Jianze Li
email: lijianze@gmail.com

Konstantin Usevich
email: konstantin.usevich@univ-lorraine.fr

Pierre Comon
email: pierre.comon@gipsa-lab.fr

On approximate diagonalization of third order symmetric tensors by orthogonal transformations

Keywords: symmetric tensors, orthogonally decomposable tensors, tensor diagonalization, Jacobi-type algorithms, maximally diagonal tensors 2010 MSC: 15A69, 65F99, 90C30

In this paper, we study the orthogonal diagonalization problem of third order symmetric tensors. We define several classes of approximately diagonal tensors, including the ones corresponding to stationary points of the problem. We study relationships between these classes, and other well-known objects, such as tensor eigenvalues and eigenvectors. We also prove a number of convergence properties for the cyclic Jacobi (or Jacobi CoM) algorithm.

Introduction

Arrays with more than two indices have become more and more important in the last two decades because of their usefulness in various fields, including signal processing, numerical linear algebra and data analysis [START_REF] Cichocki | Tensor decompositions for signal processing applications: From two-way to multiway component analysis[END_REF][START_REF] Comon | Tensors: a brief introduction[END_REF][START_REF]Handbook of Blind Source Separation[END_REF][START_REF] Kolda | Tensor decompositions and applications[END_REF][START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF]. Admitting a common abuse of language, we shall refer to them as tensors, being understood that we are considering associated multilinear forms (and hence fully contravariant tensors) [START_REF] Comon | Tensors: a brief introduction[END_REF]. Real symmetric matrices can be diagonalized by orthogonal transformation, which is a key property leading to spectral decomposition. On the other hand, the orthogonal diagonalization of symmetric tensors has also been addressed, as an exact decomposition in [START_REF] Robeva | Orthogonal decomposition of symmetric tensors[END_REF][START_REF] Kolda | Orthogonal tensor decompositions[END_REF][START_REF] Kolda | Symmetric orthogonal tensor decomposition is trivial[END_REF], or as a low-rank approximation in [START_REF] Comon | Independent component analysis, a new concept ?[END_REF]. In fact, approximate orthogonal diagonalization of third and fourth order cumulant tensors was at the core of Independent Component Analysis [START_REF] Comon | Independent Component Analysis[END_REF][START_REF] Comon | Independent component analysis, a new concept ?[END_REF][START_REF] Comon | Tensor Diagonalization, A useful Tool in Signal Processing[END_REF], and finds many applications [START_REF]Handbook of Blind Source Separation[END_REF]. However, the latter problem is much more difficult than the spectral decomposition of symmetric matrices since it is well known that not every symmetric tensor can be diagonalized by orthogonal transformation [START_REF] Robeva | Orthogonal decomposition of symmetric tensors[END_REF][START_REF] Kolda | Orthogonal tensor decompositions[END_REF].

Notation. Let R n×n×n def = R n ⊗ R n ⊗ R n be the linear space of third order real tensors and S n ⊆ R n×n×n be the set of symmetric ones, whose entries do not change under any permutation of indices [START_REF] Comon | Symmetric tensors and symmetric tensor rank[END_REF][START_REF] Qi | Tensor analysis: Spectral theory and special tensors[END_REF]. Let O n ⊆ R n×n be the orthogonal group. Let SO n ⊆ R n×n be the special orthogonal group, that is, the set of orthogonal matrices with determinant 1. We denote by • the Frobenius norm of a tensor or a matrix, or the Euclidean norm of a vector. Tensor arrays, matrices, and vectors, will be respectively denoted by bold calligraphic letters, e.g. A, with bold uppercase letters, e.g. M , and with bold lowercase letters, e.g. u; corresponding entries will be denoted by A ijk , M ij , and u i . Operator • p denotes contraction on the pth index of a tensor; when contracted with a matrix, it is understood that summation is always performed on the second index of the matrix. For instance, [A

• 1 M ] ijk =
A jk M i . When contraction is performed on vectors, the subscript p can be omitted. For A ∈ S n and a fixed set of indices {i, j}, 1 ≤ i < j ≤ n, we denote by A (i,j) the 2-dimensional subtensor obtained from A by allowing its indices to vary in {i, j} only. Similarly for 1 ≤ i < j < k ≤ n, we denote by A (i,j,k) the 3-dimensional subtensor obtained by allowing indices of A to vary in {i, j, k} only. The identity matrix of size n is denoted by I n , and its columns by e i , 1 ≤ i ≤ n, which form the canonical orthonormal basis.

Contribution. We formulate the approximate orthogonal symmetric tensor diagonalization problem as the maximization of diagonal terms [START_REF] Comon | Tensor diagonalization by orthogonal transforms[END_REF]. More precisely, let A ∈ S n , Q ∈ SO n , and

W = A • 1 Q T • 2 Q T • 3 Q T .
This problem is to find

Q * = argmax Q∈SOn f (Q), (1)
where

f (Q) def = diag{W} 2 = n i=1 W 2 iii . (2)
Methods based on Jacobi rotations (e.g., the well-known Jacobi CoM algorithm [START_REF] Comon | Independent Component Analysis[END_REF][START_REF] Comon | Independent component analysis, a new concept ?[END_REF][START_REF] Comon | Tensor Diagonalization, A useful Tool in Signal Processing[END_REF]) are widely used in practice [START_REF]Handbook of Blind Source Separation[END_REF][START_REF] Martin | A jacobi-type method for computing orthogonal tensor decompositions[END_REF] to solve problem [START_REF] Cichocki | Tensor decompositions for signal processing applications: From two-way to multiway component analysis[END_REF]. These methods aim at making a symmetric tensor as diagonal as possible by successive Jacobi rotations. They are particularly attractive due to the low computational cost of iterations. Other popular methods include Riemannian optimization methods [START_REF] Absil | Optimization algorithms on matrix manifolds[END_REF] that alternate between descent steps and retractions.

The above methods are typically known to converge (globally or locally) to stationary points [START_REF] Absil | Optimization algorithms on matrix manifolds[END_REF][START_REF] Li | Globally convergent jacobi-type algorithms for simultaneous orthogonal symmetric tensor diagonalization[END_REF], though the convergence of the original Jacobi CoM method has not been studied.

The main goal of this paper is to quantify the notion of approximate diagonality, by introducing several classes of approximately diagonal tensors and studying relationships between them. These classes include stationary diagonal tensors, Jacobi diagonal tensors, locally maximally diagonal tensors, maximally diagonal tensors, generally maximally diagonal tensors and pseudo diagonal tensors. We characterize (i) the class of Jacobi diagonal tensors by the stationary diagonal ratio, and (ii) the orbit of pseudo diagonal tensors by Z-eigenvalue and Z-eigenvectors. Moreover, we study (iii) the class of locally maximally diagonal tensors based on Riemannian Hessian. We show that this class is not equal to the class of Jacobi diagonal tensors, and thus Jacobi-type algorithms may converge to a saddle point of [START_REF] Comon | Tensors: a brief introduction[END_REF]. We also study (iv) whether a symmetric tensor is maximally diagonal if and only if it is generally maximally diagonal. Several problems related to low rank orthogonal approximation are proved to be equivalent to the fact that these two classes are equal when the dimension is greater than 2. We present a counterexample to these equivalent problems based on the decomposition of orthogonal matrices. Moreover, we prove a result that can be seen as an orthogonal analogue of the so-called Comon's Conjecture [START_REF] Zhang | Comon's conjecture, rank decomposition, and symmetric rank decomposition of symmetric tensors[END_REF]. The second goal is to study the convergence properties of the Jacobi CoM algorithm [START_REF] Comon | Tensor Diagonalization, A useful Tool in Signal Processing[END_REF].

Organization. The paper is organized as follows. In section 2, we recall basic properties of the cost function, introduce notation for derivatives, and present the scheme of Jacobi-type algorithms. In section 3, we define the classes of approximately diagonal tensors, considered in this paper. Some basic relationships between these classes are shown. The stationary diagonal ratio is introduced, and the orbit of pseudo diagonal tensors is studied. In section 4, we study the class of locally maximally diagonal tensors using Riemannian Hessian. In section 5, we study the relationship between maximally diagonal tensors and generally maximally diagonal tensors. Section 6 contains results on convergence of the Jacobi CoM algorithm. Finally, Appendix A contains long proofs.

Optimization problem: properties and algorithms

Riemannian gradient and stationary points

First, we recall that the Riemannian gradient of (2

) [17, §4.1], is, by definition, Proj∇ f (Q) = QΛ(Q), (3)
where Λ(Q) is the matrix with entries

Λ(Q) k,l = 3(W lll W llk -W lkk W kkk ). (4)
The matrix Q is a stationary point of (2) if and only if Proj∇ f (Q) = 0. A local maximum point of (2), of course, is a stationary point. A reasonable local optimization algorithm should at least converge to a stationary point.

Elementary rotations and Jacobi-type algorithms

Let (i, j) be a pair of indices with 1 ≤ i < j ≤ n. We denote the Givens rotation (by an angle θ ∈ R) matrix to be

G (i,j,θ) =              1 . . . 0 cos θ -sin θ . . . sin θ cos θ 0 . . . 1              ,
i.e., the matrix defined by

(G (i,j,θ) ) k,l =                1, k = l, k ∈ {i, j}, cos θ, k = l, k ∈ {i, j}, sin θ, (k, l) = (j, i), -sin θ, (k, l) = (i, j), 0, otherwise for 1 ≤ k, l ≤ n.
Jacobi-type algorithms proceed by successive optimization of the cost function with respect to elementary rotations, summarized in the following scheme.

Algorithm 1. Input: A ∈ S n and Q 0 = I n . Output: a sequence of iterations {Q k : k ∈ N}.

• For k = 1, 2, . . . until a stopping criterion is satisfied do:

-

Choose the pair (i k , j k ) according to a certain pair selection rule. -Compute the angle θ * k that maximizes the function

h k (θ) def = f (Q k-1 G (i k ,j k ,θ) ). (5)
- Update Q k = Q k-1 G (i k ,j k ,θ * k ) .
• End for

The algorithm is similar in spirit to block-coordinate descent. Important differences are: the coordinate system is changing at every iteration, and, for each elementary rotation, the global maximum is achieved. Recently, local and global convergence to stationary points [START_REF] Ishteva | Jacobi algorithm for the best low multilinear rank approximation of symmetric tensors[END_REF][START_REF] Li | Globally convergent jacobi-type algorithms for simultaneous orthogonal symmetric tensor diagonalization[END_REF] have been established for variants of Algorithm 1. Apart from Jacobi-type algorithms, Jacobi rotations are also very useful in computing fast retractions [16, p. 58] in Riemannian optimisation methods [START_REF] Absil | Optimization algorithms on matrix manifolds[END_REF].

Directional derivatives

First we introduce some useful notation that will be used throughout the paper.

Definition 2.1. Let A ∈ S n and 1 ≤ i < j ≤ n. Define

d i,j (A) def = A iii A iij -A ijj A jjj , ω i,j (A) def = A 2 iii + A 2 jjj -3A 2 iij -3A 2 ijj -2A iii A ijj -2A iij A jjj .
In order to simplify notation, we will consider the analogues of univariate functions [START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF] with

Q = I n : hi,j (θ) def = diag{A • 1 (G (i,j,θ) ) T • 2 (G (i,j,θ) ) T • 3 (G (i,j,θ) ) T } 2
for 1 ≤ i < j ≤ n as in [START_REF] Sidiropoulos | Tensor decomposition for signal processing and machine learning[END_REF].

Then it holds that [17, Lemma 5.7] h i,j (0) = 6d i,j (A) and h i,j (0) = -6ω i,j (A).

3. Classes of approximately diagonal tensors

Definitions of classes

In this subsection, we define several classes of third order symmetric tensors. Some of them are related to stationary points of problem [START_REF] Cichocki | Tensor decompositions for signal processing applications: From two-way to multiway component analysis[END_REF] or the points where Algorithm 1 may stop. For simplification, we look at the derivatives of (1) at Q = I n . Definition 3.1. (i) Let A, B ∈ S n . Then A is orthogonally similar [START_REF] Qi | Tensor analysis: Spectral theory and special tensors[END_REF][START_REF] Qi | Z-eigenvalue methods for a global polynomial optimization problem[END_REF]

to B if there exists Q ∈ O n such that B = A • 1 Q • 2 Q • 3 Q.
(ii) Let C ⊆ S n be a subset. Define the orbit 1 of C to be:

O(C) def = {A • 1 Q • 2 Q • 3 Q, A ∈ C, Q ∈ O n }.
Definition 3.2. We denote by D = D n the set of diagonal tensors in S n , and O(D) the set of orthogonally decomposable tensors ( referred to as "odeco" in [START_REF] Robeva | Orthogonal decomposition of symmetric tensors[END_REF]). More precisely, any A ∈ O(D) can be decomposed as

A = n k=1 λ k u k ⊗ u k ⊗ u k where λ k ∈ R and u 1 , • • • u n ∈ R n form an orthonormal basis.
Remark 3.3. In the definitions of tensor classes we will often drop the subscript, if the dimension is clear from the context. Definition 3.4. Let A ∈ S n . The class of pseudo diagonal tensors is defined to be

PD = PD n def = {A : A ijj = A iij = 0, for any 1 ≤ i < j ≤ n}. Remark 3.5. It is clear that D ⊆ PD and O(D) ⊆ O(PD).
In section 3.4, we will give characterizations of PD and O(PD) from the perspective of tensor spectral theory. Besides, it is well known that O(D) S n , that is, not every symmetric tensor can be diagonalized by orthogonal transformation [START_REF] Kolda | Orthogonal tensor decompositions[END_REF][START_REF] Robeva | Orthogonal decomposition of symmetric tensors[END_REF].

Definition 3.6. Let A ∈ S n . (i) The class of stationary diagonal tensors is defined to be SD = SD n def = {A : d i,j (A) = 0, for any 1 ≤ i < j ≤ n}.
1 Classically, the notion of orbit is defined for a single element (e.g., C ∈ Sn). In this paper, we use the word "orbit" as a shorthand for saying "the action of O on C".

(ii) The class of Jacobi diagonal tensors is defined to be

JD = JD n def = {A : 0 ∈ argmax θ∈R hi,j (θ), for any 1 ≤ i < j ≤ n}.
(iii) The class of locally Jacobi diagonal tensors is defined to be

LJD = LJD n def = {A : 0 is a local maximum point of hi,j (θ), for any 1 ≤ i < j ≤ n}. Remark 3.7. From (4), it follows that A ∈ SD if and only if Proj∇ f (I n ) = 0 in (3).
In other words, A ∈ SD if and only if I n is a stationary point of (2). This is the reason why we call the tensors in SD stationary diagonal. Moreover, it can be seen that Algorithm 1 stops at A if A ∈ JD. This is the reason why we call the tensors in JD Jacobi diagonal. [START_REF] Robeva | Orthogonal decomposition of symmetric tensors[END_REF]. Now we prove that (iii)⇒(i). We have

Lemma 3.8. Let A ∈ S n . The following are equivalent. (i) A ∈ JD. (ii) A ∈ LJD. (iii) d i,j (A) = 0 and ω i,j (A) ≥ 0 for any 1 ≤ i < j ≤ n. Proof. (i)⇒(ii) is clear. (ii)⇒(iii) follows from
h i,j (θ) -h i,j (0) = 3 (1 + x 2 ) 2 (2d i,j (A)(x -x 3 ) -ω i,j (A)x 2 ) (7)
for any 1 ≤ i < j ≤ n by [START_REF] Martin | A jacobi-type method for computing orthogonal tensor decompositions[END_REF]. Note that

h i,j (θ) -h i,j (0) ≡ 0 if d i,j (A) = ω i,j (A) = 0. If d i,j (
A) = 0 and ω i,j (A) ≥ 0, then h i,j (θ) reaches its maximum value at θ = 0, by [START_REF] Kolda | Orthogonal tensor decompositions[END_REF]. It follows that A ∈ JD.

Definition 3.9. Let A ∈ S n and f be as in [START_REF] Comon | Tensors: a brief introduction[END_REF].

(i) The class of maximally diagonal tensors is defined to be

MD = MD n def = {A : I n ∈ argmax Q∈SOn f (Q)}.
(ii) The class of locally maximally diagonal tensors is defined to be

LMD = LMD n def = {A : I n is a local maximum point of f (Q)}.
(iii) The class of generally maximally diagonal tensors is defined to be

GMD = GMD n def = {A : (I n , I n , I n ) ∈ argmax P ,Q,R∈SOn F(P , Q, R)},
where

F(P , Q, R) def = diag{A • 1 P T • 2 Q T • 3 R T } 2 . ( 8
) Remark 3.10. Note that O n ⊆ R n×n is a compact submanifold and (2) is continuous. Since (2) takes the same maximum on O n and SO n , we get that O(MD n ) = S n . Note that MD n ⊆ LMD n . It follows that O(LMD n ) = S n .
In other words, for any A ∈ S n , there exists

Q * and Q * * in SO n such that A • 1 Q T * • 2 Q T * • 3 Q T * ∈ LMD and A • 1 Q T * * • 2 Q T * * • 3 Q T * * ∈ MD,
respectively. How to find Q * or Q * * is the goal of problem (1).

Basic relations

The tensor classes defined in section 3.1 have the following relationships. The first row and column denote the corresponding orbits, i.e. arrows stand for the action of O n .

O(D) O(PD) S n O(D) D PD JD O(GMD) GMD JD SD S n S n MD LMD LJD ⊆ ⊆ ⊆ = ⊆ = ⊆ ⊆
Remark 3.11. Most of the above relationships are easy to get by Definition 3.6 and Definition 3.9. We only derive some of them for S 2 , which are not obvious.

(i) Note that SO 2 coincides with the set of Jacobi rotations. We see that

MD 2 = LMD 2 = JD 2 = LJD 2
by Lemma 3.8. It will be shown that GMD 2 = MD 2 in Theorem 5.3. It follows that

GMD 2 = MD 2 = LMD 2 = JD 2 = LJD 2 .
(ii) PD and JD will be characterized in Remark 3.13 and Theorem 3.14. It follows by these characterizations that PD 2 JD 2 .

(iii) Note that D 2 = PD 2 . It follows by (i) and (ii) that

D 2 GMD 2 .
(iv) By Theorem 3.14, we see that JD 2 SD 2 .

(v) Note that D 2 = PD 2 and O(D 2 ) S 2 by Remark 3.5. We have that O(PD 2 ) S 2 .

Stationary diagonal ratio

In this subsection, we define the stationary diagonal ratio for the tensors in SD, which can be used to characterize JD and PD. Definition 3.12. Let A ∈ SD and 1 ≤ i < j ≤ n. The stationary diagonal ratio, denoted by γ ij , is defined as follows.

γ ij def = 0, if A (i,j) = 0; ∞, if A iii = A jjj = 0 and A 2 ijj + A 2 iij = 0;
otherwise, γ ij is the (unique) number such that

A ijj A iij = γ ij A iii A jjj . Remark 3.13. Let A ∈ SD. Then A ∈ PD if and only if γ ij = 0 for any 1 ≤ i < j ≤ n. Theorem 3.14. Let A ∈ SD. Then A ∈ JD if and only if γ ij ∈ [-1, 1/3] for any 1 ≤ i < j ≤ n.
Proof. Note that A ∈ JD if and only if d i,j (A) = 0 and ω i,j (A) ≥ 0 for any 1 ≤ i < j ≤ n by Lemma 3.8. We only need to show that ω i,j (A) ≥ 0 if and only if

γ ij ∈ [-1, 1/3]. If γ ij = ∞, then ω i,j (A) < 0. If γ ij < ∞, by Definition 3.
12, we have that

-ω i,j (A) = (3γ 2 ij + 2γ ij -1)(A 2 iii + A 2 jjj ).
It follows that ω i,j (A) ≥ 0 if and only if

γ ij ∈ [-1, 1/3].

Orbit of the pseudo diagonal tensors 3.4.1. Characterization

In this subsection, we characterize the equivalence class of pseudo diagonal tensors based on the Z-eigenvalue and Z-eigenvectors defined in [START_REF] Qi | Tensor analysis: Spectral theory and special tensors[END_REF]. [START_REF] Qi | Tensor analysis: Spectral theory and special tensors[END_REF] of A. This vector is called the Z-eigenvector associated with λ.

Definition 3.15. Let A ∈ S n and λ ∈ R. If λ satisfies A • u • u = λ u for a unit vector u ∈ R n . Then λ is called a Z-eigenvalue
A = B • 1 Q T • 2 Q T • 3 Q T and A • u • u = λ u for λ ∈ R and a unit vector u ∈ R n , then B • (Qu) • (Qu) = λ Qu.
Theorem 3.17. Let A ∈ S n . We have two necessary and sufficient conditions below:

(i) A ∈ PD if and only if {e i : 1 ≤ i ≤ n} is a set of Z-eigenvectors. This is equivalent to that A • e i • e i • e j = 0
for any 1 ≤ i = j ≤ n.

(ii) A ∈ O(PD) if and only if there exists an orthonormal set of Z-eigenvectors {u i : 1 ≤ i ≤ n}. This is equivalent to that

A • u i • u i • u j = 0 for any 1 ≤ i = j ≤ n. In this case, A • 1 Q T * • 2 Q T * • 3 Q T * ∈ PD for Q * = [u 1 , • • • , u n ]. Proof. (i) By definition, A ∈ PD if and only if A • e i • e i • e j = 0 for any 1 ≤ i = j ≤ n. But if A • e i • e i is orthogonal
to every e j , j = i, it must be collinear to e i , which means:

A • e i • e i = λe i
for some nonzero λ, which turns out to be λ = A • e i • e i • e i .

(ii) The second result follows from (i) and Remark 3.16.

A • e i • e j ∈ span{e i , e j }
for any

1 ≤ i = j ≤ n. (ii) Let A ∈ O(PD n ).
Let {u i : 1 ≤ i ≤ n} be the set of orthonormal Z-eigenvectors, proved to exist in Theorem 3.17 (ii). Then A ∈ O(D n ) if and only if

A • u i • u j ∈ span{u i , u j } for any 1 ≤ i = j ≤ n.
Proof. First note that A • e i • e j ∈ span{e i , e j } for any 1 ≤ i < j ≤ n if and only if A ijk = 0 for any 1 ≤ i < j < k ≤ n. Then (i) is proved. Next, (ii) follows from (i) and Remark 3.16.

Locally maximally diagonal tensors

Even if Givens rotations span SO n , it is not obvious that a sequence of optimally chosen Givens rotations will find the optimal orthogonal transform in SO n . In other words, we know that LMD ⊆ LJD, but the converse may not be true. This motivates the comparison between LJD and LMD.

Riemannian Hessian

In this subsection, we study the conditions that a tensor in S n is locally maximally diagonal based on the Riemannian Hessian [START_REF] Absil | Optimization algorithms on matrix manifolds[END_REF][START_REF] Absil | An Extrinsic Look at the Riemannian Hessian[END_REF][START_REF] Edelman | The geometry of algorithms with orthogonality constraints[END_REF].

Lemma 4.1. Let A ∈ S n and f be as in (2). Let T Q O n be the tangent vector space at Q; it contains matrices of the form Q∆, where ∆ are skew matrices satisfying ∆ T = -∆. Next, define:

U = A • 3 Q T , V = A • 2 Q T • 3 Q T , X = V • 1 (Q∆) T , Y = U • 1 (Q∆) T • 2 (Q∆) T , Z = V • 1 (Q∆ 2 ) T .
Let Hessf (Q) be the Riemannian Hessian of

f at Q. Then Hessf (Q)(∆ 1 , ∆ 2 ) is a bilinear form defined on T Q O n .
We have:

Hessf (Q)(Q∆, Q∆) = 6 j (3X 2 jjj + 2Y jjj W jjj -Z jjj W jjj ).
Proof. By [23, eqn. (2.55)], it can be calculated that

Hessf (Q)(Q∆, Q∆) = i,j,k,l ∂ 2 f ∂Q i,j ∂Q k,l (Q∆) ij (Q∆) kl + 1 2 tr((∇f (Q)) T Q∆ 2 + ∆(∇f (Q)) T Q∆) = 6 i,j,k (3V ijj V kjj + 2W jjj U ikj )(Q∆) ij (Q∆) kj + tr((∇f (Q)) T Q∆ 2 ) = 6 i,j,k (3V ijj V kjj + 2W jjj U ikj )(Q∆) ij (Q∆) kj -6 i,j,k,l Q ik V ijj W jjj ∆ kl ∆ jl (9) = 6 j (3X 2 jjj + 2Y jjj W jjj + Z jjj W jjj ).
Hessf (I n )(∆, ∆) = 6 i,j,k (3A ijj A kjj + 2A jjj A ikj )∆ ij ∆ kj -6 i,j,k A kii A iii ∆ ij ∆ kj = 6 i,j (3A 2 ijj + 2A jjj A iij -A 2 iii )∆ 2 ij + 6 i,j,k,k =i (3A ijj A kjj + 2A jjj A ikj -A kii A iii )∆ ij ∆ kj Remark 4.3. Let A ∈ JD n . (i) If Hessf (I n )(∆, ∆) < 0 for any ∆ ∈ T In O n \ {0}, then A ∈ LMD n . (ii) If A ∈ LMD n , then Hessf (I n )(∆, ∆) ≤ 0 for any ∆ ∈ T In O n .

Euclidean Hessian matrix for S 3

Note that LMD ⊆ JD and LMD is corresponding to the local maximum point of (2). In this subsection, based on Corollary 4.2, we show how to determine whether A ∈ JD 3 is locally maximally diagonal or not.

+ 2γ 23 -1)(b 2 + c 2 ).   Theorem 4.5. Let A ∈ JD 3 . If M A is negative definite, then A ∈ LMD. If A ∈ LMD, then M A is negative semidefinite. Proof. Let ∆ =   0 u v -u 0 w -v -w 0   ∈ T I3 O 3 .
Define Φ(u, v, w) def = Hessf (I 3 )(∆, ∆). By Corollary 4.2, we have that

Φ(u, v, w) = 6[(3γ 2 12 + 2γ 12 -1)(a 2 + b 2 )u 2 + (3γ 2 13 + 2γ 13 -1)(c 2 + a 2 )v 2 + (3γ 2 23 + 2γ 23 -1)(b 2 + c 2 )w 2 + 4g(auv + cvw -bwu) + 6(γ 12 γ 13 bcuv + γ 13 γ 23 abvw -γ 23 γ 12 cawu) -2(γ 23 bcuv + γ 12 abvw -γ 13 cawu)] = 6 ξ T M A ξ, (10)
where ξ = (u, v, w) T . By Remark 4.3, the proof is complete.

Example 4.6. Let A ∈ JD 3 be such that

A 111 = A 222 = A 333 = 1, γ 12 = γ 13 = γ 23 = γ. (i) Then Φ(u, v, w) = 12(u 2 + v 2 + w 2 ) (3γ 2 + 2γ -1) -(3γ 2 -γ + 2A 123 ) uw -uv -vw u 2 + v 2 + w 2
for any (u, v, w) ∈ R 3 \{(0, 0, 0)} by [START_REF] Comon | Independent Component Analysis[END_REF]. Note that

uw -uv -vw u 2 + v 2 + w 2 ∈ [-1/2, 1].
Since 3γ 2 + 2γ -1 ≤ 0 by Theorem 3.14, it follows that A ∈ LMD if

3 2 γ - 1 2 < A 123 < - 9 2 γ 2 - 3 2 γ + 1.
Moreover, we have that A / ∈ LMD, if

A 123 < 3 2 γ - 1 2 or A 123 > - 9 2 γ 2 - 3 2 γ + 1. (ii) If γ = 0, then Φ(u, v, w) = -12(u 2 + v 2 + w 2 )[1 + 2A 123 uw -uv -vw u 2 + v 2 + w 2 ].
for any (u, v, w) ∈ R 3 \{(0, 0, 0)}. Note that

uw -uv -vw u 2 + v 2 + w 2 ∈ [-1/2, 1]. It follows that A ∈ LMD if A 123 ∈ (-1/2, 1). Moreover, if A 123 / ∈ [-1/2, 1], then A / ∈ LMD.
Example 4.7. Let A ∈ S n with n > 3. Suppose that

A (i,j,k) ∈ LMD 3
for any 1 ≤ i < j < k ≤ n. It may be interesting to wonder whether it holds that A ∈ LMD n .

In fact, the answer is negative. Let A ∈ PD 4 with

A ijk =      1, i = j = k, 3/4, i = j = k, 0, otherwise.
By Example 4.6 (ii), we see that

A (i,j,k) ∈ LMD 3 for any 1 ≤ i < j < k ≤ n. Let ∆ * =     0 1 1 1 -1 0 0 0 -1 0 0 0 -1 0 0 0     ∈ T I4 O 4 .
By Corollary 4.2, we get that Hessf (I 4 )(∆ * , ∆ * ) = 18 > 0. It follows that A / ∈ LMD 4 .

Orbit of generally maximally diagonal tensors

Equivalent problem formulations

In this subsection, we first prove that the statement O(GMD) = S n is equivalent to several other optimization problems in Proposition 5.2. Then we give a positive answer to these equivalent problems when the dimension is 2 in Theorem 5.3. Proof. We only have to prove that

GMD n = MD n if O(GMD n ) = S n . In fact, if A ∈ MD n , there exists Q * such that A • 1 Q T * • 2 Q T * • 3 Q T * ∈ GMD n .
Let f be as in [START_REF] Comon | Tensors: a brief introduction[END_REF] and F be as in [START_REF] Kolda | Symmetric orthogonal tensor decomposition is trivial[END_REF]. It follows that

f (I n ) ≥ f (Q * ) = max P ,Q,R∈SOn F(P , Q, R) ≥ f (I n ).
Then we have that A ∈ GMD n .

Proposition 5.2. Denote

GO(D)

def = {A • 1 P T • 2 Q T • 3 R T , A ∈ D, P , Q, R ∈ O n }.
The following statements are equivalent.

(i) GMD n = MD n . (ii) For any A ∈ S n , max Q∈SOn f (Q) = max P ,Q,R∈SOn F(P , Q, R),
where f be as in (2) and F be as in [START_REF] Kolda | Symmetric orthogonal tensor decomposition is trivial[END_REF].

(iii) For any A ∈ S n , it holds that

min ui⊥uj ,∀i =j, µ k ∈R A - n k=1 µ k u k ⊗ u k ⊗ u k = min xi⊥xj ,y i ⊥y j , zi⊥zj ,∀i =j,λ k ∈R A - n k=1 λ k x k ⊗ y k ⊗ z k . ( 11
)
(iv) For any A ∈ S n and for the Euclidean distance d, it holds that

d(A, O(D)) = d(A, GO(D)).
(v) Let A ∈ S n . The best rank-n orthogonal approximation can always be chosen to be symmetric, that is, there exist µ k ∈ R and orthonormal basis {u k , 1 ≤ k ≤ n} such that

A - n k=1 µ k u k ⊗ u k ⊗ u k = min xi⊥xj ,y i ⊥y j , zi⊥zj ,∀i =j,λ k ∈R A - n k=1 λ k x k ⊗ y k ⊗ z k .
Proof. (i)⇔(ii). Suppose that (i) holds and

Q * = arg max Q∈SOn f (Q). Let W * = A • 1 Q T * • 2 Q T * • 3 Q T * .
Then W * ∈ MD and thus W * ∈ GMD. It follows that (ii)⇔(iii). By [24, Proposition 5.1], [24, (5.6)] and [24, (5.23)], we get that

f (Q * ) = max P ,Q,R∈SOn F(P , Q, R).

If (ii) holds and

max Q∈SOn f (Q) = A 2 - min ui⊥uj ,∀i =j, µ k ∈R A - n k=1 µ k u k ⊗ u k ⊗ u k 2 , max P ,Q,R∈SOn F(P , Q, R) = A 2 - min xi⊥xj ,y i ⊥y j , zi⊥zj ,∀i =j,λ k ∈R A - n k=1 λ k x k ⊗ y k ⊗ z k 2 . It follows that (ii)⇔(iii). (iii)⇔(iv) is clear. (iii)⇔(v). Note that O(D) is closed, hence there exist µ k ∈ R and orthonormal basis {u * k , 1 ≤ k ≤ n} such that A - n k=1 µ k u k ⊗ u k ⊗ u k = min vi⊥vj ,∀i =j, µ k ∈R A - n k=1 µ k v k ⊗ v k ⊗ v k . Theorem 5.3. It holds that MD 2 = GMD 2 .
Proof. We only need to prove that

A ∈ GMD 2 if A ∈ MD 2 . Let W = A • 1 P T • 2 Q T • 3 R T with P , Q, R ∈ SO 2 .
These rotations can be written as

P = 1 √ 1 + x 2 1 -x x 1 , Q = 1 1 + y 2 1 -y y 1 and R = 1 √ 1 + z 2 1 -z z 1 for x, y, z ∈ R. Define F(x, y, z) def = diag{W} 2
as in [START_REF] Kolda | Symmetric orthogonal tensor decomposition is trivial[END_REF]. Denote

F(x, y, z) = a 2 + d 2 + (a 2 + d 2 )(γ + 1) (1 + x 2 )(1 + y 2 )(1 + z 2 ) σ (x, y, z). where σ(x, y, z) def =
(γ -1)(x 2 + y 2 + z 2 + x 2 y 2 + y 2 z 2 + z 2 x 2 ) + 2γ(x 2 yz + xy 2 z + xyz 2 + xy + yz + zx).
Note that F(0, 0, 0) = a 2 + d 2 . We only need to prove that σ(x, y, z) ≤ 0 for any x, y, z ∈ R. If γ ∈ [0, 1/3], then γ -1 ≤ -2γ, and thus

σ(x, y, z) ≤ -2γ[(x 2 + y 2 + z 2 + x 2 y 2 + y 2 z 2 + z 2 x 2 ) -(x 2 yz + xy 2 z + xyz 2 + xy + yz + zx)] = -γ[(x -y) 2 + (y -z) 2 + (z -x) 2 + (xy -yz) 2 + (yz -zx) 2 + (zx -xy) 2 ] ≤ 0. If γ ∈ [-1, 0), then σ(x, y, z) = -(1 + γ)(x 2 + y 2 + z 2 + x 2 y 2 + y 2 z 2 + z 2 x 2 ) + γ[(x + y) 2 + (y + z) 2 + (z + x) 2 + (xy + yz) 2 + (yz + zx) 2 + (zx + xy) 2 ] ≤ 0.

Symmetric tensors of dimension n > 2

In this subsection, we first present a counterexample to show that the equivalent problems in Proposition 5.1 and Proposition 5.2 have a negative answer when n > 2. Then we prove a related result, which can be seen as an orthogonal analogue of the Comon's conjecture.

A counterexample

Lemma 5.4. Define ρ(Q) def = Q 2 11 Q 2 12 Q 2 13 + Q 2 21 Q 2 22 Q 2 23 + Q 2 31 Q 2 32 Q 2 33 ( 12
)
for Q ∈ O 3 . We have that ρ(Q) < 1/12 for any Q ∈ SO 3 .

The proof of Lemma 5.4 can be found in Appendix A.

Example 5.5. Let A be as in Example 3.18. Let F be as in [START_REF] Kolda | Symmetric orthogonal tensor decomposition is trivial[END_REF]. Suppose that

P * =   1 0 0 0 1 0 0 0 1   , Q * =   0 0 1 1 0 0 0 1 0   , R * =   0 1 0 0 0 1 1 0 0   .
Then F(P * , Q * , R * ) = 3. However, easy calculations show that

f (Q) = 36ρ(Q) < 3,
where the last inequality follows by Lemma 5.4. Thus, we see that

f (Q) < 3 = F(P * , Q * , R * )
for any Q ∈ O 3 . It follows that Proposition 5.2 (ii) has a negative answer when n > 2.

Moreover, we have O(GMD) S n when n > 2 by Proposition 5.1.

Remark 5.6. It was proved that the best rank-1 approximation of any A ∈ S n can always be chosen to be symmetric [START_REF] Friedland | Best rank one approximation of real symmetric tensors can be chosen symmetric[END_REF][START_REF] Zhang | The best rank-1 approximation of a symmetric tensor and related spherical optimization problems[END_REF]. Example 5.5 provides a counterexample to Proposition 5.2 (v) when n > 2. It will be interesting to study whether the best rank-p (1 < p < n) orthogonal approximation can be chosen to be symmetric when n > 2, which can be seen as an orthogonal analogue of [27, Conjecture 8.7].

An orthogonal analogue of Comon's conjecture

Although Proposition 5.2 (iii) has a negative answer by Example 5.5 when n > 2, we have the following result. Proposition 5.7. Let A ∈ S n . Then for any p

min xi⊥xj ,y i ⊥y j , zi⊥zj ,∀i =j,λ k ∈R A - p k=1 λ k x k ⊗ y k ⊗ z k = 0 implies min ui⊥uj ,∀i =j, µ k ∈R A - p k=1 µ k u k ⊗ u k ⊗ u k = 0.
Proof. Suppose that p ∈ N and

A = p k=1 λ k x k ⊗ y k ⊗ z k , (13)
where λ k ∈ R/{0} and x i ⊥ x j , y i ⊥ y j , z i ⊥ z j for any i = j. We assume that x k , y k and z k are all unit vectors without loss of generality. Note that A is symmetric. Then

A • 3 z k = λ k x k ⊗ y k
is a symmetric matrix for any 1 ≤ k ≤ p. It follows that x k = ±y k . In a similar way, we can prove that y k = ±z k . The proof is complete. Remark 5.9. (i) Proposition 5.7 can be seen as an orthogonal analogue of the Comon's conjecture [START_REF] Comon | Symmetric tensors and symmetric tensor rank[END_REF][START_REF] Friedland | Remarks on the symmetric rank of symmetric tensors[END_REF][START_REF] Zhang | Comon's conjecture, rank decomposition, and symmetric rank decomposition of symmetric tensors[END_REF], which conjectured that rank and symmetric rank of a symmetric tensor are equal, that is,

min x k = y k = z k =1, λ k ∈R A- p k=1 λ k x k ⊗y k ⊗z k = 0 ⇒ min u k =1,µ k ∈R A- p k=1 µ k u k ⊗u k ⊗u k = 0
for any A ∈ S n and p ∈ N minimal.

(ii) An alternative proof of Corollary 5.8 can be found in [START_REF] Boralevi | Orthogonal and unitary tensor decomposition from an algebraic perspective[END_REF]Proposition 32].

6. Convergence results for cyclic Jacobi algorithm

Cyclic Jacobi algorithm description

In this subsection, we recall the cyclic Jacobi algorithm (also called the Jacobi CoM algorithm) given in [START_REF] Comon | Independent component analysis, a new concept ?[END_REF], which is a special case of Algorithm 1.

Algorithm 2. Input: A ∈ S n and Q 0 = I n . Output: a sequence of iterations {Q k : k ∈ N}.

• For k = 1, 2, . . . until a stopping criterion is satisfied do:

-

Choose the pair (i k , j k ) according to the following cyclic-by-row rule

(1, 2) → (1, 3) → • • • → (1, n) → (2, 3) → • • • → (2, n) → • • • → (n -1, n) → (1, 2) → (1, 3) → • • • - Compute the angle θ * k that maximizes the function h k (θ) defined in (5). - Update Q k = Q k-1 G (i k ,j k ,θ * k ) .
• End for

Derivatives and relations between them

In this subsection, we present some basic properties of Algorithm 2. More details can be found in [START_REF] Comon | Independent component analysis, a new concept ?[END_REF][START_REF] Li | Globally convergent jacobi-type algorithms for simultaneous orthogonal symmetric tensor diagonalization[END_REF]. We first give a definition.

Take the k-th iteration with pair (i k , j k ) in Algorithm 2. Let

W (k-1) = A • 1 Q T k-1 • 2 Q T k-1 • 3 Q T k-1 .
By ( 5), we have that

h k (θ) = diag{W (k-1) • 1 (G (i k ,j k ,θ) ) T • 2 (G (i k ,j k ,θ) ) T • 3 (G (i k ,j k ,θ) ) T } 2 . ( 14
)
Let x = tan(θ), and define

τ k : R → R by τ k (x) def = h k (arctan(x)).
In the rest of this subsection, with some abuse of notation, we use a shorthand notation

d k = d i k ,j k (W (k-1) ) and ω k = ω i k ,j k (W (k-1) ).
It can be calculated that [START_REF] Li | Globally convergent jacobi-type algorithms for simultaneous orthogonal symmetric tensor diagonalization[END_REF]Lemma 5.8].

τ k (x) -τ k (0) = 3 (1 + x 2 ) 2 (2d k (x -x 3 ) -ω k x 2 ), (15)
τ k (x) = 6 (1 + x 2 ) 3 (d k (1 -6x 2 + x 4 ) -ω k (x -x 3 )), (16)
τ k (x) = 6 (1 + x 2 ) 4 [2d k (-9x + 14x 3 -x 5 ) -ω k (1 -8x 2 + 3x 4 )].
d k (1 -6x * k 2 + x * k 4 ) -ω k (x * k -x * k 3 ) = 0. ( 17
) (i) If x * k -x * k 3 = 0, we get that ω k = (1 -6x * k 2 + x * k 4 ) x * k (1 -x * k 2 ) d k , and thus τ k (x * k ) -τ k (0) = 3x * k (1 -x * k 2 ) d k . (ii) If 1 -6x * k 2 + x * k 4 = 0, we get that d k = x * k (1 -x * k 2 ) (1 -6x * k 2 + x * k 4 ) ω k ,
and thus

τ k (x * k ) = -6ω k 1 -6x * k 2 + x * k 4 , τ k (x * k ) -τ k (0) = 3x * k 2 (1 -6x * k 2 + x * k 4 ) ω k . (18)

Convergence properties

In this subsection we prove some results about the convergence properties of Algorithm 2. Before that, we first quote a lemma, which could be proved easily by the fact that any multivariate polynomial is continuous. Proposition 6.2. Suppose that A ∈ S n and {Q

k : k ∈ N} ⊆ SO n are the iterations of Algorithm 2. If Q k → Q * and W * = A • 1 Q T * • 2 Q T * • 3 Q T * ,
then d i,j (W * ) = 0 and ω i,j (W * ) ≥ 0 for any 1 ≤ i < j ≤ n.

Proof. Fix any 1 ≤ i * < j * ≤ n. We choose a subsequence L ⊆ N such that

(i +1 , j +1 ) = (i * , j * )
for any ∈ L. It follows by Q k → Q * that x * +1 → 0 when ∈ L tends to infinity. Then we get that d i * ,j * (W ( ) ) → 0 by [START_REF] Li | Globally convergent jacobi-type algorithms for simultaneous orthogonal symmetric tensor diagonalization[END_REF]. Note that τ +1 (x * +1 ) ≤ 0 for any ∈ L. By ( 18), we have that ω i * ,j * (W ( ) ) ≥ 0 when ∈ L is large enough. Since Q → Q * , the result follows from continuity of the function

Q → A • 1 Q T • 2 Q T • 3 Q T . (19)
Proposition 6.3. Let A ∈ S n and {Q k : k ∈ N} ⊆ SO n be the iterations of Algorithm 2. Suppose that there are a finite number of accumulation points of {Q k : k ∈ N}.

(i) Let Q * ∈ SO n be any accumulation point and

W * = A • 1 Q T * • 2 Q T * • 3 Q T * .
Then there exists 1 ≤ i * < j * ≤ n such that d i * ,j * (W * ) = ω i * ,j * (W * ) = 0.

(ii) For any 1 ≤ i * < j * ≤ n, there exists an accumulation point Q * ∈ SO n such that

W * = A • 1 Q T * • 2 Q T * • 3 Q T * satisfies d i * ,j * (W * ) = 0 and ω i * ,j * (W * ) ≥ 0. (iii)
We have that the directional derivative (6) tends to zero:

h k+1 (0) = 6d i k+1 ,j k+1 (W (k) ) → 0.
The proof can be found in Appendix A.

Corollary 6.4. Suppose that A ∈ S n and {Q k : k ∈ N} ⊆ SO n are the iterations of Algorithm 2. Let Q * ∈ SO n be an accumulation point and

W * = A • 1 Q T * • 2 Q T * • 3 Q T * .
If ω i,j (W * ) > 0 for any 1 ≤ i < j ≤ n, then either Q k → Q * , or there exist an infinite number of accumulation points in the iterations.

Remark 6.5. By Theorem 6.2 and Lemma 3.8, we see that if the iterations of Algorithm 2 converge to Q * ∈ O n , then

W * = A • 1 Q T * • 2 Q T * • 3 Q T * satisfies W * ∈ JD.
In other words, Q * is a stationary point of (2) by Remark 3.7. By Example 4.6, we see that LMD 3 JD 3 . Note that Algorithm 2 stops at any tensor in JD. It follows that Algorithm 2 may converge to a saddle point of (2).

Conclusions

In this paper, we studied several classes of third order approximately diagonal tensors, which are closely related to Jacobi-type algorithms and the approximate diagonalization problem [START_REF] Cichocki | Tensor decompositions for signal processing applications: From two-way to multiway component analysis[END_REF]. We believe that these classes provide a better understanding of problem (1) and behaviour of optimisation algorithms; some examples in this paper can be used as test cases for the algorithms. There are some open questions left for future research, such as the global convergence of Algorithm 2 for third (or higher) order symmetric tensors is still unknown.

Appendix A. Long proofs

Proof of Lemma 5.4. Since ( 12) is invariant with respect to changes of signs of the columns of Q, it suffices to prove the statement for Q ∈ SO 3 .

Step 1. By [29, p. 10], any Q ∈ SO 3 can be decomposed as

Q T = Q 1 (x)Q 2 (y)Q 3 (z)
, where The similar result holds if z = 0. Therefore, we only need to prove that ρ(x, y, z) < 1/12 in the case that xz = 0.

Q 1 (x) = 1 √ 1 + x 2   √ 1 + x 2 0 0 0 1 -x 0 x 1   , Q 2 (y) =
Step 2. Let

u = x - 1 x and v = z - 1 z .
Note that ϕ(u, y) = 2(y 4 + y 2 + y 2 + 1 y 2 + 1y 2 )u 2 + 4y 2 + 1 (u 2 + 4) 2 (1 + y 2 for any x, y, z ∈ R, which completes the proof.

Remark 3 . 16 .

 316 Let A, B ∈ S n . If A is orthogonally similar to B, then A and B have the same Z-eigenvalues [13, Thm 2.20]. In fact, if

3. 4 . 2 . 1 + e 2 ⊗ e 1 ⊗ e 3 + e 3 ⊗ e 1 ⊗ e 2 + e 3 ⊗ e 2 ⊗ e 1 .

 421213312321 Relationship with orthogonally decomposable tensors Example 3.18. We present an example to show that O(D n ) O(PD n ) for S n . Let A = e 1 ⊗ e 2 ⊗ e 3 + e 1 ⊗ e 3 ⊗ e 2 + e 2 ⊗ e 3 ⊗ e It is easy to see that A ∈ O(PD 3 ). On the other hand, it is known [21, Prop. 3.1 and 4.3] that the symmetric tensor rank is srank{A} = 4, hence A cannot be in O(D 3 ) (otherse it would have rank at most 3). Proposition 3.19. (i) Let A ∈ PD n . Then A ∈ D n if and only if

Corollary 4 . 2 .

 42 Let Q = I n in Lemma 4.1. The tangent vector space T In O n contains the skew symmetric matrices ∆. It follows by (9) that

Definition 4 . 4 .

 44 Let A ∈ JD 3 . Let γ 12 , γ 13 and γ 23 be the stationary diagonal ratios introduced in Definition 3.12. Denote by a = A 111 , b = A 222 , c = A 333 and g = A 123 .We define the Euclidean Hessian matrix of A to be M A def

Proposition 5 . 1 .

 51 Let n ≥ 2. Then O(GMD n ) = S n if and only if GMD n = MD n .

  A ∈ MD, then I n = arg max Q∈SOn f (Q) and thus (I n , I n , I n ) = arg max P ,Q,R∈SOnF(P , Q, R),which implies that A ∈ GMD.

  a = A 111 , b = A 112 , c = A 122 , d = A 222 , γ = γ 12 the stationary diagonal ratio in Definition 3.12. Then c = γa and b = γd by definition. Moreover, γ ∈ [-1, 1/3] by Theorem 3.14. It can be calculated that

Corollary 5 . 8 .

 58 Let A ∈ S n . Then we have d(A, GO(D)) = 0 ⇒ d(A, O(D)) = 0, that is, O(D) = S n ∩ GO(D).

Remark 6 . 1 .

 61 Denote by x * k = tan(θ * k ) the optimal point of τ k (x). Note that τ k (x * k ) = 0. It follows by (16) that

  y, z ∈ R. It can be calculated that ρ(x, y, z)def = ρ(Q) = 1 (1 + x 2 ) 2 (1 + y 2 ) 3 (1 + z 2 ) 2 [(y 4 z 2 + y 2 z 2 )(x 4 + 1) + 2 y 2 + 1y 3 (z 3 -z)(x 3 -x) + (y 4 z 4 -4y 4 z 2 + y 4 + y 2 z 4 + y 2 + z 2 )x 2 ]. If x = 0, then ρ(0, y, z) = y 2 z 2 (1 + y 2 ) 2 (1 + z 2 )

) 3 ≥ 9 ) 6 * * + 18y 4 * * + 11y 2 * * - 1 .* * 28y 4 * * + 22y 2 * * - 1 , 4 .

 3964214214 ψ(u, y) for any u, y ∈ R. It is enough to prove that ϕ(u, y) < 1/12 for any u, y ∈ R. Let (u * * , y * * ) be the maximal point of ϕ(u, y). By ∂ϕ ∂u (u * * , y * * ) = ∂ϕ ∂y (u * * , y * * ) = 0, we have that y * * (4u 2 * * y 4 * * -4u 2 * * + 8y 2 * * -1) = 0, (A.7) u * * (2u 2 * * y 4 * * + 2u 2 * * y 2 * * -8y 4 * * -4y 2 * * + 1) = 0. (A.8) If y * * = 0, then u * * = 0 by (A.8). If u * * = 0, then y * * = 0 or y 2 * * = 1/8 by (A.7). It is easy to check that ϕ(u * * , y * * ) < 1/12 in all these cases. Now we assume that y * * = 0 and u * * = 0. Then eqn. (A.8) can be rewritten as By substituting (A.9) into (A.6), we get that ϕ(u * * , y * * ) = 4y 4 * * 16y Next, we substitute (A.9) into (A.7), and get that y * * should satisfy (1 -8y 2 * * )(2y 4 * * + 2y 2 * * ) = (4y 4 * * -4)(8y 4 * * + 4y 2 * * -1), After division by (y 2 * * + 1), we have 16y 6 * * -11y 2 * * + 2 = 0, (A.10) which is a 3-rd degree polynomial equation in y 2 * * ; there are two positive solutions of (A.10) given by positive roots of the polynomial, i.e. y 2 * * ≈ 0.7162 or y 2 * * ≈ 0.1921. Taking into account (A.10), we have have that ϕ(u * * , y * * ) = 4y 4 hence ϕ(u * * , y * * ) ≈ 0.0757 < 1/12 or ϕ(u * * , y * * ) ≈ 0.0653 < 1/12 in these two cases.Step Above all, we prove that ρ(x, y, z) ≤ max u,v,y∈R Φ(u, v, y) = max u,y∈R ψ(u, y) ≤ ϕ(u * * , y * * ) < 1 12

We use a simplified notation for subsequences in order to avoid multilevel indices.

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions. This work was supported by the ERC project "DECODA" no.320594, in the frame of the European program FP7/2007-2013. The first author was partially supported by the National Natural Science Foundation of China (No.11601371).

We define Φ(u, v, y) def = ρ(x, y, z) = 1 (u 2 + 4)(v 2 + 4)(1 + y 2 ) 3 [(y 4 + y 2 )u 2 + 2 y 2 + 1y 3 vu + (y 4 + y 2 )(v 2 + 4) + 1 -4y 4 ].

Let (u * , v * , y * ) be the maximal point. If y * = 0, then

Moreover, by (A.1) and (A.2), we also get

which implies that

By (A.3) and (A.4), we get that 1 -4y 4 * = 0. It follows that u * v * = 0 by (A.3), which contradicts the assumption that u * = 0. Therefore, we prove that u 2 * = v 2 * .

Step 3. Now we define

Proof of Proposition 6.3. Since there are finite number of accumulation points, there exists δ > 0 such that the δ-neighborhoods of these accumulation points have positive distance to each other. (i) Let Q * be any accumulation point. Let L ⊆ N be a subsequence such the subsequence 2 {Q , ∈ L} is located in the δ-neighborhood N (Q * , δ) and Q → Q * when ∈ L tends to infinity. Note that Q * is not the unique accumulation point. There exists a pair (i * , j * ) such that it appears for an infinite number of times in the sequence of pairs

Now we construct a subsequence P ⊆ L such that

for any p ∈ P. Note that the δ-neighborhoods of different accumulation points have positive distance to each other. There exists σ > 0 such that |x * p+1 | > σ for any p ∈ P. Note that |x * p+1 | ≤ 1 for any p ∈ P. There exists

We assume that x * p+1 → ζ 0 for simplicity. Now we prove that d i * ,j * (W (p) ) → 0 and ω i * ,j * (W (p) ) → 0 when p ∈ P tends to infinity, and thus get (i) by the continuity of [START_REF] Ishteva | Jacobi algorithm for the best low multilinear rank approximation of symmetric tensors[END_REF].

Denote by

By ( 15) and ( 17), we see that

0 when p ∈ P tends to infinity. Then M p is invertible when p is large enough. Note that ϑ p → 0. It follows that

when p ∈ P tends to infinity.

(ii) Let (i * , j * ) be any pair. There exists an accumulation point

, then (i * , j * ) appears for an infinite number of times in the sequence of pairs {(i +1 , j +1 ), ∈ L}.

(a) If it appears for an infinite number of times in

then the result follows by the same reasoning as in (i).

(b) Otherwise, it appears for an infinite number of times in

We construct the subsequence {Q p , p ∈ P} of {Q , ∈ L} such that (i p+1 , j p+1 ) = (i * , j * ) and

Note that Q p → Q * and Q p+1 → Q * when p ∈ P tends to infinity. We get that x * p+1 → 0, and eventually from the proof of Theorem 6.2: d i * ,j * (W (p) ) → 0 and ω i * ,j * (W (p) ) ≥ 0 when p ∈ P is large enough. Then we prove (ii) by the continuity of ( 19). Next, we have that d i +1 ,j +1 (W ( ) ) → 0, which follows by [START_REF] Li | Globally convergent jacobi-type algorithms for simultaneous orthogonal symmetric tensor diagonalization[END_REF] if ζ 0 = 0, and by a reasoning similar to (i) if ζ 0 = 0. Finally, note that there exist a finite number of accumulation points in {x * k }, hence the proof is complete.


Sandy Figiel

Côme Pasqualin

Fanny Bery

Véronique Maupoil

Christophe Vandier

Marie Potier-Cartereau

Isabelle Domingo

Roseline Guibon

Franck Bruyere

Karine Mahéo

Veronique Maupoil

Karine Maheo

MD Gaelle Fromont
email: gaelle.fromont-hankard@univ-tours.fr

Functional Organotypic Cultures of Prostate Tissues: A Relevant Preclinical Model That Preserves Hypoxia Sensitivity and Calcium Signaling

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

INTRODUCTION

Although prostate cancer (PCa) is the second leading cause of cancer-related death among men in developed countries, the transfer of scientific findings into medical progress is challenged by the lack of valuable preclinical models. Immortalized PCa cell lines have acquired several mutations in culture, and both their genotype and phenotype are different from those observed in native human cancer cells [START_REF] Toivanen | Breaking through a roadblock in prostate cancer research: an update on human model systems[END_REF]. Moreover, in vitro studies with cell lines do not reproduce the interactions with the prostate microenvironment, which is critical for PCa development and progression. Xenografts and genetically engineered mice represent improved models but have inherent advantages and limitations regarding the mimicking of the PCa situation in humans. These models have advanced our understanding of PCa behavior, but differences between species and donors may lead to difficulties in interpreting the results.

Additionally, their deviation from the human physiology could result in inaccurate preclinical assessment [START_REF] Chauchereau | Experimental models for the development of new medical treatments in prostate cancer[END_REF]. To address these limitations, the use of ex vivo culture of human tissues has been described for different types of cancers [START_REF] Vaira | Preclinical model of organotypic culture for pharmacodynamic profiling of human tumors[END_REF][START_REF] Maund | Optimization and comprehensive characterization of a faithful tissue culture model of the benign and malignant human prostate[END_REF][START_REF] Carranza-Torres | Organotypic culture of breast tumor explants as a multicellular system for the screening of natural compounds with antineoplastic potential[END_REF][START_REF] Koerfer | Organotypic slice cultures of human gastric and esophagogastric junction cancer[END_REF]. Culture of tissue slices is a good representative model of the in vivo situation in patients since it maintains both cancer cell heterogeneity and an intact microenvironment, which allows stromal-epithelial interactions. In the past, in vitro maintenance of prostate tissues has been challenged by rapid degradation and poor survival of epithelial cells [START_REF] Papini | Selective growth of epithelial basal cells from human prostate in a three-dimensional organ culture[END_REF][START_REF] Bläuer | A novel tissue-slice culture model for nonmalignant human prostate[END_REF]. However, the use of precision-cut slicing has led to more reproducible models, which have been reported by a few teams in preclinical studies [START_REF] Jäämaa | DNA damage recognition via activated ATM and p53 pathway in nonproliferating human prostate tissue[END_REF][START_REF] Zhang | Differential epithelium DNA damage response to ATM and DNA-PK pathway inhibition in human prostate tissue culture[END_REF][START_REF] Centenera | Evidence for efficacy of new Hsp90 inhibitors revealed by ex vivo culture of human prostate tumors[END_REF][START_REF] Schiewer | Dual roles of PARP-1 promote cancer growth and progression[END_REF]. However, in these previous studies, the culture conditions did not take into account the specific variations related to the tumor microenvironment. Essential factors such as hypoxia and calcium signaling have been shown to influence tumor progression and resistance to treatment. Hypoxia is a feature characteristic of most solid tumors, and is a major contributor to cancer progression and resistance to treatment, and it mainly exerts its effects through the process of epithelial to mesenchymal transition (EMT) [START_REF] Byrne | Androgen deprivation in LNCaP prostate tumour xenografts induces vascular changes and hypoxic stress, resulting in promotion of epithelial-to-mesenchymal transition[END_REF][START_REF] Rankin | Hypoxic control of metastasis[END_REF][START_REF] Montanari | Epithelial-mesenchymal transition in prostate cancer: an overview[END_REF]. Prostate tumors are known to grow under a hypoxic microenvironment, which comprises oxygen concentrations lower than those found in the normal prostate [START_REF] Vaupel | Detection and characterization of tumor hypoxia using pO2 histography[END_REF][START_REF] Mckeown | Defining normoxia, physoxia and hypoxia in tumours-implications for treatment response[END_REF]. In addition, Ca 2+ signaling associated with variations in extracellular Ca 2+ concentrations has been shown to play a major role in the regulation of cellular processes leading to tumor progression, including proliferation, migration, and apoptosis [START_REF] Roderick | Ca2+ signalling checkpoints in cancer: remodelling Ca2+ for cancer cell proliferation and survival[END_REF][START_REF] Prevarskaya | Calcium in tumor metastasis: new roles for known actors[END_REF][START_REF] Montheith | The calcium-cancer signalling nexus[END_REF]. Extracellular calcium concentrations in the tumor microenvironment can reach levels between 8 and 40 mmol/L, and it has been implicated in the development of bone metastasis [START_REF] Breuksch | The role of extracellular calcium in bone metastasis[END_REF]. Moreover, altered expression of specific Ca 2+ channels and pumps are characteristic features of most cancers including PCa [START_REF] Montheith | Calcium channels and pumps in cancer: changes and consequences[END_REF]. Therefore, the sensitivity of human cancer samples to extracellular calcium concentrations is a major element that needs to be assessed to demonstrate the functional value of an ex vivo prostate tissue model. Here, we described a relevant and functional model of cultured benign and malignant human prostate tissues that preserves hypoxia and calcium sensitivity. The development of this model will allow a better analysis of the intrinsic properties of PCa and, possibly, the rapid and accurate identification of targeted cancer treatment.

MATERIAL AND METHODS

Patients and tissues

Prostatic tissue samples, including malignant and normal tissues, were obtained from 26 patients undergoing radical prostatectomy for clinically localized PCa. Tumors were classified according to the International Society of UroPathology (ISUP) in group 1 to 5, and according to the pTNM stage as pT2 or pT3 (Table 1). Written informed consent was received from all patients.

Preparation of tissue slices

Radical prostatectomy specimens were obtained immediately after surgery, and 4 to 5 mm samples were dissected aseptically from the peripheral zone within the putative benign or cancer areas (Fig 1A). Tissue samples were glued on a platen against an agarose block (8% agarose) in stair-step cut and were immersed in RPMI medium supplemented with 10% FBS and 1 % penicillin-streptomycin. Briefly, the samples were immediately cut with a vibratome (Vibratome VT1200, Leica) at 200 to 300 µm thickness, with a total number of slices ranging from 6 to 10 per case. Slices were used for either ex vivo culture and biomarkers expression (15 patients) or for ex vivo culture and intracellular Ca 2+ measurements (11 patients).

Ex vivo culture of prostate tissue slices

Each slice was transferred into a 6-well cell culture plate containing DMEM medium (Lonza, Levallois-Perret, France) supplemented with 10% FBS and 1% penicillin-streptomycin, and incubated with or without 1 nM dihydrotestosterone (DHT) (D073, Sigma). Slices were incubated in a 37 °C humidified incubator with 5% CO2. Medium was changed daily.

Hypoxia experiments were performed in an hypoxia chamber (SCI-tive, Alliance Bio Expertise) at 1% O2 at 37 °C in a 5% CO2 humidified environment. Slices were first placed in normoxia for 24 hours, to avoid additional stress after cutting, and then, were incubated for 24 hours in hypoxia environment. The medium was replaced every day.

Slices were cultured between 3 to 7 days after tissue collection. Analysis of sections after hematoxylin-eosin-saffron (HES) and immunohistochemical staining was performed at baseline (Day 0), then for Day1 (D1) to D3, or D1 to D7, depending on the number of slices available. In four cases the number of slices was sufficient to analyze the morphology from D0 to D7. Analysis of biomarker expression was performed in 15 cases at D0 and after 48H of culture. The effects of DHT deprivation and hypoxia were measured at D2, in 15 samples for DHT deprivation, and in five samples for hypoxia.

Immunohistochemistry

After fixation in 10% formalin, tissue slices were embedded in paraffin, and sections were serially cut at 5 µm with a microtome. Then, sections were spread on a SuperFrost® Plus slide and dried for one hour at 56 ° C. One section was stained with HES for morphological analysis. The other slides, used for immunostaining, were deparaffinized (histosol and alcohol baths), rehydrated, and heated in citrate buffer pH 6 (citric acid monohydrate and sodium citrate) for antigenic retrieval. After blocking for endogenous peroxidase with 3% hydrogen peroxide, primary antibodies were incubated with the section. The panel of primary antibodies included p63 (Biocare, clone 4A4, dilution 1/100, 30 min), p504S (DakoCytomation, clone 13H4, 1/200, 30 min), androgen receptor (AR) (Abnova, clone 1G3, 1/500, 30 min), PSA (BioSB, clone BSB7,1/250, 30 min), Zeb1 (Abnova, clone 4C4, 1/500, 1 hour), carbonic anhydrase IX (CAIX) (Novocastra, clone TH22, 1/100, 30 min), ERG (Roche, EPR 3864, 1/1, 30 min), and the proliferation marker Ki67 (DakoCytomation, clone 39-9, 1/50, 30 min).

Immunohistochemistry was performed with either the automated BenchMark XT slide stainer (Ventana Medical Systems Inc) using OptiView Detection Kit (Ventana Medical Systems Inc) or manually (for Zeb1) using the streptavidin-biotin-peroxidase method with diaminobenzidine as the chromogen (Kit LSAB, Dakocytomation, Glostrup, Denmark). Slides were finally counterstained with haematoxylin. Negative controls were obtained after omission of the primary antibody or incubation with an irrelevant antibody.

Scoring of antibody staining

Staining for p63, p504, CAIX, and PSA was scored as positive or negative, and staining for AR, ERG, and Zeb1 was evaluated as follows: -, no stained cells; +, focal positivity; ++, diffuse positivity. The number of Ki67 positive cells was expressed as the percentage of total epithelial cells.

Intracellular Ca 2+ measurements

Intracellular Ca 2+ measurements were performed on tissue slices after 48 H of ex vivo culture.

Intracellular calcium variations were assessed with the rhod-2 calcium sensitive dye. All the next steps were done in the darkness. Briefly, human prostate slices were incubated for 30 min with 5 µM Rhod-2 AM at room temperature. Slices were placed in the bath of Macro Zoom System Microscope MVX10 and were washed for 20 min with a physiological saline solution (PSS) before experiments, using a pump in an open circuit. The dye was excited at 545 ± 20 nm with LED illumination (CoolLED pE-300 white) and fluorescence was collected at 605 ± 40 nm with a sCMOS Zyla 4.2 PLUS camera and a Macro Zoom System Microscope MVX10. Image acquisition rate was 0.1 Hz and exposure time was 600 ms. For each measurement, human prostate samples were incubated with 2 mM CaCl2 in PSS. After a stabilizing time, CaCl2 was added to the incubation bath to reach a final concentration of 5 mM Ca 2+ . Analyses were performed with the ImageJ 1.52a analysis software (https://imagej.nih.gov/ij/; last accessed November 29, 2019). The Ca 2+ fluorescence signal was normalized with the initial fluorescence signal obtained at 2mM CaCl2 to correct for differences in dye loading across the preparation. Variations of the Ca 2+ signal were then determined by calculating the difference between the basal value, at 2 mM CaCl2, and the maximal Ca 2+ value after 5 mM CaCl2 addition.

Statistical analysis

Statistical analyses were performed with StatView, version 5.0, software (Abacus Concepts, Berkeley, CA). Comparison between groups was performed using the χ2 test for categorical data and non parametric Mann-Whitney U test for continuous data. Paired data were analyzed using the non parametric Wilcoxon test.

RESULTS

Tissue morphology

Tissue morphology after culture under normoxic conditions was compared to the morphology of native tissues (baseline, at D0). For all the tested tissues, the architecture was preserved in both benign and malignant tissues from D1 to D5. No necrosis was observed between D1 and D5, but at D6 tissue samples started showing some necrosis foci, and at D7, necrosis was extensive (Fig 1).

Biomarkers expression

The expression of biomarkers after culture under normoxic conditions during 2 days was compared to native tissues (baseline, at D0). Biomarkers included markers of normal basal epithelial cells (p63), prostate cancer cells (p504S), proliferation (Ki67), EMT (Zeb1 transcription factor), hypoxia (CAIX), and of the fusion gene TMPRSS2-ERG product (ERG). In normal glands at D0, proliferation was limited and restricted to basal cells (median 1% Ki67 positive cells in the epithelial compartment, range 1% to 3%). Basal cell proliferation increased after culture (median 8% Ki67 positive cells in the epithelial compartment, range 6% to 15%) when compared to the baseline status (P < 0.02, Mann Whitney test) (Fig 2C ,2D). In tumor tissues, the proliferation rate was identical after culture (median 3%, range 2% to 5%) when compared to native tumors at D0 (median 3%, range 1% to 4%) (P = 0.8) (Fig

2E, 2F).

At baseline (D0), prostate cancer cells expressed ERG in four cases out of eight (50%). In all positive cases, ERG expression was preserved in the cultured slices, and ERG negative tumors remained negative after tissue culture. ERG expression was negative in all normal glands, at baseline and after culture. PSA was expressed in all cases of both normal and cancer tissues, before and after culture. In normal tissues at D0, epithelial cells were negative for the Zeb1 transcription factor and for CAIX expression. In cultured slices obtained from normal tissues (DHT-supplemented or not), CAIX staining remained negative, and focal expression of Zeb1 (rare positive epithelial cells) was observed in two of seven cases (without significant difference compared to baseline, P = 0.4). At D0, three malignant samples out of eight showed focal Zeb1 expression in cancer cells, without CAIX staining. After culture under normoxic conditions, CAIX remained negative. Tumors with focal Zeb1 expression at baseline remained unchanged, whereas two out of the five initially Zeb1 negative cases showed focal expression (without significant difference compared to D0, P = 0.6).

Effects of DHT deprivation

In both benign and malignant glands, absence of DHT in the medium led to decreased expression of AR (from diffuse to focal staining) and of its target, the PSA gene product, (from positive to negative staining), when compared to slices cultured in medium supplemented with DHT (P = 0.0001) (Fig 3A to 3F). In ERG positive tumors, DHT ablation was associated with decreased ERG levels in cancer cells (from diffuse to focal staining) (P < 0.03) (Fig 3G to 3I).

Effects of hypoxia

Hypoxic conditions were tested with DHT supplementation for five samples (two benign and three malignant). In normoxia, benign samples were negative for Zeb1 and CAIX, and When cultured under hypoxic conditions, all samples (benign and malignant) showed an intense and diffuse membranous staining for CAIX (P = 0.001 compared to normoxic conditions) (Fig 4B). In addition, in all cases, hypoxia also led to increased Zeb1 expression, with a diffuse positive staining (P = 0.007 compared to normoxic conditions) (Fig 4D). In contrast, hypoxia did not modify ERG expression in cancer cells, as well as AR and PSA staining that remained positive (since hypoxic conditions were tested with DHT supplementation). Cancer cell proliferation was also identical in hypoxic compared to normoxic conditions, with a proliferation rate less than 3% (Fig 4E ,4F).

Ex vivo functional Ca 2+ imaging

Finally, functional Ca 2+ signaling was evaluated in prostate tissue slices obtained from tumor and adjacent non tumor tissues from 11 patients. This study was performed by measuring the variations in basal intracellular Ca 2+ concentration in response to variation of extracellular Ca 2+ concentrations (from 2 to 5 mM and 5 from 2 mM). Interestingly, as observed in the graphical trace, exposure to supra-physiological extracellular Ca 2+ concentration (5 mM) induced a robust and rapid increase of basal cytosolic Ca 2+ concentration resulting from a Ca 2+ entry followed by a sustained plateau. Interestingly, this increase could be reversed 2).

DISCUSSION

This study demonstrates that prostate tissue slices can be cultured for up to 5 days without necrosis with a preserved morphology and more importantly with unaffected physiological properties. This longevity of prostate tissue is obtained by using precision-cut slicing and daily replacement of medium complemented with DHT, and it is identical to that previously described by others (4). The expression of classical cell type markers was maintained, and they include p63 for basal cells, and AR and PSA for luminal cells. The cancer cellassociated marker p504S was present in all malignant tissue slices. Moreover, ERG expression was maintained after tissue culture in initially positive ERG tumors. Importantly, this finding is in contrast with a previous study that reported a loss of ERG expression (4).

Since ERG positive staining in prostate cancer cells is mediated by the action of androgens on the fusion gene TMPRSS2-ERG [START_REF] Kumar-Sinha | Recurrent gene fusions in prostate cancer[END_REF], this result suggests a fully functional AR-dependent signaling pathway in our model.

In benign tissue slices, an increased proliferation of basal cells was observed after culture when compared to the native tissues (D0). This finding, that is likely to reflect an elevated glandular regeneration after culture, was also observed in a previous study [START_REF] Papini | Selective growth of epithelial basal cells from human prostate in a three-dimensional organ culture[END_REF], but not in another (4). However, in malignant samples, the proliferation rate remained identical in cancer cells after culture, as previously described [START_REF] Maund | Optimization and comprehensive characterization of a faithful tissue culture model of the benign and malignant human prostate[END_REF]. Since one of the major potential interests of ex vivo models is the testing of therapeutic agents, the important point was to maintain an identical proliferation rate in cancer cells before and after culture.

When removing DHT from the medium, decreased levels of proteins encoded by androgenregulated genes was observed. These proteins include AR, PSA, and ERG, in both malignant and benign tissues. This demonstration of androgen dependence has been reported only in one previous study (4) but without effect on ERG, and it is mandatory to reproduce ex vivo the biology of human PCa cells. In fact, the AR is a key molecule that drives both normal prostate development and PCa. To date, the study of androgen action in the human prostate has been hampered by the lack of adequate models, since most PCa cell lines used for in vitro studies are generated from long term culture in the absence of androgens [START_REF] Chauchereau | Experimental models for the development of new medical treatments in prostate cancer[END_REF]. The present ex vivo model would therefore allow studying the physiological effects of androgens in either normal or malignant human prostate. Some previous studies have reported the functionality of PCa organotypic culture. Vaira et al have used this model to demonstrate the effects of targeting PI3K signaling on cancer cell proliferation and apoptosis (3). Other reports have shown in ex vivo culture of PCa slices that cytotoxic drugs and ionizing radiations can induce DNA damages [START_REF] Jäämaa | DNA damage recognition via activated ATM and p53 pathway in nonproliferating human prostate tissue[END_REF], and that inhibition of PARP-1 can lead to AR signaling alterations [START_REF] Schiewer | Dual roles of PARP-1 promote cancer growth and progression[END_REF]. However, no previous studies have analyzed to date the ability of an ex vivo PCa model to respond to major signals of cancer microenvironment, hypoxia, and extracellular calcium.

Since the Ca 2+ signal is a phenomenon well suited for the rapid transfer of information from the microenvironment to cancer cells [START_REF] Montheith | Calcium channels and pumps in cancer: changes and consequences[END_REF], the functional value of ex vivo cultures could also be reflected by the ability of tissue slices to respond to variations in extracellular Ca 2+ concentrations. We have developed for the first time an ex vivo organotypic culture method suitable for the evaluation of live Ca 2+ signaling in both tumor and adjacent non tumor samples. This method allowed us to follow the variations of basal intracellular Ca 2+ concentration dynamics in human prostate slices, resulting from Ca 2+ entry from the extracellular side of the plasma membrane. The functionality of our model was attested by the variations of intracellular Ca 2+ concentrations in response to variation in extracellular Ca 2+ .

An increased Ca 2+ entry in malignant compared to benign tissues was also shown. These data are in agreement with previous studies reporting alterations in Ca 2+ homeostasis in cancer cells compared to normal cells [START_REF] Stewart | Altered calcium signaling in cancer cells[END_REF]. This type of Ca 2+ entry has already been described in breast cancer cell lines that was named constitutive Ca 2+ entry [START_REF] Chantôme | Pivotal role of the lipid Raft SK3-Orai1 complex in human cancer cell migration and bone metastases[END_REF]. Here, we demonstrate for the first time the existence of a constitutive Ca 2+ entry in PCa and, more interestingly, its differential level in tumor versus non tumor tissues. A greater intracellular Ca 2+ entry was also found in tissues sampled from high risk PCa compared to tumors with low or intermediate risk of recurrence, suggesting that Ca 2+ entry may contribute to cancer progression. There is substantial evidence to suggest an important role for Ca 2+ during tumorigenesis. Many mitogens and tumor growth factors elicit rapid elevations of intracellular calcium. Furthermore, Ca 2+ is involved in cellular events that regulate cell cycle progression, cell migration, angiogenesis, and apoptosis in various cancer cells [START_REF] Clapham | Calcium signaling[END_REF]. Our findings raise the possibility that PCa cells might be particularly susceptible to novel anticancer agents that target Ca 2+ homeostasis. Elucidating the nature of Ca 2+ channels involved in dysregulated constitutive Ca 2+ entry is therefore essential to better understand the mechanisms contributing to malignant transformation and cancer progression.

Hypoxia is a common phenomenon observed in solid tumors, including PCa. In human tumors, the effects of hypoxia are largely mediated by the hypoxia-inducible factors (HIFs)transcription factors that induce the expression of several proteins including the carbonic anhydrase IX (CAIX) [START_REF] Balamurugan | HIF-1 at the crossroads of hypoxia, inflammation, and cancer[END_REF]. Immunohistochemical staining of CAIX has been shown to be an endogenous surrogate marker for marked hypoxia in several types of solid tumors [START_REF] Russell | Immunohistochemical detection of changes in tumor hypoxia[END_REF][START_REF] Forker | The hypoxia marker CAIX is prognostic in the UK phase III VorteX-Biobank cohort: an important resource for translational research in soft tissue sarcoma[END_REF][START_REF] Ambrosio | Carbonic anhydrase IX is a marker of hypoxia and correlates with higher Gleason scores and ISUP grading in prostate cancer[END_REF].

Here, we demonstrated for the first time that hypoxic conditions could induce the expression of CAIX in ex vivo cultures of human prostate even in benign tissues. In PCa, the process of epithelial to mesenchymal transition (EMT) induced by hypoxia is involved in the resistance to both radiotherapy and chemotherapeutic treatment [START_REF] Montanari | Epithelial-mesenchymal transition in prostate cancer: an overview[END_REF][START_REF] Stark | Predictive value of epithelial-mesenchymal-transition (EMT) signature and PARP-1 in prostate cancer radioresistance[END_REF]. The key transcription factor of EMT, Zeb1, is associated in human PCa with cancer progression, metastases, and decreased survival [START_REF] Figiel | Clinical significance of epithelial-mesenchymal transition markers in prostate cancer[END_REF]. Here, we showed that, together with CAIX induction, hypoxia led to increased expression of Zeb1 in epithelial cells. Since EMT is a reversible process mostly driven by epigenetic mechanisms [START_REF] Wang | Epigenetic regulation of epithelial-mesenchymal transition by hypoxia in cancer: targets and therapy[END_REF][START_REF] Lee | Roles and epigenetic regulation of epithelial-mesenchymal transition and its transcription factors in cancer initiation and progression[END_REF], the use of organotypic cultures could therefore represent a valuable preclinical model to study the effects of drugs targeting epigenetics, such as histonemodifying enzymes.

In conclusion, we report for the first time an ex vivo culture model of human PCa that remains sensitive to androgen ablation, hypoxia, and extracellular Ca 2+ . This model could be used to test therapeutic agents in viable tumor samples, thereby allowing the development of a precision medicine in clinical practice. In cancer samples cultured in normoxic conditions, the hypoxia marker CAIX is negative (A), with negative to focal Zeb1 staining (C). After culture in hypoxia, cancer cells show a diffuse and membranous staining for CAIX (B), with a diffuse nuclear staining for Zeb1 (D). In contrast, hypoxia does not modify cancer cell proliferation, with less than 3% of Ki67 positive cells in both normoxic (E) and hypoxic (F) conditions. Scale bars = 100 µm.

  In non tumor tissues (n=7), all benign glands showed preservation of the p63 positive basal cell layer (Fig2A). In malignant tissue slices (n=8), cancer cells expressed p504S (Fig2B) in all cases, as observed in the native tumors.

  malignant samples were negative for CAIX (Fig 4A), and focally positive for Zeb1 (Fig 4C).

  towards basal intracellular Ca 2+ concentration when returning to a 2 mM extracellular Ca 2+ concentration (Fig 5A). The comparison of intracellular Ca 2+ entry between tumor and adjacent non tumor prostate slices shows an enhanced Ca 2+ entry in malignant compared to benign tissues (for nine cases out of 11) (Wilcoxon test, P = 0.014) (Fig 5B). In tumor tissues, intracellular Ca 2+ entry was greater in PCa from patients at high risk of recurrence (ISUP group 4 or more, or/and pT3) (n=5), compared to PCa from patients at low or intermediate risk of recurrence (ISUP group 1 to 3, and pT2) (n=6) (Mann Whitney, P = 0.045) (Table

Figure 1 :

 1 Figure 1: Tissue morphology and maintenance in culture. Radical prostatectomy specimen

Figure 2 :

 2 Figure 2: Biomarker expression in samples cultured 2 days under normoxic conditions.

Figure 3 :

 3 Figure 3: Expression of AR (A, B, C), PSA (D, E, F), and ERG (G, H, I) in native tumors

Figure 4 :

 4 Figure 4: Effects of hypoxia on the expression of CAIX, Zeb1 transcription factor, and Ki67.

Figure 5 :

 5 Figure 5: Variations in intracellular Ca 2+ in human prostate slices. Exposure to supraphysiological extracellular Ca 2+ concentration (5 mM) induces a robust and rapid Ca 2+ entry

20

 20

Table 1 :

 1 Patients and tissues characteristics.

Table 2 : Differences in calcium entry.

 2

	Risk of recurrence (d'Amico criteria)	Median ∆F/F0 (min-max)	p (Mann-Whitney)
	Low	1.08 (0.81-1.25)
			0.045
	High	1.30 (1.11-2.78)

Supported by Region Centre Val de Loire, Project "ReSCaP" Canceropole Grand Ouest, France.