Abdellatif Zeggar

On the cohomological equation of a hyperbolic automorphism

Keywords: Mathematics Subject Classification, 58D25 ; 42B05, Keywords, Hyperbolic automorphism, Fourier series

In this paper, we study the discrete cohomological equation (e) : f -f • γ = g associated with a hyperbolic affine automorphism γ : T d -→ T d of the torus T d = R d /Z d . g being a given C ∞ function and f an unknown function. More precisely, if we denote by E the Fréchet space of all C ∞ functions on T d and by δ the operator defined on E by δ(f ) = f -f • γ, we show that the space δ(E) is a closed subspace of E and we explicitly determine a continuous linear operator L : δ(E) → E such that for every element g of δ(E), the function f = L(g), given by its Fourier series, is a solution of the equation (e).

Introduction

Let M be a connected differentiable manifold. The space E = C ∞ (M ) of C ∞ functions on M is a Fréchet space for the C ∞ topology (the topology of the uniform convergence of all the derivatives on compacts subsets). A differentiable action of a discrete group Γ (supposed of finite presentation) on M induces a natural action on E given by:

∀γ ∈ Γ, ∀f ∈ E, γ.f = f • γ -1
which makes E a Γ-module. Then one can consider the cohomolgy H * (Γ, E) of the discrete group Γ with values in E. In the case Γ is generated by a single element γ we can easily show that the space H 1 (Γ, E) that we will denote H 1 (γ, E) is the Cokernel of the cobord operator:

δ : E → E, f → δ(f ) = f -f • γ
The calculation of the Cokernel E/δ(E) of δ amounts to solving the following equation [START_REF] Kacimi | Cohomologie des groupes discrets à valeurs dans un Fréchet. Prépublication UPHF[END_REF]:

f -f • γ = g where       
f ∈ E is unknown and g ∈ E is given called the cohomological equation associated with the difféomorphisme γ : M → M . We say that the automorphism γ : M → M is cohomologically C ∞ -stable when δ(E) is a closed subspace of E, that is when the topological space H 1 (γ, E) = E/δ(E) is Hausdorff [START_REF] Avila | Cohomological equations and invariant distributions for minimal circle diffeomorphisms[END_REF].

In the following, we will study this equation in the case of a hyperbolic automorphism of a torus using the fundamental tool of Fourier series.

Let A be a hyperbolic matrix of GL(d, Z) (every eigenvalue λ ∈ C satisfies |λ| = 1) and b an element of R d . The affine automorphism γ : R d → R d , x → γ(x) = Ax + b of R d induces an affine hyperbolic automorphism of the torus T d = R d /Z d that we still denote γ. The cohomological equation associated to the hyperbolic dynamical system γ is the following:

(e) f -f • γ = g where        f ∈ E := C ∞ (T d ) is unknown and g ∈ E is given
The purpose of this work is to establish the following theorem:

The main theorem

With the notations and the conditions above, we have the following result:

The range δ(E) of the cobord operator δ :

E → E, h → δ(h) = h -h • γ is a closed subspace of the Fréchet space E
and there is a continuous linear operator L : δ(E) → E such that for any element g of δ(E) the function f = L(g), given by its Fourier series, is a solution of the cohomological equation (e).

This result is a generalization of the Theorem 5.1 of [START_REF] Dehghan-Nezhad | Équations cohomologiques de flots riemanniens et de difféomorphismes d'Anosov[END_REF] proved by A. Dehghan-Nezhad and A. El Kacimi in the case b = 0 and A is a diagonalizable matrix whose all eigenvalues are real and positive.

Let us first give some examples of hyperbolic automorphisms and recall some notions that we will use in the proof of the main theorem.

Examples of hyperbolic matrix

Diagonalizable matrix with eigenvalues all real

A well-known example on the torus T 2 is the hyperbolic automorphism (called Arnold's cat) defined by the matrix A 1 = 1 1 1 2 whose eigenvalues are :

λ 1 = 3- √ 5 2 and λ 2 = 3+ √ 5 2 .
A 1 is a diagonalizable hyperbolic matrix whose eigenvalues are positive.

The matrix A 2 = 1 1 1 0 , whose eigenvalues are :

λ 1 = 1- √ 5 2 and λ 2 = 1+ √ 5
2 , defines a hyperbolic automorphism on the torus T 2 . It is diagonalizable and its eigenvalues are not all positive.

Diagonalizable matrix with complex eigenvalues

Consider the matrix

A =    1 1 1 1 0 0 0 1 0   .
Its characteristic polynomial P A (X) is given by

P A (X) = det(A -X id) = -X 3 + X 2 + X + 1
Its derivative is factorized as follows:

P A (x) = -3x 2 + 2x + 1 = (x 2 + 2x + 1) -4x 2 = (x + 1) 2 -(2x) 2 = (3x + 1)(1 -x)
A study of variation allows to see that the polynomial function P A admits only one real root µ in the interval

[ 3 2 , 2].
Denote by λ = a + ib and λ = a -ib the other two roots of P A in C (the polynomial being with real coefficients). We have : b = 0 and 1 = det(A) = µλλ = µ|λ| 2 .

So we have three eigenvalues µ, λ, λ such that µ > 1 and |λ| = |λ| = 1 √ µ < 1.

Non-diagonalizable matrix with complex eigenvalues

Consider Q(X) = (P A (X)) 2 where P A is the polynomial of the example above. Explicitly we have

Q(X) = X 6 -2X 5 -X 4 + 3X 2 + 2X + 1 ∈ Z[X]
; it is both the characteristic polynomial and the minimal polynomial of its companion matrix:

C Q =           0 0 0 0 0 -1 1 0 0 0 0 -2 0 1 0 0 0 -3 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 2          
C Q is therefore a matrix of GL(6, Z) admitting µ, λ and λ as eigenvalues each of them with multiplicity equal 2. C Q is therefore a non-diagonalizable hyperbolic matrix whose eigenvalues are not all real.

Preliminary results

Suppose that the matrix A of GL(d, Z) is hyperbolic. Then the transpose B of A -1 is also hyperbolic since its eigenvalues are the inverse of those of A.

Denote by λ 1 , ..., λ r the eigenvalues of B in C with:

       0 < |λ j | < 1 for 1 ≤ j ≤ q < r |λ j | > 1 for q + 1 ≤ j ≤ r
For j ∈ {1, ..., r}, the eigenvalue λ j and its conjugate λ j are roots of the characteristic polynomial P B (X) of B with the same modulus |λ j | and the same multiplicity k j ∈ N * . So we have :

P B (X) = S B (X) U B (X) where    S B (X) = (X -λ 1 ) k1 • • • (X -λ q ) kq ∈ R[X] U B (X) = (X -λ q+1 ) kq+1 • • • (X -λ r ) kr ∈ R[X]
The polynomials S B (X) and U B (X) are relatively prime in the ring R[X] and the kernel lemma gives the following decomposition of the vector space V = R d :

V = V s ⊕ V u where V s = KerS(B) (the stable subspace) and V u = KerU (B) (the unstable subspace)
The spaces V s and V u being invariant by B, we note by B s and B u the automorphisms induced by B on these respective spaces. The eigenvalues of the automorphism B s all have modulus < 1 and those of B u all have modulus > 1.

Moreover, we have the following proposition:

proposition

We can construct on the vector space V a norm • * adapted to the hyperbolic automorphism B in the following meaning :

(i) for all x s ∈ V s and x u ∈ V u , x s + x u * = max( x s * , x u * ). (ii) B s * < 1 and B -1 u * < 1.
Proof. Let L : W → W be a linear automorphism of a real vector subspace W of V = R d . We consider a basis (w j ) 1≤j≤m of W (dim W = m) and we denote by W the complex vector subspace of C d generated by the vectors w 1 , ..., w m : W = V ect C (w 1 , ..., w m ). The automorphism L induces on W the linear complex automorphism :

L : W -→ W , x = m j=1 x j w j -→ L(x) = m j=1 x j L(w j ) W is of complex dimension m, W ⊂ W and ∀x ∈ W, L(x) = L(x)
. Now consider a basis (v j ) 1≤j≤m of W in which the matrix T = (t i,j ) of L is upper triangular (t i,j = 0 for i > j) and for a real number α fixed in ]0, 1], we consider the new base (u j ) 1≤j≤m of W defined by: u j = α j-1 v j . We have :

∀x = m j=1 x j u j ∈ W , L(x) = m j=1 x j L(u j ) = m j=1 α j-1 x j m i=1 t i,j v i = m j=1 m i=1 α j-i x j t i,j u i = m i=1   m j=1 α j-i x j t i,j   u i Thus, for the norm . L,α : x = m j=1 x j u j → x L,α = m j=1 |x j | we have, for all x ∈ W : L(x) L,α = m i=1 m j=1 α j-i x j t i,j ≤ m i=1 m j=1 α j-i |x j ||t i,j | ≤ max 1≤i≤m |t i,i | x L,α +α   1≤i<j≤m |t i,j |   x L,α Which implies that L L,α ≤ ρ(L) + α   1≤i<j≤m |t i,j |   where ρ(L) = max 1≤i≤m |t i,i | is the spectral radius of L.
We thus obtain by restriction a norm on the real vector space W , again denoted . L,α , such that

L L,α ≤ ρ(L) + α   1≤i<j≤m |t i,j |   In addition, if ρ(L) < 1, we can choose α in ]0, 1] so that ρ(L) + α   1≤i<j≤m |t i,j |   < 1.
In this case, we will have L L,α < 1.

In this way, we obtain a norm . s on V s such that B s s < 1

(if L = B s ) and a norm . u on V u such that B -1 u u < 1 (if L = B -1 u ). The norm . * defined on V = V s ⊕ V u by: ∀x = x s + x u ∈ V = V s ⊕ V u , x * = max ( x s s , x u u )
clearly verifies the desired properties.

remark

(i) For any eigenvalue λ j of B, λ j = 1 λj is an eigenvalue of the hyperbolic automorphism B = B -1 having for multiplicity k j .

(ii) For any eigenvalue λ j of B,

(B -λ j id) kj = (-λ j ) kj B kj B -1 - 1 λ j id kj = (-λ j ) kj B kj B -λ j id kj
Consequently, we have:

       S B (B) = τ s B l U B (B ) where l = dim(V s ) and τ s ∈ R \ {0} U B (B) = τ u B d-l S B (B ) where d -l = dim(V u ) and τ u ∈ R \ {0} (iii) V u = V s , V s = V u and for all x ∈ V, x u = x s and x s = x u (iv) The norm • * is adapted to the hyperbolic automorphism B = B -1 .
We also have the following lemma

lemma

Let m ∈ Z d \ {0}. 4

The space E := C ∞ (T d )

The vector space R d being equipped with its usual scalar product < •, • > and the associated Euclidean norm • , we denote by C ∞ per (R d ) the space of C ∞ -functions ϕ : R d -→ C which are Z d -periodic, that is:

∀m ∈ Z d , ∀x ∈ R d : ϕ(x + m) = ϕ(x).
If π : R d → T d = R d /Z d is the canonical projection, we have the bijection:

C ∞ (T d ) -→ C ∞ per (R d ), h -→ h • π which identifies the elements of E = C ∞ (T d ) to those of C ∞ per (R d ).
For more detail, we can refer to [2] for instance.

Any function h ∈ E can be represented by its Fourier expansion:

h = m∈Z d h(m)Θ m
where h(m) is the m-th Fourier coefficient of h given by:

h(m) = T d h(x)e -2iπ<x,m> dx = [0,1] p h(t)e -2iπ<t,m> dt 1 • • • dt p and Θ m : R d -→ C is the Z d -periodic function defined by Θ m (x) = e 2iπ<x,m>
. It is well known that the family of complex numbers h(m) m∈Z d belongs to the vector space S(Z d , C) consisting of families (a m ) m∈Z d of complex numbers, with rapid decay, that is:

∀r ∈ N, lim m →+∞ m r |a m | = 0
Moreover, the map:

C ∞ (T d ) -→ S(Z d , C), h -→ h(m) m∈Z d
is a linear bijection identifiyng E to the space S(Z d , C) which can be also expressed as:

S(Z d , C) = r∈N W 1,r .
Here W 1,r is the space of families (a m ) m∈Z d of complex numbers satisfying the condition:

m∈Z d m r |a m | < ∞
equipped with the norm:

(a m ) 1,r = |a 0 | + m∈Z d \{0} m r |a m |
The topology of the vector space S(Z d , C) being that defined by the family of norms ( . 1,r ) r∈N .

Necessary conditions for solving equation (e)

We consider the affine hyperbolic automorphism γ :

R d -→ R d , x -→ γ(x) = Ax + b. For all k ∈ Z, we note: γ 0 = id, γ k = γ • ... • γ k times
for k > 0 and γ k = γ -1 |k| for k < 0.

Let B ∈ GL(d, Z) be the transpose of A -1 and set b k = γ k (0) for k ∈ Z. We have:

lemma

For k ∈ Z, m ∈ Z d and h ∈ E, we have:

b -k = -A -k b k and h • γ k (m) = e 2iπ<b k ,B k m> h(B k m).
Proof. We have:

0 = γ -k γ k (0) = γ -k (b k ) = A -k b k + b -k . Hence b -k = -A -k b k . We also have : h • γ k (m) = T d h • γ k (x)e -2iπ<x,m> dx = T d h γ k (x) e -2iπ<x,m> dx = T d h(u)e -2iπ<γ -k (u),m> du = T d h(u)e -2iπ<A -k (u)+b -k ,m> du = e -2iπ<b -k ,m> T d h(u)e -2iπ<A -k (u),m> du = e 2iπ<A -k b k ,m> T d h(u)e -2iπ<A -k (u),m> du = e 2iπ<b k ,B k m> T d h(u)e -2iπ<u,B k m> du = e 2iπ<b k ,B k m> h(B k m)

lemma

For m ∈ Z d \ {0} and h ∈ E, the three series:

k≥0 h • γ k (m), k<0 h • γ k (m) = l>0 h • γ -l (m) and k∈Z h • γ k (m)
are absolutely convergent and the functionals:

Φ 0 : E → C, h → Φ 0 (h) = T d h(x) dx Φ m : E → Z, h → Φ m (h) = k∈Z h • γ k (m) Φ + m : E → C, h → Φ + m (h) = k≥0 h • γ k (m) Φ - m : E → C, h → Φ - m (h) = - k<0 h • γ k (m)
are continuous.

Proof. Since the family h(α)

α∈Z d is absolutely summable, so are each of the sub-families h(B k m) k≥0 and h(B k m) k<0

. Therfore the three series:

k≥0 h • γ k (m) = k≥0 e 2iπ<b k ,B k m> h(B k m) k<0 h • γ k (m) = k<0 e 2iπ<b k ,B k m> h(B k m) k∈Z h • γ k (m) = k∈Z e 2iπ<b k ,B k m> h(B k m)
are absolutely convergent. Then, the functionals Φ m , Φ + m and Φ - m are well defined for each m ∈ Z d \ {0}. Furthermore, We have :

∀r ∈ N, ∀h ∈ E, |Φ 0 (h)| = | h(0)| ≤ α∈Z d | h(α)| = h 1,0 ≤ h 1,r ∀r ∈ N, ∀h ∈ E, |Φ m (h)| = | k∈Z e 2iπ<b k ,B k m> h(B k m)| ≤ k∈Z | h(B k m)| ≤ α∈Z d | h(α)| = h 1,0 ≤ h 1,r Also, we have: ∀r ∈ N, ∀h ∈ E, |Φ + m (h)| ≤ h 1,r and |Φ - m (h)| ≤ h 1,r .
This proves the continuity of the considered functionals.

proposition

If the cohomological equation (e) admits a solution f for the data g then:

       ∀m ∈ Z d , Φ m (g) = 0 ∀m ∈ Z d \ {0}, f (m) = Φ + m (g) = Φ - m (g)
Proof. Suppose that the cohomological equation (e) admits a solution f for the data g then:

Φ 0 (g) = T d g(x)dx = T d f (x)dx - T d f [γ(x)] dx = T d f (x)dx - T d f (t)dt = 0 (because det(A) = 1)
and if m ∈ Z d \ {0} and n ∈ N * , we have :

n k=0 g • γ k (m) = n k=0 f • γ k (m) - n k=0 f • γ k+1 (m) = f (m) -f • γ n+1 (m)
In addition, the numerical series

k≥0 f • γ k (m) is convergent. So lim n→+∞ f • γ n+1 (m) = 0 and hence k≥0 g • γ k (m) = f (m) either Φ + m (g) = f (m). Similarly, Φ - m (g) = - k<0 g • γ k (m) = f (m); so Φ m (g) = k∈Z g • γ k (m) = Φ + m (g) -Φ - m (g) = 0.
6 End of the proof of the main theorem

Proof. For all m ∈ Z d , the linear form Φ m is continuous on E. This implies that its kernel Ker(Φ m ) is a closed subspace of E. So the intersection H := ∩ m∈Z d Ker(Φ m ) is a closed subspace of E. On the other hand, if g is in the range δ(E) of the cobord operator δ : E -→ E, h -→ δ(h) = h -h • γ, the equation (e) admits at least one solution and so g ∈ H from Proposition 5.3. Then δ(E) ⊆ H. Conversely, if g ∈ H, then ∀m ∈ Z d , Φ m (g) = 0 or, in an equivalent way,

       Φ 0 (g) = 0 ∀m ∈ Z d \ {0}, Φ + m (g) = Φ - m (g)
Let us show that the sequence of complex numbers Φ + m (g) m∈Z d \{0} is fast decreasing and thus defines a function f =

m∈Z d \{0} Φ + m (g)Θ m ∈ E = C ∞ (T d ) which is a solution of (e).
For this, it suffices to prove that for all r ∈ N, lim

m →+∞ m r |Φ + m (g)| = 0 or again, in an equivalent way, lim m * →+∞ m r * |Φ + m (g)| = 0. Let r ∈ N and let m ∈ Z d \ {0}.
• If m * = m u * then, according to Lemma 3.3. the sequence ( B k m * ) k≥0 is strictly increasing; so we have: We also have (f -f • γ)(0) = 0 = g(0). Which proves that g = δ(f ) ∈ δ(E). We deduce that H ⊂ δ(E). Finally δ(E) = H which shows that this space is closed in the Fréchet space E.

m r+2 * |Φ + m (g)| = m r+2 * | k≥0 g • γ k (m) ≤ k≥0 m r+2 * | g(B k m)| ≤ k≥0 B k m r+2 * | g(B k m)| ≤ α∈Z d \{0} α r+2 * | g(α)| • If m * = m s * then,
The linear operator: L : δ(E) -→ E, g -→ f = In addition, we have: ∀g ∈ δ(E), δ • L(g) = g.

(i) m s = 0 <

 0 and m u = 0. (ii) If m * = m u * , then the numerical sequence ( B k m * ) k≥0 is strictly increasing. (iii) If m * = m s * , then the numerical sequence ( B -k m * ) k≥0 is strictly increasing. Proof. Let m ∈ Z d \ {0}. (i) If m u = 0, then lim k→+∞ B k (m) * = lim k→+∞ B k s (m s ) * = 0 ( B s * < 1) and by the equivalence of norms • and • * we also have lim k→+∞ B k (m) 2 = 0. The sequence of strictly positive integers ( B k m 2 ) is therefore zero from a certain rank. which is not possible. Hence m u = 0. Likewise m s = m u = 0 from Remark 3.2. (ii) If m * = m u * , then m * = m u * = B -1 u (Bm u ) * ≤ |B -1 Bm u * ≤ Bm * Bm s * ≤ |B s | * . m s * < m s * ≤ m * ≤ Bm u * and so Bm * = Bm u * So we have m * < Bm * and Bm * = Bm u * which allows to prove by induction that, for any integer k ∈ N, B k m * < B k+1 m * . The sequence ( B k m * ) k≥0 is then strictly increasing. (iii) If m * = m s * , by Remark 3.2. we have m * = m u * and therefore according to the previous case, the sequence ( B k m * ) k≥0 = ( B -k m * ) k≥0 is strictly increasing.

  according to Lemma 3.3. the sequence ( B -k m * ) k≥0 is strictly increasing; so we have:m r+2 * |Φ + m (g)| = m r+2 * |Φ - m (g)| = m r+2 * -k<0 g • γ k (m) = m r+2 * l>0 g • γ -l (m) ≤ l>0 m r+2 * | g(B -l m)| ≤ l>0 B -l m r+2 * | g(B -l m)| ≤ α∈Z d \{0} α r+2 * | g(α)|The norms • and • * being equivalents on V , there are real numbers η > 0 and µ > 0 such asη x * ≤ x ≤ µ x * for all x ∈ V . Consquently: ∀r ∈ N, ∀m ∈ Z d \ {0}, m r+2 * |Φ + m (g)| ≤ g 1,r+2 η r+2 and m r * |Φ + m (g)| ≤ g 1,r+2 η r+2 m 2 * where g 1,r+2 = | g(0)| =0 + α∈Z d \{0} α r+2 | g(α)| < ∞.This implies that ∀r ∈ N, limm * →+∞ m r * |Φ + m (g)| = 0. So f = m∈Z d \{0} Φ + m (g)Θ m ∈ E and according to the first point of Lemma 3.3. for all m ∈ Z d \ {0}, f • γ(m) = e 2iπ<b1,Bm> f (Bm) = e 2iπ<b1,Bm> Φ + Bm (g) = k≥0 e 2iπ<b1,Bm> g • γ k (Bm) = k≥0 g • γ k+1 (m) So: ∀m ∈ Z d \ {0}, (f -f • γ)(m) = k≥0 g • γ k (m) -k≥0 g • γ k+1 (m) = g(m).

  )Θ m is continuous. Indeed, ∀r ∈ N, ∀g ∈ δ(E), L(g) 1,r = m∈Z d \{0} m r |Φ + m (g)| ≤


Sandra Kara

Hejer Ammar

Florian Chabot

Quoc-Cuong Pham

Image Segmentation-based Unsupervised Multiple Objects Discovery

Unsupervised object discovery aims to localize objects in images, while removing the dependence on annotations required by most deep learning-based methods. To address this problem, we propose a fully unsupervised, bottom-up approach, for multiple objects discovery. The proposed approach is a two-stage framework. First, instances of object parts are segmented by using the intra-image similarity between self-supervised local features. The second step merges and filters the object parts to form complete object instances. The latter is performed by two CNN models that capture semantic information on objects from the entire dataset. We demonstrate that the pseudo-labels generated by our method provide a better precision-recall trade-off than existing single and multiple objects discovery methods. In particular, we provide state-of-the-art results for both unsupervised class-agnostic object detection and unsupervised image segmentation.

Introduction

Deep learning methods have shown tremendous advances in resolving several computer vision tasks such as object detection and image segmentation. However, massive amounts of carefully labeled images are necessary to train reliable deep learning models that can reach high performances. Due to the high cost of such manual annotations, several approaches were proposed to use only limited amounts of annotated data, such as semi-supervised learning, weakly supervised learning or few-shot learning. In this work, we address the problem of localizing objects in images without any supervision, called Unsupervised Object Discovery (UOD).

UOD can be useful for other vision tasks related to object localization. Pseudo-labels generated without supervision have been shown to provide reliable object priors for image instance retrieval in [START_REF] Zhang | Dataset-driven unsupervised object discovery for region-based instance image retrieval[END_REF]. For object detection, they can be used either to initialize an object detector without additional annotation [START_REF] Sim'eoni | Localizing objects with self-supervised transformers and no labels[END_REF], or in a semi-supervised setting, when combined with few labeled data [START_REF] Wei | Deep descriptor transforming for image co-localization[END_REF]. Providing robust pseudo-labels with limited noise is key to the success of these tasks. However, this remains a major challenge, especially in a completely unsupervised context, where no prior knowledge is provided about the semantics and localization of objects present in an image.

Many approaches solve this problem by leveraging interimage similarities [START_REF] Huy | Unsupervised image matching and object discovery as optimization[END_REF][START_REF] Huy | Toward unsupervised, multi-object discovery in large-scale image collections[END_REF] between pairs of object proposals in different images. Without carefully designed optimization mechanisms, these methods come with a computational cost and complexity that compromise their scalability. Moreover, these methods have shown to be dependent on supervised CNN features for the calculation of similarities.

Recently, vision transformers (ViT) have achieved excellent performances, outperforming CNN architectures in both supervised tasks [START_REF] El-Nouby | Training vision transformers for image retrieval[END_REF][START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF][START_REF] Carion | End-toend object detection with transformers[END_REF] and self-supervised learning [START_REF] Caron | Emerging properties in self-supervised vision transformers[END_REF][START_REF] Chen | An empirical study of training self-supervised vision transformers[END_REF]. Particularly, strong objects localization hints emerge from training ViT models using a self-distillation scheme, in DINO [START_REF] Caron | Emerging properties in self-supervised vision transformers[END_REF]. These self-supervised features were so far explored only for solving single object discovery task [START_REF] Sim'eoni | Localizing objects with self-supervised transformers and no labels[END_REF][START_REF] Wang | Self-supervised transformers for unsupervised object discovery using normalized cut[END_REF]. TokenCut [START_REF] Wang | Self-supervised transformers for unsupervised object discovery using normalized cut[END_REF] demonstrated the effectiveness of spectral-clustering applied on self-supervised ViT features, for saliency detection, and significantly improved the stateof-the-art for single object discovery.

In this work, we propose a new approach to address multiple objects discovery without any supervision. We explore self-supervised Vision Transformer (SS-ViT) features to localize and segment multiple object instances in the image. Discovering multiple objects in each image is not straightforward as it requires a clear definition of what an object is. In fact, objects are either defined as the annotated regions in the supervised setting, or as the salient region in each image, in unsupervised single object discovery approaches. To address the localization of multiple objects in a fully unsupervised way, we propose to recognise object regions using a semantic information captured at the dataset level. In other terms, an object is defined as belonging to one of the discovered semantic categories, in the im-age collection. Concretely, the semantic categories present in the dataset are discovered in an unsupervised way. This information is encoded using classification models. Object discovery is then designed as the activation of object parts in each image using SS-ViT features, and the merging of these object parts using the self-supervised classifiers, to discover complete object instances. The effectiveness of the proposed framework is demonstrated through extensive experiments on the object detection benchmarks PASCAL VOC [START_REF] Everingham | The pascal visual object classes (voc) challenge[END_REF] and MSCOCO [START_REF] Lin | Microsoft coco: Common objects in context[END_REF]. Since by design, our method provides pixel-wise mask proposals, we also show that the same framework solves the unsupervised image segmentation task.

Our contributions can be formulated as follows :

• We propose a fully unsupervised, bottom-up approach, for multiple objects discovery. We first discover object parts using intra-image similarities. Object parts are merged using a dataset-driven information, to form complete object instances. Both stages exploit selfsupervised ViT features to produce instance masks. To the best of our knowledge, this is the first work that builds on SS-ViT features to solve the multi-object discovery task.

• We generalize the saliency-based approach in Token-Cut [START_REF] Wang | Self-supervised transformers for unsupervised object discovery using normalized cut[END_REF] for the discovery of local fine semantic concepts (object parts) of multiple objects in an image.

• We propose a novel semantic object proposal method for the self-supervised learning of a region classifier. This visual model encodes the dataset-level semantic information.

• We improve the state-of-the-art in unsupervised multiple objects discovery, unsupervised class-agnostic object detection and unsupervised image segmentation, on challenging object detection benchmarks.

Related work 2.1. Unsupervised object discovery/co-localization

We can distinguish, from previous works, two distinct tasks: object discovery and object co-localization. The former consists in localizing objects in an image without any prior knowledge of the image content. This is the real object discovery task, which is much more challenging than object co-localization [START_REF] Huy | Toward unsupervised, multi-object discovery in large-scale image collections[END_REF]. On the other hand, co-localization aims at localizing common objects between images, that share the same semantic content. Algorithms in this setting are fed with perfect image clusters derived from the groundtruth. It is therefore a weakly supervised version of object discovery.

DDT [START_REF] Wei | Deep descriptor transforming for image co-localization[END_REF] addresses the co-localization task and is the first work that demonstrated the reusability of supervised CNN features for object co-localization. In DDT, objects are selected from regions of high correlation within a given cluster (semantic category).

Other methods address both tasks, and many of them leverage inter-image similarities between off-the-shelf region proposals. Cho et al. [START_REF] Cho | Unsupervised object discovery and localization in the wild: Part-based matching with bottom-up region proposals[END_REF] formulated the problem as a structure and objects discovery, by iterating part-matching and object localization. Similarly, OSD [START_REF] Huy | Unsupervised image matching and object discovery as optimization[END_REF] simultaneously localizes objects and discovers structures of the image collection. It formalizes the task as an optimization problem. Although OSD brought a large improvement, it has shown to be highly dependent on supervised proposals provided by [START_REF] Manen | Prime object proposals with randomized prim's algorithm[END_REF]. The method also suffers from overlapping region proposals, which prevents it from proposing multiple objects per image. These limitations were addressed in rOSD [START_REF] Huy | Toward unsupervised, multi-object discovery in large-scale image collections[END_REF] by providing unsupervised proposals corresponding to regions of high activations around local maxima, within CNN feature maps. rOSD also constrains the number of proposals per local maximum, and performs non-maximum suppression (NMS) [START_REF] Shaoqing Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF] post-processing, to address the problem of overlapping proposals. Note that these methods, while unsupervised, are built on supervised CNN features, from the ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] classification task. LOD [START_REF] Huy | Large-scale unsupervised object discovery[END_REF] formalized the task as a ranking problem, and focused on ensuring the scalability of the proposed approach. It also demonstrated the utility of self-supervised CNN features for single and multiple objects discovery.

Other methods [START_REF] Sim'eoni | Localizing objects with self-supervised transformers and no labels[END_REF][START_REF] Wang | Self-supervised transformers for unsupervised object discovery using normalized cut[END_REF] tackled the single object discovery problem, and showed the potential of self-supervised features, especially from ViT models, for saliency detection. LOST [START_REF] Sim'eoni | Localizing objects with self-supervised transformers and no labels[END_REF] proposed a seed expansion heuristic based on inter-patch correlation. TokenCut [START_REF] Wang | Self-supervised transformers for unsupervised object discovery using normalized cut[END_REF] investigated the use of spectral-clustering on self-supervised ViT features, which are projected into a new space that allows for a more accurate binary clustering [START_REF] Shi | Normalized cuts and image segmentation[END_REF].

In previous multi-object discovery methods, relying on inter-image similarities in the computation of an objectness score could favor the discovery of the most frequent objects. In our approach, even though we also use a datasetdriven information, we overcome this issue by training visual classification models to encode the information of semantic classes. This results in a better separation between object and non-object regions, and a better detection of under-represented classes.

Unsupervised image segmentation

Image segmentation is the task of grouping all pixels of an image into meaningful regions, where pixels sharing the same characteristics are assigned to the same region [START_REF] Kim | Unsupervised learning of image segmentation based on differentiable feature clustering[END_REF]. Due to the very high cost of such a dense annotation, weakly supervised and fully unsupervised methods were explored. In the weakly supervised setting, [START_REF] Zabari | Semantic segmentation inthe-wild without seeing any segmentation examples[END_REF] takes as input the image-level labels of the class categories present in the im-age, and utilizes a vision-language embedding model to create a rough segmentation map for each class.

Other approaches do not use any kind of supervision. On one hand, we find classic methods such as k-means [START_REF] Hartigan | A k-means clustering algorithm[END_REF], that focuses on pixels clustering based on color and texture features, and assigns each pixel to the cluster with the nearest mean. Moreover, graph-based segmentation (GS) [START_REF] Felzenszwalb | Efficient graph-based image segmentation[END_REF] generates image segments, while ensuring that these segments are not being too coarse or too detailed, based on regions comparison. More recently, methods based on unsupervised learning for image segmentation have been introduced. For example, IIC [START_REF] Ji | Invariant information clustering for unsupervised image classification and segmentation[END_REF] learns to maximize the mutual information between an image and its augmentations on a patch-level cluster. Kim et al. [START_REF] Kanezaki | Unsupervised image segmentation by backpropagation[END_REF][START_REF] Kim | Unsupervised learning of image segmentation based on differentiable feature clustering[END_REF] trains a CNN by iterating features clustering and network parameters tuning. The method is based on three criteria to maximize the features similarity between spatially continuous pixels and pixels assigned to the same cluster, while imposing a large number of clusters. The authors proposed two solutions for label assignment without any supervision (i) by superpixel extraction using simple linear iterative clustering in [START_REF] Kanezaki | Unsupervised image segmentation by backpropagation[END_REF] and (ii) by the use of a spacial continuity loss in [START_REF] Kim | Unsupervised learning of image segmentation based on differentiable feature clustering[END_REF] to address the limitation of fixed segment boundaries.

These methods discover multiple objects by proposing dense object candidates. Several other methods address the semantic segmentation task in an unsupervised way, without proposing dense object discovery. We do not consider these approaches in our study as we solve a different task.

Self-supervised vision transformers

The self-supervised setting aims at learning useful representations with no real label. It was first used for pretraining CNN models [START_REF] Hinton | Reducing the dimensionality of data with neural networks[END_REF][START_REF] Pathak | Context encoders: Feature learning by inpainting[END_REF][START_REF] Vincent | Extracting and composing robust features with denoising autoencoders[END_REF], and showed a strong generalization ability to downstream tasks. More recently, the self-attention based encoding of images using transformers for vision [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF] was proved to be effective for a large spectrum of supervised vision tasks such as classification [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF], semantic segmentation [START_REF] Strudel | Segmenter: Transformer for semantic segmentation[END_REF] and dense prediction tasks [START_REF] Rabarisoa | Self-Supervised Pre-training of Vision Transformers for Dense Prediction Tasks[END_REF]. ViT has also become a reference architectural choice of neural nets for visual representation learning. In addition to the classic masked auto-encoding paradigm inspired from NLP, MoCo-v3 [START_REF] Chen | An empirical study of training self-supervised vision transformers[END_REF] among others, demonstrated a strong potential of training ViT with a contrastive approach.

Recently, a self-distillation scheme was used in DINO [START_REF] Caron | Emerging properties in self-supervised vision transformers[END_REF] to train ViT with no labels. The choices made during training result in effective semantic separation and localglobal alignment of the learned features. In particular, the resulting attention maps strongly activate the object regions, which provide clues to the localization of objects in the image. SS-ViT features have been explored recently to perform saliency detection and single object discovery tasks [START_REF] Sim'eoni | Localizing objects with self-supervised transformers and no labels[END_REF][START_REF] Wang | Self-supervised transformers for unsupervised object discovery using normalized cut[END_REF].

To the best of our knowledge, our method is the first to exploit self-supervised ViT features in a fully unsupervised multiple object discovery pipeline. The method outputs object instance masks resolving also the unsupervised image segmentation task. Such results can be used as pseudolabels to initialize the training of a class-agnostic object detector, without any supervision.

Method

Overview

Recently, SS-ViT features showed to generalize well to saliency-based tasks [START_REF] Sim'eoni | Localizing objects with self-supervised transformers and no labels[END_REF][START_REF] Wang | Self-supervised transformers for unsupervised object discovery using normalized cut[END_REF]. In this work, we aim to demonstrate the potential of using those features for multiple objects discovery, without any supervision. We adopt a bottom-up approach, illustrated in figure 1, starting with an intra-image analysis, for the discovery of object parts. At the dataset level, two CNN models are trained in a selfsupervised manner, using carefully selected, and semantic object proposals. These models are used to merge and filter object parts, to form complete object instances.

The intra-image analysis can be seen as a generalization of TokenCut [START_REF] Wang | Self-supervised transformers for unsupervised object discovery using normalized cut[END_REF] to the multiple object discovery task. Similar to TokenCut, we perform spectral clustering using SS-ViT features, to decompose the image into eigen vectors with useful information. Different from TokenCut: (i) Since we focus on the localization of multiple objects, we look for more localization clues than just saliency. Thus we use multiple eigen vectors, as the feature space to apply local clustering, instead of only using the second eigen vector. (ii) The number of local clusters is no longer known as we try to solve the multi-object discovery task (2 clusters in saliency detection task). To manage this, we propose an algorithm for choosing an optimal number of clusters, without any knowledge about the number of objects, or semantic concepts, in each image. The algorithm is detailed in section 3.2 and aims at discovering multiple object parts, while limiting over-segmentation.

The goal of the dataset-level analysis is to build two classifiers that capture the dominant semantic classes in the image collection. One classifier is used for merging object parts, resulting from the local segmentation, and associates a confidence score to each discovered object. The second classifier separates foreground/background classes and is used to filter remaining noise after the merging phase. We perform image clustering to get pseudo-labels for training both models. Since images may contain several semantic concepts, instead of using the whole images, we apply clustering on selected object proposals from Selective Search [START_REF] Jasper Rr Uijlings | Selective search for object recognition[END_REF]. For proposals selection, we build an objectness score detailed in section 3.3. The retained top proposals are grouped into clusters, which are used for training the classifiers.

Finally, the classifiers are used in cascade to merge and denoise the discovered object parts. Both stages use selfsupervised ViT features trained using DINO [START_REF] Caron | Emerging properties in self-supervised vision transformers[END_REF]. We show in section 3.3 how these features are particularly relevant to our approach because of some properties like semantic separation, local-global alignment, and object regions activation.

Discovery of intra-image semantic concepts

In this step, we extend TokenCut [START_REF] Wang | Self-supervised transformers for unsupervised object discovery using normalized cut[END_REF] to discover multiple objects in each image, instead of solving saliency detection. TokenCut constructs a weighted graph where nodes are ViT embeddings of image patches and edges correspond to the cosine similarity between tokens. Single object discovery is then formalized as a normalized graph-cut (Ncut) problem, which is solved using spectral clustering: features are projected into a new space via eigen decomposition. In this space, the second smallest eigen vector provides a solution to the Ncut problem for binary clustering, as demonstrated by Shi and Malik [START_REF] Shi | Normalized cuts and image segmentation[END_REF]. Likewise, we create a similarity graph based on SS-ViT features. The image is then decomposed into eigen vectors with useful information. We consider N eigen vectors (N ≥ 2) for local clustering, since we aim to capture multiple objects in the image. The choice of N is studied in section 4.6. The selected N eigen vectors represent the feature space where local clustering of image pixels is performed: each pixel is represented with a new feature vector f ′ i of size N , where i varies between 1 and the total number of pixels per image n p .

Since in a fully unsupervised setting the number of semantic concepts in each image is unknown, we determine an optimal number of clusters K using an iterative process as detailed in algorithm 1. We apply k-means clustering to the image pixels, in the new feature space of eigen vectors

F = {f ′ i ; 1 ≤ i ≤ n p }.
This partitions the image into K groups, which we denote C K . We consider the background cluster as the one occupying the biggest area in the image. The background id is denoted as b id. All the remaining clusters represent the objects area. K is incremented, starting from K = 2, until no significant object area is newly activated. The goal is to activate multiple object regions in the image, while limiting over-segmentation. Examples of the results of this step are provided in figure 3, first column. In particular, we see in the last row that, in some cases, the algorithm directly outputs an optimal segmentation of the image. This shows its effectiveness compared to a simple over-segmentation, where a predefined number of clusters is used, without adapting to the content of each image.

Dataset-level semantic object proposals

As stated above, we use Selective Search [START_REF] Jasper Rr Uijlings | Selective search for object recognition[END_REF] (SeSe) region proposals as object priors to discover the semantic classes in the dataset, through proposals clustering. These proposals provide a fairly high recall. However, their rank-Algorithm 1 Iterative clustering for intra-image discovery of semantic concepts 1: Initialize:

K ← 2 C K ← Kmeans(F, K) b id ← arg max k {area(C K (k)), 1 ≤ k ≤ K} obj area ← K k=1
,k̸ =b id area(C K (k)) add semantic concepts ← T rue 2: while add semantic concepts do 3:

K ← K + 1 4: C K ← Kmeans(F, K) 5: new obj area ← K k=1,k̸ =b id area(C K (k)) 6:
if new obj area obj area > thresh then else add semantic concepts ← F alse 9: return K ing is rather naive: given an over-segmentation of the image, the regions merged first, based on color and texture similarities, are ranked first. This makes even the top proposals subject to a lot of noise. We thus propose a new ranking of SeSe proposals, to select the most relevant ones. To do this, we build an objectness score, based on assumptions about object-like regions.

Note that the objectness score is computed within each image, independently from all other images in the dataset. Concretely, we use two main measures in this computation: intersection over union (IoU) and cosine similarity between object proposals in the same image. Given M proposals (p 1 , p 2 , ..., p M ) in an image, we define u ij as the overlap rate and s ij as the similarity between p i and p j . For the latter, we use the cosine similarity between the CLS tokens from the last layer of a ViT trained using DINO. Let f i and f j be the feature vectors (CLS token) that result from passing p i and p j respectively to the SS-ViT. The cosine similarity s ij is defined as:

s ij = f i .f j ||f i ||||f j || (1)
The new objectness score for object proposals re-ranking is the weighted sum of three normalized terms:

score(p i ) = α 2 (Sim L (p i ) + Dissim G (p i )) + (1 -α)H(p i ) (2)
Each term of this score is based on a different assumption: Object-like regions have high local similarity. We define local similarity for a given proposal p i as its average similarity to its neighbouring proposals, i.e. proposals having an IoU with p i above a threshold t. We notice that these proposals correspond usually to parts of the same objects. We also recall that we are using SS-ViT features learned using DINO, with a global local alignment objective. This means that an object is close to its parts in the DINO features space.

From this we deduce that a high similarity between p i and its neighbours increases its chance of containing an object. Thus, we make all the neighbours of p i vote positively for it, in the following local similarity term:

Sim L (p i ) = M j=1 s ij , j ̸ = i, u ij ≥ t (3)
Object-like regions have high global dissimilarity. We now consider the global similarity, i.e. the average similarity between p i and all other proposals that do not overlap with p i . Given the foreground/background imbalance in real-world images, most object proposals in an image occupy the background, and have similar visual content (e.g. 'sky'). Objects, on the contrary, occupy regions that are distinct in the image. If a proposal p i contains an object, then it has low overall similarity, as all object proposals in the background vote negatively for it. Thus, p i will be highly dissimilar, hence the following global dissimilarity term:

Dissim G (p i ) = M j=1 (1 -s ij ), j ̸ = i, u ij < t (4)
Object-like regions have high entropy. The Shannon entropy of a discrete random variable is defined as:

H(p) = - x P x log(P x ) (5)
This measure is used to quantify the randomness of a variable [START_REF] Thomas | Information theory and statistics[END_REF]. In image processing, P x refers to the distribution of gray levels x in image p (or colors intensities in RGB images). The previous formula associates higher entropy to images with more details and colors variation. Inversely, homogeneous regions are characterized by a low entropy. We thus associate low-entropy proposals to background, by adding an entropy term in the final objectness score.

It can be seen from figure 2 that the proposed ranking improves the detection rate for a fixed number of proposals, compared to two modes of SeSe. This is especially true when a small number of proposals are selected. We also compare qualitatively the retained top-1 proposals with the two rankings.

We recall that the aim of this new ranking is to reduce the amount of noise in the top proposals, which will be retained for clustering, as explained in section 3.4. We choose to use SeSe proposals for its popularity. However, the proposed ranking should be valid with other proposals, provided that they have a similar distribution, i.e. bounding boxes that occupy the whole image, and thus verify the background dominance condition, discussed above.

Dataset-driven self-supervised region classification

After re-ranking the SeSe object proposals, the goal is to transform the top P object priors in each image into pseudo-labels to train multi-class classifiers. These will be used to merge and refine the discovered local semantic concepts. We use a value of P large enough to make sure that we do not only select objects but also object parts. This is important since the classifier must learn to assign the same semantic class to an object and its parts, for an accurate merging. We use k-means clustering [START_REF] Hartigan | A k-means clustering algorithm[END_REF] on the SS-ViT features of all the selected object proposals. The optimal number of clusters is chosen by finding the best silhouette score [START_REF] Rousseeuw | Silhouettes: a graphical aid to the interpretation and validation of cluster analysis[END_REF], which minimizes the mean intra-cluster distance and maximizes the mean nearest-cluster distance. With the semantic information contained in SS-ViT features, similar semantic concepts are grouped together. Moreover, each cluster contains proposals of both objects and their parts. This is especially due to the multi-crop augmentation technique used in DINO. The obtained clusters capture the global semantic information of the dataset. Note that the number of the clusters is not necessarily equal to the number of classes annotated in the dataset. However, we can still localize instances of undiscovered categories, such as 'bottle' and 'plant'.

Since some of the selected proposals may still belong to background (Bg) regions, some of the discovered pseudoclasses are Bg clusters, that we aim to identify. According to our ranking score detailed in 3.3, the proposals having the lowest scores are the ones representing most probably Bg regions. Each of these proposals is passed to a SS-ViT to extract its features. The average vector of these features is considered as a pattern of Bg regions. The clusters whose center has a distance below a threshold t bg with the Bg pattern, are considered Bg clusters.

After Foreground (Fg) and Bg groups identification, we associate to the clusters two types of labels (i) Each Fg cluster is assigned an id representing one discovered semantic class. (ii) All clusters have a binary label indicating whether it belongs to Fg or Bg. These image clusters are used to train two CNN-based classifiers, with the cluster id as a classification target. The first is a multi-class classifier trained using Fg clusters to assign objects and object parts to a specific class. The second classifier is trained using all the discovered clusters, and learns to distinguish between objects and Bg regions.

Instance segmentation using dataset-level information

In this final step, the obtained classifiers are used to merge and refine the object parts identified in the intraimage analysis. The multi-class classifier is first used on each segmented region: Image crops enclosing each object part segment are passed to the CNN-based classifier. Nearby regions assigned to the same category are merged to form complete object instances. The image crop around each merged region segment is then passed to the Fg/Bg classifier to eliminate segments classified as Bg. This binary classification is performed second to avoid incorrect classification of small object parts as Bg, if used before merging. The multi-class classifier also assigns to each object a confidence score, which is necessary for the evaluation metrics (AP@50, odAP). We provide in figure 3 illustrations for each step of the proposed framework.

Experiments

Implementation details

Following previous works [START_REF] Huy | Toward unsupervised, multi-object discovery in large-scale image collections[END_REF][START_REF] Huy | Large-scale unsupervised object discovery[END_REF][START_REF] Sim'eoni | Localizing objects with self-supervised transformers and no labels[END_REF] we conduct our experiments on three detection and localization benchmarks: VOC2007 trainval, VOC2012 trainval [START_REF] Everingham | The pascal visual object classes (voc) challenge[END_REF] and COCO20k which is composed of 19817 images randomly chosen from COCO2014 trainval dataset [START_REF] Sim'eoni | Localizing objects with self-supervised transformers and no labels[END_REF]. We specify in the following the implementation details and hyperparameters for each addressed task. Unsupervised multiple objects discovery. In the intraimage analysis, local clustering is applied on SS-ViT features learned using the DINO training scheme. Based on the conclusions from previous works [START_REF] Wang | Self-supervised transformers for unsupervised object discovery using normalized cut[END_REF][START_REF] Sim'eoni | Localizing objects with self-supervised transformers and no labels[END_REF], we use the variant ViT-S with a patch size of 16. Eigen decomposition is performed using the keys features of the last layer. To find the optimal number of local clusters, we set as convergence criterion a fraction of newly activated area of 2%, (thresh = 1.02 in algorithm 1). The number of eigen vectors used for clustering is studied in section 4.6, and showed to be invariant to the dataset: 3 eigen vectors for PASCAL VOC and COCO20k. For object proposals re-ranking, the three terms are found to have an equivalent impact on the final re-ranking, with α = 0.7, and t = 0.1. Proposals from Selective Search single mode are used in this work. For the dataset-level analysis, P = 20 top proposals in each image are selected to train the classifiers. A distance threshold t bg = 0.8 from the Bg is used to separate Fg and Bg clusters. We use ResNet50 as the backbone of the two classifiers, initialized with DINO pre-training. Unsupervised class-agnostic object detection. We follow the same configuration described in [START_REF] Sim'eoni | Localizing objects with self-supervised transformers and no labels[END_REF] for training a classagnostic Faster-RCNN, with our pseudo-labels. We also use the same batch-size and the number of training iterations, for an objective comparison with previous works. Unsupervised image segmentation. Following [START_REF] Kim | Unsupervised learning of image segmentation based on differentiable feature clustering[END_REF], this experiment is conducted on VOC2012 validation set, consisting of 1446 images. Masks resulting from multi-object discovery task are evaluated using mIOU, see section 4.2.

Metrics and evaluation settings

Different metrics are used to evaluate different tasks: Unsupervised multiple objects discovery. Most of multiple objects discovery methods are based on ranking of object proposals. This makes them able to produce a large number of object candidates. The question then arises as to how many proposals to keep for computing recall, precision, or even the classical AP50 metric, since all of these would be affected by the number of retained top proposals. [START_REF] Huy | Large-scale unsupervised object discovery[END_REF] addressed this issue and proposed an new version of AP, adapted to the object discovery task, called odAP. odAP is presented as the area under the precision-recall curve, where each precision-recall point is computed for a num-ber of retained proposals, starting from 1, to the maximum number of objects in any image in the dataset. Even though by design, our approach outputs a reduced number of proposals, we use odAP to compare with previous works. We report the odAP50 where a detection is considered correct if its overlap rate with a ground truth bounding box is above 50%. And odAP@[50 : 95], which is the average odAP for 10 values of IoU, varying from 50% to 95%. Class-agnostic unsupervised object detection. A classical class agnostic Average Precision (AP@50) is calculated. Unsupervised image segmentation. Following [START_REF] Kim | Unsupervised learning of image segmentation based on differentiable feature clustering[END_REF], we use the mean intersection over union (mIoU) to evaluate unsupervised image segmentation. mIOU is calculated as the average IOU between each ground truth mask (along with the background) and the detected mask that has the largest IOU with it, without considering any class label.

Unsupervised multiple objects discovery

We follow previous works and evaluate our method using odAP 4.2. Note that this metric is particularly adapted to the methods that propose a large number of object candidates, based on a ranking of object proposals. Since our approach is built on image segmentation, a limited number of boxes are proposed: 3 per image on average in PASCAL VOC dataset [START_REF] Everingham | The pascal visual object classes (voc) challenge[END_REF]. Hence, our approach is disadvantaged by this metric regarding the recall. Despite that, we show in table 1 the superiority of our method on both odAP@50 and the much more demanding odAP[50-95] metric.

The higher odAP[50-95] demonstrates the accuracy of our returned pseudo-boxes: Since these are generated from instance masks, they better enclose objects, and thus remain valid for a higher IoU threshold condition. Also, our method uses self-supervised features, which makes it fully unsupervised, unlike previous methods, which showed to be dependant on supervised features.

Class-agnostic unsupervised object detection

State-of-the-art multiple objects discovery methods (MOD) usually rely on a ranking of object proposals based on inter-image similarities. These methods output a large number of object candidates and the question then arises as to how many bounding boxes to keep for the initialization of an object detector. Inversely, single object discovery methods (SOD) have a very limited recall. We argue that our method provides a better precision/recall trade-off than the previous methods in both settings. To prove this, we train a class-agnostic object detector using our generated pseudo-labels. Results are presented in table 2. We notice a clear improvement with our approach compared to MOD methods on all tested datasets. The gap however gets smaller when comparing with the SOD methods on PAS-CAL VOC [START_REF] Everingham | The pascal visual object classes (voc) challenge[END_REF] dataset. This can be explained by the presence of a dataset bias in PASCAL VOC: A large proportion of images in this dataset contain one object, which gives a clear advantage to SOD methods. On the more challenging COCO20k dataset, our method exceeds both categories (MOD and SOD). This demonstrates the superiority of our pseudo-labels, even for datasets with complex scenes.

Unsupervised image segmentation

We further evaluate the performance of our method on VOC12 [START_REF] Everingham | The pascal visual object classes (voc) challenge[END_REF] validation set for unsupervised image segmentation task (see Table 3). Our method significantly outperforms previous state-of-the-art methods for discovering object masks in a fully unsupervised way. More qualitative results are provided in the supplementary material.

Method VOC12

k-means clustering [START_REF] Hartigan | A k-means clustering algorithm[END_REF], k=2 0.3166 k-means clustering [START_REF] Hartigan | A k-means clustering algorithm[END_REF], k=17 0.2383 Graph-based segmentation (GS) [START_REF] Felzenszwalb | Efficient graph-based image segmentation[END_REF], τ = 100 0.2682 Graph-based segmentation (GS) [START_REF] Felzenszwalb | Efficient graph-based image segmentation[END_REF], τ = 500 0.3647 IIC [START_REF] Ji | Invariant information clustering for unsupervised image classification and segmentation[END_REF]

Ablation study

In table 4, we provide an ablation study on different terms of the ranking score presented in 3.3. We evaluate the recall@50 (recall at IoU=50%) for different numbers of the retained top proposals. We compare the results of the overall ranking score, with the ranking obtained when one of the terms is removed from the final score. The best results are achieved by considering all 3 terms, which supports the assumptions made in section 3.3. We also compare our score with the original SeSe ranking of proposals from two settings. Using this new ranking, we ensure that the top proposals are more reliable for the classifiers training. We also provide a study of the number of eigen vectors to be used in the intra-image analysis, in order to activate multiple objects, while limiting the amount of noise. In table 5, we evaluate the AP@50 of the generated pseudo-boxes, to choose the best precision/recal trade-off. We conduct the study on PASCAL VOC and COCO since they present different distributions of objects. Considering this study, the reported results are obtained with 3 eigen vectors in the intra-image analysis, for both datasets. Table 5. AP@50 as a function of the number of eigen vectors used for local analysis

Conclusion and future work

We presented a fully unsupervised approach for multiple objects discovery. The aim of this work was to address some of the limitations observed in existing methods. Namely, low recall in saliency detection-oriented methods, and the high amount of noise when several object candidates are proposed. We have shown that formulating the problem as an unsupervised segmentation is particularly suitable for reducing the noise in the generated pseudo-boxes. This provides a better precision-recall trade-off, which leads to a better initialization of an object detector. Still in this direction, we can further investigate the use of these pseudolabels as an initial seed in a pseudo-labelling approach. Similarly, we can investigate the use of these object candidates with noise handling mechanisms.

Acknowledgements

This work benefited from the FactoryIA supercomputer financially supported by the Ile-deFrance Regional Council.

7 :

 7 obj area ← new obj area 8:

Figure 1 .

 1 Figure 1. Pipeline of the method. Top left: Intra-image analysis for the discovery of local semantic concepts. Bottom: Dataset-level analysis for the selection of semantic object proposals to train self-supervised classifiers. Top right: Using the data-driven classifiers on each image for parts merging and denoising.

Figure 2 .

 2 Figure 2. Results of ranking object proposals. Left: Comparison of the detection rate given the number of retained top proposals from two modes of SeSe, with our ranking. Right: Examples of the top1 proposal using SeSe score (top) and our score (bottom).

Figure 3 .

 3 Figure 3. Example of results. By column: results of the discovery of local concepts, segmentation result after parts merging, final instance mask segmentation, final bounding boxes.

Table 1 .

 1 Multi-object discovery performance in odAP (Average Precision for object discovery)

	Method	Features		odAP@50			odAP@[50-95]
			VOC07 VOC12 COCO20k VOC07 VOC12 COCO20k
	Kim et al. [15, 24]	Sup	9.5	11.8	3.93	2.5	3.1	0.96
	DDT+ [32, 24]	Sup	8.7	11.1	2.41	3.0	4.1	0.73
	rOSD [29, 24]	Sup	13.1	15.4	5.18	4.3	5.3	1.62
	LOD [30, 24]	Sup	13.9	16.1	6.63	4.5	5.3	1.98
	Ours	Self	15.4	17.6	5.44	6.8	8.1	2.11

Table 2 .

 2 Class-agnostic unsupervised object detection in AP50%

	Method	VOC07 VOC12 COCO20K
	Selective Search [26]	3.6	4.8	1.8
	EdgeBoxes [35]	2.9	4.2	1.6
	rOSD + CAD [29]	24.2	29.0	8.4
	LOD + CAD [30]	22.7	28.4	8.8
	LOST + CAD [24]	29.0	33.5	9.9
	TokenCut + CAD [31]	26.2	35.0	10.5
	Ours + CAD	27.9	36.2	13.8

Table 3 .

 3 Unsupervised image segmentation results in mIOU

	, k=2	0.2729
	IIC [13], k=20	0.2005
	Kim et al. with superpixels [14]	0.3082
	Kim et al. with continuity loss [16], ν = 5	0.3520
	Ours	0.4247

Table 4 .

 4 Sim L + Dissim G 13.9 23.5 34.4 44.1 Ours: Sim L + H 12.8 24.9 35.9 44.6 Ours: Overall score 15.7 27.1 36.9 45.0 Ablation study on the impact of the different terms composing the re-ranking score, evaluated on VOC07 testset

	Method		Recall@50
	Number of boxes	1	4	10	20
	SeSe Fast mode [26]	7.5	19.6 30.3 40.0
	SeSe Single mode [26]	7.9	19.9 30.7 40.9
	Ours:


Didier Cariou

Cariou

teaching and research institutions in France or abroad, or from public or private research centers.

Compte rendu publié dans : Recherches et éducation. En ligne : https://journals.openedition.org/rechercheseducations/15186 Gomes, L. (2023). Enseignement de l'histoire et esprit critique. (préface de Michel Fabre) Rennes : Presses Universitaires de Rennes, collection Paideia, 166 p.

Didier Cariou

Université de Brest / Centre de Recherches sur l'Education, les Apprentissages et la Didactique L'ouvrage de Lucie Gomes est issu de sa thèse dont le texte a été très largement revu pour cette publication. Sa parution est un événement important pour la communauté des didacticien.nes car il expose d'une façon très accessible l'épistémologie et les méthodes du Cadre de l'apprentissage par problématisation (CAP) appliquées au champ de l'histoire scolaire. mettent d'envisager ce qui pourrait constituer des enquêtes dans la classe. Certes, le but n'est pas de faire des élèves des petits historiens et des petites historiennes mais plutôt, pourrions-nous dire, de renouveler la « forme scolaire » en les conduisant à poser et à formuler eux aussi des problèmes quand ils et elles lisent des documents historiques. Cette orientation explicite le titre de l'ouvrage : l'objectif de l'autrice est de chercher à comprendre à quelles conditions peut être développé l'esprit critique des élèves. Le présupposé est que les élèves peuvent développer des compétences de lecture critique des documents proches de certaines pratiques historiennes de critique des sources. Lucie Gomes propose ici une voie originale de l'examen des apports de l'épistémologie de l'histoire à la didactique de l'histoire. Mais son propos va beaucoup plus loin par les pistes qu'elle propose pour former les élèves à une pratique de lecture qui leur permette de mettre à distance les propos auxquels ils sont confrontés également dans leur vie quotidienne.

L'ouvrage est composé de cinq chapitres qui proposent une avancée progressive dans la compréhension de la démarche de la problématisation appliquée à l'enseignement de l'histoire et à la question de la lecture des documents en classe d'histoire.

Le premier chapitre porte sur les compétences critiques ellesmêmes, enjeu essentiel mais souvent inaccessible de l'enseignement de l'histoire et qui sont dépliées dans les chapitres qui suivent. Le second chapitre éclaire les présupposés et les démarches de la problématisation envisagés par le CAP d'après les travaux de Fabre et Orange. On sait que, la plupart du temps, les savoirs enseignés en classe d'histoire sont des savoirs factuels, assertoriques, présentant peu de sens pour les élèves. Or les savoirs devraient être fondés en raison, apodictiques et, pour cela, construits par les élèves. La logique bachelardienne du CAP suppose le passage d'un type de savoirs à l'autre par une forme de rupture que crée la tension entre le registre explicatif du sens commun mobilisé initialement par des élèves et les données disponibles dans les dossiers documentaires qui leur sont fournis. Par cette tension, les élèves repèrent et formulent le problème en jeu, avant que de chercher à le résoudre. Car, suivant les analyses de Bachelard, le savoir ne se tient pas dans la bonne réponse à un problème, mais dans la façon dont on construit le problème, à l'instar des historien.nes qui problématisent leur enquête.

Pour le comprendre, le troisième chapitre expose la méthodologie retenue dans le cadre du CAP, celle des séquences forcées destinées à « forcer » ce qu'il est possible aux élèves de produire dans une logique de construction de problèmes, en les faisant aller au-delà de leurs habitudes de travail. A cet effet, Lucie Gomes, à la fois professeure de la classe et chercheure, a mené cinq séquences forcées en classe de seconde pour vérifier à quelles conditions les élèves pouvaient être amenés sur la voie de la problématisation qui les rende capables de remobiliser des compétences critiques d'une séquence à l'autre. Chaque séquence est considérée non pas comme une simple étude de documents mais comme une étude de cas dont la densité est supposée mettre en oeuvre les compétences de lecture critique des élèves.

Le quatrième chapitre, qui constitue la partie centrale de l'ouvrage, expose l'analyse de ces séquences travaillées durant une année scolaire en classe de seconde. La première concerne une forme d'émigration européenne au XIXe siècle, celle des habitants de la vallée de Barcelonnette vers le Mexique. La seconde porte sur la répression de l'hérésie cathare au Moyen Age. La troisième concerne l'arrivée du portugais Vasco de Gama en Inde, en 1498. La quatrième porte sur la guerre de Vendée, un épisode de la Révolution française. La cinquième concerne l'abolition de l'esclavage en 1848.

Ces études permettent une modélisation d'une approche de la lecture critique des documents en classe d'histoire. Dans un premier temps, les élèves mobilisent des registres explicatifs du sens commun fondés sur une lecture littérale et réaliste des documents. Dans un second temps, comme le montre l'analyse fine des transcriptions des échanges entre les élèves, la professeure les conduit à interroger le statut de l'auteur des documents étudiés et les raisons pour lesquelles il tient le propos lu par les élèves. Ce questionnement conduit à la mise en tension du registre explicatif du sens commun par les données fournies par les documents et à la formulation du problème historique en jeu. Les élèves perçoivent alors des éléments qu'ils n'avaient pas pris en compte lors de leur première lecture. Ainsi, les habitants de Barcelonnette n'avaient pas émigré pour fuir la misère car ils avaient depuis longtemps construit des réseaux de commercialisation des draps fabriqués dans la vallée. L'extension géographique de ces réseaux leur avait permis d'émigrer temporairement jusqu'au Mexique. Alors que le texte sur l'hérésie cathare, écrit par un homme d'Église, présentait les cathares comme de pauvres gens, la photographie du château actuel de Montségur, construit sur un éperon rocheux à la place d'un château cathare leur fait comprendre que, si les cathares étaient capables de construire un château à cet endroit, alors ils étaient sans doute plus puissants et plus dangereux que ne le suggérait l'homme d'Église. Le problème en jeu est alors déplacé de la dimension religieuse de la répression d'une hérésie à la question d'un affrontement entre deux puissances rivales pour la domination politique de l'espace languedocien.

Le passage d'un registre du sens commun à une analyse plus scientifique renvoie à la notion de configuration historiographique élaborée par Antoine Prost, qui désigne la façon dont les générations successives d'historien.nes ont étudié un fait historique. Le registre explicatif du sens commun des élèves rejoint la première configuration historiographique selon laquelle l'Église a violemment réagi face au développement de l'hérésie cathare qui menaçait le dogme chrétien. Dans le cadre de la configuration historiographique actuelle, les historien.nes considèrent plutôt que l'Église avait catégorisé comme hérétiques des pratiques religieuses acceptées jusque-là, afin de pouvoir les combattre et d'imposer sa domination sur les populations. Lucie Gomes montre l'utilité d'une catégorie relevant de l'épistémologie de l'histoire pour conduire et analyser les modalités d'une formulation plus scientifique d'un problème historique. Le travail de problématisation revient donc à faire rejouer dans la classe ce changement de paradigme historiographique.

Il en va de même pour l'étude de la « découverte » de l'Inde par Vasco de Gama en 1498. L'extrait d'un récit rédigé par un membre de l'équipage, évoquant la présence à Calicut de Maures de Tunis sachant parler le castillan et le génois, remet en cause la notion européocentrique de « découverte » de l'Inde et oriente les élèves vers une configuration historiographique proche de celle de l'histoire connectée. Pour parvenir à cette problématisation, les élèves sont contraints de passer par une généralisation à la période toute entière : peut-on parler de « Grandes découvertes » alors que les espaces concernés étaient déjà peuplés et connectés à l'Ancien monde par des réseaux marchands ? La tension entre le registre explicatif des élèves et les données passe ici par l'articulation entre le texte et son contexte historique plus général.

Le cas des guerres de Vendée fournit un dernier élément de cette mise en tension. Le lycée où eut lieu l'expérimentation se trouvant en Vendée, cette question est vive pour les élèves. Plutôt que de démonter le mythe du peuple vendéen catholique et monarchiste construit par divers sites touristiques locaux en lui opposant le récit des historien.nes, selon une logique de fact-checking, Lucie Gomes a préféré se pencher sur la genèse de ce récit mythique. Elle fait étudier aux élèves les mémoires d'une noble vendéenne qui décrit les exploits de son beau-frère, trente ans après les faits, sous la Restauration. Les élèves questionnent les intentions de l'autrice du texte, et articulent ce dis-cours au contexte politique général de sa production, à savoir la période de la Restauration. Les élèves prennent alors conscience que ce discours est le même que celui qui est tenu par les institutions vendéennes actuelles. On voit là toute l'intérêt de substituer, à la confrontation d'un discours idéologique à celui des historiens selon une logique de fact-checking réaliste, la généalogie de ce discours. Cette orientation est certainement très prometteuse pour l'étude critique d'autres discours qui posent question.

Le cinquième et dernier chapitre propose un bilan des expérimentations et revient sur la question des compétences critiques en histoire. Sur la base de la notion de compétence selon Bernard Rey on comprends bien qu'il ne s'agit plus pour l'enseignant de montrer une technique simple aux élèves afin que ces derniers s'en emparent par l'entraînement et la répétition, comme s'il s'agissait d'acquérir un automatisme. Il s'agit plutôt d'entraîner les élèves à reconnaître et à mobiliser des ressources déjà abordées lors des séquences précédentes et dont nous avons tenté de dresser la liste plus haut, selon le modèle des exemplars (ou exemples exemplaires) de Kuhn, que travaillent les étudiants qui souhaitent intégrer une communauté scientifique. Lucie Gomes a proposé ici une démarche candidate à l'apprentissage de l'histoire nourrie de l'épistémologie de l'histoire et supposée développer des pratiques de problématisation.

Signalons, pour conclure, une petite réticence de l'autrice quant à la méthode étudiée : à la fin de l'année scolaire, les élèves ne sont toujours pas autonomes, contrairement à ce que supposerait la notion de compétence, et la place de la professeure reste essentielle dans la mise en tension des registres explicatifs et des données. Gageons qu'une transformation de la forme scolaire de l'histoire, en direction de laquelle cet ouvrage contribue fortement, permettrait d'étaler cet apprentissage sur plusieurs années à des fins d'autonomisation des élèves.

Le propos de l'ouvrage part d'un constat largement partagé en didactique de l'histoire : face à un document historique, et souhaitant recueillir des informations fiables sur le passé, les élèves sont souvent tentés de se demander si son auteur dit vrai ou pas, selon une logique de fact-cheking. Or, le passé n'est pas directement connaissable, il se présente à nous sous la forme de traces à propos desquelles il convient de mener une enquête. Les historien.nes posent des questions non pas sur les faits du passé mais sur les documents qui en sont la trace, et cherchent à comprendre pourquoi un auteur tient tel discours sur tel fait du passé. Ce type d'enquête permet de construire la réalité du passé à partir de traces. Il existe en effet un consensus dans le champ de la didactique de l'histoire : les pratiques d'enquête historienne per-1


Theoretical

• A (roughly) Quinean approach:

➢ look at your best biological theory, and extract its ontological commitments.

• An ontological spectrum from monism to pluralism

• Focus on "overlap pluralism"

Metaphysical

• What kind of thing is a biological individual?

➢ Substance? Process? Should we care?

• A natural connection: from existence to identity ➢ "No entity without identity" (Quine 1969) ➢ The case of pluralism

• The "individuation problem": what, in x's profile (with x ∈ BI), might serve as a principle of individuation for this very entity?

Metaphysical

• What kind of thing is a biological individual?

➢ Substance? Process? Should we care?

• A natural connection: from existence to identity x B = "has a life" P = "has a physiology" E = "has evolutionary dispositions" …

x's (unique) property profile

• H1: For any x such that x ∈ BI, x's principle of individuation is located/grounded in {B, E, P}. (= sortalism) H2: For some x such that x ∈ BI, x's principle of individuation is located/grounded in C = \{B, E, P}, where C contains any property (distinct from B, E and P) that is true of x.

B = "has a life" P = "has a physiology" E = "has evolutionary dispositions" Two hypotheses

• H1: For any x such that x ∈ BI, x's principle of individuation is located/grounded in {B, E, P}. (= sortalism)

• H2: For some x such that x ∈ BI, x's principle of individuation is located/grounded in C = \{B, E, P}, where C contains any property (distinct from B, E and P) that is true of x. (i) Epistemic: to individuate some x = to single out x as a distinct object of perception/thought/reference.

➢ A fallible process.

(ii) Metaphysical: "what individuates an object, in this sense, is whatever it is that makes it the single object that it is---whatever it is that makes it one object, distinct from others, and the very object that it is" (p.76)

➢ Keep track of x throughout space and time.

• Metaphysical individuation is a precondition to any ontological inquiry:

➢For any given individual object, one should be able to provide a sufficient (and necessary) criterion of individuation for this object.

Individuation

• Two senses of "individuation (Lowe 2003):

(i) Epistemic: to individuate some x = to single out x as a distinct object of perception/thought/reference.

➢ A fallible process.

(ii) Metaphysical: "what individuates an object, in this sense, is whatever it is that makes it the single object that it is---whatever it is that makes it one object, distinct from others, and the very object that it is" (p.76)

➢ Open vs narrow individuation.

• Metaphysical individuation is a precondition to any ontological inquiry:

➢For any given individual object, one should be able to provide a sufficient (and necessary) criterion of individuation for this object.

Individuation

• Two senses of "individuation (Lowe 2003):

(i) Epistemic: to individuate some x = to single out x as a distinct object of perception/thought/reference.

➢ A fallible process.

(ii) Metaphysical: "what individuates an object, in this sense, is whatever it is that makes it the single object that it is---whatever it is that makes it one object, distinct from others, and the very object that it is" (p.76)

➢ Open vs. narrow individuation.

• Metaphysical individuation is a precondition to any ontological inquiry:

➢For any given individual object, one should be able to provide a sufficient (and necessary) criterion of individuation for this object.

Individuation

• Two methods:

1) The principle of identity of indiscernibles (unique property profile): H1 entails Argument 1

(x)(y)((Φx ↔ Φy) → (x = y))
• Argument 1: For any x ∈ BI, there is a K such that:

(i) Kx

(ii) K is a substantial property a. There are changes that x cannot survive ➢ Objection: bare particulars b. These changes are grounded in the principle of individuation associated with K.

➢ Objection: nominalism (iii) K belongs to {B, E, P} ❖Too coarse-grained: "material object" ❖Too fine-grained: species ("squirrel", "Tiger", etc.)

H1 entails Argument 1

• Argument 1: For any x ∈ BI, there is a K such that:

(i) Kx

(ii) K is a substantial property a. There are changes that x cannot survive ➢ Objection: bare particulars b. These changes are grounded in the principle of individuation associated with K.

➢ Objection: nominalism (iii) K belongs to {B, E, P} ❖Too coarse-grained: "material object" ❖Too fine-grained: species ("squirrel", "Tiger", etc.)

H1 entails Argument 1

• Argument 1: For any x ∈ BI, there is a K such that:

(i) Kx

(ii) K is a substantial property a. There are changes that x cannot survive ➢ Objection: bare particulars b. These changes are grounded in the principle of individuation associated with K.

➢ Objection: nominalism (iii) K belongs to {B, E, P} ❖Too coarse-grained: "material object" ❖Too fine-grained: species ("squirrel", "Tiger", etc.)

H1 entails Argument 1

• Argument 1: For any x ∈ BI, there is a K such that:

(i) Kx

(ii) K is a substantial property a. There are changes that x cannot survive ➢ Objection: bare particulars b. These changes are grounded in the principle of individuation associated with K.

➢ Objection: nominalism (iii) K belongs to {B, E, P} ❖Too coarse-grained: "material object" ❖Too fine-grained: species ("squirrel", "Tiger", etc.)

H1 entails Argument 1

• Argument 1: For any x ∈ BI, there is a K such that:

(i) Kx i. From x t R O y t+1 , it follows that x t is identical to y t+1 ii. From x t R EI y t+1 , it follows that x t is not identical to y t+1

PI EI

➢ Is x t identical to y t+1 ?

Interpretation 5. Both P and E refer to substantial properties 31 x t y t+1 i. From x t R PI y t+1 , it follows that x t is identical to y t+1 ii. From x t R EI y t+1 , it follows that x t is not identical to y t+1

PI EI

Interpretation 5. Both P and E refer to substantial properties 32 x t y t+1 i. From x t R PI y t+1 , it follows that x t is identical to y t+1 ii. From x t R EI y t+1 , it follows that x t is not identical to y t+1

PI EI

Interpretation 5. Both P and E refer to substantial properties

• In defence of strong pluralism: the coincidence thesis (J. Wilson 1999)

➢The statue and the clay: different persistence/modal conditions.

• The supervenience test i. Do physiological properties supervene on evolutionary properties? No.

ii. Do evolutionary properties supervene on physiological property? No.

• Wilson's proposal: a 3-place, not a 2 place, relation ➢PI(x, y) = x composes/constitutes a physiological individual y.

➢EI(x, z) = x composes/constitutes an evolutionary individual z.

Interpretation 5. Both P and E refer to substantial properties

• In defence of strong pluralism: the coincidence thesis (J. Wilson 1999)

➢The statue and the clay: different persistence/modal conditions.

• The supervenience test i. Do physiological properties supervene on evolutionary properties? No.

ii. Do evolutionary properties supervene on physiological property? No.

• Wilson's proposal: a 3-place, not a 2 place, relation Interpretation 5. Both P and E refer to substantial properties

• In defence of strong pluralism: the coincidence thesis (J. Wilson 1999)

➢The statue and the clay: different persistence/modal conditions.

• The supervenience test i. Do physiological properties supervene on evolutionary properties? No.

ii. Do evolutionary properties supervene on physiological property? No.

• Wilson's proposal: a 3-place, not a 2 place, relation ➢PI(x, y) = x composes/constitutes a physiological individual y.

➢EI(x, z) = x composes/constitutes an evolutionary individual z.

Interpretation 5. Both P and E refer to substantial properties

• In defence of strong pluralism: the coincidence thesis (J. Wilson 1999)

➢The statue and the clay: different persistence/modal conditions.

• The supervenience test i. Do physiological properties supervene on evolutionary properties? No.

ii. Do evolutionary properties supervene on physiological property? No.

• Wilson's proposal: a 3-place, not a 2 place, relation Possible ways of fleshing out "promiscuous realism"

••

  What kinds of biological individuals are there? What kind of thing is a biological individual? ➢ Substance? Process? Should we care? • A natural connection: from existence to identity ➢ "No entity without identity" (Quine 1969) ➢ The case of pluralism • The "individuation problem": what, in x's profile (with x ∈ BI), might serve as a principle of individuation for this very entity?

➢•

  "No entity without identity" (Quine 1969) ➢ The case of pluralism • The "individuation problem": what, in x's profile (with x ∈ BI), might serve as a principle of individuation for this very entity? B = "has a life" P = "has a physiology" E = "has evolutionary dispositions" … x's (unique) property profile x Metaphysical What kind of thing is a biological individual? ➢ Substance? Process? Should we care? • A natural connection: from existence to identity ➢ "No entity without identity" (Quine 1969) ➢ The case of pluralism • The "individuation problem": what, in x's unique property profile (with x ∈ BI), might serve as a principle of individuation for this very entity?

  ind /E ind /P ind = B/E/P is an individuating property Individuation • Two senses of "individuation (Lowe 2003):

••

  ❖Two issues: a) A classical dilemma b) An epistemological/practical problem 2) Look for x's individuating property: sortalism (D. Wiggins) ➢Substance sortal vs. phase sortals ➢Criterion of identity: If x and y are Ks, then x is identical to y iff xR K y.Individuation• Two methods:1) The principle of identity of indiscernibles (unique property profile):(x)(y)((Φx ↔ Φy) → (x = y))❖Two issues: a) A classical dilemma (M. Black 1952) b) An epistemological/practical problem 2) Look for x's individuating property: sortalism (D. Wiggins) ➢Substance sortal vs. phase sortals ➢Criterion of identity: If x and y are Ks, then x is identical to y iff xR K y. Individuation Two methods: 1) The principle of identity of indiscernibles (unique property profile): (x)(y)((Φx ↔ Φy) → (x = y)) ❖Two issues: a) A classical dilemma b) An epistemological/practical problem (D. Wiggins 2012) 2) Look for x's individuating property: sortalism (D. Wiggins) ➢Substance sortal vs. phase sortals ➢Criterion of identity: If x and y are Ks, then x is identical to y iff xR K y. Individuation Two methods: 1) The principle of identity of indiscernibles (unique property profile): (x)(y)((Φx ↔ Φy) → (x = y)) ❖Two issues: a) A classical dilemma b) An epistemological/practical problem 2) Look for x's individuating property: sortalism (D. Wiggins) ➢Substance sortal vs. phase sortals ➢Criterion of identity: If x and y are Ks, then x is identical to y iff xR K y.

(

  ii) K is a substantial property a. There are changes that x cannot survive ➢ Objection: bare particulars b. These changes are grounded in the principle of individuation associated with K. ➢ Objection: nominalism (iii) K belongs to {B, E, P} ❖Too coarse-grained: "material object" ❖Too fine-grained: species ("Squirrel", "Tiger", etc.) H1 H2 Legend: B ind /E ind /P ind = B/E/P is an individuating property H1 H2 Legend: B ind /E ind /P ind = B/E/P is an individuating property H1 H2 Legend: B ind /E ind /P ind = B/E/P is an individuating property Interpretation 5. Both P and E refer to substantial properties

  ➢PI(x, y) = x composes/constitutes a physiological individual y. ➢EI(x, z) = x composes/constitutes an evolutionary individual z.

  ➢P(x, y) = x composes/constitutes a physiological individual y. ➢E(x, z) = x composes/constitutes an evolutionary individual z. H1 H2 Legend: B ind /E ind /P ind = B/E/P is an individuating property ind /E ind /P ind = B/E/P is an individuating property H2: "promiscuous realism"

Dupré & Nicholson's (2018) arguments

• Criticisms of the substantialist account "The problem for substance ontology has always been that it is extraordinarily difficult to specify any such change-exempt descriptive properties that permanently characterize the essence of things. [...].

(i) First, […], a biological individual is never materially identical from one moment to the next.

(ii) Second, […], it undergoes massive morphological changes.

(iii) And, third, […], the symbiotic associations that compose and maintain it change considerably over its lifetime." (p.24)

• Dupré & Nicholson's argument "in support of" the process view:

➢ Analogy between a whirlpool and an organism.

• Two (major) problems: • Dupré & Nicholson's argument "in support of" the process view:

➢ Analogy between a whirlpool and an organism.

• Two (major) problems: • Dupré & Nicholson's argument "in support of" the process view:

➢ Analogy between a whirlpool and an organism.

• Two (major) problems: • Dupré & Nicholson's argument "in support of" the process view:

➢ Analogy between a whirlpool and an organism.

• Two (major) problems: • Dupré & Nicholson's argument "in support of" the process view:

➢ Analogy between a whirlpool and an organism.

• Two (major) problems: • Dupré & Nicholson's analogy "in support of" the process view:

➢ Analogy between a whirlpool and an organism.

• Two (major) problems: • Dupré & Nicholson's analogy "in support of" the process view:

➢ Analogy between a whirlpool and an organism.

• Two (major) problems: Thanks for your attention.