Said Hamadène

Marie-Amelie Morlais

Viscosity solutions for second order integro-differential equations without monotonicity condition: The probabilistic Approach

Keywords: AMS Classification subjects: 60H30 Integro-differential equation, Backward stochastic differential equation with jumps, Viscosity solution, Non-local term

In this paper, we establish a new existence and uniqueness result of a continuous viscosity solution for integro-partial differential equation (IPDE in short). The novelty is that we relax the so-called monotonicity assumption on the driver which is classically assumed in the literature of viscosity solution of equation with non local terms. Our method strongly relies on the link between IPDEs and backward stochastic differential equations (BSDEs in short) with jumps for which we already know that the solution exists and is unique. In the second part of the paper, we deal with the IPDE with obstacle and we obtain similar results.

Introduction

In this paper, our objective is to establish a new existence and uniqueness result of the solution in viscosity sense of the following system of integro-partial differential equations: ∀i ∈ {1, . . . , m},

                 -∂ t u i (t, x) -b(t, x) ⊤ D x u i (t, x) -1 2 Tr σσ ⊤ (t, x)D 2 xx u i (t, x) -Ku i (t, x)
-h (i) (t, x, (u i (t, x)) i=1,m , (σ ⊤ D x u i )(t, x), B i u i (t, x)) = 0, (t, x) ∈ [0, T ] × R k ;

u i (T, x) = g i (x), ∀ i ∈ {1, • • • m}, m ∈ N * (1.1)
where the operators B i and K i are defined by

B i u i (t, x) = E γ i (t,
x, e) u i (t, x + β(t, x, e))u i (t, x) λ(de) and Ku i (t, x) = E u i (t, x + β(t, x, e))u i (t, x)β(t, x, e) ⊤ D x u i (t, x) λ(de).

(1.2)

We first note that, due to the presence of B i u i and K i u i in equation (1.1), such an IPDE is called of non-local type. IPDEs with non-local terms have been considered by several authors (see e.g. [START_REF] Alvarez | Viscosity solutions of nonlinear integro-differential equations[END_REF][START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF][START_REF] Barles | On the Dirichlet problem for second-order elliptic integrodifferential equations[END_REF][START_REF] Barles | Second-Order Elliptic Integro-Differential Equations : Viscosity Solutions' Theory Revisited[END_REF][START_REF] Dumetrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF][START_REF] Harraj | Double barriers Reflected BSDEs with jumps and viscosity solutions of parabolic Integro-differential PDEs[END_REF][START_REF] Lundstrom | Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness Manuscripta Mathematica[END_REF], etc. and * Université du Maine, LMM, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9, France, e-mail: hamadene@univ-lemans.fr † Université du Maine, LMM, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9, France, e-mail: Marie -Amelie.Morlais@univlemans.fr 1 the references therein). It is by now well-known that this IPDE is connected with the following multi-dimensional backward stochastic differential equation with jumps: ∀i ∈ {1, . . . , m}, dY i;t,x s = -f (i) (s, X t,x s , (Y i;t,x s ) i=1,m , Z i;t,x s , U i;t,x s )ds + Z i;t,x s dB s + E U i;t,x s (e)μ(ds, de), s ≤ T ;

Y i;t,x T = g i (X t,x T ) (1.3)
where (t, x) ∈ [0, T ] × R k , B := (B s ) s≤T is a Brownian motion, µ an independant Poisson random measure with compensator dsλ(de) (λ is the Lévy measure of µ) and μ(ds, de) := µ(ds, de)dsλ(de).

For completeness, let us recall some already known results in the IPDE literature (and also those concerning the related BSDE with jumps). In [START_REF] Tang | Necessary conditions for optimal control of stochastic systems with random jumps[END_REF], Tang-Li have shown that this BSDE with jumps (1.3) has a unique solution while Barles et al., in [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF], have made the connection between this BSDE and the IPDE (1.1). Actually in [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF], the authors have shown that if the coefficients f (i) , i = 1, ..., m, have the following form:

f (i) (t, x, y, z, ζ) = h (i) (t,
x, y, z, E γ i (t, x, e)ζ(e)λ(de)) (1.4) and, mainly, if (i) γ i ≥ 0 (ii) q -→ h (i) (t, x, y, z, q), is non-decreasing ;

then the deterministic continuous functions (u i (t, x)) i=1,m , defined by means of the representation of Feynman Kac's type of the processes (Y i;t,x ) i=1,m , i.e., ∀i = 1, . . . , m, Y i;t,x s = u i (s, X t,x s ) for s ∈ [t, T ] and then u i (t, x) := Y i;t,x t ,

is the unique viscosity solution of (1.1) in the class of functions of polynomial growth. The two assertions (i)-(ii)

above shall be referred later as the monotonicity conditions.

Therefore and in the first part of this paper, the main objective is to deal with IPDE (1.1) without assuming the two points (i)-(ii) above related to the non local term and the functions h (i) . Actually we show that when the measure λ is finite, equation (1.1) has a unique solution. Our method relies mainly on the following points:

(a) the characterization of the jump part of the BSDE (1.3) ;

(b) the existence and uniqueness of a solution of (1.3) for general f (i) , i = 1, . . . , m, which are merely Lipschitz in (y, z, ζ) and nothing more ;

(c) the existence and uniqueness result of a solution of the IPDE (1.1) in the case when h (i) does not depend on the component ζ, which involves the jump part. This result is already obtained in [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF].

Thus, our main contribution consists in proving the following result: there exists a unique viscosity solution of the general IPDE (1.1) (uniqueness holds within the class of continuous functions with at most polynomial growth (w.r.t. x)). Besides, the solution being given by the representation (1.5) of Feynman Kac's type, we fill in the gap between existence and uniqueness results for BSDE with jumps of the form (1.3) (results which are already available for BSDEs and do not require the monotonicity conditions) and the results available in the IPDE literature. Finally, let us mention that, due to the presence of the operator K inside the functional h in IPDE (1.1) and since neither (i) or (ii) holds, one cannot prove, as it is usual in viscosity literature, the classical comparison theorem. This motivates the introduction of a new definition of viscosity solution, which coincides with the usual one when h does not depend on the jump part and which allows to apply the classical results such as those obtained in [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF].

According to the best of our knowledge and without assuming the two points (i)-(ii), such a result of existence and uniqueness of the solution of IPDE (1.1) has not been obtained so far. We should also mention here one crucial point: since we assume the Lévy measure λ is bounded, the operators B i u i are well-posed for functions which grow as polynomials w.r.t. x at infinity. Thus we naturally introduce a new definition of viscosity solution (for IPDES with or without obstacle). The main point is that even if our definitions are a bit different from the ones given in [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF][START_REF] Dumetrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF][START_REF] Harraj | Double barriers Reflected BSDEs with jumps and viscosity solutions of parabolic Integro-differential PDEs[END_REF], etc., but one can show that they coincide if (i)-(ii) above are satisfied. As a consequence, our study naturally extends the already known results in the IPDE literature.

In the second part of this paper, we consider the following IPDE with obstacle (m = 1):

       min u(t, x) -ℓ(t, x); -∂ t u(t, x) -b(t, x) ⊤ D x u(t, x) -1 2 Tr σσ ⊤ (t, x)D 2 xx u(t, x) -Ku(t, x) -h(t, x, u(t, x), (σ ⊤ D x u)(t, x), Bu(t, x)) = 0, (t, x) ∈ [0, T ] × R k ; u(T, x) = g(x) (1.6)
where the operators Bu and Ku are defined similarly as in (1.2) (just take m = 1). Once again, this IPDE with obstacle (1.6) is connected with the following reflected BSDE with jumps:

       dY t,x s = -f (s, X t,x s , Y t,x s , Z t,x s , U t,x s )ds -dK t,x s + Z t,x s dB s + E U t,x s (e)μ(ds, de), s ≤ T ; Y t,x s ≥ ℓ(s, X t,x s ), s ≤ T and T 0 (Y t,x s -ℓ(s, X t,x s ))dK t,x s = 0; Y t,x T = g(X t,x T ) (1.7)
for which Hamadène-Ouknine [START_REF] Hamadène | Reflected backward stochastic differential equation with jumps and random obstacle Electron[END_REF] provide a unique solution for general generators f by means of a fixed point theorem. The related IPDE is considered in several papers amongst one can quote ( [START_REF] Dumetrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF][START_REF] Harraj | Double barriers Reflected BSDEs with jumps and viscosity solutions of parabolic Integro-differential PDEs[END_REF], etc.). However in those papers the conditions (i)-(ii) above on γ 1 and h are assumed. Therefore our second main objective is to deal with the IPDE with obstacle (1.6) for general functions h and γ which do not satisfy (i)-(ii). Indeed, similarly to the framework without obstacle, by using reflected BSDEs with jumps, we show that equation (1.6) has a unique solution when the Lévy measure λ is bounded. This solution is also obtained with the help of the representation of Feynman Kac's formula of the unique solution of (1.7).

The outline of the paper is as follows: in the following second section, we provide all the necessary notations, assumptions and preliminary results concerning the study of general IPDEs (1.1) and related BSDEs with jumps as well. In the third and fourth sections, we proceed with the two main results of the paper: (i) we first provide the main theoretical result of the paper, i.e. the existence and uniqueness of the solution of the general non linear IPDE ; (ii) we generalize the result of the first part to IPDEs with obstacle. For completeness, usual definitions for viscosity solutions for both a non linear IPDE with and without obtacle are provided in an Appendix.

2 Preliminary results on BSDEs with jumps and their associated

IPDEs

For sake of clarity, let us give the framework of our study as well as some notations which shall be used throughout the paper. In particular, we shall deeply rely on the relationship between the viscosity solution of some IPDEs and the solution of the related BSDE with jumps. Therefore and for sake of completeness, we need to introduce the stochastic framework and then give the connection with the integro-partial differential equation we shall study.

Let (Ω, F , (F t ) t≤T , P) be a stochastic basis such that F 0 contains all P-null sets of F , and F t = F t+ := ε>0 F t+ε , t ≥ 0, and we suppose that the filtration is generated by the two mutually independant processes: (i) B := (B t ) t≥0 a d-dimensional Brownian motion and (ii) a Poisson random measure µ on R + × E, where E := R ℓ -{0} is equipped with its Borel field E (ℓ ≥ 1).

The compensator ν(dt, de) = dtλ(de) is such that {μ([0, t] × A) = (µν)([0, t] × A)} t≥0 is a martingale for all Next we denote by: (iii) P (resp. P) the field on [0, T ] × Ω of (F t ) t≤T -progressively measurable (resp. predictable) sets ;

(iv) L 2 (λ) the space of Borel measurable functions ϕ := (ϕ(e)

) e∈E from E into R such that ϕ 2 L 2 (λ) := E |ϕ(e)| 2 λ(de) < ∞ ; (v) S 2 (R ℓ ) (ℓ ∈ N *
) the space of RCLL (for right continuous with left limits) P-measurable and R ℓ -valued

processes such that E sup s≤T |Y s | 2 < ∞ ; A 2
c is its subspace of continuous non-decreasing processes (K t ) t≤T such that K 0 = 0 ;

(vi) H 2 (R ℓ×d ) the space of processes Z := (Z s ) s≤T which are P-measurable, R ℓ×d -valued and satisfying E[

T 0 |Z s | 2 ds] < ∞ ; (vii) H 2 (L 2 (λ)
) the space of processes U := (U s ) s≤T which are P-measurable, L 2 (λ)-valued and satisfying

E[ T 0 U s (ω) 2 L 2 (λ) ds] < ∞ ; (viii) Π g the set of deterministic functions ̟: (t, x) ∈ [0, T ] × R k → ̟(t, x) ∈ R of polynomial growth, i.e., for
which there exists two constants C and p such that for any (t,

x) ∈ [0, T ] × R k , |̟(t, x)| ≤ C(1 + |x| p ).
The subspace of Π g of continuous functions will be denoted by Π c g ; (ix) For any process θ := (θ s ) s≤T and t ∈ (0, T ], θ t-= lim sրt θ s and ∆ t θ = θ tθ t-. Now let b and σ be the following functions:

b : (t, x) ∈ [0, T ] × R k → b(t, x) ∈ R k σ : (t, x) ∈ [0, T ] × R k → σ(t, x) ∈ R k×d .
We assume that they are jointly continuous in (t, x) and Lipschitz continuous w.r.t. x uniformly in t, i.e., there exists a constant C such that

∀ (t, x, x ′ ) ∈ [0, T ] × R k+k , |b(t, x) -b(t, x ′ | + |σ(t, x) -σ(t, x ′ )| ≤ C|x -x ′ |.
(2.1)

Since b and σ are jointly continuous then by (2.1), we easily deduce that they are of linear growth, i.e., there exists a constant C such that

∀ (t, x) ∈ [0, T ] × R d |b(t, x)| + |σ(t, x)| ≤ C(1 + |x|). (2.2) Let β : (t, x, e) ∈ [0, T ] × R k × E → β(t,
x, e) ∈ R k be a measurable function such that for some real constant C, and for all e ∈ E,

(i) |β(t, x, e)| ≤ C(1 ∧ |e|); (ii) |β(t, x, e) -β(t, x ′ , e)| ≤ C|x -x ′ |(1 ∧ |e|); (iii) the mapping (t, x) ∈ [0, T ] × R k → β(t, x, e) ∈ R k is continuous uniformly w.r.t. e. (2.3) Next let (t, x) ∈ [0, T ] × R k and (X t,x s
) s≤T be the stochastic process solution of the following standard stochastic differential equation of diffusion-jump type:

X t,x s = x + s t b(r, X t,x r )dr + s t σ(r, X t,x r )dB r + s t E β(r, X t,x r-, e)μ(dr, de), for s ∈ [t, T ] and X t,x s = x if s ≤ t. (2.4)
Under assumptions (2.1), (2.2) and (2.3) the solution of equation (2.4) exists and is unique (see [START_REF] Fujiwara | Stochastic differential equations of jump type and Lvy processes in differomorphism group[END_REF] for more details). Moreover it satisfies the following estimates:

∀p ≥ 2, x, x ′ ∈ R k and s ≥ t, E[ sup r∈[t,s] |X t,x r -x| p ] ≤ M p (s -t)(1 + |x| p )] and E[ sup r∈[t,s] |X t,x r -X t,x ′ r -(x -x ′ )| p ] ≤ M p (s -t)|x -x ′ | p (2.5)
for some constant M p .

Once for all and throughout this paper, we assume that (2.1), (2.2) and (2.3) are satisfied.

Let us now consider the following m-dimensional backward stochastic differential equation with jumps ((t, x) ∈

[0, T ] × R k ):        Y t,x := (Y i;t,x ) i=1,m ∈ S 2 (R m ), Z t,x ∈ H 2 (R m×d ), U t,x := (U i;t,x ) i=1,m ∈ (H 2 (L 2 (λ)) m ; ∀i ∈ {1, . . . , m}, Y i T = g i (X t,x T ) and ∀s ≤ T, dY i;t,x s = -f (i) (s, X t,x s , Y t,x s , Z i;t,x s , U i;t,x s )ds -Z i;t,x s dB s -E U i;t,x
s (e)μ(ds, de).

(2.6)

where for any i ∈ {1, . . . , m},

(i) f (i) is a deterministic measurable function from [0, T ] × R k+m+m×d × L 2 (λ) into R ; (ii) Z i;t,x s is the i-th row of Z t,x s and U i;t,x s is the i-th component of U t,x s ;
(iii) g i are Borel measurable deterministic functions from R k to R.

We now consider the following assumptions:

(H1): For any i ∈ {1, . . . , m},

(i) f (i) is Lipschitz in (y, z, u) uniformly in (t, x), i.e.
, there exists a real constant C such that for any

(t, x) ∈ [0, T ] × R k , (y, p, ζ) and (y ′ , p ′ , ζ ′ ) elements of R m+d × L 2 (λ), |f (i) (t, x, y, p, ζ) -f (i) (t, x, y ′ , p ′ , ζ ′ )| ≤ C(|y -y ′ | + |p -p ′ | + ζ -ζ ′ L 2 (λ) ). (2.7)
(ii) the functions f (i) (t, x, 0, 0) and g i are of polynomial growth, i.e., belong to Π g .

(H2): For any i ∈ {1, . . . , m}:

(i) the functions g i are continuous ;

(ii) the mapping (t, x)

∈ [0, T ] × R k -→ f (i) (t, x, y, z, ζ) ∈ R is continuous uniformly w.r.t. ( y, z, ζ).
BSDEs with jumps have been already considered by Li-Tang in [START_REF] Tang | Necessary conditions for optimal control of stochastic systems with random jumps[END_REF] where they have provided the following result related to existence and uniqueness of the solution of (2.6) (see also the paper by Barles et al. [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]).

Proposition 2.1. (Tang-Li, [START_REF] Tang | Necessary conditions for optimal control of stochastic systems with random jumps[END_REF]): Assume that Assumption (H1) is fulfilled. Then for any

(t, x) ∈ [0, T ] × R k , the BSDE (2.6) has a unique solution ( Y t,x , Z t,x , U t,x ).
Next let us consider the following structure condition on the functions (f (i) ) i=1,m .

(H3): For any i ∈ {1, . . . , m}, there exists a Borel measurable deterministic function

h (i) from [0, T ] × R k+m+d+1
into R such that:

f (i) (t, x, y, z, ζ) = h (i) (t, x, y, z, E ζ(e)γ i (t, x, e)λ(de)) (2.8)
where for i = 1, . . . , m, γ i is Borel measurable and verifies:

∀(t, x, x ′ ) ∈ [0, T ] × R k+k and e ∈ E, there exists a constant C ≥ 0, (i) |γ i (t, x, e)| ≤ C(1 ∧ |e|) (ii) |γ i (t, x, e) -γ i (t, x ′ , e)| ≤ C(1 ∧ |e|)|x -x ′ | (iii) the mapping t ∈ [0, T ] -→ γ(t,
x, e)is continuous uniformly w.r.t. (x, e).

(2.9)

We then have the following result whose proof is given in Barles et

(c-i) γ i ≥ 0 ; (c-ii) for any fixed (t, x, y, z) ∈ [0, T ]×R k+m+d , the mapping q ∈ R -→ h (i) (t, x, y, z, q) ∈ R is non-decreasing; then (u i ) i=1,
       -∂ t u i (t, x) -b(t, x) ⊤ D x u i (t, x) -1 2 Tr σσ ⊤ (t, x)D 2 xx u i (t, x) -Ku i (t, x) -h (i) (t, x, (u i (t, x)) i=1,m , (σ ⊤ D x u i )(t, x), B i u i (t, x)) = 0, (t, x) ∈ [0, T ] × R k ; u i (T, x) = g i (x) (2.11)
where

B i u i (t, x) = E γ i (t, x, e){u i (t, x + β(t,
x, e))u i (t, x)}λ(de) and

Ku i (t, x) = E {u i (t, x + β(t, x, e)) -u i (t, x) -β(t,
x, e) ⊤ D x u i (t, x)}λ(de).

(2.12)

Finally, the solution (u i (t, x)) i=1,m is unique in the class of continuous functions of Π g .

Remark 2.1. By (2.10), for any i ∈ {1, . . . , m} and (t,

x) ∈ [0, T ] × R k , u i (t, x) := Y i;t,x t . (2.13)
3 The first main result: Existence and uniqueness of the solution for system of IPDEs

To begin with, we are going to deal with the link between the stochastic process U i;t,x of the BSDE (2.6) and the function u i defined in (2.13). For that, we need to assume additionally the following hypothesis on the Lévy measure λ.

(H4): The measure λ is finite, i.e., λ(E) < ∞.

Proposition 3.1. Assume that (H1)-(H4) are fulfilled. Then for any i = 1, . . . , m,

U i;t,x s (e) = u i (s, X t,x s-+ β(s, X t,x s-, e)) -u i (s, X t,x s-), ds ⊗ dP ⊗ dλ on [t, T ] × Ω × E. (3.1)
Proof. Let i be fixed. First note that since u i belongs to Π g and β is bounded then by (H4) we have

E[ T 0 E {|U i;t,x s (e)| 2 + |u i (s, X t,x s-+ β(s, X t,x s-, e) -u i (s, X t,x s-)| 2 }λ(de)ds] < ∞
and hence, due to the finiteness of λ, one has

E[ T 0 E {|U i;t,x s (e)| + |u i (s, X t,x s-+ β(s, X t,x s-, e) -u i (s, X t,x s-)|}λ(de)ds] < ∞.
Therefore and referring to [START_REF] Cont | Financial modelling with jump processes[END_REF] = u i (s, X t,x s ) and u i is continuous then

s t E (u i (r, X t,x r-+ β(s, X t,x s-, e) -u i (r, X t,x r-)µ(dr, de) = t<r≤s {Y i;t,x r -Y i;t,x r-}. (3.3)
It follows that for any s ∈ [t, T ],

s t E (u i (r, X t,x r-+ β(r, X t,x r-, e)) -u i (r, X t,x r-) -U i;t,x
r (e))µ(dr, de) = 0.

Taking now the quadratic variation of this last process and then expectation to obtain

E[ T t dr E |u i (r, X t,x r-+ β(r, X t,x r-, e)) -u i (r, X t,x r-) -U i;t,x r (e)| 2 λ(de)] = 0
which provides the desired equality.

Remark 3.1. This characterization of U i;t,x in terms of u i which is given in (3.1) plays a prominent role in the proof of our main result. It is obtained under the condition (H4) of finiteness of the Lévy measure λ. However it can also be obtained under other conditions by using e.g. Malliavin calculus (see e.g. [START_REF] Delong | Reflected Backward SDEs with jumps and their financial and actuarial applications[END_REF], pp.84). But the use of Malliavin calculus requires stringent regularity condition on the data, therefore we do not use it as we are interested in obtaining results for quite general IPDEs.

Existence and uniqueness of the solution of the system of IPDEs

We first give our meaning of the definition of the viscosity solution of system (1.1). It is not exactly the same as the one of Barles et al.'s paper (see Definition 4.1 in Appendix).

For any function φ belonging to C 1,2 ([0, T ] × R k ) and R-valued, we define L X φ by

L X φ(t, x) = 1 2 Tr[(σσ ⊤ )(t, x)D 2 xx φ(t, x)] + b(t, x) ⊤ D x φ(t, x) + Kφ(t, x), (t, x) ∈ [0, T ] × R k ,
where Kφ(t, x) is given in (2.12), and it is actually well-posed for any φ in C 1,2 ([0, T ] × R k ).

Definition 3.2. A family of deterministic functions u = (u i ) i=1,m , such that, for any i ∈ {1, . . . , m}, the map u i : (t, x) → u i (t, x) belongs to Π c g (spaces of continuous functions with at most polynomial growth w.r.t x), is said to be a viscosity sub-solution (resp. super-solution) of the IPDE (1.1) if: ∀i ∈ {1, . . . , m},

(i) ∀x ∈ R k , u i (T, x) ≤ g i (x) (resp. u i (T, x) ≥ g i (x)) ; (ii) For any (t, x) ∈ (0, T ) × R k and any function φ of class C 1,2 ([0, T ] × R k ) such that (t, x) is a global maximum (resp. minimum) point of u i -φ and (u i -φ)(t, x) = 0, one has -∂ t φ(t, x) -L X φ(t, x) -h (i) (t, x, (u i (t, x)) i=1,m , σ ⊤ (t, x)D x φ(t, x), B i (u i )(t, x)) ≤ 0, (resp. -∂ t φ(t, x) -L X φ(t, x) -h (i) (t, x, (u i (t, x)) i=1,m , σ ⊤ (t, x)D x φ(t, x), B i (u i )(t, x)) ≥ 0.)
The family u = (u i ) i=1,m is a viscosity solution of (1.1) if it is both a viscosity sub-solution and viscosity super-solution.

Let us now compare the two definitions 3.2 and 4.1 of viscosity solutions: (ii) In our Definition 3.2, we have used B i u i instead of B i φ: indeed B i u i is well posed since u i is in Π g , β is bounded and λ finite while it is replaced by B i φ (φ being the smooth test function) in Barles et al.'s definition (Definition 4.1 in Appendix). In the latter definition of Barles et al.'s this is the lack of regularity of u i which makes that B i u i is ill-posed.

(iii) We finally mention that this new definition is crucial in our proof of the existence result and more precisely to prove that our candidate is a viscosity solution: more precisely and by using this new definition, we shall be able to use classical existence results for IPDE obtained in Barles et al. [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]: the main point being that, when the non local term is frozen, we fall in Barles et al.'s framework.

The main result of this paper is the following one: Theorem 3.4. Under Assumptions (H1)-(H4), the family of functions (u i ) i=1,m which belong to Π c g defined in (2.13) is a viscosity solution of (1.1). Moreover it is unique in the class Π c g .

Proof. Let us consider the following multi-dimensional BSDE:

                           Y t,x := (Y i;t,x ) i=1,m ∈ S 2 (R m ), Z t,x ∈ H 2 (R m×d ), U t,x := (U i;t,x ) i=1,m ∈ (H 2 (L 2 (λ)) m ; ∀i ∈ {1, . . . , m}, Y i T = g i (X t,x T ) and ∀s ≤ T, dY i;t,x s = -h (i) (s, X t,x s , Y t,x s , Z i;t,x s , E γ i (s, X t,x s , e){u i (s, X t,x s + β(s, X t,x s , e)) -u i (s, X t,x s )}λ(de))ds +Z i;t,x s dB s + E U i;t,x
s (e)μ(ds, de).

(3.4)

As, for any i = 1, ..., m, u i belongs to Π c g , β(t, x, e) is bounded and verifies (2.3) and since λ is finite, the solution of this equation exists and is unique by Proposition 2.1, noting that the functions g i and (t, x, y, z) -→ h (i) (t, x, y, z, E γ i (t, x, e){u i (t, x + β(t, x, e))u i (t, x)}λ(de))

verify Assumptions (H1). Moreover, by Proposition 2.2, there exists a family of deterministic continuous functions of polynomial growth (u i ) i=1,m such that for any (t, x)

∈ [0, T ] × R k , ∀s ∈ [t, T ], Y i;t,x s = u i (s, X t,x s ).
Next by Proposition 2.2 and Remark 3.3-(i), the family (u i ) i=1,m is a viscosity solution of the following system:

       -∂ t u i (t, x) -b(t, x) ⊤ D x u i (t, x) -1 2 Tr σσ ⊤ (t, x)D 2 xx u i (t, x) -Ku i (t, x) -h (i) (t, x, (u i (t, x)) i=1,m , (σ ⊤ D x u i )(t, x), B i u i (t, x)) = 0, (t, x) ∈ [0, T ] × R k ; u i (T, x) = g i (x) (3.5)
Note that in this system (3.5), the last component of h (i) is B i u i (t, x) and not B i u i (t, x).

Next and once more, let us consider the system of BSDEs by which the family (u i ) i=1,m is defined through the Feynman Kac's formula (2.13). Such a system of BSDEs is given by

                           Y t,x := (Y i;t,x ) i=1,m ∈ S 2 (R m ), Z t,x ∈ H 2 (R m×d ), U t,x := (U i;t,x ) i=1,m ∈ (H 2 (L 2 (λ)) m ; ∀i ∈ {1, . . . , m}, Y i T = g i (X t,x
T ) and ∀s ≤ T,

dY i;t,x s = -h (i) (s, X t,x s , Y t,x s , Z i;t,x
s , E γ i (s, X t,x s , e)U i;t,x s (e)λ(de))ds +Z i;t,x s dB s + E U i;t,x s (e)μ(ds, de).

(3.6)

Since we know that, for any i in {1, • • • , m}, u i belongs to Π c g , therefore and due to Proposition 3.1, one has

U i;t,x s (e) = u i (s, X t,x s + β(s, X t,x s , e)) -u i (s, X t,x s ), ds ⊗ dP ⊗ dλ on [t, T ] × Ω × E.
Plugging now this relation in the first term of the right-hand side of the second equality of (3.6), one obtains, by uniqueness of the solution of the BSDE (3.4), that for any s ∈ [t, T ] and i ∈ {1, . . . , m}, Y i;t,x s = Y i;t,x s . Thus for any i ∈ {1, . . . , m}, u i = u i . Henceforth, the family (u i ) i=1,m is a viscosity solution of (1.1) in the sense of Definition 3.2.

Next, let us show that it is unique in the class Π c g . So let (ū i ) i=1,m be another family of Π c g which is solution of the system (1.1) in the sense of Definition 3.2.

Let us consider the following system of BSDEs:

                           Ȳ t,x := ( Ȳ i;t,x ) i=1,m ∈ S 2 (R m ), Zt,x ∈ H 2 (R m×d ), Ū t,x := ( Ū i;t,x ) i=1,m ∈ (H 2 (L 2 (λ)) m ; ∀i ∈ {1, . . . , m}, Ȳ i T = g i (X
Therefore there exists a family of deterministic continuous functions

(v i ) i=1,m of class Π g such that ∀s ∈ [t, T ], Ȳ i;t,x s = v i (s, X t,x s ).
Additionally, by Definition 4.1 and Proposition 2.2, (v i ) i=1,m is the unique solution in the subclass Π c g of continuous functions of the following system:

       -∂ t v i (t, x) -b(t, x) ⊤ D x v i (t, x) -1 2 Tr σσ ⊤ (t, x)D 2 xx v i (t, x) -Kv i (t, x) -h (i) (t, x, (v i (t, x)) i=1,m , (σ ⊤ D x v i )(t, x), B i ūi (t, x)) = 0, (t, x) ∈ [0, T ] × R k ; v i (T, x) = g i (x) (3.8)
But, the family (ū i ) i=1,m belongs to Π c g and solves system (3.8). Therefore, by the uniqueness result of Proposition 2.2 and Remark 3.3 -(i), for any i ∈ {1, . . . , m}, one deduces that ūi = v i . On the other hand, by the characterization of the jumps of Proposition 3.1, for any i ∈ {1, . . . , m}, it holds

Ū i;t,x s (e) = v i (s, X t,x s-+ β(s, X t,x s-, e)) -v i (s, X t,x s-) = ūi (s, X t,x s-+ β(s, X t,x s-, e)) -ūi (s, X t,x s-).
Next by replacing in (3.7) the quantity ūi (s, X t,x s-+ β(s, X t,x s-, e)) -ūi (s, X t,x s-) with Ū i;t,x s (e), we deduce that

∀ i ∈ {1, . . . , m}, Ȳ i;t,x = Y i;t,x
since the solution of the BSDE (3.7) is unique. Thus and for any i ∈ {1, . . . , m}, u i = ūi = v i which means that the solution of (1.1) in the sense of Definition 3.2 is unique inside the class Π c g .

Remark 3.5. Since the Lévy measure λ is assumed to be bounded then one can relax a bit the conditions (2.3) and (2.9) on β and (γ i ) i=1,m respectively.

The second main result: Generalization to IPDEs with obstacles

The previous result can be generalized to IPDEs with one (either lower or upper) obstacle. Actually assume that m = 1 and let us denote f (1) , h (1) , g 1 and γ 1 simply by f , h, g and γ respectively. Next let us consider the following IPDE with obctacle ℓ, which is a function of (t, x):

       min u(t, x) -ℓ(t, x); -∂ t u(t, x) -b(t, x) ⊤ D x u(t, x) -1 2 Tr σσ ⊤ (t, x)D 2 xx u(t, x) -Ku(t, x) -h(t, x, u(t, x), (σ ⊤ D x u)(t, x), Bu(t, x)) = 0, (t, x) ∈ [0, T ] × R k ; u(T, x) = g(x) (3.9)
where once again the operators Bu and Ku are given by: Bu(t, x) = E γ(t, x, e){u(t, x + β(t, x, e))u(t, x)}λ(de) and

Ku(t, x) = E {u(t, x + β(t, x, e)) -u(t, x) -β(t, x, e) ⊤ D x u(t, x)}λ(de).
(3.10) Note that under (H4) if u belongs to Π g then the operator Bu is well-posed.

The general reflected BSDE with jumps associated with IPDE with obstacle (3.9) is the following one:

                               Y t,x ∈ S 2 (R), Z t,x ∈ H 2 (R d ), U t,x ∈ H 2 (L 2 (λ)) and K t,x ∈ A 2 c ; dY t,x s = -f (s, X t,x s , Y t,x s , Z t,x s , U t,x s )ds -dK t,x s + Z t,x s dB s + E U t,x
s (e)μ(ds, de), s ≤ T ;

Y t,x s ≥ ℓ(s, X t,x s ), s ≤ T and T 0 (Y t,x s -ℓ(s, X t,x s ))dK t,x s = 0; Y t,x T = g(X t,x T ) (3.11) where (t, x) ∈ [0, T ] × R k is fixed.
The following result related to existence and uniqueness of a solution for this reflected BSDE with jumps Proposition 3.2. [START_REF] Hamadène | Reflected backward stochastic differential equation with jumps and random obstacle Electron[END_REF] Assume that:

(i) f is Lipschitz in (y, z, ζ) ∈ R 1+d × L 2 (
λ) uniformly w.r.t. (t, x) and the function

(t, x) ∈ [0, T ] × R k -→ f (t,
x, 0, 0, 0) belongs to Π g ;

(ii) g belongs to Π g and ℓ(T, x) ≥ g(x), ∀x ∈ R ;

(iii) ℓ is continuous and belongs to Π g .

Then the BSDE (3.11) has a unique solution (Y t,x , Z t,x , U t,x , K t,x ). Moreover is satisfies the following estimate:

E sup s≤T |Y t,x s | 2 +(K t,x T ) 2 + T 0 {|Z t,x s | 2 + U t,x s 2 L 2 (λ) }ds ≤ CE |g(X t,x T )| 2 +sup s≤T |ℓ(s, X t,x s )| 2 + T 0 |f (s, X t,x s , 0, 0, 0)| 2 ds . (3.12)
Our main objective is now to connect the solution of the RBSDE with jumps with the solution in viscosity sense of IPDE with obstacle (3.9). To begin with let us precise the definition of viscosity solution we deal with.

Definition 3.6. We say that a function u(t, x) which belongs to Π c g is a viscosity sub-solution (resp. supersolution) of the IPDE (3.9) if:

(i) ∀x ∈ R k , u(T, x) ≤ g(x) (resp. u(T, x) ≥ g(x)) ;

(ii) For any (t, x) ∈ (0, T ) × R k and any function φ of class C 1,2 ([0, T ] × R k ) such that (t, x) is a global maximum (resp. minimum) point of uφ and (uφ)(t, x) = 0, one has min u(t, x)ℓ(t, x); -∂ t φ(t, x) -L X φ(t, x)h(t, x, u(t, x), σ ⊤ (t, x)D x φ(t, x), Bu(t, x)) ≤ 0, (resp.

min u(t, x) -ℓ(t, x); -∂ t φ(t, x) -L X φ(t, x) -h(t, x, u(t, x), σ ⊤ (t, x)D x φ(t, x), Bu(t, x)) ≥ 0.)
The function u is a viscosity solution of (3.9) if it is both a viscosity sub-solution and viscosity super-solution.

Next let us introduce the following assumptions:

(H5) (i) The assumptions of Proposition 3.2 are satisfied ;

(ii) the function γ(t, x, e) verifies (2.9) ;

(iii) the function h(t, x, y, z, η) such that

f (t, x, y, z, ζ) = h(t, x, y, z, E ζ(e)γ(t, x, e)λ(de)) (3.13)
is continuous in (t, x, y, z, η) and Lipschitz in (x, y, z, η) uniformly w.r.t. t ;

(iv) the function g is continuous in x.

We then have the following result related to the solution of (3.11) which exists under (H5). Proof. Let Σ := H 2 (R) × H 2 (L 2 (λ)) and Ψ be the functional which with a pair of processes (y, v) which belongs to Σ associates Ψ(y, v) := (Y, V ) such that (Y, Z, V, K) is the solution of the following reflected BSDE with jumps:

                               Y ∈ S 2 (R), Z ∈ H 2 (R d ), V ∈ H 2 (L 2 (λ)) and K ∈ A 2 c ; dY s = -f (s, X t,x s , y s , Z s , v s )ds -dK s + Z s dB s + E V s (e)μ(ds, de), s ≤ T ; Y s ≥ ℓ(s, X t,x s ), s ≤ T and T 0 (Y s -ℓ(s, X t,x s ))dK s = 0 ; Y T = g(X t,x T ) (3.16)
where (t, x) ∈ [0, T ] × R k is fixed (we have omitted the dependance in (t, x) of (Y, V ) as there is no confusion).

The solution of this equation exists thanks to ([9], Theorem 1.2.b). Next for α ∈ R, let us define the norm . α on Σ by:

(y, v) α := E[ T 0 e αs {|y s | 2 + v s 2 L 2 (λ) }ds].
As in [START_REF] Hamadène | Reflected backward stochastic differential equation with jumps and random obstacle Electron[END_REF], Theorem 1.2.b, one can show that for an appropriate α 0 , Ψ is a contraction on (Σ, . α0 ) and thus, this mapping has a unique fixed point (Y t,x , U t,x ) which with Z t,x and K t,x gives the unique solution of (3.11). Let us now consider the following sequence of processes:

(Y 0 , V 0 ) = (0, 0) and for n ≥ 1, (Y n , V n ) = Ψ(Y n-1 , V n-1 ).
Then obviously (Y n , V n ) n≥0 converges in (Σ, . α0 ) to (Y t,x , U t,x ). Next by an induction argument we have: for any n ≥ 0:

(i) there exists a deterministic continuous function

u n : (t, x) ∈ [0, T ] × R k → u n (t, x) which belongs to Π g such that for any s ∈ [t, T ], Y n s = u n (s, X t,x s ) ; (ii) u n belongs to Π c g and V n s (e) := u n (s, X t,x s-+ β(s, X t,x s-, e)) -u n (s, X t,x s-), ds ⊗ dP ⊗ dλ on [t, T ] × Ω × E.
Indeed for n = 0, the properties (i), (ii) are valid. So suppose that they are satified for some n. Then

(Y n+1 , Z n+1 , V n+1 , K n+1 ) verifies: ∀s ∈ [t, T ],                            dY n+1 s = -f (s, X t,x s , u n (s, X t,x s ), Z n+1 s , E {u n (s, X t,x s-+ β(s, X t,x s-, e)) -u n (s, X t,x s-)}γ(s, X t,x s-, e)λ(de))ds -dK n+1 s + Z n+1 s dB s + E V n+1 s (e)μ(ds, de), ; Y n+1 s ≥ ℓ(s, X t,x s ) and T t (Y n+1 s -ℓ(s, X t,x s ))dK n+1 s = 0 ; Y n+1 T = g(X t,x T ). ( 3
.17)

Therefore the existence and continuity of u n+1 are obtained in the same way as in ( [START_REF] Dumetrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF], [START_REF] Harraj | Double barriers Reflected BSDEs with jumps and viscosity solutions of parabolic Integro-differential PDEs[END_REF]) since the generator of Y n+1 does not depend on V n+1 . Note that by (3.12) we easily deduce that u n+1 belongs to Π g . Finally the last property of (ii), i.e.,

V n+1 s (e) := u n+1 (s, X t,x s-+ β(s, X t,x s-, e)) -u n+1 (s, X t,x s-), ds ⊗ dP ⊗ dλ on [t, T ] × Ω × E.
is obtained in a similar fashion as in Proposition 3.1. The proof of the induction procedure and thus of the two claims (i) and (ii) is now complete.

It now remains to justify that the two representations (3.14) and (3.15) of both processes Y t,x and U t,x hold at the limit (when n goes to ∞). Then and to proceed we note that the following inequality holds: Therefore, the sequence of functions (u n ) n≥0 converges pointwisely in [0, T ] × R k to a deterministic function u.

T t (Y n+1;t,x s -Y n+1;t,x s )d(K n+1;t,x s -K n+1;t,x s ) ≤ 0. ( 3
-Y m+1 s ) 2 + T s |Z n+1 r -Z m+1 r | 2 dr + s<r≤T (∆ r (Y n+1 -Y m+1 r )) 2 ≤ T s (Y n+1 r -Y m+1 r )(f (r, X t,x r , Y n r , Z n+1 r , V n r ) -f (
Moreover for any (t, x) ∈ [0, T ] × R k we have ∀s ∈ [t, T ], Y t,x s = u(s, X t,x s ).

Finally the continuity of u is obtained in a similar way as in (( [START_REF] Dumetrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF]), pp.6) or ( [START_REF] Harraj | Double barriers Reflected BSDEs with jumps and viscosity solutions of parabolic Integro-differential PDEs[END_REF]), pp.45) and relying to the proof of Proposition 3.1 we obtain (3.15) since λ is finite.

We now are ready to give the main result of this subsection.

Conclusion

In this paper, we provide a new theoretical result of existence and uniqueness for solutions of some general class of non linear IPDEs. Especially and to our knowledge, there does not exist any study concerning viscosity solutions of such equations without assuming the monotonicity condition on the driver (with respect to its jump component). We note that our result deeply relies on the relationship between the solution of the non linear IPDE and the one of the related BSDE with jumps, relation given by the Feynman-Kac's formula. We also mention that since our proof is based on this relationship with some explicit BSDE (or reflected BSDE) with jumps, we obtain without additional difficulties the (existence and uniqueness) result both for the multidimensional case and for non linear IPDEs with one obstacle (see last Section 3.2). As a consequence, this enlarges the class of economic and financial optimizations and/or control problems we can deal with which naturally lead to the study of partial differential equations (or system of equations). Additionnaly , this study reinforces the interest of using probabilistic tools in order to study PDEs or system of variational inequalities related to optimization problems. Another future application we have in mind is that this new result could be applied to obtain numerical implementations of such IPDEs.

Appendix

Remark 3. 3 .

 3 (i) If for any i ∈ {1, . . . , m}, the function h(i) does not depend on its last component ζ then Definitions 3.2 and 4.1 are the same.

( 3 .

 3 11) is given in ([9], Theorem 1.2.b).

)| 2 ]

 2 dB r -2 T s E {(Y n+1 r--Y m+1 r-)(V n+1 r (e) -V m+1r (e))μ(de, dr).Then in a classical way we obtain that → 0 as n, m → ∞.

Theorem 3 . 7 .

 37 Assume that (H4), (H5) are fulfilled. Then the function u defined in(3.14) is the unique viscosity solution of (3.9) in the class Π c g .Proof. : The proof is similar to the case without obstacle and based on the following facts: (i) u is continuous and belongs to Π g ; (ii) The solution of the BSDE (3.11) exists and is unique and is connected to u by relation (3.14); (iii) The characterization of the jumps of Y t,x by relation (3.15) ; (iv) The generalization of Barles et al.'s definition (from the case without obstacle) to the case with obstacle and which is given in Appendix (Definition 4.2). This generalization coincides with our definition when the generator h does not depend on ζ. The details of the proof are almost the same as the ones of the proof of Theorem 3.4 therefore they are left to the care of the reader.

  Proposition 2.2. ([START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]): Assume that (H1), (H2) and (H3) are fulfilled. Then there exist deterministic continuous functions (u i (t, x)) i=1,m which belong to Π g such that for any (t, x) ∈ [0, T ] × R k , the solution of the BSDE(2.6)

		al. ([2], Proposition 2.5 and Theorems
	3.4, 3.5):
	verifies:
	∀i ∈ {1, . . . , m}, ∀s ∈ [t, T ], Y i;t,x s	= u i (s, X t,x s ).	(2.10)
	(c) Moreover if for any i ∈ {1, . . . , m},

  m is a continuous viscosity solution (in Barles et al.'s sense, see Definition 4.1 in Appendix) of the following system of IPDEs: ∀i ∈ {1, . . . , m},

  , pp. 60, ∀s ∈ [t, T ],On the other hand, since Y i;t,x satisfies the BSDE (2.6) then for any s ∈ [t, T ],

	s			s			s
		U i;t,x r	(e)μ(dr, de) =	U i;t,x r	(e)µ(dr, de) -	U i;t,x r	(e)λ(de)dr.
	t	E		t	E		t	E
					s
			{Y i;t,x r	-Y i;t,x r-} =		U i;t,x r	(e)µ(dr, de).	(3.2)
		t<r≤s		t	E
	Next, for any s ∈ [t, T ], Y i;t,x s

We say that a family of deterministic functions u = (u i ) i=1,m , defined on [0, T ] × R k and R-valued and such that for any i ∈ {1, . . . , m}, u i is continuous, is viscosity sub-solution (resp. super-solution) of the IPDE (1.1) if, for any i ∈ {1, . . . , m}:

The family u = (u i ) i=1,m is a viscosity solution of (1.1) if it is both a viscosity sub-solution and viscosity super-solution.

The adaptation of this definition to the case when there is an obstacle is the following (see [START_REF] Harraj | Double barriers Reflected BSDEs with jumps and viscosity solutions of parabolic Integro-differential PDEs[END_REF] or [START_REF] Dumetrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF]). Definition 4.2. We say that a deterministic continuous functions u, defined on [0, T ] × R k and R-valued, is a viscosity sub-solution (resp. super-solution) of the IPDE (3.9) if:

min u(t, x)ℓ(t, x); -∂ t φ(t, x) -L X φ(t, x)h(t, x, u(t, x), σ ⊤ (t, x)D x φ(t, x), B(φ)(t, x)) ≥ 0.)

The function u is a viscosity solution of (3.9) if it is both a viscosity sub-solution and viscosity super-solution. (ii) It is shown in [START_REF] Harraj | Double barriers Reflected BSDEs with jumps and viscosity solutions of parabolic Integro-differential PDEs[END_REF] or [START_REF] Dumetrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF] that when h is non-decreasing w.r.t. ζ and γ is moreover non-negative then the function u defined in(3.14) is the unique continuous viscosity solution of (3.9) in the sub-class of Π g of continuous functions.


Laurent R Bergé

Charles Bouveyron

Marco Corneli

Pierre Latouche

The Latent Topic Block Model for the Co-Clustering of Textual Interaction Data

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

In all aspects of everyday life, the recent digitalization of the systems has resulted in a massive generation of data, including text data. For instance, many e-commerce websites (such as Amazon or TripAdvisor) ask their clients to make comments about the products/services they bought. Similarly, major hospitals have fully numerical information systems which allow doctors to directly record surgery reports, biopsy reports or medical prescriptions about their patients. In these examples, the texts are the result of the interactions between individuals of type A (doctors, customers, etc.) and type B (patients, products, etc.). Since datasets have become larger and larger, clustering methods have been proposed as a tool to reduce the dimension and to provide a synthetic view of the available information. Taking the example of customers rating products, instead of focusing on the raw data, it is rather more relevant to look for coherent groups (also known as clusters) of both customers and goods. The task of simultaneously clustering the rows and the columns of such an array (here defined by the interactions between individuals and objects) is known as co-clustering. When the interactions are textual (e.g. a review), the text can provide significant information to perform a more realistic clustering. For instance, a group of users could review the same goods but with different arguments. Unfortunately, a large amount of the existing co-clustering methods do not account for the textual information. The aim of this paper is to tackle this issue by providing a new model-based method to co-cluster both individuals and objects while accounting for the textual content of their interactions.

Model-based co-clustering

Let us consider a binary table with M rows and P columns. A non-null entry at position (i, j) corresponds to an observed interaction between i (individual) and j (object). If no interaction occurred, the same entry is zero. Such table is called incidence matrix and can be used to summarize the observed interactions between two disjoint sets of M and P actors/objects, respectively. In the statistics and machine learning literature, methods for the co-clustering of rows and columns of an incidence matrix can be split into two main categories: deterministic approaches (see for instance [START_REF] George | A scalable collaborative filtering framework based on co-clustering[END_REF][START_REF] Banerjee | A generalized maximum entropy approach to bregman co-clustering and matrix approximation[END_REF][START_REF] Wang | Penalized nonnegative matrix tri-factorization for co-clustering[END_REF] and model-based approaches. The selection of the number of row and column clusters (see Section 3.3) is one of the most important tasks in co-clustering analysis and model-based approaches provide a well defined framework for model selection. Moreover, model-based approaches are usually very flexible: accounting for groups of different sizes, the allow to manage different types of data. These are the main reasons motivating us to adopt the model-based point of view.

Several model-based methods for co-clustering are based on the the latent block model (LBM, [START_REF] Govaert | Clustering with block mixture models[END_REF]. In its original version, LBM assumes that the rows and the columns of the incidence matrix are clustered in hidden groups. The probability that the entry (i, j) of the matrix is 1 only depends on the row cluster of i and the column cluster of j. The model was extended later to deal with counting data [START_REF] Govaert | Latent block model for contingency table[END_REF], real data [START_REF] Lomet | Sélection de modèle pour la classification croisée de données continues[END_REF], categorical data [START_REF] Keribin | Estimation and selection for the latent block model on categorical data[END_REF] and ordinal data [START_REF] Jacques | Model-based co-clustering for ordinal data[END_REF]. Several inference procedures have been proposed for LBM, including likelihood based methods [START_REF] Govaert | Block clustering with bernoulli mixture models: Comparison of different approaches[END_REF], variational inference [START_REF] Keribin | Model selection for the binary latent block model[END_REF], Bayesian inference [START_REF] Keribin | Model selection for the binary latent block model[END_REF][START_REF] Wyse | Block clustering with collapsed latent block models[END_REF] and greedy search approaches [START_REF] Wyse | Inferring structure in bipartite networks using the latent blockmodel and exact icl[END_REF]. A recent algorithm (knows as Largest Gaps) of [START_REF] Brault | Fast and consistent algorithm for the latent block model[END_REF] allows to perform clustering and model selection in LBM, only using the marginals (sum of the entries in rows and columns) of the incidence matrix, thus dramatically reducing the computational complexity of the estimation procedure. However, the algorithm is only consistent under certain assumptions concerning the degree distributions of rows and columns.

The contributions mentioned so far do not involve the analysis of textual data. For instance, if the individual i reviews the object j, the standard LBM would neglect the textual content of the review and just consider the entry (i, j) of the incidence matrix (equal to 1). However, the textual content of the review could be the key to perform a more realistic clustering. Before illustrating how we proceed to integrate the text analysis into a co-clustering model based method, we pass through some of the most well known statistical models for text analysis.

Statistical models for text corpora

One of the first contributions to the statistical modeling of texts is the work of [START_REF] Papadimitriou | Latent semantic indexing: A probabilistic analysis[END_REF], based on latent semantic indexing (LSI) [START_REF] Deerwester | Indexing by latent semantic analysis[END_REF]. LSI is known in particular for allowing the recovery of linguistic notions such as synonymy and polysemy from "term frequency -inverse document frequency" (tf-idf) data. [START_REF] Hofmann | Probabilistic latent semantic indexing[END_REF] proposed an alternative model for LSI, called probabilistic latent semantic analysis (pLSI), which models each word within a document using a mixture model. In pLSI, each mixture component is modeled by a multinomial random variable and the latent groups can be viewed as "topics". Thus, each word is generated from a single topic and different words in a document can be generated from different topics. However, pLSI has no model at the document level and may suffer from overfitting. Notice that pLSI can also be viewed has an extension of the mixture of unigrams, proposed by [START_REF] Nigam | Text classification from labeled and unlabeled documents using em[END_REF]. The model which finally concentrates most desired features was proposed by [START_REF] Blei | Latent dirichlet allocation[END_REF] and is called latent Dirichlet allocation (LDA). The LDA model has rapidly become a standard tool in statistical text analytics and is even used in different scientific fields such has image analysis [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF] or transportation research [START_REF] Côme | Spatio-temporal analysis of dynamic origin-destination data using latent dirichlet allocation. application to the vélib? bike sharing system of paris[END_REF] for instance. The idea of LDA is that documents are represented as random mixtures over latent topics, where each topic is characterized by a distribution over words. LDA is therefore similar to pLSI except that the topic distribution in LDA has a Dirichlet distribution. Several inference procedures have been proposed for LDA, including variational inference [START_REF] Blei | Latent dirichlet allocation[END_REF][START_REF] Teh | A collapsed variational bayesian inference algorithm for latent Dirichlet allocation[END_REF] and Bayesian inference, via Gibbs sampling [START_REF] Phan | Learning to classify short and sparse text & web with hidden topics from large-scale data collections[END_REF]. More recently [START_REF] Anandkumar | A spectral algorithm for latent dirichlet allocation[END_REF]; [START_REF] Podosinnikova | Rethinking lda: moment matching for discrete ica[END_REF] introduced new inference procedures based on moment matching and tensorial decomposition.

A limitation of LDA would be the inability to take into account possible topic correlations. This is due to the use of the Dirichlet distribution to model the variability among the topic proportions. To overcome this limitation, the correlated topic model (CTM) was also developed by [START_REF] Blei | Correlated topic models[END_REF]. From another point of view, other works extend LDA to account for co-clustering of words and documents. In [START_REF] Shafiei | Latent dirichlet co-clustering[END_REF], given each document's specific parameter, words within a document are no longer i.i.d. They still follow a mixture distribution over latent topics depending on the segment (paragraph) of the document. In this way, each document is partitioned in segments that are topic homogeneous. Instead, [START_REF] Wang | Latent dirichlet bayesian co-clustering[END_REF] use a Bayesian framework to simultaneously cluster documents and words of a document-term matrix, in such a way that topic proportions are no longer specific to each document but to each cluster of documents. This approach is partially related with the one we adopt as will we see in Section 2.3. More recently the work of [START_REF] Wang | Latent dirichlet bayesian co-clustering[END_REF] was generalized by [START_REF] Kumar | Co-clustering for dual topic models[END_REF] who introduced statistical dependence between row and column clusters of a document-term matrix.

Contributions and organization of the paper

The present paper introduces a new model, the latent topic block model (LTBM), along with an inference algorithm, to simultaneously cluster the rows and columns of an incidence matrix accounting for the textual information associated with the non-null entries. In other words, a non-null entry (i, j) in the incidence matrix corresponds to a textual interaction between i and j, and the text is part of the generative model we introduce. Moreover, our approach aims at estimating the topics used for the textual interactions associated with the incidence matrix, thus allowing a deeper understanding of the co-clustering. In real data applications (Section 5), each topic (or argument) is identified by a list of most representative words. These words are used to decrypt the topic and therefore the clusters.

We stress that our aim is to cluster the rows/columns of an incidence matrix and not those of a document-term matrix. In this sense, (i) the approach introduced here can be seen as an extension of LBM and (ii) the co-clustering we perform is not the same as in [START_REF] Wang | Latent dirichlet bayesian co-clustering[END_REF], who focus on the co-clustering of the rows and columns of a document-term matrix.

That said, our generative model is related to LDA, as discussed in Section 2.3. Finally, our model-based approach is related to the one introduced in [START_REF] Bouveyron | The stochastic topic block model for the clustering of vertices in networks with textual edges[END_REF] for network analysis. However, in that paper the scope is to cluster the nodes of a graph, whereas we aim here at co-clustering two disjoint sets of actors.

The present paper is organized as follows. Section 2 introduces the LTBM and details its relationship with other generative models. Section 3 describes the inference procedure adopted to estimate the model parameters. It also discusses further issues such as the initialization of the estimation algorithm and model selection. Section 4 focuses on experiments on synthetic data. The aim is to assess the capacity of the estimation algorithm to recover the true partitions (and the number of groups), when they are known. Finally, in Section 5, two realworld datasets are analysed in order to show the appeal of our methodology. The former dataset is collected from the Amazon e-commerce system, the latter is collected from the PubMed database.

The Latent Topic Block Model

In this section, we describe the latent topic block model that we introduce. The observed data is represented by an M × P incidence matrix A and a corpus of documents W . The observed connections (i.e. non null entries in A) are characterized by documents.

If A ij = 1, a set of D ij documents W d ij , d ∈ {1, . . . , D ij }
is associated with the connection between i and j (e.g. the review of i of the good j). Hence

W ij := {W d ij } d≤D ij and W = {W ij } i≤M,j≤P .

Modeling of connections

Following LBM, we assume that the rows of A (the individuals) are grouped into Q latent row clusters. An hidden M × Q binary matrix Y is introduced such that the i-th row is denoted by Y i and its entries are all zeros but one. In more details, Y iq = 1 if and only if the i-th row of A belongs to the q-th row cluster, where q ∈ {1, . . . , Q}. The rows Y 1 , . . . , Y M are assumed to be independent random vectors such that

P(Y iq = 1) = ρ q ,
∀i ∈ {1, . . . , M }, where ρ q > 0 and Q q=1 ρ q = 1. Similarly, the columns of A are grouped into L column clusters and an hidden P × L binary matrix is introduced such that the row X j is an indicator of the cluster of the j-th column of A. For all j ∈ {1, . . . , P }, we assume that

P(X jl = 1) = δ l ,
where δ l > 0 and L l=1 δ l = 1. The label matrices Y and X are assumed to be independent. Hence

p (Y, X|ρ, δ) = p (Y |ρ) × p (X|δ) = M i=1 Q q=1 ρ Y iq q × P j=1 L l=1 δ X jl l .
(1)

As in LBM, we assume that the probability of a connection between i and j only depends on their clusters. Thus, conditionally on Y and X, A ij is assumed to be a random variable following a Bernoulli distribution

p(A ij |X iq Y jl = 1) = B(A ij ; π ql ) := π A ij ql (1 -π ql ) 1-A ij , ( 2
)
where π ql ∈ [0, 1]. We denote by π the Q × L matrix of the connections probabilities.

Conditionally on Y and X, the entries of A are then all assumed to be independent

p (A|Y, X, π) = M i=1 P j=1 p(A ij |Y i , X j , π) = M i=1 P j=1   Q q=1 L l=1 B (A ij , π ql ) Y iq X jl   .
(3)

Finally, the complete data likelihood of the model detailed is

p(A, Y, X|π, ρ, δ) = p (A|Y, X, π) p(Y |ρ)p(X|δ). (4)

Modeling of documents

As pointed out previously, when

A ij = 0, a sequence of D ij documents W 1 ij , . . . , W D ij ij
is associated with the interaction between the i-th individual and the j-th object. In LTBM, a document is defined as a set of words extracted from a common dictionary with V words. The number of words in W d ij (the d-th document sent from i to j) is N d ij and the n-th word in

W d ij is denoted by W dn ij .
As in the latent Dirichlet allocation (LDA, [START_REF] Blei | Latent dirichlet allocation[END_REF] model, each word within a document follows a mixture distribution over a set of latent topics whose number K is unknown and must be estimated. However, while in LDA the topic proportions of words are specific to each document, in the model we propose, this proportions only depend on the row cluster of the i-th row of A and the column cluster of the j-th column of A.

Thus, we introduce a binary random vector Z dn ij , of length K, such that the k-th entry Z dnk ij is 1 if and only if W dn ij is sampled from the k-th topic, k ∈ {1, . . . , K}, 0 otherwise. Then, conditional on Y i and X j , Z dn ij follows a multinomial distribution

Z dn ij |{Y iq X jl A ij = 1} ∼ M (1, θ ql = (θ ql1 , . . . , θ qlK )) ,
where θ qlk ≥ 0 and K k=1 θ qlk = 1. Moreover, conditional on Y i , X j and θ, the random vectors Z d1 ij , . . . , Z

N d ij ij
are all assumed to be independent. Therefore

p (Z|A, Y, X, θ) = M i=1 P j=1 D ij d=1 p Z d ij |Y i , X j , θ A ij = M i=1 P j=1 D ij d=1   Q q=1 L l=1 p Z d ij |θ ql Y iq X jl   A ij = M i=1 P j=1 D ij d=1     Q q=1 L l=1    N d ij n=1 K k=1 θ Z dnk ij qlk    Y iq X jl     A ij , ( 5
)
where

Z d ij := (Z 1 ij , . . . , Z N d ij ij ), Z ij := {Z 1 ij , . . . , Z D ij ij } and finally Z := {Z ij } i,j .
For each pair (q, l) of row and column clusters, the model proportions θ ql are themselves assumed to be independent random vectors, each one following a Dirichlet distribution Given Z dn ij , the word W dn ij is finally assumed to be drawn from a multinomial distribution

θ ql ∼ D(α = (α 1 , . . . , α K )). (6)
W dn ij |Z dnk ij = 1 ∼ M(1, β k = (β k1 , . . . , β kV )), (7)
where β kv > 0 and V v=1 β kv = 1, for all k ∈ {1, . . . , K}. Henceforth, β denotes the K × V matrix whose k-th row is β k . Notice that, unlike θ, the matrix β depends neither on the row clusters nor on the column clusters. Moreover, note that the number K of topics is the same for all cluster pairs. Thus, the following conditional distribution is obtained by independence

p(W |Z, A, β) = M i=1 P j=1 D ij d=1 p W d ij |Z d ij , β A ij = M i=1 P j=1 D ij d=1    N d ij n=1 p W dn ij |Z dn ij , β    A ij = M i=1 P j=1 D ij d=1    N d ij n=1 K k=1 V v=1 (β kv ) W dnv ij Z dnk ij    A ij . (8)
Finally, the complete-data likelihood for the textual part of the model is obtained by conditioning

p(W, Z, θ|A, Y, X, β, α) = p(W |Z, A, β)p(Z|A, Y, X, θ)p(θ|α). ( 9
)
All the notations used for the description of LTBM are given in Table 1. A graphical representation of the model can be seen in Figure 1.

Links with related models

As pointed out previously, the sampling scheme of the documents is similar to the one of LDA, but not identical. Indeed, assuming that Y and X are known, the joint distribution in ( 9

) can θ Z ij W ij α β Y i ρ X j δ A ij π Figure 1: Graphical representation of LTBM.
be written as

p (W, Z, θ|A, Y, X, β, α) = p (θ|α) M i=1 P j=1 p (W ij , Z ij |Y i , X j , θ, β) A ij = p (θ|α) M i=1 P j=1 (p (W ij |Z ij , β) p (Z ij |Y i , X j , θ)) A ij = p (θ|α) M i=1 P j=1   p (W ij |Z ij , β)   Q q=1 L l=1 p (Z ij |θ ql ) Y iq X jl     A ij = p(θ|α) M i=1 P j=1 Q q=1 L l=1 (p(W ij |Z ij , β)p(Z ij |θ ql )) Y iq Y jl A ij ≡ Q q=1 L l=1 p (W ql |Z ql , β) p (Z ql |θ ql ) p (θ ql |α) , where W ql =: {W ij |A ij Y iq X jl = 1,
∀i, j} is the meta document of all the exchanged documents involving the pair of clusters (q, l). Similarly Z ql =: {Z ij |A ij Y iq X jl = 1, ∀i, j} is the set of all the topic labels in the exchanged documents involving the pair of clusters (q, l). Therefore, the joint distribution of W and Z, can be seen from the LDA perspective: there are Q × L independent documents with their own topic proportions. However, notice that this analogy with LDA crucially hinges on the knowledge of Y and X. Since Y and X are unknown, LTBM is more general than LDA and the related inference more difficult.

When K = 1 and a single topic is associated with all the connections, the text analysis does not bring any further information and LTBM reduces to LBM. Finally, as mentioned in Section 1.3, LTBM is related to the stochastic topic block model (STBM, [START_REF] Bouveyron | The stochastic topic block model for the clustering of vertices in networks with textual edges[END_REF]. However, while STBM characterizes textual interactions between actors/objects in the same set, LTBM models textual interactions between two disjoint sets of actors/objects.

Inference

The main goal of this section is to detail a variational expectation maximization (VEM, [START_REF] Dempster | Maximum likelihood from incomplete data via the em algorithm[END_REF][START_REF] Hathaway | Another interpretation of the em algorithm for mixture distributions[END_REF] algorithm to estimate the LTBM model parameters and to provide estimates of Y , X as well as Z. After having introduced the optimization procedure, we focus our attention on the algorithm initialization (3.2) and on the model selection (3.3) task.

Variational inference

In this section, the values of Q (number of row clusters), L (number of column clusters) and K (number of topics) are assumed to be known. The choice of Q, L and K will be discussed in Section 3.3. In this context, we aim at estimating the hidden labels Y, X, the model parameters (π, ρ, δ) as well as β. In order to simplify both the exposition and the derivation of the results, the parameter α is considered fixed and it is not part of the inference procedure. In all the experiments in Section 4, we set α 1 = • • • = α K = 1 for all pairs (q, l), thus inducing a uniform distribution over the (K -1)-simplex of all θ ql .

We focus on the following log-likelihood

log p(W, A, Y, X|π, ρ, δ, β) = log p(W |A, Y, X, β) + log p(A, Y, X|π, ρ, δ), ( 10
)
where the second term on the right hand side of the equality is detailed in ( 4) and the first one is obtained from ( 9), by integrating out Z and θ. We aim at maximizing the above log-likelihood with respect to (β, ρ, δ, π) and (X, Y ). Let us consider log p(W |A, Y, X, β) at first. Although this term is not explicitly tractable, we can take advantage of the following variational decomposition

log p (W |A, Y, X, β) = θ Z q (Z, θ) log p (W, Z, θ|A, Y, X, β) q (Z, θ) dθ - θ Z q (Z, θ) log p (Z, θ|W, A, Y, X, β) q (Z, θ) dθ, (11)
where q(Z, θ) is any distribution over the pair (Z, θ). The sum inside the integral is taken over the set of all the possible outcomes of Z. The last term on the right hand side of the above equation is the Kullback-Leibler (KL) divergence between the approximate and the true posterior distribution of the pair (Z, θ). The KL divergence is known to be positive and null if and only if q(•) is equal to p(•|W, A, Y, X, β). As a consequence, the first term on the right hand side of ( 11) is a lower bound for the integrated log-likelihood log p (W |A, Y, X, β). We denote it

L(q(•)|A, Y, X, β) : = θ Z q (Z, θ) log p (W, Z, θ|A, Y, X, β) q (Z, θ) dθ = E Z,θ log p (W, Z, θ|A, Y, X, β) q (Z, θ) , ( 12
)
where the expectation is taken with respect to (Z, θ) following q(•). Since the posterior distribution of (Z, θ) is not tractable (due to similar arguments as in [START_REF] Blei | Latent dirichlet allocation[END_REF], the following mean field variational approximation is adopted to specify q(•) q(Z, θ) := q(θ)q(Z) = q(θ)

M i=1 P j=1 D ij d=1 N d ij n=1 q(Z dn ij ), (13)
corresponding to an independence assumption over the approximate posterior distribution. The basic idea is then to replace the log-likelihood in (11) by its lower bound

log p (W |A, Y, X, β) ≥ L(q(•)|A, Y, X, β)
and to use a VEM algorithm [START_REF] Hathaway | Another interpretation of the em algorithm for mixture distributions[END_REF] to maximize the right hand side of the above inequality with respect to q(•) in ( 13) and β. We stress that this maximization corresponds to minimizing the KL divergence between the approximate and true posterior distribution of the pair (Z, θ). Thus, we can go back to (10) and notice that

log p(W, A, Y, X|π, ρ, δ, β) = log p(W |A, Y, X, β) + log p(A, Y, X|π, ρ, δ) ≥ L(q(•)|A, Y, X, β) + log p(A, Y, X|π, ρ, δ), ( 14
)
where the term log p(A, Y, X|π, ρ, δ) is independent from the variational approximation adopted.

The proposed estimation procedure is detailed in the following two steps:

1. Y and X being fixed, a VEM algorithm is applied to alternatively maximize the right hand side of ( 14) with respect to q(•) in ( 13) (E-step) and the model parameters (π, ρ, δ) and β (M-step), up to convergence.

2. The model parameters (π, ρ, δ, β) being fixed, a greedy search strategy is adopted to maximize the right hand side of ( 14) with respect to Y and X. Each row (respectively column) of A is switched from its current cluster to all the other row clusters (resp. column cluster) and the switch leading to the highest increase of the right hand side of ( 14) is finally retained.

The two steps above are iteratively repeated until convergence. The estimation algorithm detailed alternates a variational EM step with a classification step. Hence, it is referred to as the C-VEM algorithm. It was first used in [START_REF] Bouveyron | The stochastic topic block model for the clustering of vertices in networks with textual edges[END_REF] and it is built upon the Classification-EM (CEM) algorithm [START_REF] Celeux | A classification EM algorithm for clustering and two stochastic versions[END_REF]. Each step of the C-VEM algorithm is now analyzed in detail.

Variational EM step

In this section Y and X are fixed and we provide the updating formulas for q(Z, θ) and the model parameters.

Maximization of L with respect to q(Z, θ). The E step of the VEM algorithm is given by the following two propositions.

Proposition 1. The VEM update step for distribution q(Z ij n ) is given by

q(Z dn ij ) = M(Z dn ij ; 1, φ dn ij = (φ dn1 ij , . . . , φ dnK ij )),
where

φ dnk ij ∝ V v=1 β W dnv ij kv Q q=1 L l=1 exp ψ(γ qlk ) -ψ K k =1 γ qlk Y iq X jl , ∀(n, k) (15)
where φ dnk ij is the approximate posterior probability of word W dn ij being in topic k, γ qlk is defined in the following proposition and ψ(•) denotes the digamma function.

Proof. In Appendix A.1.

Proposition 2. The VEM update step for distribution q(θ) is given by

q(θ) = Q q=1 L l=1 D(θ ql ; γ ql = (γ ql1 , . . . , γ qlK )),
where

γ qlk = α k + M i=1 P j=1 D ij d=1 N d ij n=1 A ij Y iq X jl φ dnk ij , ∀(q, l).
Proof. In Appendix A.2.

Maximization of L with respect to the model parameters. The following proposition details the M step of the VEM algorithm providing the estimates of the model parameters (π, ρ, δ, β). These are obtained through the maximization of the lower bound in ( 14) .

Proposition 3. The estimates of (β, π, ρ) and δ are given by

β kv ∝ M i=1 P j=1 D ij d=1 N d ij n=1 A ij W dnv ij φ dnk ij , ∀(k, v) (16) π ql ∝ M i=1 P j=1 Y iq X jl A ij , ∀(q, l) (17) ρ q ∝ M i=1 Y iq , ∀q, (18)
δ l ∝ P j=1 X jl , ∀l. (19)
Proof. In Appendix A.4.

Maximization with respect to (Y, X)

The model parameters being estimated in the previous step, a greedy search strategy is now employed to maximize the lower bound on the right hand side of the inequality in ( 14) with respect to Y and X. Greedy search methods are generally used to solve combinatorial problems that are too computationally intensive to be treated (Q M L P combinations to test in our case). However, these methods are not guaranteed to reach a global maximum (for non-convex objective functions). One way to solve this drawback is to run the greedy search several times, with different initializations, in order to choose the estimates leading to the highest value of the objective function.

Let us consider Y at first and assume that an initial clustering of the rows of A in Q cluster is provided (see Section 3.2 for more details). If the i-th row is in the q-th row cluster, the algorithm assesses the increase/decrease in the lower bound due to switching i to the cluster q , for each q = q. The switch (if any) leading to the highest increase of the lower bound is actually performed and the entire routine is iteratively applied to all the rows of A until no further increase of the lower bound is possible. The maximization with respect to X is performed similarly.

Note that [START_REF] Wyse | Inferring structure in bipartite networks using the latent blockmodel and exact icl[END_REF] used a greedy search approach similar to the one described above to perform inference in bipartite graphs using LBM. However, the greedy search algorithm they introduced is also used to perform model selection by maximizing the complete data integrated log-likelihood of LBM. In our case, the number of clusters is assumed fixed during the maximization and a row (column) of A which is alone in its own cluster cannot be switched.

Initialization strategy

EM-like algorithms are known to be sensible to the initialization and are not guaranteed to converge toward a global maximum. A possible strategy consists in choosing several random initializations and finally retaining the estimate leading to the highest value of the lower bound of the observed data log-likelihood. Alternatively, the algorithm can be initialized relying on other (simpler) clustering algorithms (k-means, spectral clustering, etc.) hoping that the initialization provided is not so far from the global optimum. See [START_REF] Biernacki | Choosing starting values for the em algorithm for getting the highest likelihood in multivariate gaussian mixture models[END_REF] for a further inspection of this point.

In our case, we need to get initial estimates of Y and X in order to perform the variational step detailed in Section 3.1.1. When testing the C-VEM algorithm on synthetic data, in Section 4, several initializations are compared. One of them proves to work very well and it is now described. Henceforth, this initialization is referred to as the spectral initialization strategy. Consider at first the label vector Y :

1. LDA is run on the whole set of documents W . We recall that each set of documents W d ij corresponds to a non null entry of A. Via LDA, we can estimate the main topic discussed in each document set

W ij = {W 1 ij , . . . , W D ij ij }. Hence, an M × P matrix T is obtained such that T ij = k if A ij = 1 and k is the main topic appearing in W ij . If A ij = 0, then T ij = 0.
2. An M × M similarity matrix S is created such that its entry (i, i ) is defined as

S i,i = P j=1 A ij A i j 1 {T ij =T i j } ,
where 1 I denotes the indicator function on a set I. The above equation states that if i and i have a common connection j and they share the same main topic associated with this connection, then their similarity increases.

3. The spectral clustering algorithm can be used to produce an estimate of Y based on the graphs Laplacians associated with S (see von Luxburg, 2007, for a detailed review of graphs Laplacians and the spectral clustering algorithm).

An initial estimate of X can be obtained similarly and the VEM step (Section 3.1.1) can be implemented.

Model selection

The following proposition details a model selection criterion to estimate the number Q of row clusters, the number L of column clusters as well as the number K of topics, from the data. Proof. In Appendix A.5.

Proposition 4. A ICL criterion for LTBM can be obtained

ICL LT BM = max β L(q(•)|A, Y, X, β) -K V -1 2 log(QL) + max π,ρ,δ log p(A, Y, X|π, ρ, δ, Q, L) - QL 2 log(M P ) - Q -1 2 log M - L -1 2 log P. ( 20
)
This result relies on two Laplace approximations, the Stirling formula and a variational estimation. In particular, the first term L(q(•)|A, Y, X, β) on the right hand side of the equality is defined in (12). For a detailed description of the Laplace and Stirling approximations to obtain ICL we recommend the original paper of [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF].

Experiments on synthetic data

In this section, we carry out some experiments on simulated datasets to test the estimation strategy detailed in the previous section.

Synthetic datasets

Three different setups are considered. In each setup, an individual i connects with an object j either with a higher probability of 30% or with a lower probability of 2%. As detailed in Section 1.1, these probabilities only depend on the clusters of the pair (i, j). Once i and j are connected, a document is sampled and associated with the connection. The words in each document are extracted from three texts from BBC news. One text is about black holes in astrophysics (A), the second is about the birth of Princess Charlotte (B) and the third one focuses on UK politics (C). Notice that the three texts have been chosen different enough and specific, such that they can be considered as "pure" topics. The number of words in a document is drawn from a Poisson distribution of mean 100. The topic proportions in each text then only depend on the clusters of the corresponding pair (i, j).

The three setups are illustrated in more details in Figure 2. For instance, the first picture on the left hand side represents the connectivity matrix π in the first setup, where the number of row clusters is equal to the number of column clusters (Q = L = 3). Denser colored grids on the diagonal represent the higher connection probabilities of 30%. Two colors are q q q q q q q q q q q q q q q 2 4 6 8 10 12 14

Reorganized Adjacency Matrix

Column-Clusters (Objects)

Row-Clusters (Individuals)

Figure 3: LTBM is fitted on a dataset simulated according to Setup 3. The evolution of the lower bound during the optimization process can be seen on the left. In the colored adjacency matrix on the right, rows and columns are reorganized according to the final clustering provided by the C-VEM algorithm.

associated with two different topics: (A) in red and (B) in blue. Notice that, relying solely on the textual analysis, it would be impossible to distinguish the three row clusters. Similarly, the first and the second column cluster would look as a single group.

An illustration. We now consider the third setup. An incidence matrix with M = 150 individuals and P = 120 objects is simulated according to the parametrization in Figure 2. The C-VEM algorithm is provided with the true values of K as well as Q and L, used to assign each row (respectively column) to its row (column) cluster. An illustration can be seen in Figure 3. On the left hand side, we can see the lower bound at each step of the C-VEM algorithm. The right pane of the same figure shows the reorganized adjacency matrix, in which rows and columns are permuted according to the clustering provided by the algorithm. As it can be seen, this matrix looks very similar (up to label switching) to the one on the right hand side of Figure 2.

Initialization performance

The aim of the present section is to compare the clustering results provided by the C-VEM algorithm when different initializations are adopted. The datasets we use are sampled according to the setups illustrated in the previous section. The C-VEM algorithm is always provided with the actual values of Q, L as well as K, and four initialization techniques are tested:

1. Random. Each row (resp. column) of the incidence matrix is randomly assigned to a row (column) cluster.

2. k-means. The k-means algorithm is applied to the rows (resp. columns) of the incidence matrix to estimate Y (X).

3. Spectral. The initialization strategy described in Section 3.2.

4. LBM. LBM is fitted on the incidence matrix to provide initial estimates of Y and X.

The quality of the estimates provided by the C-VEM algorithm is assessed via the adjusted Rand index (ARI, [START_REF] Rand | Objective criteria for the evaluation of clustering methods[END_REF]. This index compares two partitions (for example the true Y and its estimate Ŷ provided by C-VEM) and measures how close they are. More specifically, the ARI takes real values in [0, 1], where a value of 1 means that the two classifications are identical (up to label switching).

Twenty datasets are simulated for each setup. The results are illustrated in Figure 4 which also reports statistics concerning the lower bound. The LBM initialization works well in situations where all the available information is encoded in π and the analysis of the textual content does not bring any further insight. In Setup 1, for example, the row and column clusters can be detected by solely looking at the interaction frequency and LBM is perfectly able to uncover the hidden partitions. In Setup 2, the column clusters 1 and 2 would be indistinguishable based on the interaction frequency. However, despite the LBM initialization, C-VEM does not remain trapped into local maxima and the ARI is still good in rows and columns. However, in Setup 3, the LBM initialization penalizes heavily the C-VEM algorithm.

Interestingly, the random initializations can lead to good results in all scenarios. However, the variance of the clustering results remain high. Finally, we clearly see that the spectral initialization outperforms its competitors: the ARI is 1 most of the time.

Model selection

So far, we have assumed that Q, L and K were known in advance. In this section, several datasets are sampled according to the different setups and the C-VEM algorithm is run on each dataset for different values of (Q, L, K). The model selection in ( 20) is then used to estimate the number of clusters/topics and the aim of this section is to assess how well it works.

For each setup, 50 independent datasets are generated. The C-VEM is run for all values of Q, L and K ranging from 1 to 10. Table 2 reports the results of the model selection. As it can be seen, the criterion selects the actual model most of the time and when it fails, the number of clusters/topics is misclassified by one unit (except for one case in Setup 1).

Real data

In the following, in order to illustrate the appeal of the methodology outlined in this paper, LTBM is fitted on two real datasets.

Amazon Fine Foods

This section focuses on a real dataset consisting of reviews of fine foods from Amazon. The dataset can be freely downloaded at the following address https://snap.stanford.edu/ data/web-FineFoods.html. A time horizon of 10 years is considered, up to October 2012. The number of reported reviews is 568,464 and in the original dataset, each row corresponds to one review. Some additional information is reported for each review: the user/product numerical identifiers, a summary of the review and a rating attributed to the product by the user. The rating is expressed via an integer number spanning from 1 (very bad) to 5 (very good). The original dataset was preprocessed as follows. To focus on the most meaningful part of the data, we only considered the users reviewing more than 20 times and the products being reviewed more than 50 times. Each review was preprocessed in a classical way: very short words (less than three characters) and stop words were removed and punctuation and q q q q q q q q q q q q q q q q q q q q q q q q q ARI Row Initialization Type --Setup 1 0 0.25 0.5 0.75 1 Random K-means Spectral LBM q q q q q q q q q q q q q q q q q q q q q q q q q q q ARI Column Initialization Type --Setup 1 0 0.25 0.5 0.75 1 Random K-means Spectral LBM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Lower Bound Initialization Type --Setup 1 Random K-means Spectral LBM -260000

-250000

-240000

-230000

-220000

-210000 q q q q q q q q q q q q ARI Row Initialization Type --Setup 2 0 0.25 0.5 0.75 1 Random K-means Spectral LBM q q q q q q q q q q q q q q q q q q q q q q ARI Column Initialization Type --Setup 2 0 0.25 0.5 0.75 1 Random K-means Spectral LBM q q q q q q q q q q q q q q q q q Lower Bound Initialization Type --Setup 2 Random K-means Spectral LBM -550000 -540000 -530000 -520000 -510000 -500000 -490000 Figure 5: The Reorganized incidence matrix where nearby rows/columns belong to the same row/column cluster (delimited by grey lines). The colors of the cells mark the main topic used for the corresponding reviews. One colored dash marks an interaction/review between the corresponding pair.

numbers ignored. Moreover, the vocables appearing less than 10 times in the whole corpus were neglected. The main features of the resulting dataset are summarized in Table 3 selected Q = 12 row clusters, L = 14 column clusters and K = 12 topics. The estimates of Y and X allowed us to permute the rows and the columns of the incidence matrix of the dataset, in such a way that nearby rows/columns belong to the same row/column cluster. The reorganized incidence matrix can be observed in Figure 5. The grey grid (horizontal and vertical lines) delimits the blocks estimated by the C-VEM algorithm. In the matrix, each colored dash corresponds to one user reviewing the corresponding object. The color of each block marks the main topic used for the reviews inside the block. A list of the most representative words of each topic can be seen in Figure 6. So, for instance the orange block at position (1, 9) in the reorganized incidence matrix corresponds to the reviews that the users in the row cluster q = 1 made about the objects in the column cluster l = 9. The main topic used for such reviews is the orange one, namely Topic 8 whose key words are "coffee", "pods", "pod", etc. Figure 7 displays the estimated topic proportions θ ql for each meta document associated with the pair (q, l). For instance, the entry (12, 11) is blue and green, meaning that reviewers in row cluster q = 12 used Topics 2 and 4 (respectively) to review the items in column cluster l = 11. Topic 4 is the most used. In general, the denser the colored grids inside a cell, the higher the interaction frequency. We now move to some (non-exhaustive) remarks about the results shown in the three figures.

1. The buyers in the row cluster q = 2 are mainly involved in reviewing a specific brand of cat food. Indeed, all the reviewed products in the column cluster l = 3 belong to the same brand. The mean rating for this pair is 2.51 but the mode is 1, corresponding to 289 very bad ratings. All the negative reviews are from different reviewers and contain the following sentence "Filler food is empty, leaves your cat always needing more". Notice that the most representative words of Topic 12 (brown) include "cat", "food" and "hungry". Since it is not credible that 289 different users adopt the very same sentence, we think that they are robots or anyway a fake.

2. In a similar fashion, the buyers in the row cluster q = 6 are mainly involved in reviewing a specific brand of tea, whose name "alvita" appears between the most representative Distribution of topics between groups (θ qr )

Object Individual L=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=11 L=13 Q=1 Q=2 Q=3 Q=4 Q=5 Q=6 Q=7 Q=8 Q=9 Q=10 Q=11 Q=12
Figure 7: A graphical representation of the estimated topic proportions θ ql for each metadocument. The denser the colored grids, the higher the frequency of the interactions/reviews. words of Topic 9 (light violet). The products of this brand are grouped in the column cluster l = 4. The reviews are very positive and the most common rating is 5 (60 % of times). In this case, distinct users provide different reviews for each product of the brand, but strangely each reviewer gives the very same review for each product.

3. The column cluster l = 11 groups dog food items of the same brand "wellness". These products are only reviewed by the users in the row cluster q = 12, with very positive ratings. In this case too, the reviews for the same product are all different but the single review of each user is spread to all products of the brand. As it can be seen in Figure 7 there are two topics involved in these reviews: Topic 2 and Topic 4. Both of them contain the two key words "dog" and "food". Moreover, Topic 4 contains the key word "wellness" which is the brand of the dog food. Interestingly the words "cat" appears among the most representative words of Topic 4. This is due to the presence of some reviews in which reviewers discuss about differences between the dogs and the cats diet.

4. The first row cluster (q = 1) consists of 277 buyers mainly reviewing the products in the ninth column cluster (l = 9). The reviews associated with this pair of clusters are the 40% of the total number of reviews. The scores are globally positive with a mean value of 3.88. The reviewed products are foods and beverages. The most commented three items are an Italian style coffee drink, an energy drink and a Ginger lemon beverage drink. Topic 8 (orange) is the main topic associated with this pair of clusters and it contains some key words like "coffee pods", "drink", "flavor", "taste", "like" and "fair". Interestingly, the items in the column cluster l = 9 are plenty reviewed by the users in row cluster q = 8. This cluster is made of 850 actors who globally behave as the users in cluster q = 1 except for the main topic used to review goods in columns cluster l = 9. Indeed, the main topic used for these reviews by the users in the row cluster q = 1 is Topic 8 whereas the main topic used by the reviewers in the row cluster q = 8 is Topic 11. These two topics are quite similar. However, Topic 11 seems to be more food oriented. For example the two key words "granola" and "popchips" are related with two brands producing chocolate and chips, respectively. These two words are not key words in Topic 8, which conversely has "marley" has a key word, which is a coffee brand. Notice that, if colors were removed the row clusters 1 and 8 would be indistinguishable and a LBM could fail to detect them both. This is an useful application of LTBM.

5. The row cluster q = 9 groups 34 users. This group is peculiar for two reasons i) It is the only group to massively use Topic 6 (red).

ii) Its users are the main reviewers of the items in the column cluster l = 13.

All the items in the column cluster l = 13 are foods of the same brand. More precisely, the items are packs of beef "jerky" (dried up) meat. There are 11 distinct reviews of any single product, but each review is repeated for all the products. The reviewers in group q = 9 also reviews the items in the column cluster l = 2. The main items that they review are packs of almonds of the same brand, whose name is "Blue Diamond". Notice that the three words "almond", "blue" and "diamond" appear as key words in Topic 6 (Figure 6). Globally, the reviews formulated by the users in the row cluster q = 9 are very positive.

6. The reviewers in the row cluster q = 11 are the main users of Topic 10. The first key words of this topic are "cheese", "potato", "kettle". Indeed, the most reviewed items in the column cluster l = 7 are the Kettle chips (several items of the same brand). Interestingly, the same objects are also reviewed by the users in the row cluster q = 4. This explains why the same key words appear on top of Topic 3 in Figure 6. However, the buyers in the row cluster q = 11 also review chocolate cookies (for instance), which is not the case for the row cluster q = 4. Indeed, "cookies" and "chocolate" are key words of Topic 10 but not of Topic 3.

PubMed data

This section focuses on a dataset extracted from the National Center for Biotechnology Information (NCBI) databases, via the free search engine PubMed (https:www.ncbi.nlm.nih. gov/pubmed/). The CRAN package RISmed (https://cran.r-project.org/package= RISmed) was used by the statistical software R to query PubMed about the keyword "colorectal", over a time horizon spanning from 2012 to 2016. As a result, we obtained a list of scientific articles, published during the selected time horizon, about colorectal pathologies. In order to capture the most significant information, the list of articles was pre-processed in order to remove the authors submitting less than 10 articles and the journals publishing less than 20 times over the considered time period. Moreover, only the abstracts written in English were taken into account. Finally, we obtained a table of 111 117 rows and 3 columns. The data in the table previously described can be modelled via LTBM in the following way. Each author corresponds to an individual and one journal to an object. Overall, there are Figure 8: The Reorganized incidence matrix where nearby rows/columns belong to the same row/column cluster (delimited by grey lines). The colors of the cells mark the main topic used for the corresponding article. One coloured dash marks an interaction/article involving the corresponding pair.

Q = 4918 authors/individuals and P = 1367 objects/journals. If the i-th author published (one or more articles) on the j-th journal, the corresponding entry (i, j) of a Q × P incidence matrix was set to one, zero otherwise. The abstracts of the articles for the pair (i, j) were collected into W ij and used for the textual analysis. Overall, 74 817 abstracts were analysed, corresponding to V = 21 014 different vocables. The C-VEM algorithm for LTBM was run on the data for several values of Q, L and K and several initialisations. Finally the algorithm selected Q = 8 row clusters, L = 14 column clusters and K = 9 topics. The estimates provided for Y and X were used to permute the rows and columns of the incidence matrix in order to obtain the reorganized incidence matrix in Figure 8, whose interpretation is the same as discussed in the previous section. Similarly, the most representative words for each topic can be seen in Figure 9. Some remarks about these figures can be done starting from the Topic 1. Containing some key words like "colonoscopy", "screening" and "polyps" this topic is clearly related with prevention. Moreover it is widely correlated with the articles published in journals belonging to the column cluster l = 10. In particular it is the main topic related with the articles of the cluster pair (q = 6, l = 10).

Topic 4. The key words are "surgery", "resection", "postoperative" and it is related with surgical treatment, preparation and monitoring for colorectal diseases. As in can be seen in Figure 8, Topic 4 is mainly associated with the column cluster l = 5 which contains journals like "Annals of surgical oncology", "Surgical endoscopy" and "Annals of surgery".

Topics 6 and 8. Topic 6 focuses on chemoterapy and other remedies for metastatic colorectal cancer (mCRC). In particular, "Bevacizumab", "Folfiri" and "Folfox" are names of specific drugs used to treat mCRC. Topic 8 also exhibits the words "chemoterapy" and "metastasis" and seems to be very closed to Topic 6 but no specific drug name is reported and this seems to be the main difference between the two topics. Moreover, Topic 6 is often the main topic in the journals in column cluster l = 8 (including for instance the "British journal of cancer", "Annals of oncology" and other American or European journals) whereas Topic 8 is central in column cluster l = 7 ("Oncology reports", "Anticancer research" and other Asiatic journals).

Topics 5 and 9. These two topics characterize articles on fundamental biology, with experiments "in vitro" on cells ("cell", "lines"). However, while Topic 5 seems to focus on signaling pathways in cells ("signal", "activation", "growth", "proliferation", "expression"), Topic 9 is more involved the role of mi-RNA ("mir") and some experiments are carried out on "mice". It is interesting to observe that these two topics are mainly used by authors in row cluster q = 5 but in different journal clusters: Topic 5 in column cluster l = 2 and Topic 9 in column cluster l = 3 (see Figures 8 and10).

Topic 7. This topic is certainly related with genetics, which play a crucial role in colorectal diseases, especially the colorectal cancer (CRC). Not only the words "genes", "gene", "genetic" appear as key words, but also "msi" and "mlh" are technical abbreviations related with gene expressions or DNA repair. Topic 7 is central in the articles submitted by authors in row cluster q = 1 and published in journals in column cluster l = 11 (Figure 8). However, unlike Topic 4, is is not associated with one particular column cluster.

Conclusion

This paper introduces a novel model-based co-clustering method to simultaneously cluster the rows and columns of an array of textual interaction data. The peculiarity of the model that we propose, called latent topic block model (LTBM) is the ability to account for both the presence/absence of interactions (as in a standard LBM) and the textual content associated with the interactions. We detailed an inference procedure to estimate the model parameters and obtained a model selection criterion to select the number of row clusters, columns clusters and discussed topics. Experiments on both simulated and real datasets were used to assess the appeal of the proposed methodology. In particular, we showed that LTBM allows a fine and improved understanding of two complex real-world datasets such as the Amazon reviews and the PubMed scientific publications.

A Appendix

A.1 Proof of Proposition 1

For all pairs (i, j) such that A ij = 1, the update step for q(Z dn ij ) (for all n) is given by

log q(Z dn ij ) = E Z \i,

A.2 Proof of Propostion 2

The VEM update step for q(θ) can be obtained as follows A ij Y iq X jl φ dnk ij .

A.3 Computing the lower bound

In this section, we compute the expectation in (12). In the following, the expectation is denoted by E Z,θ and it is taken with respect to the probability distribution defined in (13), conditionally on A, X, Y being known.

L(q(•)|A, Y, X, β) : = E Z,θ log p(W, Z, θ|A, Y, X, β) q(Z, θ) = E Z [log p(W |A, Z, β)] + E Z,θ [log p(Z|A, Y, X, θ)] + E θ [log p(θ)] -E Z [log q(Z)] -E θ [log q(θ)] (21)

A.4 Proof of Proposition 3

In order to obtain the stationary point in ( 16), we need to maximize the lower bound in where K Lagrange multipliers are introduced to account for the constraint V v=1 β kv = 1. Setting the derivative of g(•) with respect to β kv equal to zero, we immediately obtain (16).

The optimal π ql in (17) can be obtained by observing that the distribution p(A|Y, X, π) in ( 4) is the only one involved with π. Hence, taking the log we get log p(A|Y, X, π)

= N i=1 P j=1 Q q=1 L l=1 Y iq X jl (A ij log π ql + (1 -A ij ) log(1 -π ql ))
and setting the derivative with respect to π ql equal to zero, we obtain (17). Equations ( 19) and ( 18) are obtained in a very similar fashion.

A.5 Proof of Proposition 4

This section outlines the main steps needed to obtain (20). We preliminarily recall that the topic proportions θ ql are assumed to be i. where p(W |β, A, Y, X) appeares in (11), and p(β) denotes a prior density function on β. Following the approach adopted in [START_REF] Bouveyron | The stochastic topic block model for the clustering of vertices in networks with textual edges[END_REF] and [START_REF] Than | Fully sparse topic models[END_REF], the number of i.i.d. observations is equal to the number of meta documents (conditionally to {β, A, Y, X}). Therefore, we adopt the following BIC criterion to approximate log p(W |A, Y, X)

log p(W |A, Y, X) ≈ max β p(W |β, A, Y, X) -K V -1 2 log(QL), (22)
where the first term on the right hand side of the equality is replaced by the lower bound in (12) evaluated at β provided by ( 16) and K(V -1) accounts for the number of free parameters in β. We now focus on the LBM part of the log-likelihood (23)

The same criterion was already introduced in the literature for model selection in binary LBM (see for instance [START_REF] Keribin | Model selection for the binary latent block model[END_REF]. Equation ( 20) follows immediately from ( 22) and ( 23).

Figure 2 :

 2 Figure 2: Schematic representation of the model parameters in each setup. Each image represents the parameters π and θ. The density of the coloured grids represents the probability of connections. All the individuals/objects are uniformly split into the row/column clusters. Different colors are associated with different topics.

Figure 4 :

 4 Figure 4: Four initialization techniques are compared on datasets sampled according to Setups 1,2 and 3. Twenty datasets are sampled for each setup.

Figure 6 :

 6 Figure 6: A list of the most representative words of each topic.

Figure 9 :

 9 Figure 9: A list of the most representative words of each topic.

Figure 10 :

 10 Figure10: A graphical representation of the estimated topic proportions θ ql for each metadocument. The denser the colored grids, the higher the frequency of the interactions/published articles.

Y

  j,d,n ,θ [log p(W |A, Z, β) + log p(Z|A, Y, X, θ)] + const = iq X jl ψ(γ qlk ) -ψ term includes everything not depending on Z dnk ij and ψ(•) denotes the digamma function.The functional form of a multinomial distribution can be recognized in the above equation. Henceq(Z dn ij ) = M(Z dn ij ; 1, φ dn ij = (φ dn1 ij , . . . , φ dnK ij



  log q(θ) = E Z [log p(Z|A, Y, X, θ)] + p(θ|α) + const = log θ qlk + const,where the constant term includes everything not depending on θ. The functional form of factorizing Dirichlet distributions can be detected in the above equation. Thereforeqql ; γ ql = (γ ql1 , . . . , γ qlK )),whereγ qlk = α k +

  -1) ψ(γ qlk ) -ψ

  (21) with respect to β kv . Hence, all terms in (21) depending on β are grouped in the following objective function g(β, λ)

  i.d. following a symmetric Dirichlet distribution with parameter α = 1. First of all, we look for a BIC approximation of the following term log p(W |A, Y, X) = log β p(W |β, A, Y, X)p(β)dβ ,

  log p(A, Y, X|Q, L) = log π,ρ,δ p(A, Y, X|π, ρ, δ, Q, L)p(π, ρ, δ)dπdρdδ,where p(π, ρ, δ) denotes any prior distribution over (π, ρ, δ). An ICL criterion is adopted to approximate the above log-likelihood:log p(A, Y, X|Q, L) ≈ max π,ρ,δ log p(A,Y, X|π, ρ, δ, Q, L)

Table 1 :

 1 Notations in LTBM.

		Indexes		Variables
	i, j Row/column indicators	Y i	The row cluster indicator of the i-th row of A
	q, l Row/column clusters	X j	The column cluster indicator of the j-th column of A
	k	Topics	A ij	The entry (i, j) of the M × P incidence matrix A
	n	Word indicator	W dn ij	The n-th word in the d-th document W d ij
	d	Document indicator	Z dn ij	The topic of W dn ij
	v	Word v of the vocabulary	θ ql	The topic proportions for the pair of groups (q, l)
		Numbers		Parameters
	Q	Number of row clusters	ρ	Multinomial distribution parameters (Y )
	L	Number of column clusters	δ	Multinomial distribution parameters (X)
	K	Number of topics	π ql	Connection probability between classes q and l
	M	Number of individuals/rows	β k	Multinomial distribution probabilities (k-th topic)
	P	Number of objects/columns	α	Dirichlet distribution parameters (θ)
	N d ij	Number of words in W d ij
	V	Number of vocables in the vocabulary

Table 3 :

 3 . Amazon fine food dataset statistics.The C-VEM algorithm was run on the dataset for several initializations and we tested all values of Q, L and K in a range from 1 to 20. The model selection criterion in (20) finally

	Number of users (M )	1644
	Number of objects (P )	1733
	Number of reviews	32836
	Number of vocables (V ) 11151