Yating Liu
email: liu@ceremade.dauphine.fr.

Gilles Pagès
email: gilles.pages@sorbonne-universite.fr.

Convergence Rate of Optimal Quantization and Application to the Clustering Performance of the Empirical Measure

Keywords: Clustering performance, Convergence rate of optimal quantization, Distortion function, Empirical measure, Optimal quantization

We study the convergence rate of the optimal quantization for a probability measure sequence (µn) n∈N * on R d converging in the Wasserstein distance in two aspects: the first one is the convergence rate of optimal quantizer x (n) ∈ (R d ) K of µn at level K; the other one is the convergence rate of the distortion function valued at x (n) , called the "performance" of x (n) . Moreover, we also study the mean performance of the optimal quantization for the empirical measure of a distribution µ with finite second moment but possibly unbounded support. As an application, we show that the mean performance for the empirical measure of the multidimensional normal distribution N (m, Σ) and of distributions with hyper-exponential tails behave like O( log n √ n ). This extends the results from [BDL08] obtained for compactly supported distribution. We also derive an upper bound which is sharper in the quantization level K but suboptimal in n by applying results in [FG15].

Introduction

The K-means clustering procedure in the unsupervised learning area was first introduced by [START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF], which consists in partitioning a data set of observations {η 1 , ..., η N } ⊂ R d into K classes G k , 1 ≤ k ≤ K with respect to a cluster center x = (x 1 , ..., x K ) in order to minimize the quadratic distortion function D K,η defined by

x = (x 1 , ..., x K ) ∈ (R d ) K → D K,η (x) := 1 N N n=1 min k=1,...,K d(η n , x k ) 2 , (1.1)
where d denotes a distance on R d . The classification of the observations {η 1 , ..., η N } ⊂ R d in [START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF] can be described as follows

G 1 = η n ∈ {η 1 , ..., η N } : d(η n , x 1 ) ≤ min 2≤j≤K d(η n , x j ) G 2 = η n ∈ {η 1 , ..., η N } : d(η n , x 2 ) ≤ min 1≤j≤K,j =2 d(η n , x j ) \ G 1 • • • G K = η n ∈ {η 1 , ..., η N } : d(η n , x K ) ≤ min 1≤j≤K-1 d(η n , x j ) \ G K-1 ∪ • • • ∪ G 1 . (1.2)
If a cluster center x * = (x * 1 , ..., x * K ) satisfies D K,η (x * ) = inf y∈(R d ) K D K,η (y), we call x * an optimal cluster center (or K-means) for the observation η = (η 1 , ..., η N ). Such an optimal cluster center always exists but is generally not unique.

K-means clustering has a close connection with quadratic optimal quantization, originally developed as a discretization method for the signal transmission and compression by the Bell laboratories in the 1950s (see [IEE82] and [START_REF] Gersho | Vector quantization and signal compression[END_REF]). Nowadays, optimal quantization has also become an efficient tool in numerical probability, used to provide a discrete representation of a probability distribution. To be more precise, let |•| denote the Euclidean norm on R d induced by the canonical inner product •|• and let X be an R d -valued random variable defined on (Ω, F, P) with probability distribution µ having a finite second moment. The quantization method consists in discretely approximating µ by using a K-tuple x = (x 1 , ..., x K ) ∈ (R d ) K and its weight w = (w 1 , ..., w K ) as follows,

µ µ x := K k=1 w k δ x k ,
where δ a denotes the Dirac mass at a, the weights w k are computed by w k = µ C k (x) , k = 1, ..., K, and C k (x) 1≤k≤K is a Voronoï partition induced by x, that is, a Borel partition on R d satisfying

C k (x) ⊂ V k (x) := ξ ∈ R d |ξ -x k | = min 1≤j≤K |ξ -x j | , k = 1, ..., K.
(1.3)

The value K in the above description is called the quantization level and the K-tuple above x = (x 1 , ..., x K ) is called a quantizer (or quantization grid, codebook in the literature). Moreover, we define the (quadratic) quantization error function e K,µ of µ (or of X) at level K by .

x = (x 1 , ..., x K ) ∈ (R d ) K -→ e K,µ (
(1.4)

The set argmin e K,µ is not empty (see e.g. [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF][see Theorem 4.12]) and any element x * = (x * 1 , ..., x * K ) in argmin e K,µ is called a (quadratic) optimal quantizer for the probability distribution µ at level K. Moreover, we call e * K, µ = inf y=(y1,...,y K )∈(R d ) K e K,µ (y) (1.5) the optimal (quadratic) quantization error (optimal error for short) at level K.

The connection between K-means clustering and quadratic optimal quantization is the following: if the distance d in (1.1) and (1.2) is the Euclidean distance and if we consider the empirical measure μN of the dataset {η 1 , ..., η N } defined by μN := 1 N N n=1 δ ηn , (1.6) then the distortion function D K,η defined in (1.1) is in fact e 2 K,μ N and argmin D K,η = argmin e K,μ N . That is, an optimal quantizer of μN is in fact an optimal cluster center for the dataset {η 1 , ..., η N }.

In Figure 1, we show an optimal quantizer and its weights for the standard normal distribution N 0, I 2 in R 2 at level 60, where I d denotes the identity matrix of size d × d. The color of the cells in the figure represents the weight of each point x k in the quantizer x = (x 1 , ..., x K ). In Figure 2, we show an optimal cluster center at level K = 20 for an i.i.d simulated sample {η 1 , ..., η 500 } of the N (0, I 2 ) distribution. (1.7)

The space P p (R d ) equipped with the Wasserstein distance W p is a Polish space, i.e. is separable and complete (see [START_REF] Bolley | Separability and completeness for the Wasserstein distance[END_REF]). If µ, ν ∈ P p (R d ), then for any q ≤ p, W q (µ, ν) ≤ W p (µ, ν).

With a slight abuse of notation, we define the distortion function for the optimal quantization as follows.

Definition 1.1 (Distortion function). Let K ∈ N * be the quantization level. Let µ ∈ P 2 (R d ). The (quadratic) distortion function D K, µ of µ at level K is defined by x = (x 1 , ..., x K ) ∈ (R d ) K -→ D K, µ (x) = random vectors (see (1.10) below). The empirical measure of non i.i.d. random vectors appears for example when dealing with the particle method associated to the McKean-Vlasov equations (see [START_REF] Liu | Optimal quantization: limit theorems, clustering and simulation of the McKean-Vlasov equation[END_REF][Section 7.1 and Section 7.5]) or the simulation of the invariant measure of the diffusion process (see [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] and [START_REF] Lemaire | Estimation récursive de la mesure invariante d'un processus de diffusion[END_REF][Chapter 4]). This leads us to study the consistency and the convergence rate of the optimal quantization for a W p -converging probability distribution sequence (µ n ) n≥1 .

There exist several studies in the literature. The consistency of the optimal quantizers was first proved in [START_REF] Pollard | Quantization and the method of k-means[END_REF].

Theorem (Pollard's Theorem). Let µ n ∈ P 2 (R d ), n ∈ N * ∪ {∞} with W 2 (µ n , µ ∞ ) → 0 as n → +∞. Assume card supp(µ n ) ≥ K, for n ∈ N * ∪ {+∞}. For n ≥ 1, let x (n) = x (n) 1 , ..., x (n) K
be a K-optimal quantizer for µ n , then the quantizer sequence (x (n) ) n≥1 is bounded in R d and any limiting point of (x (n) ) n≥1 , denoted by x (∞) , is an optimal quantizer of µ ∞ . (1) Let n) denote an optimal quantiser of µ n . There are two ways to study the convergence rate of the optimal quantizers. The first way is to directly evaluate the distance between x (n) and argmin D K,µ∞ . The second way is called the quantization performance, defined by

µ n ∈ P 2 (R d ), n ∈ N ∪ {∞} with W 2 (µ n , µ ∞ ) → 0 as n → +∞. Let x (
D K,µ∞ (x (n) ) -inf x∈(R d ) K D K,µ∞ (x).
(1.9)

This quantity describes the distance between the optimal error of µ ∞ and the quantization error of x (n) considered as a quantizer of µ ∞ (even x (n) is obviously not "optimal" for µ ∞ ). Several results of convergence rate exist in the framework of the empirical measure. Let X 1 (ω), ..., X n (ω), ... be i.i.d random vectors with probability distribution µ ∈ P 2 (R d ) and let

µ ω n := 1 n n i=1 δ Xi(ω) (1.10)
be the empirical measure of µ. The almost sure convergence of W 2 (µ ω n , µ) has been proved in [START_REF] Pollard | Quantization and the method of k-means[END_REF][Theorem 7]. Let x (n),ω denotes an optimal quantizer of µ ω n at level K. In [START_REF] Pollard | A central limit theorem for k-means clustering[END_REF], the author has proved that if µ has a unique optimal quantizer x at level K, then the convergence rate (convergence in distribution) of x (n),ω -x is O(n -1/2 ) under appropriate conditions. Moreover, if µ has a support contained in B(0, R), where B(0, R) denotes the ball in R d centered at 0 with radius R, an upper bound of the mean performance has been proved in [START_REF] Biau | On the performance of clustering in Hilbert spaces[END_REF], shown as follows,

E D K, µ (x (n),ω ) -inf x∈(R d ) K D K, µ (x) ≤ 12K • R 2 √ n .
In this paper, we extend the convergence results in [START_REF] Pollard | A central limit theorem for k-means clustering[END_REF] and in [START_REF] Biau | On the performance of clustering in Hilbert spaces[END_REF] in two perspectives: first, we give an upper bound of the quantization performance

D K,µ∞ (x (n) ) -inf x∈(R d ) K D K,µ∞ (x).
(1.11) and that of related optimal quantizers for any probability distribution sequence (µ n ) n≥1 converging

(1) In [START_REF] Pollard | Quantization and the method of k-means[END_REF][see Theorem 9], the author used

µ K ∈ P(K) := ν ∈ P 2 (R d ) such that card supp(ν) ≤ K
to represent a "quantizer" at level K. Such a quantizer µ K is called "quadratic optimal" for a probability measure

µ if W 2 (µ K , µ) = e * K, µ .
We propose an alternative proof in Appendix A by using the usual representation of the quantizer x ∈ (R d ) K but still call this theorem "Pollard's Theorem".

in the Wasserstein distance. Then, we generalize the clustering performance results in [START_REF] Biau | On the performance of clustering in Hilbert spaces[END_REF] to empirical measures in P 2 (R d ) possibly having an unbounded support.

Our main results are as follows. We obtain in Section 2 a non-asymptotic upper bound for the quantization performance: for every n ∈ N * ,

D K, µ∞ (x (n) ) -inf x∈(R d ) K D K, µ∞ (x) ≤ 4e * K,µ∞ W 2 (µ n , µ ∞ ) + 4W 2 2 (µ n , µ ∞ ).
(1.12) Moreover, if D K,µ∞ is twice differentiable at

F K := x = (x 1 , ..., x K ) ∈ (R d ) K x i = x j , if i = j (1.13)
and if the Hessian matrix H D K,µ∞ of D K,µ∞ is positive definite in the neighboorhood of every optimal quantizer x (∞) ∈ G K (µ ∞ ) having the eigenvalues lower bounded by a λ * > 0, then, for n large enough,

d x (n) , G K (µ ∞ ) 2 ≤ 8 λ * e * K,µ∞ • W 2 (µ n , µ ∞ ) + 8 λ * • W 2 2 (µ n , µ ∞ ),
where d(ξ, A) := min a∈A |ξ -a| denotes the distance between a point ξ ∈ R d and a set A ⊂ R d .

Several criterions for the positive definiteness of the Hessian matrix H D K,µ of the distortion function D K,µ are established in Section 3. We show in Section 3.1 the conditions under which the distortion function D K,µ is twice differentiable in every x ∈ F K and give the exact formula of the Hessian matrix H D K,µ . Moreover, we also discuss several sufficient and necessary conditions for the positive definiteness of the Hessian matrix in dimension d ≥ 2 and in dimension 1.

In Section 4, we give two upper bounds for the clustering performance

E D K, µ (x (n), ω ) - inf x∈(R d ) K D K, µ (x)
, where x (n), ω is an optimal quantizer of µ ω n defined in (1.10). If µ ∈ P q (R d ) for some q > 2, a first upper bound is established in Proposition 4.1

E D K, µ (x (n),ω ) -inf x∈(R d ) K D K, µ (x) ≤ C d,q,µ,K ×        n -1/4 + n -(q-2)/2q if d < 4 and q = 4 n -1/4 log(1 + n) 1/2 + n -(q-2)/2q if d = 4 and q = 4 n -1/d + n -(q-2)/2q if d > 4 and q = d/(d -2)
, where C d,q,µ,K is a constant depending on d, q, µ and the quantization level K. This result is a direct application of the non-asymptotic upper bound (1.12) combined with results in [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF] about the mean convergence rate of the empirical measure for the Wasserstein distance. If d ≥ 4 and q > 2d d-2 , this constant C d,q,µ,K is roughly decreasing as K -1/d (see Remark 4.1). This upper bound is sharper in K compared with the upper bound (1.14) below, although it suffers from the curse of dimensionality.

Meanwhile, we establish another upper bound for the clustering performance in Theorem 4.1, which is sharper in n but increasing faster than linearly in K. This upper bound is

E D K, µ (x (n),ω ) -inf x∈(R d ) K D K, µ (x) ≤ 2K √ n r 2 2n + ρ K (µ) 2 + 2r 1 r 2n + ρ K (µ) , (1.14)
where r n := max 1≤i≤n |X i | 2 and ρ K (µ) is the maximum radius of optimal quantizers for µ, defined by

ρ K (µ) := max max 1≤k≤K |x * k | , (x * 1 , ..., x * K
) is an optimal quantizer of µ at level K .

(1.15)

In particular, we give a precise upper bound for µ = N (m, Σ), the multidimensionnal normal distribution

E D K, µ (x (n),ω ) -inf x∈(R d ) K D K, µ (x) ≤ C µ • 2K √ n 1 + log n + γ K log K 1 + 2 d , (1.16)
where lim sup K γ K = 1 and

C µ = 12 • 1 ∨ log 2 R d exp( 1 4 |ξ| 4 )µ(dξ) . If µ = N (0, I d ), C µ = 12(1 + d 2 ) • log 2.
We start our discussion with a brief review on the properties of optimal quantization.

Properties of the Optimal Quantization

Let G K (µ) = argmin D K,µ denote the set of all optimal quantizers at level K of µ and let e * K,µ denote the optimal quantization error of µ defined in (1.5).

Proposition 1.1. Let K ∈ N * . Let µ ∈ P 2 (R d ) and card supp(µ) ≥ K.

(i) If K ≥ 2, then e * K, µ < e * K-1, µ .
(ii) (Existence and boundedness of optimal quantizers) The set G K (µ) is nonempty and compact so that ρ K (µ) defined in (1.15) is finite for any fixed K. Moreover, if x = (x 1 , ..., x K ) is an optimal quantizer of µ, then x ∈ F K , where F K is defined in (1.13).

(iii) If the support of µ, denoted by supp(µ), is a compact, then for every optimal quantizer x = (x 1 , ..., x K ) ∈ G K (µ), its elements x k , 1 ≤ k ≤ K are contained in the closure of convex hull of supp(µ), denoted by H µ := conv supp(µ) .

For the proof of Proposition 1.1-(i) and (ii), we refer to [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF][see Theorem 4.12] and for the proof of (iii) to Appendix B.

Theorem. (Non-asymptotic Zador's Theorem, see [START_REF] Luschgy | Functional quantization rate and mean regularity of processes with an application to Lévy processes[END_REF] and [START_REF] Pagès | Numerical Probability: An Introduction with Applications to Finance[END_REF][Theorem 5.2]) Let η > 0. If µ ∈ P 2+η (R d ), then for every quantization level K, there exists a constant C d,η ∈ (0, +∞) which depends only on d and η such that

e * K,µ ≤ C d,η • σ 2+η (µ)K -1/d , (1.17)
where for r ∈ (0, +∞), σ r (µ) = min

a∈R d R d |ξ -a| r µ(dξ) 1/r .
When µ has an unbounded support, we know from [PS12] that lim K ρ K (µ) = +∞. The same paper also gives an asymptotic upper bound of ρ K when µ has a polynomial tail or a hyperexponential tail.

Theorem. ([PS12]

[see Theorem 1.2]) Let µ ∈ P p (R d ) be absolutely continuous with respect to the Lebesgue measure λ d on R d and let f denote its density function.

(i) Polynomial tail. For p ≥ 2, if µ has a c-th polynomial tail with c > d + p in the sense that there exists τ > 0, β ∈ R and

A > 0 such that ∀ξ ∈ R d , |ξ| ≥ A =⇒ f (ξ) = τ |ξ| c (log |ξ|) β , then lim K log ρ K log K = p + d d(c -p -d)
.

(1.18)

(ii) Hyper-exponential tail. If µ has a (ϑ, κ)-hyper-exponential tail in the sense that there exists τ > 0, κ, ϑ > 0, c > -d and

A > 0 such that ∀ξ ∈ R d , |ξ| ≥ A =⇒ f (ξ) = τ |ξ| c e -ϑ|ξ| κ , then lim sup K ρ K (log K) 1/κ ≤ 2ϑ -1/κ 1 + 2 d 1/κ . (1.19) Furthermore, if d = 1, lim K ρ K (log K) 1/κ = 3 ϑ 1/κ .
We give now the definition of the radially controlled distribution, which will be useful to control the convergence rate of the density function f (x) to 0 when x converges in every direction to infinity. Definition 1.2. Let µ ∈ P 2 (R d ) be absolutely continuous with respect to the Lebesgue measure λ d on R d having a continuous density function f . We call µ is k-radially controlled on R d if there exists A > 0 and a continuous non-increasing function g : R

+ → R + such that ∀ξ ∈ R d , |ξ| ≥ A, f (ξ) ≤ g(|ξ|)
and

R+ x d-1+k g(x)dx < +∞.
Note that the c-th polynomial tail with c > k + d and the hyper-exponential tail are sufficient conditions to satisfy the k-radially controlled assumption. A typical example of hyper-exponential tail is the multidimensional normal distribution N (m, Σ).

For µ, ν ∈ P 2 (R d ) and for every K ∈ N * , we have

e K,µ -e K,ν sup := sup x∈(R d ) K |e K, µ (x) -e K, ν (x)| ≤ W 2 (µ, ν), (1.20)
by a simple application of the triangle inequality for the L 2 -norm (see e.g. [GL00] Formula (4.4) and Lemma 3.4). Hence, if (µ n ) n≥1 is a sequence in P 2 (R d ) converging for the W 2 -distance to µ ∞ ∈ P 2 (R d ), then for every K ∈ N * , e K, µn -e K, µ∞ sup ≤ W 2 (µ n , µ ∞ ) n→+∞ -----→ 0.

(1.21)

General Case

In this section, we first establish in Theorem 2.1 a non-asymptotic upper bound of the quantization performance

D K, µ∞ (x (n) ) -inf x∈(R d ) K D K,µ∞ (x)
. Then we discuss the convergence rate of the optimal quantizer sequence in Theorem 2.2.

Theorem 2.1 (Non-asymptotic upper bound for the quantization performance). Let K ∈ N * be the quantization level. For every n ∈ N * ∪ {∞}, let

µ n ∈ P 2 (R d ) with card supp(µ n ) ≥ K. Assume that W 2 (µ n , µ ∞ ) → 0 as n → +∞.
For every n ∈ N * , let x (n) be an optimal quantizer of µ n . Then

D K,µ∞ (x (n) ) -inf x∈(R d ) K D K,µ∞ (x) ≤ 4e * K,µ∞ W 2 (µ n , µ ∞ ) + 4W 2 2 (µ n , µ ∞ ),
where e * K,µ∞ is the optimal error of µ ∞ at level K defined in (1.5).

Proof of Theorem 2.1. Let x (∞) be an optimal quantizer of µ ∞ . Remark that here we do not need that x (∞) is the limit of x (n) . First, we have (see e.g. Corollary 4.1 in [START_REF] Györfi | Principles of nonparametric learning[END_REF])

e K,µ∞ (x (n) ) -e * K,µ∞ = e K,µ∞ (x (n) ) -e K,µn (x (n) ) + e K,µn (x (n) ) -e K,µ∞ (x (∞) ) ≤ 2 e K,µ∞ -e K,µn sup ≤ 2W 2 (µ n , µ ∞ ), (2.1)
where the first inequality is due to the fact that for any µ, ν ∈ P 2 (R d ) with respective K-level optimal quantizers x µ and x ν , if e K,µ (x µ ) ≥ e K,ν (x ν ), we have

|e K,µ (x µ ) -e K,ν (x ν )| = e K,µ (x µ ) -e K,ν (x ν ) ≤ e K,µ (x ν ) -e K,ν (x ν ) ≤ e K,µ∞ -e K,µn sup .
If e K,µ (x µ ) ≤ e K,ν (x ν ), we have the same inequality by the same reasoning.

Moreover,

D K,µ∞ (x (n) ) -inf x∈(R d ) K D K,µ∞ (x) = D K,µ∞ (x (n) ) -D K,µ∞ (x (∞) ) ≤ e K,µ∞ (x (n) ) + e K,µ∞ (x (∞) ) e K,µ∞ (x (n) ) -e K,µ∞ (x (∞) ) ≤ 2 e K,µ∞ (x (n) ) -e K,µ∞ (x (∞) ) + 2e K,µ∞ (x (∞) ) • W 2 (µ n , µ ∞ ) by (2.1) ≤ 4 W 2 (µ n , µ ∞ ) + e * K,µ∞ • W 2 (µ n , µ ∞ ) by (2.1) ≤ 4e * K,µ∞ W 2 (µ n , µ ∞ ) + 4W 2 2 (µ n , µ ∞ ).
Let B(x, r) denote the ball centered at x with radius r. Recall that

F K := x = (x 1 , ..., x K ) ∈ (R d ) K x i = x j , if i = j . Remark that if x ∈ F K , then every y ∈ B x, 1 3 min 1≤i,j≤K,i =j |x i -x j | still lies in F K .
In the following theorem, we give an estimate of the convergence rate of the optimal quantizer sequence x

(n) , n ∈ N * . Theorem 2.2 (Convergence rate of optimal quantizers). Let K ∈ N * be the quantization level. For every n ∈ N * ∪ {∞}, let µ n ∈ P 2 (R d ) with card supp(µ n ) ≥ K. Assume that W 2 (µ n , µ ∞ ) → 0 as n → +∞.
For every n ∈ N * , let x (n) be an optimal quantizer of µ n and let G K (µ ∞ ) denote the set of all optimal quantizers of µ ∞ . If the following assumptions hold (a) the distortion function D K,µ∞ is twice differentiable at every ) having eigenvalues lower bounded by some λ * > 0, then, for n large enough,

x ∈ F K ; (b) card G K (µ ∞ ) < +∞; (c) for every x (∞) ∈ G K (µ ∞ ), the Hessian matrix of D K,µ∞ , denoted by H D K,µ∞ , is positive definite in the neighbourhood of x (∞
d x (n) , G K (µ ∞ ) 2 ≤ 8 λ * e * K,µ∞ • W 2 (µ n , µ ∞ ) + 8 λ * • W 2 2 (µ n , µ ∞ ).
Remark 2.1. Section 3 provides a detailed discussion of the conditions in Theorem 2.2 and their relation between each other.

(1) First, in Section 3, we establish that if µ ∞ is 1-radially controlled, then its distortion function D K,µ∞ is twice continuously differentiable at every x ∈ F K and give an exact formula of the Hessian matrix H D K,µ∞ (x) in Proposition 3.1. Thus, one may obtain Condition (c) either by an explicit computation or by numerical methods. Moreover, if H D K,µ is positive definite at x ∈ F K , it is also positive definite in its neighbourhood. In Section 3.2, we establish several sufficient conditions for the positive definiteness of the Hessian matrix

H D K,µ∞ in the neighbourhood of x (∞) ∈ G K (µ ∞ ) in one dimension.
(2) If the distribution µ ∞ is 1-radially controlled, a necessary condition for Condition (c) is Condition (b) (see Lemma 3.1). Thus, if card G K (µ ∞ ) = +∞, it is more reasonable to consider the non-asymtotic upper bound of the performance (Theorem 2.1) to study the convergence rate of the optimal quantization. A typical example is the standard multidimensional normal distribution µ ∞ = N (0, I d ): it is 1-radially controlled and any rotation of an optimal quantizer x is still optimal so that card G K (µ ∞ ) = +∞.

Proof of Theorem 2.2. Since the quantization level K is fixed throughout the proof, we will drop the subscripts K and µ of the distortion function D K, µ and we will denote by D n (respectively, D ∞ ) the distortion function of µ n (resp. µ ∞ ).

After Pollard's theorem, (x (n) ) n∈N * is bounded and any limiting point of x (n) lies in G K (µ ∞ ). We may assume that, up to the extraction of a subsequence of x (n) , still denoted by x (n) , we have

x (n) → x (∞) ∈ G K (µ ∞ ). Hence d x (n) , G K (µ ∞ ) ≤ x (n) -x (∞) . Proposition 1.1 implies that x (∞) ∈ F K . As D ∞ is twice differentiable at x (∞) , the second order Taylor expansion of D ∞ at x (∞) reads: D ∞ (x (n) ) = D ∞ (x (∞) ) + ∇D ∞ (x (∞) ) | x (n) -x (∞) + 1 2 H D∞ (ζ (n) )(x (n) -x (∞) ) ⊗2 ,
where

H D∞ denotes the Hessian matrix of D ∞ , ζ (n) lies in the geometric segment (x (n) , x (∞)
) and for a matrix A and a vector u, Au ⊗2 stands for u T Au.

As

x (∞) ∈ G K (µ ∞ ) = argmin D ∞ and card supp(µ ∞ ) ≥ K, one has ∇D ∞ (x (∞) ) = 0. Hence D ∞ (x (n) ) -D ∞ (x (∞) ) = 1 2 H D∞ (ζ (n) )(x (n) -x (∞) ) ⊗2 . (2.2)
It follows from Theorem 2.1 that

H D∞ (ζ (n) )(x (n) -x (∞) ) ⊗2 = 2 D ∞ (x (n) ) -D ∞ (x (∞) ) ≤ 8e * K,µ∞ W 2 (µ n , µ ∞ ) + 8W 2 2 (µ n , µ ∞ ). (2.3)
By Condition (c), H D∞ is assumed to be positive definite in the neighbourhood of all x (∞) ∈ G K (µ ∞ ) having eigenvalues lower bounded by some λ * > 0. As ζ (n) lies in the geometric segment (x (n) , x (∞) ) and x (n) → x (∞) , there exists an n 0 (x (∞) ) such that for all n ≥ n 0 ,

H D∞ (ζ (n) ) is a positive definite matrix. It follows that, for n ≥ n 0 , λ * x (n) -x (∞) 2 ≤ H D∞ (ζ (n) )(x (n) -x (∞) ) ⊗2 ≤ 8e * K,µ∞ W 2 (µ n , µ ∞ ) + 8W 2 2 (µ n , µ ∞ ).
Thus, one can directly conclude by multiplying at each side of the above inequality by 1 λ * .

Based on conditions in Theorem 2.2, if we know the exact limit of the optimal quantizer sequence x (n) , we have the following result whose proof is similar to that of Theorem 2.2. ∞) . If the Hessian matrix of D K, µ∞ is positive definite in the neighbourhood of x (∞) , then, for n large enough,

Corollary 2.1. Let K ∈ N * be the quantization level. For every n ∈ N * ∪ {∞}, let µ n ∈ P 2 (R d ) with card supp(µ n ) ≥ K. Assume that W 2 (µ n , µ ∞ ) → 0 as n → +∞. Let x (n) ∈ argmin D K, µn such that lim n x (n) → x (
x (n) -x (∞) 2 ≤ C (1) µ∞ • W 2 (µ n , µ ∞ ) + C (2) µ∞ • W 2 2 (µ n , µ ∞ ),
where C

(1) µ∞ and C

(2) µ∞ are real constants only depending on µ ∞ .

3 Hessian matrix H D K, µ of the distortion function D K, µ

Let µ ∈ P 2 (R d ) with card supp(µ) ≥ K and let x * be an optimal quantizer of µ at level K. In Section 3.1, we show conditions under which the distortion function D K,µ is twice differentiable and give the exact formula of its Hessian matrix H D K, µ . In Section 3.2, we give several criterions for the positive definiteness of the Hessian matrix H D K, µ in the neighbourhood of an optimal quantizer x * in dimension 1.

Hessian matrix

H D K, µ on R d
If µ is absolutely continuous with respect to the Lebesgue measure λ d on R d with the density function f , then the distortion function D K, µ is differentiable (see [START_REF] Pagès | A space quantization method for numerical integration[END_REF]) at all point

x = (x 1 , ..., x K ) ∈ F K with ∂D K, µ ∂x i (x) = 2 Vi(x) (x i -ξ)f (ξ)λ d (dξ), for i = 1, ..., K. (3.1)
In the following Proposition, we give a criterion for the twice differentiability of the distortion function D K,µ .

Proposition 3.1. Let µ ∈ P 2 (R d ) be absolutely continuous with respect to the Lebesgue measure λ d on R d with a continuous density function f . If µ is 1-radially controlled, then (i) the distortion function D K, µ is twice differentiable at every x ∈ F K and the Hessian matrix

H D K,µ (x) = ∂ 2 D K, µ ∂xj ∂xi (x)
1≤i≤j≤K is defined by

∂ 2 DK, µ ∂xj∂xi (x) = 2 V i (x)∩V j (x) (xi -ξ) ⊗ (xj -ξ) • 1 |xj -xi| f (ξ)λ ij x (dξ), if j = i, (3.2) ∂ 2 DK, µ ∂x 2 i (x) = 2 µ Vi(x) I d - i =j 1≤j≤K V i (x)∩V j (x) (xi -ξ) ⊗ (xi -ξ) • 1 |xj -xi| f (ξ)λ ij x (dξ) , (3.3)
where in (3.2) and

(3.3), u ⊗ v := [u i v j ] 1≤i,j≤d for any two vectors u = (u 1 , ..., u d ) and v = (v 1 , ..., v d ) in R d ;
(ii) every element

∂ 2 D K, µ ∂xj ∂xi of the Hessian matrix H D K,µ is continuous at every x ∈ F K .
The proof of Proposition 3.1 is postponed to Appendix C. The following lemma shows that under the condition of Proposition 3.1, Condition (c) in Theorem 2.2 implies Condition (b).

Lemma 3.1. Let µ ∈ P 2 (R d ) be absolutely continuous with the respect to the Lebesgue measure λ d on R d with a continuous density function f . If µ ∞ is 1-radially controlled and card G K (µ ∞ ) = +∞, then there exists a point x ∈ G K (µ ∞ ) such that the Hessian matrix H D K,µ∞ of D K,µ∞ at x has an eigenvalue 0.

Proof of Lemma 3.1. We denote by H D∞ instead of H D K,µ∞ to simplify the notation. Proposition 1.

1 implies that G K (µ ∞ ) is a compact set. If card G K (µ ∞ ) = +∞, there exists x, x (k) ∈ G K (µ ∞ ), k ∈ N * such that x (k) → x when k → +∞. Set u k := x (k) -x |x (k) -x| , k ≥ 1, then we have |u k | = 1 for all k ∈ N * . Hence, there exists a subsequence ϕ(k) of k such that u ϕ(k) converges to some u with | u| = 1.
The Taylor expansion of D K,µ∞ at x reads:

D K,µ∞ (x ϕ(k) ) = D K,µ∞ (x) + ∇D K,µ∞ (x) x ϕ(k) -x + 1 2 H D∞ (ζ ϕ(k) )(x ϕ(k) -x) ⊗2 , where ζ ϕ(k) lies in the geometric segment (x ϕ(k) , x). Since x, x (k) , k ∈ N * ∈ G K (µ ∞ ), then ∇D K,µ∞ (x) = 0 and for any k ∈ N * , D K,µ∞ (x ϕ(k) ) = D K,µ∞ (x). Hence, for any k ∈ N * , H D∞ (ζ ϕ(k) )(x ϕ(k) -x) ⊗2 = 0. Consequently, for any k ∈ N * , H D∞ (ζ ϕ(k) ) x ϕ(k) -x x ϕ(k) -x ⊗2 = 0.
Thus we have H D∞ (x) u ⊗2 = 0 by letting k → +∞, so that H D∞ (x) has an eigenvalue 0.

A criterion for positive definiteness of

H D∞ (x * ) in 1-dimension Let µ ∈ P 2 (R) with card supp(µ) ≥ K.
Assume that µ is absolutely continuous with respect to the Lebesgue measure having a density function f . In the one-dimensional case, it is useful to point out a sufficient condition for the uniqueness of optimal quantizer. A probability distribution µ is called strongly unimodal if its density function f satisfies that I = {f > 0} is an open (possibly unbounded) interval and log f is concave on

I. Let F + K := x = (x 1 , ..., x K ) ∈ R K | -∞ < x 1 < x 2 < ... < x K < +∞ .
Lemma 3.2. For K ∈ N * , if µ is strongly unimodal with card supp(µ) ≥ K, then there is only one stationary (then optimal) quantizer of level K in F + K .

We refer to [START_REF] Kieffer | Uniqueness of locally optimal quantizer for log-concave density and convex error weighting function[END_REF], [START_REF] Trushkin | Sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions[END_REF], [START_REF] Bouton | Self-organization and a.s. convergence of the onedimensional Kohonen algorithm with non-uniformly distributed stimuli[END_REF] and [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF][see Theorem 5.1] for the proof of Lemma 3.2 and for more details.

Given a K-tuple x = (x 1 , ..., x K ) ∈ F + K , the Voronoi region V i (x) can be explicitly written:

V 1 (x) = (-∞, x1+x2 2 ], V K (x) = [ x K-1 +x K 2 , +∞) and V i (x) = [ xi-1+xi 2 , xi+xi+1 2
] for i = 2, ..., K -1. For all x ∈ F + K , D K, µ is differentiable at x and by (3.1) and

∇D K, µ (x) = ñ Vi(x) 2(x i -ξ)f (ξ)dξ ô i=1,...,K . (3.4)
Therefore, as ∇D K, µ (x * ) = 0, one can solve the optimal quantizer x * ∈ F + K as follows,

x * i = Vi(x * ) ξf (ξ)dξ µ V i (x * ) , for i = 1, ..., K. (3.5)
For any x ∈ F + K , the Hessian matrix H D K,µ of D K, µ at x is a tridiagonal symmetry matrix and can be calculated as follows,

HD K,µ (x) =          A1 -B1,2 -B1,2 . . . -Bi-1,i Ai -Bi-1,i -Bi,i+1 -Bi,i+1 . . . -BK-1,K AK -BK-1,K          , (3.6)
where

A i = 2µ C i (x) for 1 ≤ i ≤ K and B i,j = 1 2 (x j -x i )f ( xi+xj 2 ) for 1 ≤ i < j ≤ K. Let F µ
be the cumulative distribution function of µ, then

A 1 = 2µ C 1 (x) = 2F µ x 1 + x 2 2 , A i = 2µ C i (x) = 2 F µ x i+1 + x i 2 -F µ x i-1 + x i 2 , for i = 2, ..., K -1, A K = 2µ C K (x) = 2 1 -F µ x K-1 + x K 2 .
Then the continuity of each term in the matrix H D K,µ (x) can be directly derived from the continuity of F µ .

For 1

≤ i ≤ K, we define L i (x) := K j=1 ∂ 2 D K, µ ∂x i ∂x j (x).
The following proposition gives sufficient conditions to obtain the positive definiteness of H D K,µ (x * ).

Proposition 3.2. Let µ ∈ P 2 (R) with card supp(µ) ≥ K. Assume that µ is absolutely continuous with respect to the Lebesgue measure having a density function f . Any of the following two conditions implies the positive definiteness of H D K,µ (x * ), (i) µ is the uniform distribution, (ii) f is differentiable and log f is strictly concave.

In particular, (ii) also implies that L i (x * ) > 0, i = 1, ..., K.

Proposition 3.2 is proved in Appendix D. Remark that, under the conditions of Proposition 3.2, µ is strongly unimodal so that there is exactly one optimal quantizer in F +

K for µ at level K. The conditions in Proposition 3.2 directly imply the following convergence rate results.

Theorem 3.1. Let K ∈ N * be the quantization level. For every n ∈ N * ∪{∞}, let µ n ∈ P 2 (R) with card supp(µ n ) ≥ K be such that W 2 (µ n , µ ∞ ) → 0 as n → +∞. Assume that µ ∞ is absolutely continuous with respect to the Lebesgue measure, written µ ∞ (dξ) = f (ξ)dξ. Let x (n) be an optimal quantizer of µ n converging to x (∞) . Then any one of the following two conditions (i) µ ∞ is the uniform distribution (ii) f is differentiable and log f is strictly concave implies the existence of constants C

(1) µ∞ and C

(2) µ∞ only depending on µ ∞ such that for n large enough,

x (n) -x (∞) 2 ≤ C (1) µ∞ • W 2 (µ n , µ ∞ ) + C (2) µ∞ • W 2 2 (µ n , µ ∞ ).
Proof. Let D K, µ∞ denote the distortion function of µ ∞ and let H D∞ denote the Hessian matrix of D K, µ∞ .

(i) Let g k (x) be the k-th leading principal minor of H D∞ (x) defined in (3.6), then g k (x), k = 1, ..., K, are continuous functions in x since every element in this matrix is continuous. Proposition 3.2 implies g k (x (∞) ) > 0, thus there exists r > 0 such that for every x ∈ B(x (∞) , r), g k (x (∞) ) > 0 so that H D∞ (x) is positive definite. What remains can be directly proved by Corollary 2.1.

(ii) The function

L i (x) := K j=1 ∂ 2 D K, µ∞ ∂x i ∂x j (x
) is continuous on x and Proposition 3.2 implies that

L i (x (∞) ) > 0.
Hence, there exists r > 0 such that ∀x ∈ B(x (∞) , r), L i (x) > 0. From (3.6), one can remark that the i-th diagonal elements in H D∞ (x) is always larger than L i (x) for any x ∈ R K , then after Gershgorin Circle theorem, we derive that H D∞ (x) is positive definite for every x ∈ B(x (∞) , r). What remains can be directly proved by Corollary 2.1.

Empirical measure case

Let K ∈ N * be the quantization level. Let µ ∈ P 2+ε (R d ) for some ε > 0 and card supp(µ) ≥ K.

Let X be a random variable with distribution µ and let (X n ) n≥1 be a sequence of independent identically distributed R d -valued random variables with probability distribution µ. The empirical measure is defined for every n ∈ N * by

µ ω n := 1 n n i=1 δ Xi(ω) , ω ∈ Ω, (4.1)
where δ a is the Dirac mass at a. For n ≥ 1, let x (n),ω be an optimal quantizer of µ ω n . The superscript ω is to emphasize that both µ ω n and x (n),ω are random and we will drop ω when there is no ambiguity. We cite two results of the convergence of W 2 (µ ω n , µ) among so many researches in this topic: the a.s. convergence in [START_REF] Pollard | Quantization and the method of k-means[END_REF][see Theorem 7] and the L p -convergence rate of W p (µ ω n , µ) in [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF].

Theorem. ([FG15][see Theorem 1]

) Let p > 0 and let µ ∈ P q (R d ) for some q > p. Let µ ω n denote the empirical measure of µ defined in (4.1). There exists a constant C only depending on p, d, q such that, for all n ≥ 1,

E W p p (µ ω n , µ) ≤ CM p/q q (µ) ×        n -1/2 + n -(q-p)/q if p > d/2 and q = 2p
n -1/2 log(1 + n) + n -(q-p)/q if p = d/2 and q = 2p n -p/d + n -(q-p)/q if p ∈ (0, d/2) and q = d/(d -p) , (4.2)

where M q (µ) = R d |ξ| q µ(dξ).

Let

., c K ) ∈ (R d ) K , D K, µ (c) = E min 1≤k≤K |X -c k | 2 = E |X| 2 + min 1≤k≤K -2 X|c k + |c k | 2 , and D K, µn (c) = 1 n n i=1 min 1≤k≤K |X i -c k | 2 = 1 n n i=1 |X i | 2 + min 1≤k≤K - 2 n n i=1 X i |c k + |c k | 2 .
The a.s. convergence of optimal quantizers for the empirical measure has been proved in [START_REF] Pollard | Strong consistency of k-means clustering[END_REF]. We give a first upper bound of the clustering performance by applying directly Theorem 2.1 and (4.2).

Proposition 4.1. Let K ∈ N * be the quantization level. Let µ ∈ P q (R d ) for some q > 2 with card(supp(µ)) ≥ K and let µ ω n be the empirical measure of µ defined in (4.1). Let x (n),ω be an optimal quantizer at level K of µ ω n . Then for any n > K,

E D K, µ (x (n),ω ) -inf x∈(R d ) K D K, µ (x) ≤ C d,q,µ,K ×        n -1/4 + n -(q-2)/2q if d < 4 and q = 4 n -1/4 log(1 + n) 1/2 + n -(q-2)/2q if d = 4 and q = 4 n -1/d + n -(q-2)/2q if d > 4 and q = d/(d -2) . (4.3)
where C d,q,µ,K is a constant depending on d, q, µ and the quantization level K.

The reason why we only consider n > K is that for a fixed n ∈ N * , the empirical measure µ n defined in (4.1) is supported by n points, which implies that, if n ≤ K, the optimal quantizer of µ n at level K, viewed as a set, is in fact supp(µ n ). This makes the above bound of no interest. Following the remark after Theorem 1 in [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF], one can remark that if the probability distribution µ has sufficiently large moments (namely if q > 4 when d ≤ 4 and q > 2d/(d -2) when d > 4), then the term n -(q-2)/2q is negligible and can be removed.

Proof of Proposition 4.1. For every ω ∈ Ω and for every n > K, Theorem 2.1 implies that

D K,µ (x (n),ω ) -inf x∈(R d ) K D K,µ (x) ≤ 4e * K,µ W 2 (µ ω n , µ) + 4W 2 2 (µ ω n , µ).
Thus,

E D K,µ (x (n),ω ) -inf x∈(R d ) K D K,µ (x) ≤ 4e * K,µ E W 2 (µ ω n , µ) + 4 E W 2 2 (µ ω n , µ).
It follows from (4.2) applied with p = 2 that

E W 2 2 (µ ω n , µ) ≤ C d,q,µ ×        n -1/2 + n -(q-2)/q if d < 4 and q = 4 n -1/2 log(1 + n) + n -(q-2)/q if d = 4 and q = 4 n -2/d + n -(q-2)/q if d > 4 and q = d/(d -2) , (4.4)
where

C d,q,µ = C•M 2/q q (µ) and C is the constant in (4.2). Moreover, as E W 2 (µ ω n , µ) ≤ EW 2 2 (µ ω n , µ) 1/2 and √ a + b ≤ √ a + √ b for any a, b ∈ R + , Inequality (4.2) also implies E W 2 (µ ω n , µ) ≤ C 1/2 d,q,µ ×       
n -1/4 + n -(q-2)/2q if d < 4 and q = 4 n -1/4 log(1 + n) 1/2 + n -(q-2)/2q if d = 4 and q = 4 n -1/d + n -(q-2)/2q if d > 4 and q = d/(d -2)

.

Consequently,

E D K,µ (x (n),ω ) -inf x∈(R d ) K D K,µ (x) ≤ 4e * K,µ E W 2 (µ ω n , µ) + 4 E W 2 2 (µ ω n , µ). ≤ 8(C 1/2 d,q,µ e * K,µ ∨ C d,q,µ )×        n -1/4 + n -(q-2)/2q if d < 4 and q = 4 n -1/4 log(1 + n) 1/2 + n -(q-2)/2q if d = 4 and q = 4 n -1/d + n -(q-2)/2q if d > 4 and q = d/(d -2) . (4.5)
One can conclude by setting C d,q,µ,K := 8(C

1/2 d,q,µ e * K,µ ∨ C d,q,µ ). Remark 4.1. When d ≥ 4, if q-2 q > 2 d i.e. q > 2d d-2
, Inequality (4.4) can be upper bounded as follows,

E W 2 2 (µ ω n , µ) ≤ 2C d,q,µ n -1/d × n -1 4 log(1 + n) if d = 4 and q = 4 n -1 d if d > 4 and q = d/(d -2) ≤ 2C d,q,µ K -1/d × n -1 4 log(1 + n) if d = 4 and q = 4 n -1 d if d > 4 and q = d/(d -2)
, since we consider only n ≥ K and if q > 2d d-2 , the term n -(q-2)/2q becomes negligible as n grows.

Consequently, (4.5) can be bounded by

E D K,µ (x (n),ω ) -inf x∈(R d ) K D K,µ (x) ≤ 4e * K,µ E W 2 (µ ω n , µ) + 4 E W 2 2 (µ ω n , µ). ≤ 8(C 1/2 d,q,µ e * K,µ ∨ 2C d,q,µ K -1/d )× n -1 4 (log(1 + n)) 1 2 + log(1 + n) if d = 4 and q = 4 2n -1 d if d > 4 and q = d/(d -2)
.

(4.6)

By the non-asymptotic Zador theorem (1.17), one has

e * K,µ ≤ C d,q (µ)σ q (µ)K -1/d
with σ q (µ) = min a∈R d R d |ξ -a| q µ(dξ) 1/q . Thus, Inequality (4.6) can be upper-bounded as follows,

E D K,µ (x (n),ω ) -inf x∈(R d ) K D K,µ (x) ≤ 4e * K,µ E W 2 (µ ω n , µ) + 4 E W 2 2 (µ ω n , µ). ≤ 8K -1/d C 1/2 d,q,µ C d,q (µ)σ q (µ) ∨ 2C d,q,µ × n -1 4 (log(1 + n)) 1 2 + log(1 + n) if d = 4 and q = 4 2n -1 d if d > 4 and q = d/(d -2)
, from which one can remark that the constant C d,q,µ,K in Proposition 4.1 is roughly decreasing as

K -1/d .
A second upper bound of the clustering performance is provided in the following theorem.

Theorem 4.1. Let K ∈ N * be the quantization level. Let µ ∈ P 2 (R d ) with card (supp(µ)) ≥ K and let µ ω n be the empirical measures of µ defined in (4.1), generated by i.i.d observations X 1 , ..., X n , .... We denote by x (n),ω ∈ (R d ) K an optimal quantizer of µ ω n at level K. Then, (a) General upper bound of the performance.

E D K, µ (x (n),ω ) -inf x∈(R d ) K D K, µ (x) ≤ 2K √ n r 2 2n + ρ K (µ) 2 + 2r 1 r 2n + ρ K (µ) , (4.7)
where r n := max 1≤i≤n |X i | 2 and ρ K (µ) is the maximum radius of optimal quantizers of µ, defined in (1.15).

(b) Asymptotic upper bound for distribution with polynomial tail. For p > 2, if µ has a c-th polynomial tail with c > d + p, then

E D K, µ (x (n),ω ) -inf x∈(R d ) K D K, µ (x) ≤ K √ n C µ,p n 2/p + 6K 2(p+d) d(c-p-d) γ K ,
where C µ,p is a constant depending µ, p and lim K γ K = 1.

(c) Asymptotic upper bound for distribution with hyper-exponential tail. Recall that µ has a hyper-exponential tail if µ = f • λ d and there exists τ > 0, κ, ϑ > 0, c > -d and

A > 0 such that ∀ξ ∈ R d , |ξ| ≥ A ⇒ f (ξ) = τ |ξ| c e -ϑ|ξ| κ . If κ ≥ 2,
we can obtain a more precise upper bound of the performance

E D K, µ (x (n),ω ) -inf x∈(R d ) K D K, µ (x) ≤ C ϑ,κ,µ • K √ n 1 + (log n) 2/κ + γ K (log K) 2/κ 1 + 2 d 2/κ ,
where C ϑ,κ,µ is a constant depending ϑ, κ, µ and lim sup K γ K = 1.

In particular, if µ = N (m, Σ), the multidimensional normal distribution, we have

E D K, µ (x (n),ω ) -inf x∈(R d ) K D K, µ (x) ≤ C µ • K √ n 1 + log n + γ K • (log K) 1 + 2 d ,
where lim sup K γ K = 1 and C µ = 24 • 1 ∨ log 2Ee |X| 2 /4 where X is a random variable with distribution µ. Moreover, when µ = N (0,

I d ), C µ = 24(1 + d 2 ) • log 2.
The proof of Theorem 4.1 relies on the Rademacher process theory. A Rademacher sequence (σ i ) i∈{1,...,n} is a sequence of i.i.d random variables with a symmetric {±1}-valued Bernoulli distribution, independent of (X 1 , ..., X n ) and we define the Rademacher process

R n (f ), f ∈ F by R n (f ) := 1 n n i=1 σ i f (X i ).
Remark that the Rademacher process R n (f ) depends on the sample {X 1 , ..., X n } of the probability measure µ.

Theorem (Symmetrization inequalites). For any class F of µ-integrable functions, we have

E µ n -µ F ≤ 2E R n F ,
where for a probability distribution ν, ν

F := sup f ∈F |ν(f )| := sup f ∈F R d f dν and R n F := sup f ∈F |R n (f )|.
For the proof of the above theorem, we refer to [START_REF] Koltchinskii | Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems[END_REF][see Theorem 2.1]. Another more detailed reference is [START_REF] Aad | Weak Convergence[END_REF][see Lemma 2.3.1]. We will also introduce the Contraction principle in the following theorem and we refer to [START_REF] Boucheron | Concentration Inequalities: A nonasymptotic theory of independence[END_REF][see Theorem 11.6] for the proof.

Theorem (Contraction principle). Let x 1 , ..., x n be vectors whose real-valued components are indexed by T , that is, x i = (x i,s ) s∈T . For each i = 1, ..., n, let ϕ i : R → R be a Lipschitz function such that ϕ i (0) = 0. Let σ 1 , ..., σ n be independent Rademacher random variables and let c L = max 1≤i≤n sup x,y∈R x =y ϕi(x)-ϕi(y)

x-y be the uniform Lipschitz constant of the function ϕ i . Then

E sup s∈T n i=1 σ i ϕ i (x i,s ) ≤ c L • E sup s∈T n i=1 σ i x i,s . (4.8)
Remark that, if we consider random variables (Y 1,s , ..., Y n,s ) s∈T independent of (σ 1 , ..., σ n ) and for all s ∈ T and i ∈ {1, ..., n}, Y i,s is valued in R, then (4.8) implies that

E sup s∈T n i=1 σ i ϕ i (Y i,s ) = E E sup s∈T n i=1 σ i ϕ i (Y i,s ) | (Y 1,s , ..., Y n,s ) s∈T ≤c L • E E sup s∈T n i=1 σ i Y i,s | (Y 1,s , ..., Y n,s ) s∈T ≤ c L • E sup s∈T n i=1 σ i Y i,s . (4.9)
The proof of Theorem 4.1 is inspired by that of Theorem 2.1 in [START_REF] Biau | On the performance of clustering in Hilbert spaces[END_REF].

Proof of Theorem 4.1. (a) In order to simplify the notation, we will denote by D (respectively D n ) instead of D K, µ (resp. D K, µn ) the distortion function of µ (resp. µ n ). For any c = (c 1 , ..., c K ) ∈ (R d ) K , note that the distortion function D(c) of µ can be written as

D(c) = E min 1≤k≤K |X -c k | 2 = E |X| 2 + min 1≤k≤K (-2 X|c k + |c k | 2 ) .
We define D(c

) := min 1≤k≤K -2 X|c k + |c k | 2 .
Similarly, for the distortion function D n of the empirical measure µ n ,

D n (c) = 1 n n i=1 min 1≤k≤K |X i -c k | 2 = 1 n n i=1 |X i | 2 + min 1≤k≤K - 2 n n i=1 X i |c k + |c k | 2 , we define D n (c) := min 1≤k≤K -2 n n i=1 X i |c k + |c k | 2 .
We will drop ω in x (n),ω to alleviate the notation throughout the proof. Let x ∈ argmin D K,µ . It follows that

E D(x (n) ) -D(x) = E D(x (n) ) -D(x) = E D(x (n) ) -D n (x (n) ) + E D n (x (n) ) -D(x) ≤ E D(x (n) ) -D n (x (n) ) + E D n (x) -D(x) . (4.10) Define for η, x ∈ R d , f η (x) = -2 η|x + |η| 2 . Part (i): Upper bound of E[D(x (n) ) -D n (x (n) )]. Let R n (ω) := max 1≤i≤n |X i (ω)|.
Remark that for every ω ∈ Ω, R n (ω) is invariant with the respect to all permutations of the components of (X 1 , ..., X n ). Let B R denote the ball centred at 0 with radius R. Then, owing to Proposition

1.1-(iii), x (n) = (x (n) 1 , ..., x (n) K ) ∈ B K Rn . Hence, E [D(x (n) )-D n (x (n) )] ≤ E sup c ∈B K Rn D(c) -D n (c) = E sup c ∈B K Rn E min 1≤k≤K f c k (X) - 1 n n i=1 min 1≤k≤K f c k (X i ) = E sup c ∈B K Rn E 1 n n i=1 min 1≤k≤K f c k (X i ) - 1 n n i=1 min 1≤k≤K f c k (X i ) X 1 , ..., X n , (4.11)
where X 1 , ..., X n are i.i.d random variable with the distribution µ, independent of (X 1 , ..., X n ).

Let

R 2n := max 1≤i≤n |X i | ∨ |X i |, then (4.11) becomes E [D(x (n) )-D n (x (n) )] ≤ E sup c ∈B K R 2n E 1 n n i=1 min 1≤k≤K f c k (X i ) - 1 n n i=1 min 1≤k≤K f c k (X i ) X 1 , ..., X n ≤ E E sup c ∈B K R 2n 1 n n i=1 min 1≤k≤K f c k (X i ) - 1 n n i=1 min 1≤k≤K f c k (X i ) X 1 , ..., X n = E sup c ∈B K R 2n 1 n n i=1 min 1≤k≤K f c k (X i ) -min 1≤k≤K f c k (X i ) .
(4.12)

The distribution of (X 1 , ..., X n , X 1 , ..., X n ) and that of R 2n are invariant with the respect to all permutation of the components in (X 1 , ..., X n , X 1 , ..., X n ). Hence,

E [D(x (n) )-D n (x (n) )] = E sup c ∈B K R 2n 1 n n i=1 σ i min 1≤k≤K f c k (X i ) -min 1≤k≤K f c k (X i ) ≤ E sup c ∈B K R 2n 1 n n i=1 σ i min 1≤k≤K f c k (X i ) + E sup c ∈B R K 2n 1 n n i=1 σ i min 1≤k≤K f c k (X i ) = 2E sup c ∈B K R 2n 1 n n i=1 σ i min 1≤k≤K f c k (X i ) . (4.13)
In the second line of (4.13), we can change the sign before the second term since -σ i has the same distribution of σ i , and we will continue to use this property throughout the proof. Let

S K = E sup c ∈B K R 2n 1 n n i=1 σ i min 1≤k≤K f c k (X i )

and we provide an upper bound for S K by induction on

K in what follows.

For K = 1,

S 1 = E sup c ∈B R 2n 1 n n i=1 σ i min 1≤k≤K f c (X i ) = E sup c ∈B R 2n 1 n n i=1 σ i -2 c|X i + |c| 2 ≤ 2 E sup c ∈B R 2n 1 n n i=1 σ i c|X i + E sup c ∈B R 2n 1 n n i=1 σ i |c| 2 ≤ 2 n E sup c ∈B R 2n c| n i=1 σ i X i + 1 n E n i=1 σ i • |R 2n | 2 ≤ 2 n E sup c ∈B R 2n n i=1 σ i X i • |c| + 1 n E n i=1 σ i • E |R 2n | 2
(by Cauchy-Schwarz inequality and independence of σ i and X i )

≤ 2 n n i=1 σ i X i 2 • R 2n 2 + 1 n n i=1 σ i 2 2 • R 2n 2 2 ≤ 2 n √ n X 1 2 • R 2n 2 + 1 √ n R 2n 2 2 ≤ R 2n 2 √ n 2 X 1 2 + R 2n 2 . (4.14)
The first inequality of the last line of (4.14

) follows from E | n i=1 σ i X i | 2 = E n i=1 σ 2 i X 2 i = nEX 2 1 since the (σ 1 , ..., σ n ) is independent of (X 1 , ..., X n ) and E σ i = 0. For n ∈ N * , define r n := max 1≤i≤n |Y i | 2
, where Y 1 , ..., Y n are i.i.d random variables with probability distribution µ. Hence, r 2n = R 2n 2 , since (Y 1 , ..., Y 2n ) has the same distribution as (X 1 , ..., X n , X 1 , ..., X n ). Therefore,

S 1 ≤ r 2n √ n 2 X 1 2 + r 2n .
For K = 2,

S 2 = E sup c=(c1,c2)∈B 2 R 2n 1 n n i=1 σ i f c1 (X i ) ∧ f c2 (X i ) = 1 2 E sup c ∈B 2 R 2n 1 n n i=1 σ i f c1 (X i ) + f c2 (X i ) -|f c1 (X i ) -f c2 (X i )| (as a ∧ b = a + b 2 - |a -b| 2 ) ≤ 1 2 E sup c ∈B 2 R 2n 1 n n i=1 σ i f c1 (X i ) + f c2 (X i ) + E sup c ∈B 2 R 2n 1 n n i=1 σ i |f c1 (X i ) -f c2 (X i )| ≤ 1 2 2S 1 + E sup c ∈B 2 R 2n 1 n n i=1 σ i f c1 (X i ) -f c2 (X i ) by (4.9) ≤ 1 2 2S 1 + E sup c1∈B R 2n 1 n n i=1 σ i f c1 (X i ) + E sup c2∈B R 2n 1 n n i=1 σ i f c2 (X i ) ≤ 2S 1 . (4.15)
Next, we will show by induction that S K ≤ KS 1 for every K ∈ N * . Assume that S K ≤ KS 1 , for K + 1,

S K+1 = E sup c ∈B K+1 R 2n 1 n n i=1 σ i min 1≤k≤K+1 f c k (X i ) = E sup c ∈B K+1 R 2n 1 n n i=1 σ i min 1≤k≤K f c k (X i ) ∧ f c K+1 (X i ) ≤ 1 2 E sup c ∈B K+1 R 2n 1 n n i=1 σ i min 1≤k≤K f c k (X i ) + f c K+1 (X i ) -min 1≤k≤K f c k (X i ) -f c K+1 (X i ) ≤ 1 2 E sup c ∈B K+1 R 2n 1 n n i=1 σ i min 1≤k≤K f c k (X i ) + f c K+1 (X i ) + sup c ∈B K+1 R 2n 1 n n i=1 σ i min 1≤k≤K f c k (X i ) -f c K+1 (X i ) ≤ 1 2 (S K + S 1 + S K + S 1 ) ≤ S K + S 1 ≤ (K + 1)S 1 . (4.16)
Hence,

E [D(x (n) ) -D n (x (n) )] ≤ 2S K ≤ 2KS 1 ≤ 2K • r 2n √ n 2 X 1 2 + r 2n .
(4.17)

Part (ii): Upper bound of E [D n (x) -D(x)]. As x = (x 1 , ..., x K ) is an optimal quantizer of µ, we have max 1≤k≤K |x k | ≤ ρ K (µ) owing to the definition of ρ K (µ) in (1.15). Consequently, E D n (x) -D(x) ≤ E sup c ∈B K ρ K (µ) D n (c) -D(c)
By the same reasoning of Part (I), we have

E D n (x) -D(x) ≤ 2K √ n ρ K (µ) 2 X 1 2 + ρ K (µ) . Hence E D(x (n) ) -D(x) ≤ 2K √ n r 2n 2 X 1 2 + r 2n + 2K √ n ρ K (µ) 2 X 1 2 + ρ K (µ) ≤ 2K √ n r 2 2n + ρ 2 K (µ) + 2r 1 r 2n + ρ K (µ) . (4.18)
The proof of (b) and (c) is postponed in Appendix E.

Appendix

Appendix A: Proof of Pollard's Theorem

Proof of Pollard's Theorem. Since the quantization level K is fixed, in this proof, we drop the subscript K of the distortion function and denote by D n (respectively, D ∞ ) the distortion function of µ n (resp. µ ∞ ).

We know x (n) ∈ argmin D n owing to Proposition 1.1, that is, for all y ∈ (y 1 , ..., y K ) ∈ (R d ) K , we have D n (x (n) ) ≤ D n (y). For every fixed y = (y 1 , ..., y K ), we have D n (y) → D ∞ (y) after (1.21) so that lim sup

n D n (x (n) ) ≤ inf y∈(R d ) K D ∞ (y).
(5.1)

Assume that there exists an index set I ⊂ {1, ..., K} and I c = ∅ such that (x

(n) i ) i∈I,n≥1 is bounded and (x (n) i ) i∈I c ,n≥1
is not bounded. Then there exists a subsequence ψ(n) of n such that

   x ψ(n) i → x (∞) i , i ∈ I, x ψ(n) i → +∞, i ∈ I c . After (1.21), we have D ψ(n) (x (ψ(n)) ) 1/2 ≥ D ∞ (x (ψ(n)) ) 1/2 -W 2 (µ ψ(n) , µ ∞ ). Hence, lim inf n D ψ(n) (x (ψ(n)) ) 1/2 ≥ lim inf n D ∞ (x (ψ(n)) ) 1/2 so that lim inf n D ψ(n) (x (ψ(n)) ) 1/2 ≥ lim inf n D ∞ (x (ψ(n)) ) 1/2 = lim inf n min i∈{1,...,K} x (ψ(n)) i -ξ 2 µ ∞ (dξ) 1/2 ≥ lim inf n min i∈{1,...,K} x (ψ(n)) i -ξ 2 µ ∞ (dξ) 1/2 = min i∈I x (∞) i -ξ 2 µ ∞ (dξ) 1/2 , (5.2)
where we used Fatou's Lemma in the third line. Thus, (5.1) and (5.2) imply that

min i∈I x (∞) i -ξ 2 µ ∞ (dξ) ≤ inf y∈(R d ) K D ∞ (y). (5.3)
This implies that I = {1, ..., K} after Proposition 1.1 (otherwise, (5.3) implies that e |I|, * (µ ∞ ) ≤ e K, * (µ ∞ ) with |I| < K, which is contradictory to Proposition 1.1-(i)). Therefore, (x (n) ) is bounded and any limiting point

x (∞) ∈ argmin x∈(R d ) K D ∞ (x).

Appendix B: Proof of Proposition 1.1 -(iii)

We define the open Voronoï cell generated by x i with respect to the Euclidean norm | • | by

V o xi (x) = ξ ∈ R d |ξ -x i | < min 1≤j≤K,j =i |ξ -x j | . (5.4) It follows from [GL00][see Proposition 1.3] that intV xi (x) = V o xi (x)
, where intA denotes the interior of a set A. Moreover, if we denote by λ d the Lebesgue measure on R d , we have λ d ∂V xi (x) = 0, where ∂A denotes the boundary of A (see [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF][Theorem 1.5]). If µ ∈ P 2 (R d ) and x * is an optimal quantizer of µ, even if µ is not absolutely continuous with the respect of λ d , we have µ ∂V xi (x * ) = 0 for all i ∈ {1, ..., K} (see [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF][Theorem 4.2]).

Proof. Assume that there exists an

x * = (x * 1 , ..., x * K ) ∈ G K (µ) in which there exists k ∈ {1, ..., K} such that x * k / ∈ H µ . Case (I): µ V o x * k (Γ * ) ∩ supp(µ) = 0.
The distortion function can be written as

D K, µ (x * ) = K i=1 Cx i (x) |ξ -x * i | 2 µ(dξ) = K i=1 V o x i (x) |ξ -x * i | 2 µ(dξ) (since x * is optimal and |•| is Euclidean, µ ∂V xi (Γ * ) = 0 and intV xi (Γ) = V o xi (Γ)) = K i=1,i =k V o x i (x) |ξ -x * i | 2 µ(dξ) = D K, µ ( x), (5.5)
where x = (x * 1 , ..., x * k-1 , x * k+1 , ..., x * K ). Therefore, Γ = {x * 1 , ..., x * k-1 , x * k+1 , ..., x * K } is also a K-level optimal quantizer with card( Γ) < K, contradictory to Proposition 1.1 -(i).

Case (II

): µ V o x * k (Γ * ) ∩ supp(µ) > 0. Since x * k = H µ ,
-x * k |x * k -x * k = 0. Hence, |x * k -b| 2 = |x * k -b| 2 + |x * k -x * k | 2 > |x * k -b| 2 . Therefore, |z -x * k | ≤ |z -b| + |b -x * k | < |z -b| + |x * k -b| = |z -x * k |. Let B(x, r) denote the ball on R d centered at x with radius r. Since µ V o x * k (Γ * ) ∩ supp(µ) > 0, there exists α ∈ V o x * k (Γ * ) ∩ supp(µ) such that ∃ r ≥ 0, µ B(α, r) > 0 (when r = 0, B(α, r) = {r}). Moreover, ∀β ∈ B(α, r), |β -x * k | < |β -x * k | < min i =k |β -x * i | .
(5.6)

Let x := (x * 1 , ..., x * k-1 , x * k , x * k+1 , ..., x * K ), (5.6) implies D K, µ (x) < D K, µ (x *
). This is contradictory with the fact that x * is an optimal quantizer. Hence, x * ∈ H µ .

Appendix C: Proof of Proposition 3.1

We use Lemma 11 in [START_REF] Fort | On the as convergence of the Kohonen algorithm with a general neighborhood function[END_REF] to compute the Hessian matrix H D K, µ of D K, µ .

Lemma 5.1 (Lemma 11 in [START_REF] Fort | On the as convergence of the Kohonen algorithm with a general neighborhood function[END_REF]). Let ϕ be a countinous R-valued function defined on [0, 1] d . For every x ∈ D

K := y ∈ [0, 1] d K | y i = y j if i = j , let Φ i (x) := Vi(x) ϕ(ω)dω. Then Φ i is continuously differentiable on D K and ∀i = j, ∂Φ i ∂x j (x) = Vi(x)∩Vj (x) ϕ(ξ) 1 2 - → n ij x + 1 |x j -x i | × ( x i + x j 2 -ξ) λ ij x (dξ) (5.7) and ∂Φ i ∂x i (x) = - 1≤j≤K,j =i ∂Φ j ∂x i (x), (5.8)
where -

→ n ij x := xj -xi |xj -xi| , M x ij := u ∈ R d | u - x i + x j 2 | x i -x j = 0 (5.9)
and λ ij x (dξ) denotes the Lebesgue measure on the affine hyperplane M x ij .

Note that one can simplify the result of Lemma 5.1 as follows,

∀i = j, ∂Φ i ∂x j (x) = Vi(x)∩Vj (x) ϕ(ξ) 1 2 x j -x i |x j -x i | + 1 |x j -x i | ( x i + x j 2 -ξ) λ ij x (dξ) = Vi(x)∩Vj (x) ϕ(ξ) 1 |x j -x i | x j -x i 2 + x i + x j 2 -ξ λ ij x (dξ) = Vi(x)∩Vj (x) ϕ(ξ) 1 |x j -x i | (x j -ξ)λ ij x (dξ).
(5.10)

Proof of Proposition 3.1. (i) Set ϕ i,M (ξ) = (x i -ξ)f (ξ)χ M (ξ) with χ M (ξ) :=        1 |ξ| ≤ M M + 1 -|ξ| M < |ξ| ≤ M + 1 0 |ξ| > M + 1 . Set Φ M i (x) = Vi(x) ϕ i,M (
ξ)dξ and Φ i (x) = Vi(x) (x i -ξ)f (ξ)dξ for i = 1, ..., K. Then (3.1) implies that ∂D K, µ ∂xi = 2Φ i , i = 1, ..., K.

For j = 1, ..., K and j = i, it follows from (5.10) that

∂Φ M i ∂x j (x) = Vi(x)∩Vj (x) (x i -ξ) ⊗ (x j -ξ) • 1 |x j -x i | f (ξ)χ M (ξ)λ ij x (dξ), (5.11)
and for i = 1, ..., K,

∂Φ M i ∂xi (x) = V i (ξ) f (ξ)χM (ξ)dξ I d - i =j 1≤j≤K V i (x)∩V j (x) (xi -ξ) ⊗ (xi -ξ) • 1 |xj -xi| f (ξ)χM (ξ)λ ij x (dξ) ,
(5.12)

where in (5.11) and (5.12), u ⊗ v := [u i v j ] 1≤i,j≤d for any two vectors u = (u 1 , ..., u d ) and v = (v 1 , ..., v d ) in R d .

We prove now the differentiability of Φ i in three steps.

Step 1 : We prove in this part that for every x ∈ F K ,

h ij (x) := Vi(x)∩Vj (x) (x i -ξ) ⊗ (x j -ξ) • 1 |x j -x i | f (ξ)λ ij x (dξ) < +∞. If V i (x) ∩ V j (x) = ∅, it is obvious that h ij (x) = 0 < +∞. Now we assume that V i (x) ∩ V j (x) = ∅.
Without loss of generality, we assume that V 1 (x) ∩ V 2 (x) = ∅ and we prove in the following h 12 is well defined i.e. (h 12 (x) ∈ R.

Let α(x, ξ) := (x 1 -ξ) ⊗ (x 2 -ξ) • 1 |x 2 -x 1 | f (ξ).
(5.13)

Then h 12 (x) = V1(x)∩V2(x)
α(x, ξ)λ 12 x (dξ).

Let (e 1 , ..., e d ) denote the canonical basis of R d . Set u x = x1-x2 |x1-x2| . As x 1 = x 2 , there exists at least one i 0 ∈ {1, ..., d} s.t. u x | e i0 = 0. Then (u x , e i , 1 ≤ i ≤ d, i = i 0 ) forms a new basis of R d . Applying the Gram-Schmidt orthonormalization procedure, we derive the existence of a new orthonormal basis (u x 1 , ..., u x d ) of R d such that u x 1 = u x . Moreover, the Gram-Schmidt orthonormalization procedure also implies that u

x i , 1 ≤ i ≤ d is continuous in x.
With respect to this new basis (u x 1 , ..., u x d ), the hyperplane M x 12 defined in (5.9) can be written by

M x 12 = x 1 + x 2 2 + span u x i , i = 2, ..., d ,
where span(S) denotes the vector subspace of R d spanned by S. Moreover, note that

V 1 (x) ∩ V 2 (x) = ξ ∈ M x 12 min k=3,...,K |x k -ξ| ≥ |x 1 -ξ| = |x 2 -ξ| .
Then, for every fixed ξ /

∈ ∂ V 1 (x) ∩ V 2 (x) , the function x → 1 V1(x)∩V2(x) (ξ) is continuous in x ∈ F K and λ 12 x ∂ V 1 (x) ∩ V 2 (x) = 0 (5.14) since V 1 (x) ∩ V 2 (x) is a polyhedral convex set in M x 12 . Now by a change of variable ξ = x1+x2 2 + d i=2 r i u x i , h 12 (x) = R d-1 1 V12(x) (r 2 , ..., r d ) α x, x 1 + x 2 2 + d i=2 r i u x i dr 2 ...dr d , (5.15)
where

V 12 (x) := (r 2 , ..., r d ) ∈ R d-1 min 3≤k≤K x k - x 1 + x 2 2 - d i=2 r i u x i ≥ x 1 -x 2 2 - d i=2 r i u x i .
(5.16)

Let ∂V 12 (x) be the boundary of V 12 (x) given by

∂V 12 (x) := (r 2 , ..., r d ) ∈ R d-1 min 3≤k≤K x k - x 1 + x 2 2 - d i=2 r i u x i = x 1 -x 2 2 - d i=2 r i u x i .
(5.17

α(x, x 1 + x 2 2 + d i=2 r i u x i ) = x1-x2 2 - d i=2 r i u x i ⊗ x2-x1 2 - d i=2 r i u x i |x 2 -x 1 | f x 1 + x 2 2 + d i=2 r i u x i
(5.18) can be upper-bounded by

x1-x2 2 - d i=2 r i u x i x2-x1 2 - d i=2 r i u x i |x 2 -x 1 | f x 1 + x 2 2 + d i=2 r i u x i ≤ x1-x2 2 + d i=2 r i u x i 2 |x 2 -x 1 | f x 1 + x 2 2 + d i=2 r i u x i ≤ C x (1 + d i=2 r 2 i )f x 1 + x 2 2 + d i=2 r i u x i (5.19)
where C x > 0 is a constant depending only on x.

The distribution µ is assumed to be 1-radially controlled i.e. there exist a constant A > 0 and a continuous and decreasing function g : R + (5.20)

→ R + such that ∀ξ ∈ R d , |ξ| ≥ A, f (
Now let K := 1 2 |x 1 + x 2 | ∨ A and let r := d i=2 r i u x i . As g is a non-increasing function, it follows that C x (1 + d i=2 r 2 i )f x 1 + x 2 2 + d i=2 r i u x i ≤ C x (1 + |r| 2 ) sup ξ∈B(0,3K) f (ξ)1 {|r|≤2K} + C x (1 + |r| 2 )g x (n) 1 + x (n) 2 2 + d i=2 r i u x i 1 {|r|≥2K} . ≤ C x (1 + |r| 2 ) sup ξ∈B(0,3K) f (ξ)1 {|r|≤2K} + C x (1 + |r| 2 )g |r| -K 1 {|r|≥2K} .
(5.21)

Switching to polar coordinates, one obtains by letting s = |r|

R d-1 C x |r| 2 g |r| -K 1 {|r|≥2K} dr 2 ...dr d ≤ C x,d R+ s 2 g(s -K)1 {s≥2K} s d-2 ds ≤ C x,d ∞ K (s + K) d g(s)ds ≤ 2 d C x,d ∞ K (K d + s d )g(s)ds < +∞,
where the last inequality follows from (5.20). Thus one obtains

R d-1 C x (1 + |r| 2 ) sup ξ∈B(0,3K) f (ξ)1 {|r|≤2K} + C x (1 + |r| 2 )g |r| -K 1 {|r|≥2K} dr 2 ...dr d < +∞.
Hence h 12 is well-defined since

V1(x)∩V2(x) |α(x, ξ)| λ 12 x (dξ) < +∞.
(5.22)

Step 2 : Now we prove that for any x ∈ F K , sup y∈B(x, εx)

∂Φ M i ∂x j (y) -h ij (y) M →+∞ -----→ 0, (5.23)
where ε x = 1 3 min 1≤i<j≤K |x i -x j | and (5.23) means every term in the matrix converges to 0. First, for every fixed y ∈ B(x, ε x ), the absolute value of every term in the following matrix

∂Φ M i ∂x j (y) -h ij (y) = Vi(y)∩Vj (y) (y i -ξ) ⊗ (y j -ξ) |y j -y i | f (ξ) 1 -χ M (ξ) λ ij y (dξ)
can be upper bounded by

f M (y) := Vi(y)∩Vj (y)∩ R d \B(0,M +1) |y i -ξ||y j -ξ| |y j -y i | f (ξ)λ ij y (dξ).
(5.24)

Moreover, the inequality (5.22) implies that f M (y) converges to 0 for every y ∈ B(x, ε x ) as M → +∞. As (f M ) M is a monotonically decreasing sequence, one can obtain sup y∈B(x,ε)

f M (y) → 0 owing to Dini's theorem, which in turn implies the convergence in (5.23).

Step 3 : It is obvious that Φ M i (x) converges to Φ i (x) for every x ∈ R d as M → +∞ since µ ∈ P 2 (R d ). Hence ∂Φ1 ∂x2 (x) = h 12 (x). Then one can directly obtain (3.2) since ∂D K,µ ∂xj xi = 2 ∂Φi ∂xj = 2h ij by applying (3.1). The proof for (3.3) is similar.

(ii) We will only prove the continuity of

∂ 2 D K,µ ∂x1∂x2 and ∂ 2 D K,µ ∂x 2 1 at a point x ∈ F K . The proof for ∂ 2 D K,µ
∂xi∂xj for others i, j ∈ {1, ..., K} is similar. We take the same definition of α(x, ξ) in (5.13), then

∂ 2 D K,µ ∂x 1 ∂x 2 (x) = 2 V1(x)∩V2(x) α(x, ξ)λ 12 x (dξ)
and by the same change of variable (5.15) as in (i), we have

∂ 2 D K,µ ∂x 1 ∂x 2 (x) = 2 R d-1 1 V12(x) (r 2 , ..., r d ) α x, x 1 + x 2 2 + d i=2 r i u x i dr 2 ...dr d
with the same definition of V 12 (x) as in (5.16).

Let us now consider a sequence x (n) = (x

(n) 1 , ..., x (n)
K ) ∈ (R d ) K converging to a point x = (x 1 , ..., x K ) ∈ F K satisfying that for every n ∈ N * , x (n) -x ≤ δ x := 1 3 min 1≤i,j≤K,i =j |x i -x j | , (5.25) so that x (n) ∈ F K for every n ∈ N * . For a fixed (r 2 , ..., r d ) ∈ R d-1 , the continuity of x → α(x, x1+x2 2 + d i=2 r i u x i
) in F K can be obtained by the continuity of (x, ξ) → α(x, ξ) and the continuity of Gram-Schmidt orthonormalization procedure.

By the same reasoning as in (5.19), the absolute value of every term in the matrix

α x (n) , x (n) 1 + x (n) 2 2 + d i=2 r (n) i u x (n)
i can be upper bounded by

x (n) 1 -x (n) 2 2 + d i=2 r i u x (n) i 2 x (n) 2 -x (n) 1 f x (n) 1 + x (n) 2 2 + d i=2 r (n) i u x (n) i ,
where there exists a constant C x depending only on x such that

x (n) 1 -x (n) 2 2 + d i=2 r i u x (n) i 2 x (n) 2 -x (n) 1 ≤ C x (1 + d i=2 r 2 i )
since by (5.25), one can get

∀n ∈ N * , ∀i, j ∈ {1, ..., K} with i = j, δ x ≤ x (n) i -x (n) j ≤ max 1≤i,j≤K |x i -x j | + 2δ x .
Moreover, if we take

K := 1 2 sup n x (n) 1 + x (n) 2 ∨ A and take r n := d i=2 r i u x (n) i , then C x (1 + d i=2 r 2 i )f x (n) 1 + x (n) 2 2 + d i=2 r i u x (n) i ≤ C x (1 + |r| 2 ) sup ξ∈B(0,3K) f (ξ)1 {|r|≤2K} + C x (1 + |r| 2 )g x (n) 1 + x (n) 2 2 + d i=2 r i u x (n) i 1 {|r|≥2K} . ≤ C x (1 + |r| 2 ) sup ξ∈B(0,3K) f (ξ)1 {|r|≤2K} + C x (1 + |r| 2 )g |r| -K 1 {|r|≥2K} .
(5.26)

By the same reasoning as in (i)-Step 1, we have

R d-1 C x (1 + |r| 2 ) sup ξ∈B(0,3K) f (ξ)1 {|r|≤2K} + C x (1 + |r| 2 )g |r| -K 1 {|r|≥2K} dr 2 ...dr d < +∞, which implies ∂ 2 D K,µ ∂x1∂x2 (x (n) ) → ∂ 2 D K,µ ∂x1∂x2 (x
) as n → +∞ by applying Lebesgue's dominated convergence theorem. Thus

∂ 2 D K,µ ∂x1∂x2 is continuous at x ∈ F K . It remains to prove the continuity of x → µ V 1 (x) = R d 1 V1(x) (ξ)f (ξ)λ d (dξ) to obtain the continuity of ∂ 2 D K,µ ∂x 2 1 defined in (3.3). Remark that V 1 (x) = ξ ∈ R d |ξ -x 1 | ≤ min 1≤j≤K |ξ -x j | ,
and by

[GL00][Proposition 1.3], ∂V 1 (x) = ξ ∈ R d |ξ -x 1 | = min 1≤j≤K |ξ -x j | .
Then for any ξ /

∈ ∂V 1 (x), the function x → 1 V1(x) (ξ) is continuous. As the norm |•| is the Euclidean norm, then λ d (∂V i (x)) = 0 (see [GL00]
[Proposition 1.3 and Theorem 1.5]). For any

x ∈ F K and a sequence x (n) converging to x, we have 1 Proof. (i) We will only deal with the uniform distribution U ([0, 1]). The proof is similar for other uniform distributions.

V1(x (n) ) (ξ)f (ξ) ≤ f (ξ) ∈ L 1 (λ d ). Thus the continuity of x → µ V 1 (x) =
In [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF][see Example 4.17 and 5.5] and [START_REF] Benaïm | Convergence of the onedimensional Kohonen algorithm[END_REF], the authors show that Γ * = { 2i-1 2K : i -1, ..., K} is the unique optimal quantizers of U ([0, 1]). Let x * = ( 1 2K , ..., 2i-1 2K , ..., 2K-1 2K ), then one can compute explicitly H D (x * ): The matrix H D (x * ) is tridiagonal. If we denote by f k (x * ) its k-th leading principal minor and we define f 0 (x * ) = 1, then

H D (x * ) =         
f k (x * ) = 1 K f k-1 (x * ) - 1 4K 2 f k-2 (x * ) for k = 2, ..., K -1,
(5.28)

∂ϕ i ∂u i+1 (u) = 2 ui+1 ui f (ξ)dξ f (u i+1 ) + f (u i+1 ) ui+1 ui (ξ -u i+1 )f (ξ)dξ -f (u i )f (u i+1 )(u i+1 -u i ) ∂ϕ i ∂u l
(u) = 0, for all l = i and l = i + 1.

(5.33)

The second derivatives of ϕ i are

∂ 2 ϕ i ∂u i+1 ∂u i (u) = ∂ 2 ϕ i ∂u i ∂u i+1 (u) = -f (u i+1 )f (u i ) + (u i+1 -u i ) f (u i )f (u i+1 ) -f (u i )f (u i+1 )
∂ 2 ϕ i ∂u l ∂u i (u) = ∂ 2 ϕ i ∂u i ∂u l (u) = 0 for all l = i and l = i + 1.

(5.34)

If log f is strictly concave, then (log f ) = f f is strictly decreasing. For u ∈ F + K+1 , we have

u i+1 > u i , then f (u i+1 ) f (u i+1 ) - f (u i ) f (u i ) = f (u i+1 )f (u i ) -f (u i+1 )f (u i ) f (u i )f (u i+1 ) < 0.
Thus f (u i+1 )f (u i ) -f (u i+1 )f (u i ) < 0 and from which one can get ∂ 2 ϕ i ∂u i+1 ∂u i (u) < 0.

In fact, ϕ i , ∂ϕi ∂ui , ∂ϕi ∂ui+1 and ∂ 2 ϕi ∂ui+1∂ui only depend on the variables u i and u i+1 .

(a) For 1 ≤ i ≤ K, ϕ i (u i+1 , u i+1 ) = 0. After the Mean value theorem, there exists γ ∈ (u i , u i+1 ) such that 1 u i -u i+1 ϕ i (u i , u i+1 ) -ϕ i (u i+1 , u i+1 ) = ∂ϕ i ∂u i (γ, u i+1 ).

( (b) After the Mean value theorem, there exists γ ∈ (u 1 , u 2 ) such that

∂ 2 ϕ 1 ∂u 1 ∂u 2 (u 1 , γ ) = 1 u 2 -u 1 ∂ϕ 1 ∂u 1 (u 1 , u 2 ) - ∂ϕ 1 ∂u 1 (u 1 , u 1 ) .
As ∂ 2 ϕ 1 ∂u 1 ∂u 2 (u 1 , γ ) < 0 and ∂ϕ 1 ∂u 1 (u 1 , u 1 ) = 0, one can get ∂ϕ 1 ∂u 1 (u 1 , u 2 ) < 0.

(c) In the same way, there exists ζ ∈ (u K , u K+1 ) such that

∂ 2 ϕ K ∂u K ∂u K+1 (ζ , u K+1 ) = 1 u K -u K+1 ∂ϕ K ∂u K+1 (u K , u K+1 ) - ∂ϕ K ∂u K+1 (u K+1 , u K+1 ) .
As ∂ 2 ϕ K ∂u K ∂u K+1 (ζ , u K+1 ) < 0 and ∂ϕ K ∂u K+1 (u K+1 , u K+1 ) = 0, one gets ∂ϕ K ∂u K+1 (u K , u K+1 ) > 0.

Proof of Proposition 3.2, continuation. We set x * ,M := (-M, x * 2 , ..., x * K , M ) with M large enough such that x * ,M ∈ F + K+1 , then for 2 ≤ i ≤ K -1, L i (x * ) = The real constant κ is assumed to be greater than or equal to 2. Let X be a random variable with probability distribution µ. Therefore, for every λ ∈ (0, ϑ), E e λ|X| κ < +∞ and where the last line of (5.38) is due to the fact that X 1 , ..., X n have the same distribution than X.

r n =
Under the same assumption as before, it follows from (1.19) that ρ K (µ) ≤ γ K (log K) 1/κ • 2ϑ -1/κ 1 + 2 d 1/κ with lim sup K→+∞ γ K ≤ 1.

(5.39) Moreover, it follows from (4.18) that

E D(x (n) ) -D(x) ≤ 2K √ n 3r 2 2n + (2m 2 ) ∨ ρ K (µ) • ρ K (µ)
since r 2n ≥ m 2 after the definitions of r 2n and m 2 . In addition, (5.39) implies that ρ K (µ) → +∞ as K → +∞ and, for large enough K, ρ K (µ) ≥ 2m 2 . Therefore,

E D(x (n) ) -D(x) ≤ 2K √ n 3 • 1 ∨ log 2E e λ|X| κ 2/κ 1 λ 2/κ (log n) 2/κ + 1 + 4ϑ -2/κ γ K (log K) 2/κ 1 + 2 d 2/κ .
(5.40) Inequality (5.40) is true for all λ ∈ (0, ϑ). We may take λ = ϑ 2 . It follows that

E D(x (n) ) -D(x) ≤ C ϑ,κ,µ • K √ n 1 + (log n) 2/κ + γ K (log K) 2/κ 1 + 2 d 2/κ , (5.41)
where C ϑ,κ,µ = 6 2 ϑ 2/κ • (1 ∨ log 2E e ϑ|X| κ /2 ) ∨ 8ϑ -2/κ and lim sup K γ K = 1.

Figure 1 :

 1 Figure 1: An optimal quantizer for N 0, I 2 at level 60.

Figure 2 :

 2 Figure 2: An optimal cluster center (blue points) for an observation {η 1 , ..., η 500 } i.i.d ∼ N (0, I 2 ) (grey points).

  D K, µ denote the distortion function of µ and let D K, µn denote the distortion fuction of µ ω n for any n ∈ N * . Recall by Definition 1.1 that for c = (c 1 , ..

  there exists a hyperplane H strictly separating x * k and H µ . Let x * k be the orthogonal projection of x * k on H. For any z ∈ H µ , let b denote the point in the segment joining z and x * k which lies on H, then b

  ) Then (5.14) implies that λ R d-1 ∂V 12 (x) = 0 where λ R d-1 denotes the Lebesgue measure of the subspace spanu x i , i = 2, ..., d . It is obvious that for any a = (a 1 , ..., a d ), b = (b 1 , ..., b d ) ∈ R d , we have |a i b j | ≤ |a| |b| , 1 ≤ i, j ≤ d.Thus the absolute value of every term in the matrix

  ξ) ≤ g(|ξ|) and R+ x d g(x)dx < +∞.

R d 1

 1 V1(x) (ξ)f (ξ)λ d (dξ) is a direct application of Lebesgue's dominated convergence theorem. 5.4 Appendix D: Proof of Proposition 3.2

2µ(

  Vi(x * )) ϕ i ( x * ,M ). Thus L i (x * ) > 0, i = owing to (1.18). It follows from (4.18) thatE D(x (n) ) -D(x) ≤ 2K √ n 3r 2 2n + (2m 2 ) ∨ ρ K (µ) • ρ K (µ)since r 2n ≥ m 2 after the definitions of r 2n and m 2 . In addition, (5.37) implies that ρ K (µ) → +∞ as K → +∞ and, for large enough K, ρ K (µ) ≥ 2m 2 . Therefore,E D(x (n) ) -D(x) ≤ p n 2/p + 6K p+d d(c-p-d) γ K ,whereC µ,p = 6 • 2 2/p X 2 p and lim K γ K = 1. (c) The distribution µ is assumed to have a hyper-exponential tail, that is, µ = f • λ d with f (ξ) = τ |ξ|c e -ϑ|ξ| κ for |ξ| large enough with c > -d.

  R n 2 2 = E max(|X 1 | , ..., |X n |) 2 = E max(|X 1 | κ , ..., |X n | κ ) 2/κ

	=E	1 λ	log max(e λ|X1| κ	, ..., e λ|Xn| κ	)	2/κ	≤	Å 1 λ	ã 2/κ	log E max(e λ|X1| κ	, ..., e λ|Xn| κ	)	2/κ
	≤	Å 1 λ	ã 2/κ	log E	n i=1	e λ|Xi| κ 2/κ	=	Å 1 λ	ã 2/κ	log(nE e λ|X| κ	)	2/κ
	=	Å 1 λ	ã 2/κ	log E e λ|X| κ	+ log n	2/κ ,			(5.38)

 [START_REF] El-Mikkawy | A note on a three-term recurrence for a tridiagonal matrix[END_REF]). One can solve from the three-term recurrence relation that

(5.30)

In fact, (5.29) is true for k = 1. Suppose (5.29) holds for k ≤ K -2, then owing to (5.28)

(5.31)

In order to study the positivity of L i (x * ), we define a function ϕ i (u) for any i ∈ {1, ..., K} and for any u = (u 1 , ..., u K+1 ) ∈ F + K+1 by

Lemma 5.2. If f is positive and differentiable and if log f is strictly concave, then for all u = (u 1 , ..., u K+1 ) ∈ F + K+1 , we have the following results for ϕ i (u) defined in (5.32), (a) for every i = 1, ..., K, ϕ i (u) > 0;

(b) ∂ϕ1 ∂u1 (u) < 0;

Proof of lemma 5.2. For a fixed i ∈ {1, ..., K}, the partial derivatives of ϕ i are

2, ..., K -1 owing to Lemma 5.2-(a).

For i = 1,

For all M such that -M < x * 2 , f (-M )

). We also have ϕ 1 ( x * ,M1 ) > 0 by applying Lemma 5.2-(a). It follows that

The proof of L K (x * ) is similar by applying Lemma 5.2-(c). Thus H D (x * ) is positive definite owing to Gershgorin circle theorem. R d ). Let X, X 1 , ..., X n be i.i.d random variable with probability distribution µ. Then,

where the last line is due to the fact that X 1 , ..., X n have the same distribution as X. Moreover, we have

Multi-dimensional normal distribution is a special case of hyper-exponential tail distribution, i.e. if µ = N (m, Σ), we have κ = 2, ϑ = 1 2 and c = 0. By the same reasoning as before,

where C µ = 24• 1∨log 2E e |X| 2 /4 . When µ = N (0, I d ), C µ = 24(1+ d 2 )•log 2, since E e |X| 2 /4 = 2 d/2 by the moment-generating function of a χ 2 distribution.


PhD Camille Ouvrard
email: camille.ouvrard@u-bordeaux.fr

MD PhD b Claudine Berr

MSc Céline Meillon

PhD Céline Ribet

MD PhD Marcel Goldberg

MD PhD Marie Zins

PhD Hélène Amieva

Norms for standard neuropsychological tests from the French CONSTANCES cohort

Keywords: Norms, neuropsychological tests, older adults Online only: Figure S1; Tables S1, S2, S3, S4

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Neuropsychological assessment plays a key role in a variety of clinical contexts. Indeed, neuropsychological examination is essential in the diagnosis of dementia as well as in most neurological disorders such as stroke, traumatic brain injury or multiple sclerosis. Identification of low performances relies on the use of appropriate normative scores which allow clinicians determining whether a score is in the expected range. Such norms are available for a wide panel of tests for different age ranges. Nevertheless, the middle-aged group is rarely represented. Yet, dementia does not exclusively affect people over the age of 65. Early-onset dementia refers to people diagnosed with dementia under 65. A review relying on 11 studies estimated that between 6.9% to 45.3% of all patients diagnosed with dementia had early-onset dementia [START_REF] Vieira | Epidemiology of early-onset dementia: a review of the literature[END_REF].

In particular, frontotemporal dementia is diagnosed in 10% of cases before 45 years of age, and in 60% of cases between 45 and 64 [START_REF] Bang | Frontotemporal dementia[END_REF]. Furthermore, such normative data are very important to document cognitive decline that may be present long before dementia onset [START_REF] Amieva | Compensatory mechanisms in higher-educated subjects with Alzheimer's disease: a study of 20 years of cognitive decline[END_REF]. Beyond dementia, other neurological disorders may be experienced by middle aged adults. For instance, persistent cognitive deficits have been reported to affect over 20% of adults one year post-brain injury [START_REF] Barker-Collo | Neuropsychological outcome and its correlates in the first year after adult mild traumatic brain injury: A population-based New Zealand study[END_REF] and from 11% to 30% of adults one year poststroke [START_REF] Snaphaan | Poststroke memory function in nondemented patients: a systematic review on frequency and neuroimaging correlates[END_REF]. Despite this matter of fact, most studies having computed norms for neuropsychological tests did not include persons below 65 [START_REF] Lechevallier-Michel | Normative data for the MMSE, the Benton visual retention test, the Isaacs's set test, the digit symbol substitution test and the Zazzo's cancellation task in subjects over the age 70: results from the PAQUID Study[END_REF].

Two other major issues limit the relevance of numerous published norms. First, several studies rely on a relatively small number of participants [START_REF] Malek-Ahmadi | Age-and education-adjusted normative data for the Montreal Cognitive Assessment (MoCA) in older adults age 70-99[END_REF][START_REF] Periáñez | Trail Making Test in traumatic brain injury, schizophrenia, and normal ageing: sample comparisons and normative data[END_REF], which is a substantial limitation for a study aiming at addressing the inter-individual variability of cognitive performances in general population. The need of accounting for this huge variability is particularly outstanding in elderly population [START_REF] Wilson | Individual differences in rates of change in cognitive abilities of older persons[END_REF] or patients with neurological disorders [START_REF] Lazar | Variability in language recovery after first-time stroke[END_REF] for whom cognitive performances variability is greater than for any other group population.

Second, for a reliable clinical interpretation, the use of updated norms is essential. Indeed, cognitive abilities improve from one generation to the next [START_REF] Trahan | The Flynn effect: a meta-analysis[END_REF]. Such improvement is known as the "Flynn effect" and is explained by the general improvement of education and environmental conditions.

The lack of norms designed for the middle-aged population, the lack of up-to-date norms as well as norms gathered from small-sized samples lead clinicians to use inadequate norms and increase the risk of diagnosis misclassification.

The aim of this study was to compute updated and reliable norms for five teststhe Mini-

Methods

Study design

The methodology of the national CONSTANCES cohort has been detailed elsewhere [START_REF] Zins | The French CONSTANCES populationbased cohort: design, inclusion and follow-up[END_REF].

Since January 2012, 176,000 volunteers aged 18 and over had been randomly selected among S1, online only). Five age groups ([45-50[; [50-55[; [55-60[; [60-65[; [65-70]) and three educational levels: no diploma (<5 years of education); primary, vocational or high school degree (5-13 years of education); college degree (≥14 years of education) were considered.

Neuropsychological testing

The following tests were administered:

1.

MMSE [START_REF] Folstein | Mini-mental state ». A practical method for grading the cognitive state of patients for the clinician[END_REF] assessing general cognitive functioning (maximum score=30).

2.

FCSRT [START_REF] Grober | Screening for dementia by memory testing[END_REF] assesssing verbal episodic memory. After the learning phase consisting in memorizing 16 words belonging to 16 semantic categories, the volunteer is invited to recall as many words as possible in two minutes. The category for each word that has not been recalled is provided. This procedure is repeated three times. The delayed recall takes place after a 20minute delay. Four scores were considered: the three free recall trials (maximum score=48); the three free and cued recalls (maximum score=48); the delayed free recall (maximum score=16); and the delayed free and cued recall (maximum score=16) were considered.

3.

TMT [START_REF] Reitan | Validity of the Trail Making Test as an Indicator of Organic Brain Damage[END_REF] assessing executive functions. This test consists of 25 circles displayed on a sheet of paper. In part Aassessing attention and psychomotor speedthe circles are numbered from 1 to 25 and have to be to connected in ascending order by drawing a line. In part Bassessing planning abilities and mental flexibilitythe circles contain numbers or letters to be connected in ascending order by alternating numbers and letters. In case of error, correction is provided. The score was the time (in seconds) to complete the task.

4.

Fluency tasks assess language skills in particular, lexico-semantic access, and word search strategy. The tasks consisted in naming in one minute as many words as possible belonging to animals (semantic fluency) or starting by letter R (phonological fluency). The score was the number of generated words.

5.

DSST [START_REF] Wechsler | The Wechsler Adult Intelligence Scale-Revised[END_REF] assessing psychomotor speed. On the top of the page, a coding matrix contains the digits 1 to 9, each of them paired with a symbol. Underneath, a series of digits with a blank space for sketching the symbol is presented. The participant has to assign as many symbols as possible to the respective digits in 90 seconds. The score was the number of correct symbols.

In addition to cognitive testing, depressive symptomatology was assessed by the Center for Epidemiologic Studies Depression Scale [START_REF] Fuhrer | La version française de l'échelle CES-D (Center for Epidemiologic Studies-Depression Scale). Description et traduction de l'échelle d'autoévaluation[END_REF].

Statistical analysis

We first decribed sociodemographic and health caracteristics of participants. Age, gender, and education effect was assessed with univariate and multiple linear regression analysis to determine the stratification variables. The 5th, 10th, 25th, 50th, 75th, 90th and 95th percentiles stratified on age, gender, and education were reported. Statistical analyses were performed with R statistical software.

Results

The mean age of the whole study sample was 57.7 (SD=7.1). Of the 51,879 participants, 52%

were women, 3.4% had low education (<5 years), 49.2% had medium educational level (5-13 years) and 47.4% had higher education (≥14 years). The CESD mean score was 10.5 (SD=9)

(missing data=4790).

The results from the full multiple linear regression models revealed that age and education were associated with MMSE (full model: F(7, 51783)=924. Norms are presented in Tables 1 to 6. The 10th percentile corresponds to the threshold under which 10% of participants get the worst performances. The 75th percentile corresponds to the threshold above which 25% of participants get the best performances except for TMT (the lower score, the better performance).

MMSE was available for 51,791 participants (mean score=28.1, SD=1.9) (Table 1). As may be seen, scores vary little with age and gender and more substantially with education.

Free recall of the FCSRT was available for 50,566 participants (mean score=32.4, SD=5.6) (Table 2). Scores slitghly decrease with age and increase with level of education. Score are slitghly higher in women.

Total FCSRT recall score was available for 50,526 participants (mean score=46.3, SD=2.3) (Table S1, online only). There is a ceiling effect with a median score of 47 or 48 for the majority of groups.

Delayed FCSRT free recall was available for 50,465 participants (mean score=12.7, SD=2.

(Table S2, online only) and show the same trends as those observed for the free recall.

Total FCSRT delayed recall was available for 50,459 participants (mean score=15.7, SD=1.1) (Table S3, online only). There is a slight effect of education for all categories and an effect of gender for the less-educated category. Both effects are probably minimized by a ceiling effect.

TMT-A was available for 51,597 participants (mean score=34.4, SD=12.3) (Table S4, online only) and TMT-B for 51,267 participants (mean score=69.0, SD=31.7) (Table 3). There is an overall effect of education. Scores substantially decrease with age, in particular for TMT-B.

We also observe a gender effect, more pronounced for the TMT-B.

Semantic fluency was available for 51,361 participants (mean score=23.4, SD=6.0) (Table 4).

Scores appear to vary little with age and gender, and more substantially with education.

Phonological fluency was available for 51,195 participants (mean score=15.1, SD=4.8) (Table 5). We observe the same trends as on semantic fluency, with a stronger effect of gender.

DSST was available for 51,486 participants (mean score=65.4, SD=14.8) (Table 6). Scores decrease with age and increase with education and female gender.

Discussion

This study provides updated and reliable norms for five neuropsychological tests broadly used in clinical practice for adults aged 45-70.

First, the results illustrate the effect of age in almost all tests, and more particularly in tests involving speed processing, such as TMT and DSST, illustrating age-related speed processing decline considered as one the cardinal markers of normal cognitive aging [START_REF] Salthouse | The processing-speed theory of adult age differences in cognition[END_REF]. The impact of age can be observed including in tight age ranges categories (5 years) as may be seen for the DSST: median scores are of 52 and 57, respectively for men and women aged [50-55[, and of 48 and 53, respectively for men and women aged [55-60[. Therefore, it seems important to rely on narrow age categories, especially for time-based tests for which we observe scores differences for a 5-year interval.

The results also illustrate the well-known effect of education on neuropsychological performances [START_REF] Hall | Education delays accelerated decline on a memory test in persons who develop dementia[END_REF][START_REF] Carret | The effect of education on cognitive performances and its implication for the constitution of the cognitive reserve[END_REF][START_REF] Stern | Influence of education and occupation on the incidence of Alzheimer's disease[END_REF]. The link between cognition and education is one of the founding arguments of the concept of cognitive reserve [START_REF] Stern | What is cognitive reserve? Theory and research application of the reserve concept[END_REF] postulating that a high educational level confers to individuals greater compensatory capacities in dealing with age-related cognitive decline. Therefore providing education-stratified normative data is fundamental to properly assess individuals' performances.

The results also show an effect of gender, in particular for FCRST, phonological fluency and DSST. This finding is consistent with previous studies that have shown that women perform better on tests involving verbal material such as verbal fluency tasks [START_REF] Carret | The effect of education on cognitive performances and its implication for the constitution of the cognitive reserve[END_REF] or FCRST [START_REF] Grober | Screening for dementia by memory testing[END_REF][START_REF] Amieva | Cued and uncued memory tests: norms in elderly adults from the 3 Cities epidemiological study[END_REF] and in tests involving processing speed [START_REF] Roivainen | Gender Differences in Processing Speed: A Review of Recent Research[END_REF] such as DSST.

The main goal of this study was to take advantage of the large-sized CONSTANCES cohort to derive norms for the middle-aged population, providing precise age categories of five years, allowing apprehending the heterogeneity of cognitive performances [START_REF] Vieira | Epidemiology of early-onset dementia: a review of the literature[END_REF]. Adequate normative data are needed not only for older adults but also for middle-aged people who can experience various neurological disorders including early-onset dementia, multiple sclerosis, stroke or brain injury. Even though numerous normative data for people aged 45 and over have been published, most of them presented a small number of participants in that age category [START_REF] Roos | Normative data for the Tygerberg Cognitive Battery and Mini-Mental Status Examination in a South African population[END_REF][START_REF] Zimmermann | Brazilian norms and effects of age and education on the Hayling and Trail Making Tests[END_REF].

Moreover, some studies lacked of stratification on age and education [START_REF] Roos | Normative data for the Tygerberg Cognitive Battery and Mini-Mental Status Examination in a South African population[END_REF] or relied on very large age categories (i.e., 20 years) [START_REF] Zimmermann | Brazilian norms and effects of age and education on the Hayling and Trail Making Tests[END_REF]. Such studies may have too methodological limitations to properly address the inter-individual variability of cognitive performances in elderly but also middle-aged population. As an example, the present study showed that men aged 45 with a medium level of education (5-13 years of education) had a median free recall FCSRT score of 32 versus 28 for individuals aged 65 with the same education, that is, four words less.

This study has limitations. The norms provided in our study concern only five tests.

Nevertheless, these tests cover different cognitive functions and are commonly used in diagnostic process [START_REF] Grober | Temporal unfolding of declining episodic memory on the Free and Cued Selective Reminding Test in the predementia phase of Alzheimer's disease: Implications for clinical trials[END_REF][START_REF] Hensel | Does a reliable decline in Mini Mental State Examination total score predict dementia? Diagnostic accuracy of two reliable change indices[END_REF][START_REF] Holtzer | Within-person acrossneuropsychological test variability and incident dementia[END_REF]. Another limitation is the number of participants enrolled in the lowest category of education (i.e., 143 to 199 participants). However, this limitation can be mitigated in the context of the overall increase in educational attainment which explains the low number of people with less than five years of education.

The main strength of this study obviously relies on the very large sample allowing accessing a wide population variability in terms of age, sex and education.

To conclude, cognitive deficits are frequent in numerous neurological disorders which can affect not only older adults but also middle-aged ones. In order to track changes in cognitive functioning, tailor personalised rehabilitation strategies taking into account individual's cognitive ressources and weaknesses, neuropsychological testing is an essential tool which has to be interpreted in the light of reliable and updated norms. In this perspective, the normative scores provided in this study should help physicians and psychologists in their daily clinical practice.

(

  TMT) assessing executive functions, semantic and phonological fluency tasks assessing language and the Digit Symbol Substitution Test (DSST) assessing psychomotor speedin a population aged 45 to 70, from the large-sized CONSTANCES study.

French

  adults covered by the National health insurance. We considered only the participants aged 45 and over who received a standardized neuropsychological testing. CONSTANCES obtained the authorization of the National Data Protection Authority and was approved by the Institutional Review Board of the National Institute for Medical Research. All participants gave informed consent. Sample 68,285 participants aged 45 and over have been recruited. Participants aged over 70, those who did not provide information on education or did not complete the tests were excluded (see fig.

  2; p<.001) and semantic fluency (full model: F(7, 51353)=586.6; p<.001). Gender, age and education were associated with the free recall of the FCSRT (full model: F(7, 50558)=1486; p<.001), the total FCSRT recall (full model: F(7, 50518)=611.7; p<.001), the delayed FCSRT free recall (full model: F(7, 50457)=1058; p<.001), the total FCSRT delayed recall (full model: F(7, 50451)=244.7; p<.001), the TMT-A (full model: F(7, 51589)=800.2; p<.001) and TMT-B (full model: F(7, 51259)=1188; p<.001), phonological fluency (full model: F(7, 51187)=957.7; p<.001) and DSST (full model: F(7, 51478)=2161; p<.001).

  Mental State Examination (MMSE) assessing global cognitive functioning, the Free and Cued Selective Reminding Test (FCSRT) assessing verbal episodic memory, the Trail Making Test

Table 1

 1 Normative data stratified by age, gender and education for the MMSE, CONSTANCES cohort (n = 51,791).

					Education
	Age groups	No diploma a	Primary, vocational or high school degree b	College degree c
		Men	Women	Men	Women	Men	Women
	[45-50[
	N	187	140	1750	1790	2359	2967
	5%	21	21	24	24	26	26
	10%	22	22	25	25	27	27
	25%	24	24	27	27	28	28
	50%	27	26	28	28	29	29
	75%	28	28	29	29	30	30
	90%	29	29	30	30	30	30
	95%	30	29	30	30	30	30
	[50-55[
	N	194	225	2216	2301	2168	2827
	5%	21	22	24	24	26	26
	10%	23	22	25	25	27	27
	25%	25	24	27	27	28	28
	50%	27	26	28	28	29	29
	75%	29	28	29	29	30	30
	90%	29	29	30	30	30	30
	95%	30	30	30	30	30	30
	[55-60[
	N	199	195	2847	2756	2191	2766
	5%	21	20	24	24	26	26
	10%	22	21	25	25	27	27
	25%	24	24	27	27	28	28
	50%	27	26	28	28	29	29
	75%	28	28	29	29	30	30
	90%	29	29	30	30	30	30
	95%	30	30	30	30	30	30
	[60-65[
	N	172	142	3010	3051	2269	2423
	5%	21	20	24	24	26	26
	10%	21	21	25	25	27	27
	25%	24	23	27	27	28	28
	50%	25	25	28	28	29	29
	75%	27	28	29	29	30	30
	90%	29	29	30	30	30	30
	95%	29	29	30	30	30	30
	[65-70]
	N	168	155	2782	2969	2393	2179
	5%	20	21	24	24	26	26
	10%	21	22	25	25	27	27
	25%	24	24	27	27	28	28
	50%	26	26	28	28	29	29
	75%	28	28	29	29	30	30
	90%	29	29	30	30	30	30
	95%	30	29	30	30	30	30
	a <5 years of education.

b 5-13 years of education. c ≥14 years of education.

Table 2

 2 Normative data stratified by age, gender and education for the three free recalls of the FCSRT (number of words retrieved), CONSTANCES cohort (n = 50,566).

					Education
	Age		No diploma a	Primary, vocational or high	College degree c
	groups			school degree b
		Men	Women	Men	Women	Men	Women
	[45-50[
	N	165	123	1681	1754	2306	2922
	5%	20	21	23	26	26	29
	10%	22	26	25	28	28	31
	25%	26	29	28	31	31	33
	50%	30	33	32	34	34	37
	75%	34	35	35	37	37	39
	90%	37	37	38	40	40	42
	95%	38	38	39	41	41	43
	[50-55[
	N	171	212	2137	2254	2130	2795
	5%	18	22	22	25	25	28
	10%	20	25	24	28	27	30
	25%	25	28	28	31	30	33
	50%	30	32	31	34	34	36
	75%	33	35	34	37	37	39
	90%	37	38	37	39	40	41
	95%	38	40	39	41	41	42
	[55-60[
	N	171	172	2775	2726	2147	2722
	5%	19	21	21	24	24	27
	10%	21	24	23	27	26	29
	25%	24	27	27	30	29	32
	50%	28	31	30	33	33	35
	75%	31	34	34	37	36	38
	90%	35	38	37	39	39	40
	95%	37	39	38	41	40	42
	[60-65[
	N	148	124	2941	2984	2228	2391
	5%	16	18	20	24	22	26
	10%	19	21	22	25	24	28
	25%	22	25	26	29	28	31
	50%	27	30	29	32	31	35
	75%	30	33	33	36	35	38
	90%	34	36	36	38	38	40
	95%	35	38	37	40	39	42
	[65-70]
	N	143	148	2693	2911	2337	2155
	5%	16	20	19	22	21	25
	10%	19	22	21	25	23	27
	25%	22	25	24	28	27	30
	50%	26	29	28	32	31	34
	75%	31	32	32	35	34	37
	90%	34	36	34	38	37	39
	95%	36	38	36	39	39	41

a <5 years of education. b 5-13 years of education. c ≥14 years of education.

Table 4

 4 Normative data stratified by age, gender and education for the semantic fluency task (number of words provided in one minute), CONSTANCES cohort (n = 51,361).

					Education
	Age	No diploma a	Primary, vocational or high	College degree c
	groups			school degree b
		Men	Women	Men	Women	Men	Women
	[45-50[
	N	178	131	1722	1780	2345	2946
	5%	11	10	14	14	16	16
	10%	13	12	16	15	18	18
	25%	16	15	18	18	21	21
	50%	20	19	22	22	25	25
	75%	25	24	26	26	29	29
	90%	28	27	30	29	33	33
	95%	30	28	32	31	35	35
	[50-55[
	N	182	222	2190	2287	2153	2811
	5%	11	12	14	14	17	16
	10%	13	14	15	16	18	18
	25%	16	17	19	19	21	21
	50%	20	20	23	22	25	25
	75%	25	22	27	26	29	29
	90%	29	27	30	30	33	33
	95%	31	29	32	32	36	36
	[55-60[
	N	191	186	2826	2751	2180	2748
	5%	11	10	14	14	16	16
	10%	14	14	16	16	18	18
	25%	17	16	19	19	21	21
	50%	20	20	22	22	24	25
	75%	23	25	26	26	29	29
	90%	26	29	30	30	33	33
	95%	28	32	32	32	35	36
	[60-65[
	N	162	137	2981	3027	2260	2412
	5%	10	10	14	14	16	16
	10%	12	11	15	15	17	18
	25%	14	14	18	18	20	21
	50%	18	18	22	22	24	25
	75%	22	21	26	25	28	29
	90%	27	24	30	29	32	32
	95%	29	27	32	31	35	35
	[65-70]
	N	160	153	2752	2949	2372	2167
	5%	9	10	13	13	15	16
	10%	12	12	15	15	17	18
	25%	15	15	17	18	20	20
	50%	18	18	21	21	24	24
	75%	22	21	25	25	28	28
	90%	25	25	29	29	32	32
	95%	27	27	31	31	34	34
	a <5 years of education.

b 5-13 years of education. c ≥14 years of education.

Table 5

 5 Normative data stratified by age, gender and education for the phonological fluency task (number of words provided in one minute), CONSTANCES cohort (n = 51,195).

					Education
	Age		No diploma a	Primary, vocational or high	College degree c
	groups			school degree b
		Men	Women	Men	Women	Men	Women
	[45-50[
	N	174	127	1710	1771	2342	2941
	5%	5	5	7	8	9	10
	10%	6	6	8	9	11	11
	25%	8	9	11	12	13	14
	50%	11	12	14	14	16	17
	75%	15	15	17	17	19	20
	90%	17	18	20	20	22	23
	95%	22	19	22	22	24	25
	[50-55[
	N	177	220	2183	2282	2149	2807
	5%	5	5	7	8	9	10
	10%	6	6	8	9	11	12
	25%	8	9	11	12	13	14
	50%	11	11	14	15	16	17
	75%	14	15	17	18	19	20
	90%	17	17	20	21	22	23
	95%	20	20	21	23	24	25
	[55-60[
	N	188	180	2815	2748	2176	2743
	5%	4	5	6	7	9	10
	10%	5	6	8	9	11	12
	25%	8	9	10	12	13	14
	50%	10	12	13	15	16	17
	75%	14	15	16	18	19	20
	90%	17	19	19	21	23	23
	95%	19	20	21	23	24	25
	[60-65[
	N	153	129	2975	3019	2255	2407
	5%	3	4	6	7	9	10
	10%	4	5	8	8	10	11
	25%	6	7	10	11	13	14
	50%	9	9	13	14	16	17
	75%	12	13	16	17	19	20
	90%	16	16	19	20	22	23
	95%	18	17	21	22	24	25
	[65-70]
	N	153	149	2742	2948	2368	2164
	5%	3	4	6	7	9	10
	10%	4	5	7	8	10	11
	25%	6	7	10	11	12	14
	50%	8	10	13	14	15	16
	75%	12	13	16	17	18	19
	90%	15	16	19	20	21	22
	95%	19	20	20	21	23	24
	a <5

years of education. b 5-13 years of education. c ≥14 years of education.

Table S1

 S1 Normative data stratified by age, gender and education for the three free and cued recalls of the FCSRT (number of words retrieved), CONSTANCES cohort (n = 50,526).

					Education
	Age		No diploma a	Primary, vocational or high	College degree c
	groups			school degree b
		Men	Women	Men	Women	Men	Women
	[45-50[
	N	165	123	1681	1754	2305	2921
	5%	39	41	41	43	43	45
	10%	40	42	43	45	44	46
	25%	43	45	45	46	46	47
	50%	46	47	47	47	47	48
	75%	47	48	48	48	48	48
	90%	48	48	48	48	48	48
	95%	48	48	48	48	48	48
	[50-55[
	N	171	213	2134	2254	2128	2795
	5%	34	42	40	43	43	45
	10%	38	43	43	45	44	46
	25%	44	45	45	46	46	47
	50%	46	47	47	47	47	48
	75%	47	48	48	48	48	48
	90%	48	48	48	48	48	48
	95%	48	48	48	48	48	48
	[55-60[
	N	171	171	2773	2726	2145	2721
	5%	36	40	39	43	42	44
	10%	39	43	42	44	44	45
	25%	43	45	44	46	46	47
	50%	45	47	46	47	47	48
	75%	47	48	48	48	48	48
	90%	48	48	48	48	48	48
	95%	48	48	48	48	48	48
	[60-65[
	N	147	124	2938	2978	2225	2390
	5%	35	40	39	43	42	44
	10%	37	42	41	44	43	45
	25%	41	45	44	46	45	47
	50%	45	46	46	47	47	48
	75%	47	48	47	48	48	48
	90%	48	48	48	48	48	48
	95%	48	48	48	48	48	48
	[65-70]
	N	143	147	2689	2907	2335	2152
	5%	35	40	39	42	41	44
	10%	38	41	41	44	43	45
	25%	42	44	44	46	45	46
	50%	45	46	46	47	47	48
	75%	47	47	47	48	48	48
	90%	48	48	48	48	48	48
	95%	48	48	48	48	48	48
	a <5

years of education. b 5-13 years of education. c ≥14 years of education.

Table S2

 S2 Normative data stratified by age, gender and education for the delayed free recall of the FCSRT (number of words retrieved), CONSTANCES cohort (n = 50,465).

					Education
	Age	No diploma a	Primary, vocational or high	College degree c
	groups			school degree b
		Men	Women	Men	Women	Men	Women
	[45-50[
	N	165	121	1680	1753	2303	2918
	5%	8	8	9	10	10	11
	10%	8	10	10	11	11	12
	25%	10	11	11	12	12	13
	50%	12	13	13	14	14	14
	75%	13	14	14	15	15	15
	90%	14	15	15	16	15	16
	95%	15	16	15	16	16	16
	[50-55[
	N	170	213	2133	2252	2127	2792
	5%	6	9	8	10	9	11
	10%	7	9	9	11	11	12
	25%	10	11	11	12	12	13
	50%	12	12	12	14	13	14
	75%	13	14	14	15	15	15
	90%	14	15	15	15	15	16
	95%	15	15	15	16	16	16
	[55-60[
	N	171	171	2771	2722	2143	2718
	5%	6	8	8	9	9	10
	10%	7	9	9	10	10	11
	25%	9	11	11	12	12	13
	50%	11	13	12	13	13	14
	75%	13	14	14	14	14	15
	90%	14	15	15	15	15	16
	95%	15	15	15	16	16	16
	[60-65[
	N	147	124	2934	2971	2223	2388
	5%	6	8	7	9	8	10
	10%	7	8	8	10	9	11
	25%	9	10	10	11	11	12
	50%	10	12	12	13	13	14
	75%	13	14	13	14	14	15
	90%	14	15	14	15	15	16
	95%	15	15	15	16	15	16
	[65-70]
	N	143	147	2685	2902	2328	2150
	5%	6	7	7	8	8	9
	10%	6	9	8	9	9	10
	25%	8	10	10	11	11	12
	50%	11	12	11	13	12	13
	75%	13	13	13	14	14	15
	90%	14	14	14	15	15	15
	95%	15	15	15	15	15	16
	a <5 years of education.

b 5-13 years of education. c ≥14 years of education.

Table S3

 S3 Normative data stratified by age, gender and education for the delayed free and cued recall of the FCSRT (number of words retrieved), CONSTANCES cohort (n = 50,459).

					Education
	Age	No diploma a	Primary, vocational or high	College degree c
	groups			school degree b
		Men	Women	Men	Women	Men	Women
	[45-50[
	N	165	121	1679	1753	2303	2918
	5%	13	14	14	15	15	15
	10%	13	15	15	15	15	16
	25%	14	16	15	16	16	16
	50%	16	16	16	16	16	16
	75%	16	16	16	16	16	16
	90%	16	16	16	16	16	16
	95%	16	16	16	16	16	16
	[50-55[
	N	170	212	2132	2252	2127	2791
	5%	11	14	14	15	15	15
	10%	13	15	15	15	15	16
	25%	15	16	15	16	16	16
	50%	16	16	16	16	16	16
	75%	16	16	16	16	16	16
	90%	16	16	16	16	16	16
	95%	16	16	16	16	16	16
	[55-60[
	N	171	171	2771	2722	2142	2718
	5%	12	14	14	15	14	15
	10%	13	14	14	15	15	16
	25%	15	16	15	16	16	16
	50%	16	16	16	16	16	16
	75%	16	16	16	16	16	16
	90%	16	16	16	16	16	16
	95%	16	16	16	16	16	16
	[60-65[
	N	147	124	2933	2971	2223	2388
	5%	11	13	13	15	14	15
	10%	13	14	14	15	15	16
	25%	14	15	15	16	16	16
	50%	15	16	16	16	16	16
	75%	16	16	16	16	16	16
	90%	16	16	16	16	16	16
	95%	16	16	16	16	16	16
	[65-70]
	N	143	147	2685	2902	2328	2150
	5%	12	14	13	14	14	15
	10%	13	14	14	15	15	15
	25%	15	15	15	16	15	16
	50%	16	16	16	16	16	16
	75%	16	16	16	16	16	16
	90%	16	16	16	16	16	16
	95%	16	16	16	16	16	16
	a <5 years of education.

b 5-13 years of education. c ≥14 years of education.

Table S4

 S4 Normative data stratified by age, gender and education, for the TMT-A (time in seconds), CONSTANCES cohort (n = 51,597).

					Education
	Age	No diploma a	Primary, vocational or high	College degree c
	groups			school degree b
		Men	Women	Men	Women	Men	Women
	[45-50[
	N	187	138	1742	1789	2353	2963
	5%	19	19	18	18	16	17
	10%	22	22	20	20	18	19
	25%	26	26	24	23	21	22
	50%	33	33	29	29	26	27
	75%	45	45	37	36	32	33
	90%	58	57	45	45	40	40
	95%	73	67	53	52	46	45
	[50-55[
	N	190	223	2204	2296	2162	2829
	5%	20	20	19	19	18	18
	10%	24	23	21	21	19	20
	25%	28	28	26	25	23	24
	50%	36	34	31	31	28	29
	75%	46	45	39	38	34	36
	90%	59	58	49	47	42	43
	95%	67	74	55	54	48	49
	[55-60[
	N	194	189	2834	2759	2187	2758
	5%	22	22	20	20	18	19
	10%	26	24	22	22	21	21
	25%	31	29	27	27	25	25
	50%	38	37	33	32	31	31
	75%	49	51	41	40	38	38
	90%	66	69	51	49	46	45
	95%	74	78	58	55	52	52
	[60-65[
	N	163	139	2995	3041	2262	2417
	5%	26	26	22	22	21	21
	10%	28	29	25	23	23	23
	25%	34	35	29	28	27	27
	50%	44	43	36	34	33	33
	75%	54	53	45	42	41	41
	90%	68	66	55	52	48	49
	95%	79	82	62	60	55	56
	[65-70]
	N	164	152	2763	2956	2377	2171
	5%	28	23	24	23	23	22
	10%	30	27	27	25	25	24
	25%	36	35	32	30	29	29
	50%	44	43	38	37	36	35
	75%	55	56	47	45	44	43
	90%	76	68	59	56	54	52
	95%	85	84	68	63	61	58
	a <5 years of education.

b 5-13 years of education. c ≥14 years of education.

Acknowledgments

CONSTANCES is an "Infrastructure nationale en Biologie et Santé". It is supported by the "Caisse nationale d'assurance maladie des travailleurs salariés", a grant from ANR (ANR-11-INBS-0002) and is also partly funded by MSD, AstraZeneca and Lundbeck.

We thank the "Caisse nationale d'assurance maladie des travailleurs salariés", the "Centres d'examens de santé" of the French Social Security, the "Caisse nationale d'assurance vieillesse", ClinSearch, Asqualab and Eurocell.

Disclosure of conflict of interest

None.

Study approval

All confidentiality, safety and security procedures were approved by the French legal

Supporting Information

Additional Supporting Information may be found in the online version of this article:

Table and figure captions