Giuseppe Serra
email: giuseppe.serra@eurecom.fr

Photios A Stavrou
email: fotios.stavrou@eurecom.fr

Marios Kountouris
email: marios.kountouris@eurecom.fr

Computation of Rate-Distortion-Perception Function under f -Divergence Perception Constraints

In this paper, we study the computation of the ratedistortion-perception function (RDPF) for discrete memoryless sources subject to a single-letter average distortion constraint and a perception constraint that belongs to the family of fdivergences. For that, we leverage the fact that RDPF, assuming mild regularity conditions on the perception constraint, forms a convex programming problem. We first develop parametric characterizations of the optimal solution and utilize them in an alternating minimization approach for which we prove convergence guarantees. The resulting structure of the iterations of the alternating minimization approach renders the implementation of a generalized Blahut-Arimoto (BA) type of algorithm infeasible. To overcome this difficulty, we propose a relaxed formulation of the structure of the iterations in the alternating minimization approach, which allows for the implementation of an approximate iterative scheme. This approximation is shown, via the derivation of necessary and sufficient conditions, to guarantee convergence to a globally optimal solution. We also provide sufficient conditions on the distortion and the perception constraints which guarantee that our algorithm converges exponentially fast. We corroborate our theoretical results with numerical simulations, and we draw connections with existing results.

I. INTRODUCTION

The theoretical framework of rate distortion theory subject to a single-letter average distortion constraint stems from the seminal work of Shannon in [START_REF] Shannon | A mathematical theory of communication[END_REF], [START_REF]Coding theorems for a discrete source with a fidelity criterion[END_REF]. Therein, Shannon described, for the first time, the fundamental trade-offs between the desired bit rate used for a compressed representation of the source messages and the associated achievable distortion criterion attained between the source message and its reconstructed representation. The mathematical representation in rate distortion theory is manifested by the rate-distortion function (RDF). Rate distortion theory has been the foundational basis to study and develop lossy compression algorithms for various multimedia applications.

In recent years, it has been shown through multiple works, spanning from machine learning and computer vision to multimedia applications [START_REF] Minnen | Joint autoregressive and hierarchical priors for learned image compression[END_REF]- [START_REF] Kudo | GAN-based image compression using mutual information maximizing regularization[END_REF], that focusing exclusively on distortion minimization does not necessarily imply a good perceptual quality of the reconstructed signal, where perceptual quality refers to the property of a sample to appear visually pleasing from a human perspective. Motivated by the need for a general characterization of RDF to enable encompassing the perceptual quality of the sample, Blau and Michaeli in [START_REF] Blau | Rethinking lossy compression: The ratedistortion-perception tradeoff[END_REF] introduced a generalization of the single-letter RDF, coined rate-distortion-perception function (RDPF). The specific information theoretic characterization complements the classical single-letter distortion constraint between a source message and its reconstruction, which is inherent in RDF formulation, with a divergence constraint between the induced distributions of the source and its estimated value. The divergence constraint in RDPF is precisely used as a proxy of the human perception, measuring the degree of satisfaction in the consumption of the data from a human perspective. The divergence constraint can also be viewed as a semantic quality metric, which measures the degree of relevance of the reconstructed source from the perspective of the observer, a point first hinted at in [START_REF] Kountouris | Semantics-empowered communication for networked intelligent systems[END_REF].

Another relevant yet different setup is the recently introduced robust source coding framework, see, e.g., [START_REF] Stavrou | The role of fidelity in goal-oriented semantic communication: A rate distortion approach[END_REF] (and references therein), in which in place of the perception quality, there exists an additional distortion criterion.

Since its conception, RDPF has received substantial interest from the information theory community. Theis and Wagner in [START_REF] Theis | A coding theorem for the rate-distortionperception function[END_REF] proved that the RDPF can be achieved via stochastic, variable-length codes by making use of a strong functional representation lemma [START_REF] Li | Strong functional representation lemma and applications to coding theorems[END_REF], whereas Chen et al. in [START_REF] Chen | On the Rate-Distortion-Perception function[END_REF] studied the role of stochasticity in the encoder/decoder structure and proved the achievability of the RDPF by deterministic codes (except certain extreme cases) in the asymptotic regime. Wagner in [START_REF] Wagner | The rate-distortion-perception tradeoff: The role of common randomness[END_REF] considered the case of perfect perceptual quality in the definition of RDPF and devised a coding theorem that allows for a specified amount of common randomness between the encoder and decoder.

It should be emphasized that similarly to the classical RDF, there are no known closed-form representations of the RDPF for general alphabet sources. In fact, there are only a few examples in the literature where analytical expressions are provided for the RDPF, such as the case of binary sources subject to Hamming distortion and total variation distance [START_REF] Blau | Rethinking lossy compression: The ratedistortion-perception tradeoff[END_REF], [START_REF] Qian | A rate-distortion-perception theory for binary sources[END_REF] or the case of a Gaussian source subject to a mean-squared error distortion and a 2-Wasserstein distance [START_REF] Zhang | Universal rate-distortionperception representations for lossy compression[END_REF].

In this paper, we focus on the computation of the RDPF for discrete memoryless sources subject to a single-letter average distortion constraint and a perception constraint that belongs to the class of f -divergences 1 . In particular, we leverage the fact that under mild regularity conditions on the perception constraint (i.e., convexity on the second argument) the RDPF forms a convex program. This enables us to derive a parametric characterization of the optimal solution of the RDPF (Lemma 1), which is subsequently utilized to construct an alternating minimization procedure2 for which we also establish convergence guarantees (Theorem 2). The resulting structure of the iterations in Theorem 2 prohibits the implementation of a generic Blahut-Arimoto (BA) algorithm similar to what is already known for the classical rate distortion theory for i. i. d sources and single-letter distortions [START_REF] Blahut | Computation of channel capacity and rate-distortion functions[END_REF]. To overcome this technical difficulty, we introduce a new relaxed formulation of the structure of the iterations in Theorem 2, which results in a variant of an approximate BA algorithm (Theorem 3). Additionally, in Theorem 3, we derive necessary and sufficient conditions to ensure that our approximation algorithm converges to a globally optimal solution. In light of our result in Theorem 3, we also provide sufficient conditions on the structure of the distortion and the perception constraints based on which our algorithm converges exponentially fast (Theorems 6, 7). We corroborate our theoretical findings with numerical simulations and draw comparisons with existing results in the literature (Section V).

Notation: We denote with p x (x) the probability distribution over the source alphabet X evaluated on the symbol x, and by q x(x) any arbitrary marginal probability on the output alphabet X evaluated on the symbol x. Q x|x (x|x) denotes the entry (x, x) of the transition matrix Q x|x while Q x(x) denotes the marginal on X induced by p x and Q x|x , evaluated on the symbol x. We denote by E[•] the expectation operator, and by E q [•] the probability distribution q on which the expectation operator is applied. We denote by p 1 [p 2 ] the explicit form of the functional dependence of some probability distribution p 1 functionally dependent on another probability distribution p 2 . We denote by D(•||•) any divergence measure and by

D f (•||•), D KL (•||•), D χ 2 (•||•), T V (•||•)
, the class of f -divergence, the Kullback-Leibler divergence, the chi-squared divergence and the total variation distance, respectively. We denote by f ∈ C 0 , the continuous function defining the f -divergence and ∂f denotes its sub-gradient [START_REF] Rockafellar | Variational Analysis[END_REF]Definition 8.3]. For a continuous and twice differentiable function f ∈ C 2 , we denote by f ′′ (•) the second derivative with respect to its argument.

II. PRELIMINARIES ON THE RDPF

In this section, we consider finite alphabets sources and deterministic encoders/decoders pairs and define the minimum achievable rates subject to average per-letter distortion and average per-letter perception constraints. Our analysis herein stems from recent results in [START_REF] Chen | On the Rate-Distortion-Perception function[END_REF].

We assume that we are given an i. i. d sequence of n-length random variables X n that induce the probability distribution p x . The source sequence is received by an encoder (e) that generates the index e(X n ) ∈ Z, Z = {1, 2, . . . , 2 nR }, whereas at the decoder (g), the system reconstructs an estimate of Xn . Formally, the encoder and decoder are deterministic mappings with e : X n → Z and g : Z → X n , respectively.

We let d : X × X → R + 0 to denote a single-letter distortion function and D : P × Q → R + 0 to denote a divergence measure. Moreover, we define the fidelity criterion ∆ as the expected per-symbol distortion and the fidelity criterion Φ as the expected per-symbol divergence 3 as follows:

∆ ≜ E 1 n n i=1 d(x i , xi ) Φ ≜ 1 n n i=1 D(p x ||q xi ).
We are now ready to introduce the definition of achievability and that of infimum of all achievable rates.

Definition 1. (Achievability) Given a distortion level D > 0 and a perception constraint P > 0, a rate R is said to be (D, P )-achievable if for an arbitrary ϵ > 0, there exists, for large enough n, a deterministic lossy source code (n, M, ∆, Φ) with M ≤ 2 n(R+ϵ) such that ∆ ≤ D + ϵ and Φ ≤ P + ϵ.

Then, we define

R nr (D, P ) ≡ inf{R : (R, D, P ) is achievable}. (1)
Next, we give the definition of the information theoretic characterization of the RDPF (see [START_REF] Blau | Rethinking lossy compression: The ratedistortion-perception tradeoff[END_REF]) assuming that D > 0 and P > 0.

Q x|x I(X, X) s.t. E[d(x, x)] ≤ D, D(p x ||Q x) ≤ P (2)
where D ∈ [D min , D max ] ⊂ (0, ∞), P ∈ [P min , P max ] ⊂ (0, ∞) and

I(X, X) = D KL (p x Q x|x , p x Q x) ≜ I(p x , Q x|x ) (3)
where I(p x , Q x|x ) highlights the dependency on {p x , Q x|x }.

We stress the following technical remarks on Definition 2.

Remark 1. (On Definition 2) Following [START_REF] Blau | Rethinking lossy compression: The ratedistortion-perception tradeoff[END_REF], it can be shown that (2) has some useful properties, under mild regularity conditions. In particular, [9, Theorem 1] shows that R(D, P )

is (i) monotonically non-increasing function in both D ∈ [D min , D max ] ⊂ [0, ∞) and P ∈ [P min , P max ] ⊂ [0, ∞); (ii) convex if the divergence D(•||•) is convex in its second argument.
In the sequel, we consider that in (2) the perception constraint is an f -divergence, i.e., D(

•||•) = D f (•||•), which is known to be convex in both arguments [22, Lemma 4.1].
Hence in light of the discussion in Remark 1, R(D, P ) forms a convex programming problem.

We conclude this section by providing a theorem that connects R nr (D, P ) with R(D, P ) for finite alphabets, as long as D > 0 and P > 0.

Theorem 1. For |X | < ∞, D > 0, P > 0, we obtain R nr (D, P ) = R(D, P ).

(4)

Proof: This is a consequence of [14, Assumption 2] that results into [14, Theorem 2].

It should be noted that in the previous analysis, we excluded the extreme case where P = 0. This is because, in that scenario, deterministic encoders and decoders do not achieve R(D, 0) and one instead requires common [START_REF] Theis | A coding theorem for the rate-distortionperception function[END_REF]Theorem 3] or private [START_REF] Chen | On the Rate-Distortion-Perception function[END_REF]Theorems 4] randomness to achieve it.

III. MAIN RESULTS

In this section, we present our main results. We start by reformulating (2) into a double minimization problem.

Q x|x min q x s 1 E[d(x, x)] + s 2 D f (p x ||Q x) + D KL (p x Q x|x ||p x q x) (5)
where

D = E Q * x|x [d(x, x)], P = D f (p x ||Q * x), with Q * x|x
achieving the minimum. Furthermore, for fixed Q x|x (x|x), the right side of (5) is minimized by

q x(x) = x∈X p x (x)Q x|x (x|x). (6)
For fixed q x, the right side of (5) is minimized by

Q x|x (x|x) = q x(x) • A(x,x,s) i∈X q x (i)A(x,i,s) (7)
where

A(x, x, s) = exp{-s 1 d(x, x) -s 2 g(p x , Q x, x)} (8)
g(p x , Q x, x) = f px(x) Q x(x) -px(x) Q x (x) ∂f px(x) Q x (x) .
We note that Lemma 1 differs from [23, Theorem 6.3.3] in [START_REF] Shaham | Deformation aware image compression[END_REF]. Specifically, the presence of the additional perception constraint D f (•||•) in (2) changes the functional properties of the parametric solution [START_REF] Shaham | Deformation aware image compression[END_REF], as it requires an additional exponential term, i.e., s 2 g(p x , Q x, x), with g(•) depending on the induced marginal Q x.

The next corollary is a consequence of Lemma 1.

Corollary 1. R(D, P ) in (5) can be reformulated as follows:

R(D s , P s ) = -s 1 D s -s 2 P s + min q x s 2 x∈ X p x (x)∂f px(x) Q x [q x](x) - x∈X p x (x) log   x∈ X q x(x)A(x, x, s)   (9)
where q *

x achieves the minimum and

D s = (x,x)∈X × X p x (x) q * x(x)A(x,x,s) i∈ X q *
x (i)A(x,i,s) d(x, x)

P s = D f (p x ||Q x[q * x]).
Next, we proceed to construct an alternating minimization procedure and show its convergence to a point of R(D, P ).

Theorem 2. (Alternating minimization) Let s 1 ≥ 0, s 2 ≥ 0 be given, with s = (s 1 , s 2 ), and let A[q x] be such that

A[q x](x, x, s) = exp{-s 1 d(x, x) -s 2 g(p x , Q x[q x], x)} g(p x , Q x[q x], x) = f px(x) Q x[q x ](x) -px(x) Q x[q x ](x) ∂f px(x) Q x[q x ](x) .
Let q (0) x denote any probability vector with nonzero components and let q

(n+1) x ≡ Q x[q (n) x ] and Q (n+1) x|x ≡ Q x|x [q (n)
x ] be defined as functions of the previous iteration q (n) x as follows:

Q (n+1) x|x (x|x) = q (n) x (x) A (n) (x,x,s) i∈ X q (n) x (i)A (n) (x,i,s) q (n+1) x (x) = q (n) x (x) x∈X px(x)A (n) (x,x,s) i∈ X q (n) x (i)A (n) (x,i,s) (10)
where

A (n) (x, x, s) = A[q (n)
x ](x, x, s). Then, as n -→ ∞, we obtain

D(Q (n) x|x ) - → D s , P (Q (n) x|x ) - → P s , I(p x , Q (n) x|x ) - → R(D s , P s ).
Despite being optimal, the alternating minimization scheme of Theorem 2 does not allow the implementation of a BA type of algorithmic embodiment. Due to the structure of the iterations in [START_REF] Kountouris | Semantics-empowered communication for networked intelligent systems[END_REF], a non-reversible dependency of q (n+1) x on itself appears once A (n) (x, x, s) is substituted therein, thus requiring the knowledge of q (n+1) x to evaluate q (n+1) x itself. To circumvent this technical difficulty, we introduce a relaxation in the structure of the iterations of Theorem 2. This results in a variant of an approximate alternating minimization scheme, for which we derive necessary and sufficient conditions that ensure its convergence to a globally optimal point. The aforementioned relaxation and the necessary and sufficient conditions that lead to a globally optimal solution are stated in the following theorem, which is a major result of this paper. Theorem 3. (Approximate alternating minimization) Let s 1 ≥ 0, s 2 ∈ [0, s 2,max ] be given with s = (s 1 , s 2 ) and let Ã[q x](x, x, s) be

Ã[q x](x, x, s) = exp{-s 1 d(x, x) -s 2 g(p x , v[q x], x)} g(p x , v[q x], x) = f ( px(x) v [q x ](x) ) -px(x) v [q x ](x) ∂f ( px(x) v [q x ](x) )
where v[q x] is any probability vector. Let q(0)

x be any probability vector with nonzero components and let q(n+1)

x = Q x[ q(n) x ] and Q(n+1) x|x = Q x|x [q (n)
x ] be defined as functions of the past iteration q(n)

x as follows:

Q(n+1) x|x = q(n) x (x) Ã(n) (x,x,s) i∈ X q (n) x (i) Ã(n) (x,i,s) q(n+1) x = q(n) x (x) x∈X px(x) Ã(n) (x,x,s) i∈ X q(n) x (i) Ã(n) (x,i,s)
where

Ã(n) (x, x, s) = Ã[q (n)
x ](x, x, s). Then, as n -→ ∞, we obtain

D( Q(n) x|x ) - → D s , P ( Q(n) x|x ) - → P s , I(p x , Q(n) x|x ) - → R(D s , P s ) if and only if lim n→∞ q(n+1) x v [q (n) x ]
→ 1 with at least linear rate of convergence.

Theorem 3 enables the implementation of the alternating minimization scheme by introducing an auxiliary variable v[q

(n)
x ] (approximation), designed as a function of only the current iteration of q (n)

x . Nevertheless, depending on v, this approximation may incur restrictions on the domain of the Lagrangian multiplier s 2 in order to have convergence guarantees. The implementation of Theorem 3 is illustrated in Algorithm 1.

We conclude this section, with a lemma where we provide necessary and sufficient conditions for the optimal solution of the studied problem.

Lemma 2. Let D f (•||•) be such that f ∈ C 1 (0, ∞) continuous
and differentiable on (0, ∞). Then, a probability vector q x yields a point on the R(D, P ) curve via the transition matrix

Q x|x [q x](x) = q x(x) px(x) Ã(x,x,s) i∈ X q x (i) Ã(x,i,s) if and only if, ∀x ∈ X , c(x) = x∈X px(x) Ã(x,x,s) i∈ X qx (i) Ã(x,i,s) ≤ 1,
holding with equality for any x for which q x(x) is nonzero.

IV. ANALYSIS OF ALGORITHM 1

In this section, we study a stopping criterion and the convergence rate of Algorithm 1.

Before obtaining stopping conditions for Algorithm 1, we first derive a useful lemma which is needed to obtain the stopping conditions. Lemma 3. Let s 1 ≥ 0, s 2 ∈ [0, s 2,max ) be given with s = (s 1 , s 2 ) and let Q x|x be a transition matrix included in the set L (D,P ) defined as follows:

L (D,P ) = {Q x|x : E Q x|x [d(x, x)] ≤ D ∧ D f (p x ||Q x) ≤ P }. Then, ∀λ ∈ Λ s,v [Q x ] , with Λ s,v [Q x ] = λ ∈ R |X | : ∀x ∈ X , λ(x) ≥ 0 ∧ ∀x ∈ X , x∈X p x (x)λ(x) Ã(x, x, s) ≤ 1 , we obtain R(D, P ) ≥ - x∈X p x (x) log 1 λ(x) -s 1 D -s 2 (x,x)∈X × X p x (x)Q x|x (x, x)g(x).
x . 1: n ← 0 2: flag ← 0 3: while flag == 0 do 4:

g(x) ← f px(x) v (n) (x) -px(x) v (n) (x) ∂f px(x) v (n) (x)
5:

Ã(n) (x, x, s) ← exp[-s 1 d(x, x) + s 2 g(p x , v[q x], x)] 6: c (n) (x) ← x∈X px(x) Ã(n) (x,x,s) i∈X q(n) x (i) Ã(n) (x,i ,s) 7:
q

(n+1) x ← q (n) x • c (n) 8: ω ← log c (n) max (x) -x∈ X q (n) x c (n) (x) log(c (n) (x)) 9:
if ω ≤ ϵ then n ← n + 1 13: end while Ensure:

Q x|x = q (n) x (x) Ã(n) (x,x,s) i∈ X q (n) x (i) Ã(n) (x,i,s) , D s = E pxQ x|x [d(x, x)], P s = D f (p x , q (n) x ), R(D s , P s ) = W [q (n) x ] -s 1 D s -s 2 P s - x∈ X q (n) x c(x) log(c(x)), W [•] = (13).
where W [q x] is given by

W [q (n) x ] = - x∈X p x (x) log   x∈ X q(n) x (x) Ã(n) (x, x, s)   + s 2 x∈ X p x (x) px(x) v (n) (x) ∂f px(x) v (n) (x) + s 2 x∈ X q(n+1) x f px(x) q (n+1) x (x) -f px(x) v (n) (x) . ( 13
)
Next, we study the convergence rate of Algorithm 1. This is done by first studying the convergence rate of the alternating minimization procedure of Theorem 2 and then, using it as a reference to analyze the convergence rate of the approximate alternating minimization procedure of Theorem 3.

Note that based on the structure of the parametric solution of q (n+1) x at the previous iteration n, we can similarly define a vector function S :

R |X | → R |X | as S[q x](i) = q x(i)•c[q x](i).
Using Lemma 2, a distribution q *

x that achieves the RDPF is a fixed point of S(q x). Following [START_REF] Nakagawa | Analysis of the convergence speed of the Arimoto-Blahut algorithm by the second-order recurrence formula[END_REF], we can analyze the convergence rate of the alternating minimization procedure in both Theorems 2 and 3. Using the first order Taylor expansion of S[q x] around a fixed point q *

x, we obtain S[q x] = S[q * x] + J(q *

x) • (q x -q * x) + o(||q x -q * x||), where J(q x) is the Jacobian matrix of S(q x) with entries J i,j (q x) ≜ ∂S[q x ](i) ∂q x (j) , (i, j) ∈ X × X . The next theorem provides the functional form of the Jacobian for the case of Theorem 2.

Theorem 5. (Jacobian form) The Jacobian J(q x) computed at the fixed point q *

x is given as

J(q * x) = (I -M )(I -ΓJ(q * x)) (14)
where

M ≜ q * x(i) x∈X p x (x) A(x,i,s)A(x,j,s) ( k∈ X q * x (k)A(x,k,s)) 2 (i,j)∈X × X (15) Γ ≜ s 2 • diag q * x(i) • ∂ 2 ∂q(i) 2 D f (p x ||q) q * x i∈X . (16)
Next, we introduce two lemmas, in which we use the structure of ( 14) to identify properties of matrix M . x, we have that λ i ≤ 1, ∀i ∈ 1 : |X |. Using Lemmas 4 and 5, we can now characterize the interval that contains the set of eigenvalues of J(q *

x) and subsequently the convergence rate of Theorem 2. Theorem 6. (Convergence rate of Theorem 2) Let {θ i } i∈X be the eigenvalues of J(q * x). Then, 0 ≤ {θ i } i∈X < 1.

Moreover, let γ ∈ [θ max , 1). Then, there exists δ > 0 and K > 0 such that if q 0 x ∈ {q x : ||q x -q * x|| ≤ δ}, we obtain

||q (n) x -q * x|| < K • ||q (0) x -q * x|| • γ n (17)
i.e., the iterations converge exponentially.

Following similar steps that led to Theorem 6, we can now write the Jacobian J a (q *

x) for Theorem 3 considering a specific form of the auxiliary variable v[q

(n) x ] = q (n)
x . This results into the following structure of the Jacobian matrix J a (q *

x) = (I -M )(I -Γ) where M and Γ are given by ( 15) and ( 16), respectively. Unlike Theorem 2, where the structure of ( 14) bounds its own eigenvalues, in this case we need to bound the Lagrangian multiplier s 2 , hence matrix Γ, to guarantee exponential convergence of the algorithm. This is proved in the following theorem.

Theorem 7. For a given s 1 ≥ 0, let I s2 = [0, s 2,max ] be the domain of s 2 , {θ a,i } i∈X the set of eigenvalues of J a (q *

x) and θ max the maximum eigenvalue of J(q *

x) in [START_REF] Chen | On the Rate-Distortion-Perception function[END_REF]. Define the set I ϵ s2 = [0, s 2,max -ϵ] for 0 < ϵ < s 2,max . Then, there exists an ϵ ′ such that if s 2 ∈ I ϵ ′ s2 then 0 ≤ {θ a,i } i∈X < 1. Theorem 7 guarantees exponential convergence for Theorem 3 only for s 2 ∈ I ϵ s2 which means that we are able to consider P ∈ [P min (ϵ), P max ], depending on the characteristics of the specific problem.

V. NUMERICAL RESULTS

In this section, we provide numerical results to demonstrate the utility of Algorithm 1. In Fig. 1a we compare the theoretical results of [9, Equation 6] with the numerical results obtained using Algorithm 1. We observe that Algorithm 1 achieves exactly the theoretical solution of [9, Equation 6] as long as D ≤ D max = 0.15.

In Fig. 1b and Figure 1c

Definition 2 .

 2 (RDPF) For a given finite alphabet source distribution p x , a single-letter distortion d(•, •) and a divergence D(•||•), the RDPF is characterized as follows: R(D, P ) = min

Lemma 1 .

 1 (Double minimization) Let s 1 ≥ 0, s 2 ≥ 0, and define s = (s 1 , s 2 ). Moreover, let D > 0, P > 0 and let D(•||•) = D f (•||•). Then (2) can be expressed as a double minimum as follows: R(D, P ) = -s 1 D -s 2 P + min

Theorem 4 .Algorithm 1

 41 (Stopping criterion) Let Qx|x and qx be defined as in Theorem 3, c(x) be as defined as in Lemma 2 and c max = Implementation of Theorem 3 Require: source distribution p x ; Lagrangian multipliers s = (s 1 , s 2 ) with s 1 ≥ 0 and s 2 ∈ [0, s 2,max ]; error tolerance ϵ; divergence measure D f (•||•); distortion measure d(•, •); initial assignment q

  max x∈ X c(x). Then, at the point D = E Qx|x [d(x, x)], and P = D f (p x || Qx )), the following bounds hold R(D, P ) ≥ -s 1 D -s 2 P + W [q x] -log(c max ) (11) R(D, P ) ≤ -s 1 D -s 2 P + W [q x] -x∈ X qx c(x) log(c(x))

Lemma 4 .

 4 Let {λ i } i=1:|X | be the set of eigenvalues of M . Then, given a distortion function d : X × X → R + 0 that induces a full-rank matrix D = [e -s1d(i,j) ] (i,j)∈X × X , then λ i > 0,∀i ∈ [1 : |X |], i.e., M has only positive eigenvalues. Remark 2. (On Lemma 4) We note that a popular example that satisfies the assumptions imposed on Lemma 4 is the Hamming distortion denoted hereinafter as d H [25]. Lemma 5. Let {λ i } i=1:|X | be the set of eigenvalues of M . Then, at fixed point q *

Example 1 .

 1 Suppose that X = X = {0, 1} and letp x ∼ Ber(0.15) with d(•, •) = d H (•, •) and perception constraint chosen to be either (a) D f (p x ||q x) = T V (p x ||q x) = 1 2 i∈X |p x (i) -q x(i)|, or (b) D f (•||•) = D KL (•||•), or (c) D f (•||•) = D χ 2 (•||•).

  we use Algorithm 1 to compute R(D, P ) for D f (•||•) = D KL (•||•), and for D f (•||•) = D χ 2 (•||•), respectively.

Fig. 1 :

 1 Fig. 1: R(D, P ) for a Bernoulli source under a Hamming distortion and various perception constraints.

For a mathematical background on f -divergences and their properties we refer the reader to[START_REF] Sason | On f -divergences: Integral representations, local behavior, and inequalities[END_REF].

For details on the alternating minimization procedure, see e.g.,[START_REF] Csiszar | Information geometry and alternating minimization procedures[END_REF],[START_REF] Yeung | Information theory and network coding[END_REF].

Following for instance[START_REF] Chen | On the Rate-Distortion-Perception function[END_REF] Remark 3], one can choose the perception fidelity criterion to be D(px||q xi ), i = 1, . . . , n, or D(px|| n i=1 q xi ). Both types of fidelity criteria result in the same operational quantity for finite alphabets in the asymptotic regime.

ACKNOWLEDGEMENT This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation programme (Grant agreement No. 101003431).


G Mounajed

H Boussa

F Grondin

A Menou

MICRO STRUCTURAL ORIGIN OF THE APPARENT THERMAL TRANSIENT CREEP OF CONCRETE AT HIGH TEMPERATURE

Keywords: transient thermal creep, thermal damage, multi-scale model, high temperature, micro macro

This research has been conducted within the framework of the study of concrete behaviour at high temperature. The aim of this study is to model the thermal damage mechanisms of concrete and to investigate the behaviour of concrete specimens when they are simultaneously subjected to a constant compressive load and to high temperature. A new approach, the multiphase Digital Concrete Model and the damage model MODEV, both implemented on finite-element software SYMPHONIE, has been used to model the thermal damage of concrete. In this approach, 2 main categories of concrete thermal damage mechanisms are considered:

-Mechanical thermal damage as a result of restrained thermal strain on a macroscopic and mesoscopic scale: This damage can be due, on one hand, to the temperature gradient or the boundary conditions, or even the geometric shape on a macroscopic scale. It can be due, on other hand, to the differential expansion between the cement paste and the aggregates on a mesoscopic scale.

-Purely thermal damage as a result of the different chemical transformations, which occur mainly in cement paste: dehydration, important mass loss beyond 120°C, and other chemical transformations.

In order to study the behaviour of concrete specimens when they are subjected such loading, numerical simulations of the experimental tests carried out by Holst [4] have been performed. The "Digital Concrete" model is used to scrutinise the structural mesoscopic effects and to quantify the damage induced. Material and thermal characteristics are randomly distributed accordingly to the material mixture: granulates, sand and cement paste. Theses simulations show that the « transient thermal strain», noticed during experimental tests [5]), is a structural effect and is due to heterogeneous nature of concrete at mesoscopic scale.

INTRODUCTION

This research has been conducted within the framework of the study of concrete behaviour at high temperature. The experimental studies show an important influence of the temperature on concrete behaviour. The objective is to understand and to simulate the strain mechanisms at high temperature. It particularly focuses on the evolution of the thermal strain of concrete specimens when they are simultaneously subjected to a constant compressive load and to high temperature. In fact, experimental tests show that the thermal strain is strongly influenced by the compressive load. Figure 1a shows the total strain of concrete samples heated under a constant compressive load given by Schneider [START_REF] Schneider | Concrete at high temperatures-a general review[END_REF], several levels of compressive load were tested, from 0% up to 60% of the compressive strength of concrete. Several authors explain these observations by an additional strain, called transient thermomechanical interaction strain or transient creep strain [START_REF] Heinfling | Thermo-elastic-plastic model for concrete at elevated temperatures including cracking and thermo-mechanical interaction strains[END_REF], [START_REF] Khoury | Transient thermal strain of concrete : litterature review, conditions with specimen and behaviour of individual constituents[END_REF], [START_REF] Nechnech | An elasto-plastic damage model for plain concrete subjected to high temperatures[END_REF], [START_REF] Thelandersson | Modeling of combined thermal and mechanical action in concrete[END_REF]. This additional strain is considered as a function of both the temperature and the stress. Literature review is available in Khoury & al. [5].

In our opinion, these tests results do not reflect an intrinsic material behaviour, but should be considered as a structural behaviour of concrete due to the microscopic thermal damage. To show the structural effect of such loading, numerical simulations of the experimental tests carried out by Holst et al [START_REF] Khoury | Transient thermal strain of concrete : litterature review, conditions with specimen and behaviour of individual constituents[END_REF] have been achieved. The microscopic multi-phase Digital-Concrete F.E. model (DC) (Mounajed [7]), as well as the macroscopic deviatoric damage model MODEV (Mounajed & al. [8], Ung [START_REF] Quoc | Théories de dégradation du béton et développement d'un nouveau modèle d'endommagement en formulation incrémentale et tangente[END_REF]) have been used to evaluate thermal damage at macroscopic and microscopic level. These two models are implemented in the general F.E. code SYMPHONIE.

Principle of the multi-scale identification of thermal damage

The thermal damage of concrete presents a specific mechanism directly associated to the nature of this heterogeneous material. The analysis of the different standard macroscopic models shows the insufficiency of this approach regarding the representation of the complex mechanisms involving the thermal damage. All concrete macroscopic models assume the homogeneity of this material, thus a free displacement specimen, subjected to uniform thermal loading, expands without stresses and so without damage. However, experiments conducted in laboratories show that the process of thermal damage of concrete begins early, even at relatively moderate and homogeneous temperatures.

Here, we propose a new approach to study thermal damage of concrete. This approach is based on the multi scale Homogenization of concrete using the Digital Concrete model. Several scales of modelling are taken into account, ranging from the cement paste to the concrete material [START_REF] Mounajed | Multi-scale approach of Thermal damage: Applications to concrete at high temperature[END_REF]. Thus, the concrete is considered as a combination of three materials: cement, mortar, and concrete. Each material is considered on its specific scale, and homogenisation is needed on each scale to go to the following upper one. Figure 2b shows the basis of this approach. The mechanical and thermal characteristics of cement paste and sand aggregates are conjointly introduced into the Digital Concrete DC model in order to homogenize the mortar behaviour. This process is repeated for concrete homogenization by introducing in the DC model, the characteristics of the homogenized mortar, obtained in the previous step, and the characteristics of aggregates. This step enables to obtain the thermo-mechanical behaviour of concrete. The local DC model considers a random approach to generate the internal structure of a heterogeneous material as concrete, mortar or cement paste on a smaller scale. The proposed approach considers a multi-phase material with successions of n phases spatially distributed in a random manner. In the present simulations, to represent concrete, the following phases are taken into account: 1 st phase: Solid skeleton of the matrix cement P 1 ; 2 nd phase: A random distribution of aggregates with the possibility of analyzing this phase in n sub-phases to represent different sizes and natures of aggregates P 21 , P 22 … P 2n . 1. Mechanical damage of thermal origin accompanied with strains and due to the restrained thermal strains on a macroscopic and microscopic scale: The original mechanical damage can be the temperature gradient or the boundary conditions, or even the geometric shape on a macroscopic scale, and the differential expansion between the cement paste and the aggregates on a microscopic scale. This damage is decomposed into mechanical macroscopic damage D macro and mechanical microscopic damage D micro.

The mechanical thermal damage of concrete is modelled with the deviatoric damage model MODEV which is based on the damage mechanics approach and on the thermodynamics. This model uses a non-symmetrical criterion in strain. MODEV introduces 2 scalar equivalent strains representing respectively local sliding and crack opening. They are calculated respectively from deviatoric and spherical parts of the strain tensor. MODEV model considers also 2 scalar damage variables, corresponding respectively to sliding and crack opening degradation mechanisms. Each damage variable has its own evolution law [START_REF] Quoc | Théories de dégradation du béton et développement d'un nouveau modèle d'endommagement en formulation incrémentale et tangente[END_REF]. The damage variables are independent from the temperature and can be obtained directly from the mechanical behaviour law.

2-Purely thermal damage without strains due to the different chemical transformations, which occurs mainly in cement paste: dehydration, important mass loss beyond 120°C, and other chemical transformations. This damage has been identified from experimental tests performed on cement samples [6] [10]. Figure 1b shows this damage function, obtained by fitting experimental results [START_REF] Menou | Etude du comportement thermomécanique des bétons à haute température. Approche multi échelles de l'endommagement thermique[END_REF] and introduced into the DC model. 3. SIMULATION OF CONCRETE SPECIMENS SUBJECTED TO COMPRESSIVE LOAD AT HIGH TEMPERATURE To study the thermal damage of concrete specimens when they are heated under compressive load, the experimental tests carried out by Holst [START_REF] Holst | Mechanical properties of high strength concrete at high temperatures[END_REF] have been simulated. Cylindrical specimens (68 x132mm 2 ), have been subjected to several constant compressive loads and heated up to 800°C with a heating rate of 4,98 °C/min.

The tests are modelled in 2D, with an axisymetrical model. A constant compressive load is applied at the top of the specimen at the beginning of the heating stage and is maintained constant during the test. The load represents 0%, 20%, 40% or 60% of the compressive strength of the HSC (108 MPa). Figure 4 (a) shows the geometry, the mesh and the boundary conditions adopted for the computations. The thermo mechanical characteristics of the HSC are given by Holst [START_REF] Holst | Mechanical properties of high strength concrete at high temperatures[END_REF]. To perform the computations with the Digital Concrete model, it is also necessary to know the properties of cement paste and aggregates separately (tensile and compressive strengths, thermal expansion coefficient, fracture energy, Young modulus...) most of theses information, not available in [START_REF] Holst | Mechanical properties of high strength concrete at high temperatures[END_REF], are chosen from literature [START_REF] Diederichs | Behaviour of high temperatures[END_REF], [START_REF] Menou | Etude du comportement thermomécanique des bétons à haute température. Approche multi échelles de l'endommagement thermique[END_REF]. Materials parameters used in the simulation are presented in table 1, where E is the Young modulus, f t the tensile strength, G f the fracture energy and ν the Poisson's ratio. Bc and α, are two parameters of the damage model MODEV. The aggregate size distribution of the concrete is described in figure 3.

E (Mpa) ν f t (Mpa) G (Mpa) G f (N/mm) α B c
Cement The thermal damage in the specimen, obtained after one hour of heating, is presented in Figure 4 (b). The damage computed, using a macroscopic model for such test, is often very small compared to the damage obtained at the constituent's scale, using the DC micromechanical approach. This damage is mainly due to the strong difference between the coefficients of thermal expansion of cement paste and aggregates; a uniform temperature variation generates strain gradients in the material which is now analysed as a structure. These strain gradients are responsible of the induced damage zones at the mesoscopic level. Simulations and experimental results are in good agreement. Figure 4 presents for simulations and experiments, the evolution of the total strain versus the temperature for the 4 loading levels (0, 20, 40 and 60%). These results confirm that, the evolution of the total strain is mainly due to the thermal degradation of concrete stiffness at high temperature. No additional strain component such transient or creep strain [START_REF] Heinfling | Thermo-elastic-plastic model for concrete at elevated temperatures including cracking and thermo-mechanical interaction strains[END_REF], [START_REF] Khoury | Transient thermal strain of concrete : litterature review, conditions with specimen and behaviour of individual constituents[END_REF], [START_REF] Nechnech | An elasto-plastic damage model for plain concrete subjected to high temperatures[END_REF], [START_REF] Thelandersson | Modeling of combined thermal and mechanical action in concrete[END_REF] is needed to explain the evolution of the total strain versus temperature. 4. CONCLUSION In order to understand the mechanisms leading to the deterioration of concrete under the effect of combined compressive and thermal loading, and to show the structural effect of such loading, numerical micromechanical simulations of the experimental tests carried out by Holst [START_REF] Holst | Mechanical properties of high strength concrete at high temperatures[END_REF] have been achieved at mesoscopic scale. These Simulations, show that the experimental "transient strain of concrete" do not reflect an intrinsic material behaviour, but should be considered as structural behaviour due to both restrained boundary conditions and microscopic thermal damage. These simulation results confirm that, the evolution of the total strain of concrete specimens when subjected to such loading is mainly due the thermo mechanical degradation of concrete properties at high temperature. So, no additional strain component such transient or creep strain is needed to obtain the evolution of the total strain versus temperature.

Figure 1 :

 1 Figure 1: (a) total strain vs temperature [12], (b): Physical and chemical thermal damage origin

Figure 2 :

 2 Figure 2: (a): thermal damage approach (b): Multi scale homogenization

Figure 3 .

 3 Figure 3. Aggregate size distribution adopted for the simulations.

Figure 4 :

 4 Figure 4: Mesh and boundaries conditions; Thermal damage after one hour

Figure 5 :

 5 Figure 5: Model response for coupled thermal and mechanical loads compared to tests results (up to 500°C)

Table 1 .

 1 Materials parameters defined for the cement paste and for concrete aggregates.The evolution of the thermal expansion coefficients (α th ) of concrete components (cement paste and aggregates) with temperature is given in table 2, and table 3.

	Paste	20000	0.2	4	15	0.1	0.3	80
	Aggregates	80000	0.28	10	20	0.15	0.3	70
	T (°C)	α th (°C -1 )				T (°C)	α th (°C -1 )
	20	10 e-6				20	3 e-6
	120	15 e-6				200	9 e-6
	400	-5 e-6				500	21 e-6
	1200	-25 e-6				800	57 e-6
	Table 2. Thermal expansion coefficient of
	cement paste.

Table 3 .

 3 Thermal expansion coef. of aggregates.