, using for instance as a kernel the one buit in [START_REF] Bachoc | Gaussian process forecast with multidimensional distributional entries[END_REF].

Figure 1 :

 1 Figure 1:In the case of n t = 500, fixing a value γ = 1/2, we run λ > 0 separated by 30 values from 0.005 to 30, l > 0 separated by 25 values from 0.005 to 20. We see the two graphs, one follows values of λ in the left side and l in the right side. RMSE will be minimized, in which it is lower than 0.08, with 0 < λ < 15 and l big enough. We note that when 0 < l < 1 RMSE is quite a big value for all λ > 0, so we avoid to chose these values of l.

Figure 3 :

 3 Figure 3: In the case of n t = 500, fixing a value γ = 10, we run λ > 0 separated by 30 values from 0.005 to 30, l > 0 separated by 25 values from 0.005 to 20. RMSE in this case looks bigger than two above cases of γ = 1/2 and γ = 1.

Figure 5 :

 5 Figure 5: Regression function under exact and estimated function.The green line presents an exact function, which is many more variations, we desire find the optimal parameters to obtain a more smooth curve. From Figure4, we can chose these parameters following RMSE, however, in some cases it happens over-smoothing and under-smoothing. See in this Figure, the blue line looks like have some desirable properties when we chose the big enough values of λ and l.

Figure 6 :

 6 Figure 6: Oto-emission curves. 48 TEOAE curves following to frequencies ranging from 0Hz to 10kHz.

Figure 7 :

 7 Figure 7: Histogram of real age in a human population. The age distribute diversity from 15 to 30, however, there is a few of individual of age from 35 to 40 and 45 to 50. And there exists no individual have age from 40 to 45.

Figure 8 :

 8 Figure 8: Histogram of difference between real and predicted age for OAE. In the first column, in which the difference between real and estimate age is very small closing to zero, this means more accuracy between real and predicted age. Almost the ages from 20 to 35 lie in this column.

Figure 9 :

 9 Figure 9: Real and Predicted Age. By using the optimized parameters of γ = 1, l = 10 and λ > 0 depending on age class, we obtain almost the exacted ages belonging to the age class [20, 30] with λ around 15. For instance, we can predict very well the exact ages 20, 21, 23, 24, 25, 27, 29 corresponding to the predicted ages 19.50, 21.02, 22.83, 23.87, 24.89, 27.15, 28.76.

Table 2 :

 2 TEOAE data. 48 individuals are considered in human population, recorded in South Africa, with last names in first column, their exact ages in second column and the others describe the responses of the cochlea at several frequencies ranging from 0Hz to 10kHz.

	• • • 1171.88 1210.94 • • • 9765.62 9804.69

a regression model to link between the age and these distributions representing the response of the cochlea at frequencies ranging from 0Hz to 10kHz. More precisely, we estimate the age for each level of response normalized and treated as a distribution µ by using our proposed function as follows

where F -1 µ (U ) defined as [START_REF] Preda | Regression models for functional data by reproducing kernel hilbert spaces methods[END_REF] and the value of αj is chosen by optimal parameter λ in [START_REF] Whitt | Bivariate distributions with given marginals[END_REF]. We estimate the integral in [START_REF] Eggermont | Hearing Loss: Causes, Prevention, and Treatment[END_REF] by following formula

where we can understand each F µ i is an experimental distribution function of µ i and M is the number of discretized frequencies. As far as we know, each individual is associated with a curve, which after normalization without lost relationship among original data, it is considered as a distribution µ i . To calculate F -1

, we arrange each curve in ascending order, for instance we denote

. Hence, we write again the formula ( 22)

where X µ i is a curve X following to distribution µ i .

In our simulation, we choose γ = 1, the value of l = 10 and the value of λ > 0. We aim to study the age in relation with its TEOAE curve of 48 subjects, recorded on human population in South Africa, with the range of frequency from 0Hz to 10kHz. See the Figure 7 to show the differences between the age of 15 to 50 years old. Following the estimated function in [START_REF] Eggermont | Hearing Loss: Causes, Prevention, and Treatment[END_REF], we take 47 distributions {µ j } 47 j=1 for training set to calculate estimation value of αj and try to estimate real age of a remaining individual µ i with i = j. And the results are showed clearly in the Figure 8 and Figure 9 about the exact age and predicted age.

Hence in figure 7 and Figure 9, we applied effectively our proposed estimation function in predicting age from its TEOAE data. By choosing the optimal parameters γ, l and λ we could predict very well the exact ages belonging to the age class [START_REF] Smola | Learning with kernels[END_REF]30] and negligible errors in other age classes. This is quite reasonable when seeing in the Figure 7 that the age distributed diversity almost from 20 to 30 years old, so our proposed estimation function learnt very well to predict age in this age class. Thus by using the distribution regression model, we investigated the relationship between the evoked responses from clicks covering the frequencies range up to 10kHz and its evolutionary ages.


Sascha Cording

Jasna Medvedovic

Emelyne Lecuyer

Tegest Aychek

Gerard Eberl
email: gerard.eberl@pasteur.fr

Gérard Eberl

Gérard Eberl

Gérard Eberl

Control of pathogens and microbiota by innate lymphoid cells

Keywords: Innate lymphoid cells, microbiota, pathogens, homeostasis, mucosal immunity

I. Introduction

ILCs derive from a lymphoid cell precursor common with T cells, with whom they share phenotypes and functions [START_REF] Eberl | Innate lymphoid cells: a new paradigm in immunology[END_REF][START_REF] Serafini | Transcriptional regulation of innate lymphoid cell fate[END_REF]. However, T cells are antigen-specific, carry immunological memory, and are selected by their cognate antigen in lymph nodes and Peyer's patches before expansion. Once activated, T cells recirculate through the blood to reach organs where their effector functions are required, eventually committing apoptosis or remaining in tissues as resident memory T (T RM ) cells [START_REF] Mueller | Tissue-resident memory T cells: local specialists in immune defence[END_REF]. Nevertheless, the process of T cell selection, activation and recirculation requires several days, leaving the early phase of the immune response without T cell-mediated orchestration of the response best adapted to the type of threat faced by the individual. However, early orchestration is directed by ILCs, which perform similar regulatory functions and provide prompt effector responses to infection and injury. ILCs also play important roles early in life, when the adaptive immune system is not yet in place [START_REF] Eberl | Innate lymphoid cells: a new paradigm in immunology[END_REF].

An individual faced with intracellular threats, such as tumors, viruses and particular bacteria, responds with type 1 immune responses characterized by the production of type I interferons, the inducer cytokine IL-12, the effector cytokine IFNγ, the release of cytotoxic oxygen radicals and proteins, and the elimination of the transformed or infected cells (Figure 1). Extracellular microbes, such as bacteria and fungi, elicit type 3 responses that are characterized by the release of the inducer cytokine IL-23 and the production of the effector cytokines IL-17 and IL-22, which lead to the reinforcement of mucosal barriers and the recruitment of polymorphonuclear neutrophilic phagocytes that target, ingest and destroy the microbes. In contrast, when faced with large parasites, such as worms, the individual develops type 2 responses that lead to the release of fluids and the production of mucus at mucosal surfaces, and the deposition of collagen to resist parasite penetration. The type 2 inducer cytokines are IL-25, IL-33 and TSLP, which lead to the production of the effector cytokines IL-4, IL-5 and IL-13. Myeloid cells and stromal cells produce the appropriate inducer cytokines in response to a specific type of threat, activating ILCs and T cells to differentiate into ILC1s and Th1, ILC2s and Th2, or ILC3s and Th17, which produce the effector cytokines characteristic of type 1, 2 and 3 immunity [START_REF] Eberl | Innate lymphoid cells: a new paradigm in immunology[END_REF][START_REF] Eberl | Immunity by equilibrium[END_REF].

Importantly, these immune responses are not only engaged in response to pathogens and tissue injury, but also in response to the symbiotic microbiota [START_REF] Eberl | A new vision of immunity: homeostasis of the superorganism[END_REF]. Microbes are present at all mucosal surfaces, as well as within tissues. An estimated 10 14 bacteria reside in the intestine, together with viruses, fungi, protists and occasional worms, while smaller yet significant collections of microbes are present in the oral cavity, genitourinary system and skin [START_REF] Proctor | The landscape ecology and microbiota of the human nose, mouth, and throat[END_REF][START_REF] Consortium | Structure, function and diversity of the healthy human microbiome[END_REF]. Furthermore, the list of viruses found within our organism is expanding [START_REF] Virgin | Redefining chronic viral infection[END_REF], and

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bacteria have been identified in lymphoid tissues [START_REF] Sonnenberg | Innate lymphoid cells promote anatomical containment of lymphoid-resident commensal bacteria[END_REF][START_REF] Obata | Indigenous opportunistic bacteria inhabit mammalian gut-associated lymphoid tissues and share a mucosal antibody-mediated symbiosis[END_REF] and placenta [START_REF] Aagaard | The placenta harbors a unique microbiome[END_REF]. As a consequence, the immune system is constantly activated by the presence of these diverse symbiotic microbes. We have therefore argued that the primary function of the immune system is to maintain homeostasis of the host with its microbiota [START_REF] Eberl | A new vision of immunity: homeostasis of the superorganism[END_REF], a microbiota that is necessary for digestion, production of metabolites and defense. In contrast, if microbes are associated with injury of the host's cells and tissues, the immune reaction will lead to the elimination of the pathogens.

In this review, we discuss the role of ILCs in the control of the symbiotic or pathogenic microbiota (Figure 2 and Table 1). From recent data, it emerges that ILCs play unique roles in microbiota control, observations that may lead to a new understanding of how chronic inflammatory pathologies emerge when such control is lost, and create new opportunities for prevention and therapy of infectious diseases. We will not discuss the role of NK cells, the oldest member of the ILC family, which has been extensively studied since 40 years and best reviewed elsewhere [START_REF] Trinchieri | Natural killer cells wear different hats: effector cells of innate resistance and regulatory cells of adaptive immunity and of hematopoiesis[END_REF].

II. ILCs in the control of intestinal and hepatic infections

ILC3s are the oldest and best characterized ILC family member (not considering NK cells). In the late 90's, non-B non-T lymphoid cells, termed lymphoid tissue inducer (LTi) cells, have been described that colonize developing secondary lymphoid tissues, the lymph nodes and the Peyer's patches, and shown to be required for their development [START_REF] Mebius | Developing lymph nodes collect CD4 + CD3 -LTb + cells that can differentiate to APC, NK cells, and follicular cells but not T or B cells[END_REF][START_REF] Adachi | Three distinctive steps in Peyer's patch formation of murine embryo[END_REF]. LTi cells form clusters in fetal lymphoid tissues and the intestinal lamina propria [START_REF] Bouskra | Lymphoid tissue genesis induced by commensals through NOD1 regulates intestinal homeostasis[END_REF], and activate stromal cells to initiate organogenesis [START_REF] Honda | Molecular basis for hematopoietic/mesenchymal interaction during initiation of Peyer's patch organogenesis[END_REF]. Ten years later, it was shown that LTi cells are part of a larger family of ILC3s that depend on the hormone receptor and transcription factor RORγt [START_REF] Eberl | An essential function for the nuclear receptor RORγt in the generation of fetal lymphoid tissue inducer cells[END_REF], and express the type 3 cytokines IL-17, IL-22 and lymphotoxin (LT) [START_REF] Satoh-Takayama | Microbial flora drives interleukin 22 production in intestinal NKp46+ cells that provide innate mucosal immune defense[END_REF][START_REF] Cella | A human natural killer cell subset provides an innate source of IL-22 for mucosal immunity[END_REF][START_REF] Luci | Influence of the transcription factor RORgammat on the development of NKp46+ cell populations in gut and skin[END_REF][21]. Non-LTi ILC3s do not cluster and are not involved in the development of lymphoid tissues. Rather, these cells resemble more freely moving lymphoid effector cells, and together with LTi cells, play a critical role early in defense against enteric pathogens.

Using mouse models that lack all lymphoid cells or only B and T cells, it was reported that ILC3s are required early in the control of intestinal infection by Proteobacteria, such as Citrobacter rodentium, the murine homologue of human enteropathogenic Escherichia coli.

The production of IL-22 is paramount for this protective function, and induces the expression of anti-microbial peptides (AMPs) by epithelial cells, such as Reg3γ [START_REF] Satoh-Takayama | Microbial flora drives interleukin 22 production in intestinal NKp46+ cells that provide innate mucosal immune defense[END_REF][21][START_REF] Sonnenberg | CD4(+) Lymphoid tissue-inducer cells promote innate immunity in the gut[END_REF][START_REF] Guo | Induction of innate lymphoid cell-derived interleukin-22 by the transcription factor STAT3 mediates protection against intestinal infection[END_REF]. Also involved in increasing epithelial defense is membrane-bound LT, which bind its receptor

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LTβR on epithelial cells and induces expression of the neutrophils chemoattractants CXCL1

and CXCL2 [START_REF] Wang | Lymphotoxin beta receptor signaling in intestinal epithelial cells orchestrates innate immune responses against mucosal bacterial infection[END_REF]. Furthermore, ILC3s are involved in the response to infection by Salmonella enterica through the production of IL-17 and IL22, as well as IFNγ [START_REF] Klose | A T-bet gradient controls the fate and function of CCR6-RORgammat+ innate lymphoid cells[END_REF]. The production of IL-17 by ILC3s and the recruitment of neutrophils are essential in the response to infection by E.coli K1 and Klebsiella pneumoniae and to prevent sepsis [START_REF] Deshmukh | The microbiota regulates neutrophil homeostasis and host resistance to Escherichia coli K1 sepsis in neonatal mice[END_REF]. The protective role of ILC3s against pathogens extends to parasites and viruses. Infection by Toxoplasma gondii leads to more severe inflammatory pathology in the absence of ILC3s, presumably because of a failure to contain the parasites early [START_REF] Wagage | The group 3 innate lymphoid cell defect in aryl hydrocarbon receptor deficient mice is associated with T cell hyperactivation during intestinal infection[END_REF]. Furthermore, IL-22 production by ILC3s potentiates the activity of IFNλ in epithelial cells and thus increases resistance to rotavirus [START_REF] Hernandez | Interferon-lambda and interleukin 22 act synergistically for the induction of interferonstimulated genes and control of rotavirus infection[END_REF]. However, the pro-inflammatory activity of ILC3s may also lead to pathology through the production of IFNγ in the context of S. enterica infection [START_REF] Klose | A T-bet gradient controls the fate and function of CCR6-RORgammat+ innate lymphoid cells[END_REF], as well as intestinal fibrosis through the expression of IL-17 and IL-22 in the context of S. typhimurium infection [START_REF] Lo | The orphan nuclear receptor ROR alpha and group 3 innate lymphoid cells drive fibrosis in a mouse model of Crohn's disease[END_REF].

ILC2s were first reported in the context of intestinal infection with the helminth

Nippostrongylus brasiliensis, which evokes a vigorous expansion of "non-B non-T cells" expressing IL-4, IL-5 an IL-13 [START_REF] Moro | Innate production of T(H)2 cytokines by adipose tissue-associated c-Kit(+)Sca-1(+) lymphoid cells[END_REF][START_REF] Fallon | Identification of an interleukin (IL)-25-dependent cell population that provides IL-4, IL-5, and IL-13 at the onset of helminth expulsion[END_REF]. ILC2 are also involved in defense against

Heligmosomoides polygyrus [START_REF] Shimokawa | Mast cells are crucial for induction of group 2 innate lymphoid cells and clearance of helminth infections[END_REF], Strongyloides venezuelensis [START_REF] Mukai | Differences in the Importance of Mast Cells, Basophils, IgE, and IgG versus That of CD4(+) T Cells and ILC2 Cells in Primary and Secondary Immunity to Strongyloides venezuelensis[END_REF], Trichinella spiralis [START_REF] Angkasekwinai | ILC2s activated by IL-25 promote antigen-specific Th2 and Th9 functions that contribute to the control of Trichinella spiralis infection[END_REF] and T. muris [START_REF] Spencer | Adaptation of innate lymphoid cells to a micronutrient deficiency promotes type 2 barrier immunity[END_REF], and are diminished in the blood of children infected with the blood dweller Schistosoma haematobium [START_REF] Nausch | Group 2 innate lymphoid cell proportions are diminished in young helminth infected children and restored by curative anti-helminthic treatment[END_REF]. Both inducer cytokines IL-25 and IL-33 are expressed upon helminth infection and activate ILC2s, noting that ILC2s responding to IL-33 have been suggested to be precursors of ILC2s responding to IL-25 [START_REF] Huang | IL-25-responsive, lineagenegative KLRG1(hi) cells are multipotential 'inflammatory' type 2 innate lymphoid cells[END_REF]. While IL-33 is expressed by different types of stromal cells, as well as by mast cells [START_REF] Shimokawa | Mast cells are crucial for induction of group 2 innate lymphoid cells and clearance of helminth infections[END_REF], the source of IL-25 was only recently identified, in the intestine, as Tuft epithelial cells [START_REF] Howitt | Tuft cells, taste-chemosensory cells, orchestrate parasite type 2 immunity in the gut[END_REF][START_REF] Gerbe | Intestinal epithelial tuft cells initiate type 2 mucosal immunity to helminth parasites[END_REF][START_REF] Von Moltke | Tuft-cell-derived IL-25 regulates an intestinal ILC2-epithelial response circuit[END_REF]. Interestingly, Tuft cells appear to detect helminth infection through chemosensory signaling via the G-protein coupled receptor Trpm5 [START_REF] Howitt | Tuft cells, taste-chemosensory cells, orchestrate parasite type 2 immunity in the gut[END_REF]. The activity of ILC2s is also promoted, during helminth infection, by the neuropeptide neuromedin U that is expressed by cholinergic neurons in the intestine [START_REF] Klose | The neuropeptide neuromedin U stimulates innate lymphoid cells and type 2 inflammation[END_REF][START_REF] Cardoso | Neuronal regulation of type 2 innate lymphoid cells via neuromedin U[END_REF].

ILC1s are distinguished from NK cells by their lack of expression of, and requirement for, the transcription factor Eomesodermin [START_REF] Serafini | Transcriptional regulation of innate lymphoid cell fate[END_REF]. ILC1s provide early protection to the liver from mouse cytomegalovirus infection, producing IFNγ before NK cells are engaged [START_REF] Weizman | ILC1 confer early host protection at initial sites of viral infection[END_REF], but at the same time prevent the recruitment of NK cells and CD8 + T cells that optimally fight liver infection with adenovirus [START_REF] Krueger | Murine liverresident group 1 innate lymphoid cells regulate optimal priming of anti-viral CD8+ T cells[END_REF]. Mouse hepatitis virus inoculated orally is also cleared from the intestine by ILC1s, which are activated by IL-15 released from infected stromal cells [START_REF] Gil-Cruz | Fibroblastic reticular cells regulate intestinal inflammation via IL-15-mediated control of group 1 ILCs[END_REF]. Futhermore, ILC1s and their production of IFNγ are engaged in the defense against

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6 bacteria, such as Clostridium difficile [START_REF] Abt | Innate immune defenses mediated by two ILC subsets are critical for protection against acute Clostridium difficile infection[END_REF]. Nevertheless, ILC1s and NK cells have largely overlapping function, even though ILC1s are defined as non-cytotoxic. As both cells types react promptly to activators such as IL-12, their relative role is mostly dependent on their tissue distribution before infection.

III. The ILC crosstalk with intestinal microbiota

The large intestinal microbiota has been best characterized, so far, at the level of its bacteriome and of its cross-talk between symbiotic bacteria and ILC3s. In contrast, the interaction between the virome and ILCs remains largely unexplored, and ILC2s are not known to cross-talk with microbes, even though some bacteria have been reported to induce the expansion of ILC2s at the steady state [START_REF] Sawa | RORgt+ innate lymphoid cells regulate intestinal homeostasis by integrating negative signals from the symbiotic microbiota[END_REF]. Therefore, this chapter will discuss mostly our knowledge on the cross-talk between bacterial symbionts and ILC3s.

In 2008, we reported that peptidoglycan released by proliferating Gram -bacteria in the intestine induce the activation of LTi cells clustered in so-called cryptopatches [START_REF] Bouskra | Lymphoid tissue genesis induced by commensals through NOD1 regulates intestinal homeostasis[END_REF], which are found near the base of crypts in the small intestine [START_REF] Kanamori | Identification of novel lymphoid tissues in murine intestinal mucosa where clusters of c-kit + IL-7R + Thy1 + lympho-hemopoietic progenitors develop[END_REF]. LTi cells in turn activate underlying stromal cells to release chemokines and recruit CCR6 + B cells to form isolated lymphoid follicles (ILFs). ILFs generate microbiota-specific IgA-producing B cells, in a T-cell independent way, and play an important role in intestinal homeostasis [START_REF] Bouskra | Lymphoid tissue genesis induced by commensals through NOD1 regulates intestinal homeostasis[END_REF][START_REF] Tsuji | Requirement for lymphoid tissue-inducer cells in isolated follicle formation and T cell-independent immunoglobulin a generation in the gut[END_REF]. The colonizing microbiota is therefore involved in the development of the intestinal immune system. At the same time, it provides a negative feedback on the number and activity of ILC3s, which include LTi cells, by inducing epithelial cells to produce IL-25 [START_REF] Sawa | RORgt+ innate lymphoid cells regulate intestinal homeostasis by integrating negative signals from the symbiotic microbiota[END_REF]. Of note, in both these phenomena, epithelial cells translate the recognition of microbes into signals that regulate ILCs.

Myeloid cells are nevertheless the major relay between microbiota and ILC3s through the production of the type 3 inducer cytokines IL-23 and IL-1β. In the context of C. rodentium infection, CX 3 CR1 + macrophages produce higher amounts of these cytokines than do conventional CD103 + DCs, and thus, more efficiently induce the expression by ILC3s of IL-22 [START_REF] Longman | CX(3)CR1(+) mononuclear phagocytes support colitis-associated innate lymphoid cell production of IL-22[END_REF] and GM-CSF [START_REF] Mortha | Microbiota-dependent crosstalk between macrophages and ILC3 promotes intestinal homeostasis[END_REF]. In addition, CD11b + DCs, which include CX 3 CR1 + cells, are obligate sources of IL-23 for host survival during C. rodentium infection [START_REF] Satpathy | Notch2-dependent classical dendritic cells orchestrate intestinal immunity to attaching-and-effacing bacterial pathogens[END_REF], and TNFα produced by CD11b + DCs during Helicobacter typhlonius infection synergizes with IL-23 for the expression of IL-17 by ILC3s [START_REF] Powell | The transcription factor T-bet regulates intestinal inflammation mediated by interleukin-7 receptor+ innate lymphoid cells[END_REF]. CD11b + DCs are also activated by microbiota-derived ATP to produce IL-23 during the steady state [START_REF] Atarashi | ATP drives lamina propria T(H)17 cell differentiation[END_REF]. Other members of the symbiotic microbiota induce the expression of IL-23 by myeloid cells and as a consequence, IL-22 by ILC3s, such as SFB [START_REF] Sano | An IL-23R/IL-22 circuit regulates epithelial serum amyloid A to promote local effector Th17 responses[END_REF] and Lactobacilli species [START_REF] Nakamoto | Commensal lactobacillus controls immune tolerance during acute liver injury in mice[END_REF]. ILC3s feedback positively on the [START_REF] Tumanov | Lymphotoxin controls the IL-22 protection pathway in gut innate lymphoid cells during mucosal pathogen challenge[END_REF]. Unexpectedly, intestinal glial cells sense microbiota through a Myd88-dependent pathway and produce neurotrophic factors that activate the Ret receptor on ILC3s and promote their expression of IL-22 [START_REF] Ibiza | Glialcell-derived neuroregulators control type 3 innate lymphoid cells and gut defence[END_REF].

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production of IL-23 by myeloid cells through their expression of membrane-bound LT and activation of LTβR on DCs

At the steady state, ILC3s are the major source of IL-22 [START_REF] Sawa | RORgt+ innate lymphoid cells regulate intestinal homeostasis by integrating negative signals from the symbiotic microbiota[END_REF]. IL-22, as well as IL-17, induce the production of AMPs by epithelial cells [START_REF] Liang | Interleukin (IL)-22 and IL-17 are coexpressed by Th17 cells and cooperatively enhance expression of antimicrobial peptides[END_REF], and therefore, are critical for the containment not only of pathogens, but also of the symbiotic microbiota. IL-22 also regulates the fucosylation of epithelial cells, a dietary carbohydrate for many symbionts, and thereby increases resistance to infection by Salmonella typhimurium [START_REF] Goto | Innate lymphoid cells regulate intestinal epithelial cell glycosylation[END_REF] and C. rodentium [START_REF] Pickard | Rapid fucosylation of intestinal epithelium sustains host-commensal symbiosis in sickness[END_REF].

Furthermore, membrane-bound (LTα 1 β 2 ) and soluble (LTα 3 ) LT expressed by ILC3s play an important role in the containment of the microbiota [START_REF] Zhang | Type 3 innate lymphoid cell-derived lymphotoxin prevents microbiota-dependent inflammation[END_REF] through the activation of epithelial cells [START_REF] Wang | Lymphotoxin beta receptor signaling in intestinal epithelial cells orchestrates innate immune responses against mucosal bacterial infection[END_REF], as well as of DCs and T cells that lead to the production of anti-microbial IgA [START_REF] Kruglov | Nonredundant function of soluble LTalpha3 produced by innate lymphoid cells in intestinal homeostasis[END_REF].

The containment of microbiota through the enhancement of epithelial defenses and adaptive immunity must nevertheless be controlled to avoid exaggerate responses to the symbiotic microbiota and consequent pathologic inflammation. To this end, ILCs express MHC class II molecules, as well as the machinery required to process proteins for presentation of peptides onto class II molecules, allowing ILCs to dampen the CD4 + T cell response to microbiota [START_REF] Hepworth | Innate lymphoid cells regulate CD4+ T-cell responses to intestinal commensal bacteria[END_REF] and to SFB in particular [START_REF] Goto | Segmented filamentous bacteria antigens presented by intestinal dendritic cells drive mucosal Th17 cell differentiation[END_REF]. Reciprocally, the presence of CD4 + T cells in the intestinal lamina propria decreases the expression of IL-22 by ILC3s [START_REF] Sawa | RORgt+ innate lymphoid cells regulate intestinal homeostasis by integrating negative signals from the symbiotic microbiota[END_REF][START_REF] Korn | Conventional CD4+ T cells regulate IL-22-producing intestinal innate lymphoid cells[END_REF], presumably through competition for survival and inducing cytokines such as IL-23.

Bacterial symbionts of the Alcaligenes genus have been identified that are present in Peyer's patches. ILC3s and IL-22 are critical for the containment of Alcaligenes, as the depletion of ILCs using an anti-CD90 antibody leads to the presence of the bacteria in spleen and liver [START_REF] Sonnenberg | Innate lymphoid cells promote anatomical containment of lymphoid-resident commensal bacteria[END_REF]. The depletion experiments were performed in Rag-deficient mice, as well as in mice reconstituted with mature B and T cells, indicating that adaptive immunity is not necessary for the containment of Alcaligenes. It remains nevertheless possible that, in normal mice, adaptive immunity develops over time that contributes to the containment of these symbionts through the generation of antibodies. As a matter of fact, Alcaligenes promotes the generation of IgA, which contribute to the construction of its niche [START_REF] Obata | Indigenous opportunistic bacteria inhabit mammalian gut-associated lymphoid tissues and share a mucosal antibody-mediated symbiosis[END_REF].

IV. Microbe-ILC interactions in the lungs

The impact of ILCs on microbes (and parasites) at mucosal sites other than the intestine has been best studied in the lungs. ILC2s are the dominant ILC subset in the lungs at the steady state, as a consequence of the production of IL-33 and TSLP by the alveolar epithelium [START_REF] Mckenzie | Innate lymphoid cells in inflammation and immunity[END_REF].

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 8
The production of IL-5 and IL-13 by ILC2s promotes the recruitment of eosinophils, the differentiation of monocyte into M2 (or alternatively activated) macrophages, and the differentiation of T cells into Th2 cells [START_REF] Molofsky | Innate lymphoid type 2 cells sustain visceral adipose tissue eosinophils and alternatively activated macrophages[END_REF][START_REF] Nussbaum | Type 2 innate lymphoid cells control eosinophil homeostasis[END_REF][START_REF] Halim | Group 2 innate lymphoid cells are critical for the initiation of adaptive T helper 2 cellmediated allergic lung inflammation[END_REF]. ILC2s contribute to the resistance to lung parasites, such as Strongyloides venezuelensis [START_REF] Yasuda | Importance of both innate immunity and acquired immunity for rapid expulsion of S. venezuelensis[END_REF], Nippostrongylus brasiliensis [START_REF] Bando | Type 2 innate lymphoid cells constitutively express arginase-I in the naive and inflamed lung[END_REF][START_REF] Bouchery | ILC2s and T cells cooperate to ensure maintenance of M2 macrophages for lung immunity against hookworms[END_REF],

Litomosoides sigmodontis [START_REF] Boyd | Pleural cavity type 2 innate lymphoid cells precede Th2 expansion in murine Litomosoides sigmodontis infection[END_REF], as well as to tissue repair [START_REF] Turner | IL-9mediated survival of type 2 innate lymphoid cells promotes damage control in helminthinduced lung inflammation[END_REF]. However, type 2 immunity induced by ILC2s also leads to increased susceptibility to fungal [START_REF] Flaczyk | IL-33 signaling regulates innate and adaptive immunity to Cryptococcus neoformans[END_REF][START_REF] Piehler | T1/ST2 promotes T helper 2 cell activation and polyfunctionality in bronchopulmonary mycosis[END_REF] and bacterial infection [START_REF] Saluzzo | First-breathinduced type 2 pathways shape the lung immune environment[END_REF], by mechanisms of immunological cross-regulation [START_REF] Eberl | Immunity by equilibrium[END_REF].

Type 2 immunity promoted by ILC2s plays an important role in the repair phase of immune responses, which occurs, for example, after influenza A virus infection through the production of the EGFR ligand amphiregulin [START_REF] Monticelli | Innate lymphoid cells promote lung-tissue homeostasis after infection with influenza virus[END_REF]. This repair response is nevertheless inhibited by IFNγ that is induced by the virus infection [START_REF] Califano | IFN-gamma increases susceptibility to influenza A infection through suppression of group II innate lymphoid cells[END_REF]. Furthermore, the ILC2-initiated repair responses, induced by virus-induced tissue damage, can lead to exacerbated type 2 responses and, as a consequence, asthma. Such an association, and the mechanisms driving this association, have been reported in the context of infection with rhinovirus [START_REF] Hong | Neonatal rhinovirus induces mucous metaplasia and airways hyperresponsiveness through IL-25 and type 2 innate lymphoid cells[END_REF][START_REF] Jackson | IL-33-dependent type 2 inflammation during rhinovirus-induced asthma exacerbations in vivo[END_REF][START_REF] Han | The innate cytokines IL-25, IL-33, and TSLP cooperate in the induction of type 2 innate lymphoid cell expansion and mucous metaplasia in rhinovirus-infected immature mice[END_REF] and respiratory syncytial virus [START_REF] Saravia | Respiratory Syncytial Virus Disease Is Mediated by Age-Variable IL-33[END_REF][START_REF] Stier | Respiratory syncytial virus infection activates IL-13-producing group 2 innate lymphoid cells through thymic stromal lymphopoietin[END_REF]. Similar pathological consequences are also induced by the fungi Alternaria alternata [START_REF] Stier | IL-33 promotes the egress of group 2 innate lymphoid cells from the bone marrow[END_REF] and Cryptococcus neoformans [START_REF] Piehler | The IL-33 receptor (ST2) regulates early IL-13 production in fungus-induced allergic airway inflammation[END_REF].

V. Concluding remarks

We have discussed recent data showing the critical role played by ILC early in the immune responses to pathogens, as these cells can respond promptly to inducer cytokines provided by myeloid and epithelial cells upon infection and injury. Of note, T RM cells, as well as particular subsets of invariant Tαβ or Tγδ cells, can play similar roles as their activation state alleviates the requirement for antigen-specific TCR activation. The prompt activation of ILCs is also important in response to changes in the symbiotic microbiota, in order to maintain to include bacteria, viruses, fungi and parasites, as well as viruses in the liver. Given their prompt activation, ILCs play a critical role early in the effector response to perturbations in the microbiota and to pathogens, and in the regulation of adaptive immunity. In some cases, however, the reactivity of ILCs contributes to pathology.

  ILC3s also play an important role in the lungs in response to bacterial and fungal infections. IL-22 produced by ILC3s is instrumental in lung immunity to Streptococcus pneumoniae [88] and Pseudomonas aeruginosa [89]. Interestingly, neonatal colonization of mice with intestinal bacteria induces the influx of IL-22-producing ILC3s into the lung, which confer life-saving resistance to early infection by Streptococcus pneumoniae [90].

  homeostasis.The current knowledge on the role of ILCs in defense and homeostasis has been inferred form mouse models and association studies in human. It is now possible to design strategies to harness the central role of ILCs in homeostasis, defense and immunoregulation for the prevention and therapy of infectious diseases, and of pathological consequences ofM A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 9 microbiota dysbiosis.It is also possible that novel vaccination strategies benefit from an early manipulation of ILCs, in order to force the response to specific antigens into the type most adapted to fight the pathogen. The manipulation of ILCs may involve the administration of inducer cytokines, or other molecules, yet to be discovered, that specifically activate subsets of ILCs. Another possibility is the isolation of ILC precursors from the blood of patients, and their expansion in vitro into the desired subset before re-administration and induction of type 1, 2 or 3 immune responses. We are still in the early days of ILC biology, but also at the exciting transition phase when clinical applications become possible.

Figure 1 .

 1 Figure 1. The activation of innate lymphoid cells. The three types of ILCs are promptly activated by distinct types of threats. Such threats are detected by myeloid and nonhematopoietic cells, which express inducer cytokines (in boxes). In reaction to inducer cytokines, ILCs express effector cytokines that both activate defense mechanisms and regulate immunity.

Figure 2 .

 2 Figure 2. The control of microbiota and pathogens by ILCs. The response of ILCs to symbiotic microbes and pathogens have been extensively described in the intestine and lung,

Table 1

 1

	ILC type	Role Microbe type Microbe species	Organ	References
	ILC1 +	Bacteria	Clostridium difficile	Intestine	[46]
		+	Virus	Adenovirus	Liver	[44]
		+		Mouse hepatitis virus	Liver	[45]
		+		Cytomegalovirus	Liver	[43]
	ILC2 -	Bacteria	Streptococcus pneumoniae	Lung	[78]
		-	Virus	Rhinovirus	Lung	[81-83]
		-		Respiratory syncytial virus	Lung	[84,85]
		+/-		Influenza	Lung	[79,80]
		+/-	Fungi	Cryptococcus neoformans	Lung	[76,77,87]
		-		Alternaria altnernata	Lung	[86]
		+	Parasites	Trichinella spiralis	Intestine	[34]
		+		Trichuris muris	Intestine	[35]
		+		Heligmosomoides polygyrus Intestine	[32]
		+		Nippostrongylus brasiliensis Intestine/lung	[30,31,72,73]
		+		Strongyloides venezuelensis	Intestine/lung	[33,71]
		+		Litomosoides sigmodontis	Lung	[74]
		+		Schistosoma haematobium	Blood	[36]
	ILC3 +	Bacteria	Commensal microbiota	Intestine	[15,24,47,49,54,62-64]
		+		Citrobacter rodentium	Intestine	[18,21-23,50,52,61]
		+		Escherichia coli K1	Intestine	[26]
		+		SFB	Intestine	[55,65]
		+		Lactobacilli	Intestine	[56]
		+		Alcaligenes	Intestine	[9,10]
		+/-		Salmonella enterica	Intestine	[25]
		+/-		Salmonella typhimurium	Intestine	[29,60]
		+/-		Helicobacter typhlonius	Intestine	[53]
		+		Streptococcus pneumoniae	Lung	[88,90]
		+		Pseudomonas aeruginosa	Lung	[89]
		+		Klebsiella pneumoniae	Blood	[26]
		+	Virus	Rotavirus	Intestine	[28]
		+	Fungi	Candida albicans	Lung	[89]
		+	Parasites	Toxoplasma gondii	Intestine	[27]

Conflict of interests 243

We declare no conflicts of interests