Jimmy Garnier

Thomas Giletti

François Hamel

Lionel Roques

Inside dynamics of pulled and pushed fronts

We investigate the inside structure of one-dimensional reaction-diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established.

Introduction

In this paper, we explore the spatial structure of traveling wave solutions of some reaction-diffusion equations. Namely, we consider the following one-dimensional reaction-diffusion model:

∂ t u(t, x) = ∂ 2 x u(t, x) + f (u(t, x)), t > 0, x ∈ R, (1.1)
where u(t, x) ∈ [0, 1]. This equation arises in various scientific domains of application, namely, population dynamics where the unknown quantity u typically stands for a population density [START_REF] Britton | Reaction-Diffusion Equations and their Applications to Biology[END_REF][START_REF] Cosner | Spatial Ecology via Reaction-Diffusion Equations[END_REF][START_REF] Grindrod | Theory and Applications of Reaction-Diffusion Equations[END_REF][START_REF] Murray | Mathematical Biology. Third Edition[END_REF][START_REF] Shigesada | Biological Invasions: Theory and Practice[END_REF], chemistry [START_REF] Billingham | The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form of traveling waves[END_REF][START_REF] Fife | Mathematical Aspects of Reacting and Diffusing Systems[END_REF], and combustion [START_REF] Berestycki | Quelques aspects mathématiques de la propagation des flammes prémélangées[END_REF]. In the context of population dynamics, f (u) corresponds to the population's growth rate. The nonlinear growth term f in (1.1) is assumed to satisfy

f ∈ C 1 ([0, 1]), f (0) = f (1) = 0, 1 0 f (s) ds > 0 (1.2)
and to be either of the monostable, bistable or ignition type:

(A) Monostable f is monostable if it satisfies (1.2), f ′ (0) > 0, f ′ (1) < 0 and f > 0 in (0, 1).

In this case, the growth rate f (u) is always positive on (0, 1). A classical monostable example is f (u) = u(1 -u)(1 + au), with a ≥ 0. If a ≤ 1, the per capita growth rate f (u)/u (defined by f ′ (0) at u = 0) is decreasing over the interval [0, 1] which means that f is of the KPP (for Kolmogorov-Petrovsky-Piskunov [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF]) type. On the other hand, if a > 1, the maximum of the per capita growth rate is not reached at u = 0. In population dynamics, this corresponds to a so-called weak Allee effect [START_REF] Turchin | Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants[END_REF].

(B) Bistable f is bistable if it satisfies (1.2), f ′ (0) < 0, f ′ (1) < 0 and there exists ρ ∈ (0, 1) such that f < 0 in (0, ρ) and f > 0 in (ρ, 1).

This hypothesis means that the growth rate f (u) is negative at low densities, which corresponds to a strong Allee effect [START_REF] Lewis | Allee dynamics and the spread of invading organisms[END_REF][START_REF] Turchin | Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants[END_REF]. The parameter ρ corresponds to the so-called "Allee threshold" below which the growth rate becomes negative. For instance, the cubical function f (u) = u(1 -u)(u -ρ), where ρ belongs to (0, 1/2), is a bistable nonlinearity.

(C) Ignition f is of ignition type if it satisfies (1.2), f ′ (1) < 0 and there exists ρ ∈ (0, 1) such that f = 0 in (0, ρ) and f > 0 in (ρ, 1). This reaction term occurs in combustion problems, where u corresponds to a temperature and ρ is the ignition temperature [START_REF] Berestycki | Quelques aspects mathématiques de la propagation des flammes prémélangées[END_REF][START_REF] Berestycki | Traveling wave solutions to combustion models and their singular limits[END_REF][START_REF] Roques | Study of the premixed flame model with heat losses the existence of two solutions[END_REF].

The equation (1.1) has been extensively used to model spatial propagation of elements in interaction, in parts because it admits traveling wave solutions. These particular solutions keep a constant profile U and move at a constant speed c. Aronson and Weinberger [START_REF] Aronson | Nonlinear diffusion in population genetics, combustion and nerve propagation[END_REF][START_REF] Aronson | Multidimensional non-linear diffusion arising in population genetics[END_REF] and Kanel' [START_REF] Kanel | Certain problems of burning-theory equations[END_REF] have proved that equation (1.1) with monostable, bistable or ignition nonlinearities admits traveling waves solutions of the form u(t, x) = U (x -ct), where c ∈ R and the profile U is a C 3 (R) function which satisfies the following nonlinear elliptic equation: U ′′ (y) + c U ′ (y) + f (U (y)) = 0, y ∈ R, U (-∞) = 1, U (+∞) = 0 and 0 < U < 1 on R.

(1.3)

On the one hand, if f is of the monostable type (A), there exists a minimal speed c * ≥ 2 f ′ (0) > 0 such that equation (1.3) admits a solution if and only if c ≥ c * . The solution associated to the minimal speed c * is called the critical front, while those associated to speeds c > c * are called super-critical fronts. On the other hand, if f is of the bistable (B) or ignition (C) types there exists a unique speed c > 0 such that equation (1.3) admits a solution. In all cases, if the front (c, U ) exists, the profile U is a decreasing function and it is unique up to shift (see e.g. [START_REF] Aronson | Multidimensional non-linear diffusion arising in population genetics[END_REF][START_REF] Berestycki | Traveling wave solutions to combustion models and their singular limits[END_REF][START_REF] Fife | The approach of solutions of nonlinear diffusion equations to traveling front solutions[END_REF]). The asymptotic behavior of U (y) as |y| → +∞ is also known (see Section 2).

The sensitivity of the fronts to noise or a fixed perturbation has also been extensively studied, see e.g. [START_REF] Bramson | Convergence of Solutions of the Kolmogorov Equation to Travelling Waves[END_REF][START_REF] Eckmann | The nonlinear stability of front solutions for parabolic differential equations[END_REF][START_REF] Fife | The approach of solutions of nonlinear diffusion equations to traveling front solutions[END_REF][START_REF] Lau | On the nonlinear diffusion equation of Kolmogorov, Petrovsky and Piscounov[END_REF][START_REF] Sattinger | On the stability of waves of nonlinear parabolic systems[END_REF][START_REF] Sattinger | Weighted norms for the stability of traveling waves[END_REF][START_REF] Uchiyama | The behaviour of solutions of some non-linear diffusion equations for large time[END_REF]. In the monostable case, the stability studies lead to a classification of the fronts into two types: "pulled" fronts and "pushed" fronts [START_REF] Rothe | Convergence to pushed fronts[END_REF][START_REF] Stokes | On two types of moving front in quasilinear diffusion[END_REF][START_REF] Van Saarloos | Front propagation into unstable states[END_REF]. A pulled front is either a critical front (c * , U ) such that the minimal speed c * satisfies c * = 2 f ′ (0), or any super-critical front. In the critical case the name pulled front comes from the fact that the front moves at the same speed as the solution of the linearized problem around the unstable state 0, which means that it is being pulled along by its leading edge. This denomination is not so immediate for super-critical fronts. A pushed front is a critical front (c * , U ) such that the minimal speed c * satisfies c * > 2 f ′ (0). The speed of propagation of a pushed front is determined not by the behavior of the leading edge of the distribution, but by the whole front. This means that the front is pushed from behind by the nonlinear growth rate in the nonlinear front region itself. A substantial analysis which is not restricted to the monostable case and which relies on the variational structure and the exponential decay of the fronts for more general gradient systems has been carried out in [START_REF] Lucia | Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium[END_REF][START_REF] Muratov | Front propagation in infinite cylinders. I. A variational approach[END_REF].

In the present paper, we use a completely different and new approach, which we believe to be simpler and more intuitive, by analyzing the dynamics of the inside structure of the fronts. The results we obtain on the large-time behavior of the components of the fronts in the moving frame and in the whole real line (the precise statements will be given below) shed a new light on and are in keeping with the pulled/pushed terminology in the monostable case as well as with the fact that the bistable or ignition fronts can be viewed as pushed fronts. Even if more general equations or systems could have been considered, we present the results for the one-dimensional equation (1.1) only, for the sake of simplicity and since this simple one-dimensional situation is sufficient to capture the main interesting properties of the spatial structure of the fronts (however, based on the results of the present paper and on some recent notions of generalized transition waves, we propose in Section 2.2 some definitions of pulled and pushed transition waves in a more general setting).

Let us now describe more precisely the model used in this paper. Following the ideas described in [START_REF] Hallatschek | Gene surfing in expanding populations[END_REF][START_REF] Hallatschek | Life at the front of an expanding population[END_REF][START_REF] Vlad | Enhanced (hydrodynamic) transport induced by population growth in reaction-diffusion systems with application to population genetics[END_REF], we assume that the fronts are made of several components and we study the behavior of these components. Namely, we consider a traveling wave solution

u(t, x) = U (x -ct)
of (1.1), where the profile U satisfies (1.3) and c is the front speed, and we assume that u is initially composed of different groups (υ i 0 (x)) i∈I such that, for every i ∈ I,

υ i 0 ≡ 0 and 0 ≤ υ i 0 (x) ≤ U (x) for all x ∈ R, (1.4)
where I is a subset of N and

u(0, x) = U (x) = i∈I υ i 0 (x) for all x ∈ R.
Moreover, all groups υ i are assumed to share the same characteristics in the sense that they diffuse and grow with the same manner inside the front u(t, x), see [START_REF] Hallatschek | Gene surfing in expanding populations[END_REF][START_REF] Hallatschek | Life at the front of an expanding population[END_REF][START_REF] Vlad | Enhanced (hydrodynamic) transport induced by population growth in reaction-diffusion systems with application to population genetics[END_REF]. This means that the diffusion coefficient of each group is equal to 1 and that the per capita growth rate of each group depends only on the entire population and is the same as that of the global front, namely

g(u(t, x)) := f (u(t, x)) u(t, x)
for all t ≥ 0 and x ∈ R.

In other words, the groups (υ i (t, x)) i∈I satisfy the following equation:

∂ t υ i (t, x) = ∂ 2 x υ i (t, x) + g(u(t, x))υ i (t, x), t > 0, x ∈ R, υ i (0, x) = υ i 0 (x), x ∈ R. (1.5)
Of course it follows from the uniqueness of the solution that

u(t, x) = i∈I υ i (t,
x) for all t > 0 and x ∈ R, which implies that the per capita growth rate g(u(t, x)) = g i∈I υ i (t, x) could be viewed as a coupling term in the system (1.5). The following inequalities also hold from maximum principle 0 < υ i (t, x) ≤ u(t, x) < 1 for all t > 0, x ∈ R and i ∈ I.

(1.6)

Because the components υ i in (1.5) have identical growth and dispersal characteristics, we only have to focus on the behavior of one arbitrarily chosen component υ i -we call it υ in the sequel -to understand the behavior of all the components. This is in sharp contrast with standard competitive systems such as the model of competition between a resident species and an invading species for large open space mentioned in [START_REF] Shigesada | Biological Invasions: Theory and Practice[END_REF] (see Section 7.2), where usually one of the elements is in some sense stronger than the other one and thus governs the propagation.

Even if the equation (1.1) is homogeneous and the system (1.5) is linear, one of the main difficulties in our study comes from the fact that a space-time heterogeneity is present in the per capita growth rate g(u(t, x)) of each element. It turns out that this heterogeneity does not fulfill any periodicity or monotonicity property. Comparable problems have been studied in [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF][START_REF] Hamel | Reaction-diffusion problems in cylinders with no invariance by translation. Part I: Small perturbations[END_REF][START_REF] Hamel | Reaction-diffusion problems in cylinders with no invariance by translation. Part II: Monotone perturbations[END_REF]. In these papers, the authors have considered a reaction term of the form f (x-ct, υ), where the function υ → f (x, υ) is of the monostable or bistable type for every x and is nonpositive for υ large enough uniformly in x. These properties are not fulfilled here by the function (t, x, υ) → g(u(t, x)) υ. For instance, if the reaction term f is of type (A), then g(u(t, x)) is always positive. Actually, we prove that the behavior of the groups υ i mainly depends on the type of f , as well as on the initial condition.

The next section is devoted to the statement of our main results. We begin by recalling some asymptotic properties of the solution U (y) of (1.3), as y → +∞. Then, the evolution of the density of a group υ solving (1.5) is described in two theorems. Theorem 1 deals with the monostable pulled case and Theorem 2 deals with the monostable pushed case and the bistable and ignition cases. These results show striking differences between the composition of the fronts in the pulled and pushed cases. They lead us to propose new notions of pulled and pushed transition waves in a general setting. The proofs of our results are detailed in Sections 3 and 4.

Main Results

Let u(t, x) = U (x-ct) be a traveling wave solution of (1.1) associated to a front (c, U ) solving (1.3), where f is either of type (A), (B) or (C). In order to understand the dynamics of a component υ solving (1.4)-(1.5), inside the traveling wave solution, it is natural to make the following change of variables: υ(t, x) = υ(t, x + ct) for all t ≥ 0 and x ∈ R.

The function υ corresponds to the solution υ in the moving reference at speed c and it obeys the following equation:

∂ t υ(t, x) = ∂ 2 x υ(t, x) + c∂ x υ(t, x) + g(U (x))υ(t, x), t > 0, x ∈ R, υ(0, x) = υ 0 (x), x ∈ R. (2.7)
Thus, the equation (1.5) which contains a space-time heterogeneous coefficient reduces to a the reactiondiffusion equation with a spatially heterogeneous coefficient g(U (x)), which only depends on the profile U of the front. As we will see, the leading edge of U, and therefore its asymptotic behavior as x → +∞, plays a central role in the dynamics of the solutions of (2.7). Before stating our main results, we recall some useful known facts about the asymptotic behavior of the fronts.

Monostable case (A).

On the one hand, a pulled critical front (c * , U ), whose speed c * satisfies c * = 2 f ′ (0), decays to 0 at +∞ as follows [START_REF] Aronson | Nonlinear diffusion in population genetics, combustion and nerve propagation[END_REF][START_REF] Aronson | Multidimensional non-linear diffusion arising in population genetics[END_REF]:

U (y) = (Ay + B) e -c * y 2 + O e -( c * 2 +δ)y as y → +∞, (2.8)
where δ > 0, A ≥ 0, and B > 0 if A = 0. If f is of the particular KPP type (that is g(s) = f (s)/s ≤ f ′ (0)) then A > 0. On the other hand, a pushed critical front (c * , U ), whose speed c * is such that c * > 2 f ′ (0), satisfies the following asymptotic property:

U (y) = A e -λ+(c * )y + O e -(λ+(c * )+δ)y as y → +∞, (2.9)
where δ > 0, A > 0 and λ

+ (c * ) = (c * + (c * ) 2 -4f ′ (0))/2 > c * /2.
Thus, the asymptotic behavior of a monostable critical front does depend on its pulled/pushed nature. Lastly, a super-critical front (c, U ), where c satisfies c > c * , also decays at an exponential rate slower than c/2:

U (y) = A e -λ-(c)y + O e -(λ-(c)+δ)y as y → +∞, (2.10)
where δ > 0, A > 0 and λ

-(c) = (c -c 2 -4f ′ (0))/2 < c/2.

Bistable case (B).

If follows from [START_REF] Aronson | Nonlinear diffusion in population genetics, combustion and nerve propagation[END_REF][START_REF] Aronson | Multidimensional non-linear diffusion arising in population genetics[END_REF][START_REF] Fife | The approach of solutions of nonlinear diffusion equations to traveling front solutions[END_REF] that the unique front decays to 0 at +∞ as follows:

U (y) = A e -µ y + O e -(µ+δ)y as y → +∞, (2.11)
where δ > 0, A > 0, and

µ = (c + c 2 -4f ′ (0))/2 > c > c/2.

Ignition case (C).

The unique front decays to 0 at +∞ as follows:

U (y) = A e -c y for y > 0 large enough, (2.12)

where A > 0.

We notice that the asymptotic behaviors as y → +∞ of the fronts in the monostable pushed critical case and the bistable and ignition cases are quite similar. In all cases, the exponential decay rate is faster than c/2, where c is the speed of the front. Let us now state our main results.

The inside structure of the fronts

We first investigate the case where the nonlinearity f is of the monostable type (A) and (c, U ) is a pulled front.

Theorem 1 (Pulled case) Let f be of the monostable type (A), let (c, U ) be a pulled front, that is either In other words, any single component υ of the pulled front u, which initially decays faster than the front itself, in the sense of (2.13), cannot follow the propagation of the front. In particular, the formula (2.14) implies that υ(t, x + ct) → 0 uniformly on compacts as t → +∞.

c = c * = 2 f ′ (0) or c > c * ,
(2.15)

The conclusion of Theorem 1 holds if υ 0 is of the type υ 0 ≡ U 1 (-∞,a) or more generally if υ 0 satisfies (1.4) and its support is included in (-∞, a) for some a ∈ R. This means that the propagation of the traveling wave u(t, x) = U (x -ct) is due to the leading edge of the front. This characterization agrees with the definition of pulled fronts proposed by Stokes [START_REF] Stokes | On two types of moving front in quasilinear diffusion[END_REF]. It is noteworthy that pulled critical fronts and supercritical fronts share the same inside structure. Note that (2.14) also implies that υ cannot propagate to the right with a positive speed, in the sense that max x≥εt υ(t, x) → 0 as t → +∞ for all ε > 0. Actually, under some additional assumptions on υ 0 , which include the case where υ 0 is compactly supported, a stronger uniform convergence result holds:

Proposition 1 Under the assumptions of Theorem 1, and if υ 0 satisfies the additional condition

υ 0 (x) → 0 as x → -∞, or υ 0 ∈ L p (R) for some p ∈ [1, +∞), (2.16
)

then υ(t, •) → 0 uniformly on R as t → +∞.
(2.17)

In the pushed case, the dynamics of υ is completely different, as shown by the following result:

lim sup t→+∞ max x≥α √ t |υ(t, x) -p(υ 0 )U (x -ct)| → 0 as α → +∞, (2.18)
where p(υ 0 ) ∈ (0, 1] is given by

p(υ 0 ) = R υ 0 (x) U (x) e cx dx R U 2 (x) e cx dx . (2.19) Moreover, lim inf t→+∞ min α √ t≤x≤x0+ct υ(t, x) > 0 for all α ∈ R and x 0 ∈ R. (2.20)
From (2.9), (2.11) and (2.12), p(υ 0 ) is a well defined positive real number. Theorem 2 is in sharp contrast with Theorem 1. Indeed, formula (2.18) in Theorem 2 implies that any small group inside a pushed front is able to follow the traveling wave solution in the sense υ(t, x + ct) → p(υ 0 ) U (x) uniformly on compacts as t → +∞.

(2.21)

The conclusion (2.21) holds even if υ 0 is compactly supported. This formula means that an observer who moves with a speed c will see the component υ approach the proportion p(υ 0 ) of the front U . Thus, at large times, the front is made of all its initial components υ i 0 defined in (1.4), each one with proportion p(υ i 0 ). In other words the front is pushed from the inside. Theorem 2 also shows that the inside structure of the pushed monostable critical fronts and of the bistable and ignition fronts share the same dynamics.

The second formula (2.20) in Theorem 2 shows that the left spreading speed of the group υ inside the front is at least equal to 0 in the reference frame. More precisely, the group spreads over intervals of the type (α √ t, x 0 + ct) for all α ∈ R and x 0 ∈ R. In fact, the next proposition proves that, if the initial condition υ 0 is small at -∞, then the solution υ spreads to the left with a null speed in the reference frame in the sense that υ is asymptotically small in any interval of the type (-∞, α √ t) for -α > 0 large enough:

Proposition 2 Under the assumptions of Theorem 2, if υ 0 satisfies the additional assumption (2.16), then

lim sup t→+∞ max x≤α √ t υ(t, x) → 0 as α → -∞. (2.22)
Notice that without the condition (2.16), the conclusion (2.22) may not hold. Take for instance υ 0 ≡ U, then υ(t, x) = U (x -ct) for all t ≥ 0 and x ∈ R, and sup x≤α √ t υ(t, x) = 1 for all α ∈ R and t ≥ 0.

Remark 1 a) One can observe that in the pulled case the function x → U (x)e cx/2 does not belong to L 2 (R), from (2.8) and (2.10). Thus, we can set p(υ 0 ) = 0 for any compactly supported initial condition υ 0 satisfying (1.4). From Theorems 1 and 2, we can say with this convention that for any monostable reaction term f and any compactly supported υ 0 fulfilling (1.4), the solution υ of (1.5) is such that υ(t, x + ct) → p(υ 0 )U (x) uniformly on compacts as t → +∞, where p(υ 0 ) is defined by (2.19)

if x → U (x)e cx/2 is in L 2 (R) (the pushed case) and p(υ 0 ) = 0 otherwise (the pulled case).
b) Let us consider the family of reaction terms (f a ) a≥0 of the monostable type (A), defined by

f a (u) = u(1 -u)(1 + au) for all u ∈ [0, 1] and a ≥ 0.
The minimal speed c * a is given by [START_REF] Hadeler | Travelling fronts in nonlinear diffusion equations[END_REF] c

* a =    2 if 0 ≤ a ≤ 2, 2 a + a 2 if a > 2. Thus, if a ∈ [0, 2], the critical front U a associated with f a is pulled (c * a = 2 = 2 f ′ a (0)) while if a > 2 the critical front is pushed (c * a > 2 = 2 f ′ a (0)).
Up to shift, one can normalize U a so that U a (0) = 1/2 for all a ≥ 0. A direct computation shows that if a ≥ 2 then the profile of U a is then given by

U a (x) = 1 1 + e κax for all x ∈ R, where κ a = a/2. It is easy to check that, if a > 2, then the function x → U a (x)e c * a x/2 is in L 2 (R) (a general property shared by all pushed fronts) and R U 2 a (x) e c * a x dx ≥ (κ a -κ -1 a ) -1 /4. Then, consider a fixed compactly supported initial condition υ 0 satisfying (1.4) and whose support is included in [-B, B] with B > 0. Let p(υ 0 , a) be defined by (2.19) with a > 2, U = U a and c = c * a . It follows that 0 < p(υ 0 , a) = R υ 0 (x) U a (x) e c * a x dx R U 2 a (x) e c * a x dx ≤ 4(κ a -κ -1 a ) B -B U 2 a (x) e c * a x dx ≤ 8(κ a -κ -1 a ) cosh (c * a B) c * a .
Finally, since κ a → 1 + and c * a → 2 + as a → 2 + , we get that p(υ 0 , a) → 0 as a → 2 + . Thus, with the convention p(υ 0 ) = 0 in the pulled case, this shows the proportion p(υ 0 , a) is right-continuous at a = 2, which corresponds to the transition between pushed fronts and pulled fronts.

Notions of pulled and pushed transition waves in a more general setting

Our results show that the fronts can be classified in two categories according to the dynamics of their components. This classification agrees with the pulled/pushed terminology introduced by Stokes [START_REF] Stokes | On two types of moving front in quasilinear diffusion[END_REF] in the monostable case and shows that the bistable and ignition fronts have same inside structure as the pushed monostable fronts. This classification also allows us to define the notion of pulled and pushed transition waves in a more general framework. Let us consider the following reaction-dispersion equations:

∂ t u(t, x) = D(u(t, x)) + f (t, x, u(t, x)), t > 0, x ∈ R, (2.23)
where f (t, x, u) is assumed to be of class C 0,β (with

β > 0) in (t, x) locally in u ∈ [0, +∞), locally Lipschitz- continuous in u uniformly with respect to (t, x) ∈ (0, +∞) × R, f (•, •, 0) = 0,
(u) = ∂ x (a(t, x)∂ x u)
where a(t, x) is of class C 1,β ((0, +∞) × R) and uniformly positive, the fractional Laplacian, and the integrodifferential operators D(u) = J * u -u where J * u(x) = R J(x -y)u(y)dy for all x ∈ R and J is a smooth nonnegative kernel of mass 1. Before defining the notion of pulled and pushed waves, we recall from [START_REF] Berestycki | Generalized transition waves and their properties[END_REF] the definition of transition waves, adapted to the Cauchy problem (2.23). Let p + : (0, +∞) × R → [0, +∞) be a classical solution of (2.23). A transition wave connecting p -= 0 and p + is a positive solution u of (2.23) such that 1) u ≡ p ± , 2) there exist n ∈ N and some disjoint subsets (Ω ± t ) t>0 and (Γ t

) t>0 = ({x 1 t , . . . , x n t }) t>0 of R where Γ t = ∂Ω ± t , Ω - t ∪ Ω + t ∪ Γ t = R and sup d(x, Γ t ) | x ∈ Ω - t = sup d(x, Γ t ) | x ∈ Ω + t
= +∞ for all t > 0, and 3) for all ε > 0 there exists M > 0 such that for all t ∈ (0, +∞) and

x ∈ Ω ± t , d(x, Γ t ) ≥ M ⇒ |u(t, x) -p ± (t, x)| ≤ ε ,
where d is the classical distance between subsets of R. The wave u for problem (2.23) is also assumed to have a limit u(0, •) at t = 0 (usually, it is defined for all t ∈ R, and f and p + are defined for all t ∈ R as well). In the case of Theorems 1 and 2, the travelling front u(t, x) = U (x -ct) is a transition wave connecting 0 and p + = 1, the interface Γ t can be reduced to the single point Γ t = {x t } = {ct} and the two subsets Ω ± t can be defined by Ω - t = (x t , +∞) and Ω + t = (-∞, x t ) for all t > 0.

Definition 1 (Pulled transition wave)

We say that a transition wave u connecting 0 and p + is pulled if for any subgroup υ satisfying

∂ t υ(t, x) = D(υ(t, x)) + g(t, x, u(t, x))υ(t, x), t > 0, x ∈ R, υ(0, x) = υ 0 (x), x ∈ R, (2.24)
where g(t, x, s) = f (t, x, s)/s and

υ 0 is compactly supported, 0 ≤ υ 0 ≤ u(0, •) and υ 0 ≡ 0, ( 2

The description of pulled fronts

We first prove the annihilation of υ in the moving frame, that is property (2.15). Then we prove the result (2.14) of Theorem 1 and the result (2.17) described in Proposition 1 under the additional assumption (2.16). The proof of (2.15) draws its inspiration from the front stability analyzes in [START_REF] Sattinger | On the stability of waves of nonlinear parabolic systems[END_REF][START_REF] Sattinger | Weighted norms for the stability of traveling waves[END_REF][START_REF] Van Saarloos | Front propagation into unstable states[END_REF] and especially from the paper of Eckmann and Wayne [START_REF] Eckmann | The nonlinear stability of front solutions for parabolic differential equations[END_REF]. It is based on some integral estimates of the ratio r = υ/U in a suitable weighted space. The proofs of (2.14) and (2.17) are based on the convergence result (2.15) and on the maximum principle together with the construction of suitable super-solutions.

Local asymptotic extinction in the moving frame: proof of (2.15)

Let f be of type (A) and let (c, U ) denote a pulled front satisfying

(1.3), that is c is such that either c = c * = 2 f ′ (0) or c > c * .
Let υ be the solution of (1.4)-(1.5) satisfying the condition (2.13) and let us set υ(t, x) = υ(t, x + ct) for all t ≥ 0 and x ∈ R. The function υ solves (2.7), while (1.6) implies that 0 < υ(t, x) ≤ U (x) < 1 for all t > 0 and x ∈ R.

(3.26)

Then, let us define the ratio

r(t, x) = υ(t, x) U (x) for all t ≥ 0 and x ∈ R. (3.27)
The function r is at least of class C 1 with respect to t and of class C 2 with respect to x in (0, +∞) × R. It satisfies the following Cauchy problem:

     ∂ t r(t, x) + Lr(t, x) = 0, t > 0, x ∈ R, r(0, x) = υ 0 (x) U (x) , x ∈ R, (3.28)
where L = -∂ 2 x -ψ ′ (x)∂ x and ψ(x) = cx + 2 ln(U (x)) for all x ∈ R.

Lemma 1 There exists a constant k > 0 such that

|ψ ′ (x)| + |ψ ′′ (x)| ≤ k for all x ∈ R.
Proof. The proof uses standard elementary arguments. We just sketch it for the sake of completeness. If we set q = U and p = U ′ , then ψ ′ = c + 2p/q and ψ ′′ = -2cp/q -2(p/q) 2 -2g(q). Here we use that q ′ = p and p ′ = -cp -g(q)q. Clearly g(q) is bounded. Thus we only need to bound p/q. Proposition 4.4 of [START_REF] Aronson | Multidimensional non-linear diffusion arising in population genetics[END_REF] implies that either p/q → -c/2 at +∞ if c = c * = 2 f ′ (0), or p/q → -λ -(c) at +∞ if c > c * . Moreover, since p/q → 0 at -∞ and p/q is continuous, we conclude that p/q is bounded, which proves Lemma 1.

Let us now define a weight function σ as follows:

σ(x) = U 2 (x)e cx for all x ∈ R.
Since U satisfies the asymptotic properties (2.8) or (2.10), one has lim inf x→+∞ σ(x) > 0. A direct computation shows that σ ′ (x) -ψ ′ (x)σ(x) = 0 for all x ∈ R.

(3.29)

In order to lighten the proof, we introduce some norms associated to the weight function σ:

w 2 = R σ(x) w(x) 2 dx 1/2 and w ∞ = sup x∈R |σ(x)w(x)|,
where the supremum is understood as the essential supremum, and we define the standard L 2 (R) and L ∞ (R) norms as follows:

|w| 2 = R w(x) 2 dx 1/2 and |w| ∞ = sup x∈R |w(x)|.
Notice that the hypothesis (2.13) implies that r(0, •) is in the weighted space

L 2 σ (R) = w ∈ L 2 (R) | w 2 < ∞ ,
which is a Hilbert space endowed with the inner product

(w, w) = R w(x) w(x) σ(x) dx for all w, w ∈ L 2 σ (R).

Lemma 2

The solution r of the linear Cauchy problem (3.28) satisfies the following properties:

d dt 1 2 r(t, •) 2 2 ) = -∂ x r(t, •) 2 2 for all t > 0 (3.30)
and, for any constant

K satisfying K ≥ |ψ ′′ | ∞ + 1, d dt K 2 r(t, •) 2 2 + 1 2 ∂ x r(t, •) 2 2 ≤ -∂ x r(t, •) 2 2 + ∂ 2 x r(t, •) 2 2 for all t > 0.
Proof. We first set some properties of the operator L in the Hilbert space L 2 σ (R). We define its domain D(L) as

D(L) = H 2 σ (R) = w ∈ L 2 σ (R) | w ′ , w ′′ ∈ L 2 σ (R)
, where w ′ and w ′′ are the first-and second-order derivatives of w in the sense of distributions. The domain

D(L) is dense in L 2 σ (R) and since C ∞ c (R) is also dense in H 2 σ (R) (with the obvious norm in H 2 σ (R)), it follows that (Lw, w) = - R w ′′ wσ - R ψ ′ w ′ wσ = R w ′ (wσ) ′ - R ψ ′ w ′ wσ = R (w ′ ) 2 σ + R w ′ w(σ ′ -ψ ′ σ) = w ′ 2 2 ≥ 0 for all w ∈ D(L), that is L is monotone. Furthermore, L is maximal in the sense that ∀ f ∈ L 2 σ (R), ∃ w ∈ D(L), w + Lw = f.
This equation can be solved by approximation (namely, one can first solve the equation

w n + Lw n = f in H 2 (-n, n) ∩ H 1 0 (-n, n), then show that the sequence ( w n H 2 σ (-n,n) )
n∈N is bounded and thus pass to the limit as n → +∞ to get a solution w). Moreover, the operator L is symmetric since

(Lw, w) = R w ′ ( wσ) ′ - R ψ ′ w ′ wσ = R w ′ w ′ σ = (w, L w)
for all w, w ∈ D(L). Since L is maximal, monotone and symmetric, it is thus self-adjoint. Then, since υ 0 is in L 2 σ (R), Hille-Yosida Theorem implies that the solution r of the linear Cauchy problem (3.28) is such that

r ∈ C k (0, +∞), H l σ (R) ∩ C [0, +∞), L 2 σ (R) for all k, l ∈ N, (3.31)
where H l σ (R) is the set of functions in L 2 σ (R) whose derivatives up to the l-th order are in L 2 σ (R). We can now define two additional functions W and Z. First we set,

W (t) = 1 2 R σ(x)r 2 (t, x)dx = 1 2 r(t, •) 2 2 for all t ≥ 0.
Let K be a positive constant satisfying K ≥ |ψ ′′ | ∞ + 1. We define the function Z as follows:

Z(t) = KW (t) + 1 2 R σ(x) (∂ x r) 2 (t, x) dx = K 2 r(t, •) 2 2 + 1 2 ∂ x r(t, •) 2 2 for all t > 0.
The functions W and Z are of the class C ∞ on (0, +∞) and W is continuous on [0, +∞). Since r satisfies (3.28) and (3.31) and σ obeys (3.29), we get that

W ′ (t) = d dt 1 2 r(t, •) 2 2 = 1 2 d dt R σ(x) r 2 (t, x) dx = -Lr(t, •), r(t, •) = -∂ x r(t, •) 2 2
for all t > 0. On the other hand, since r satisfies (3.31), there holds

Z ′ (t) = KW ′ (t) + 1 2 d dt R σ(x) (∂ x r) 2 (t, x) dx = KW ′ (t) + R ∂ x (σ∂ x r)(t, x) Lr(t, x) dx (3.32)
for all t > 0. One can also observe that, for all t > 0,

- R ∂ x (σ∂ x r)(t, x) ∂ 2 x r(t, x) dx = - R σ(x) (∂ 2 x r) 2 (t, x) dx - 1 2 R σ ′ (x) ∂ x ((∂ x r) 2 )(t, x) dx = - R σ(x) (∂ 2 x r) 2 (t, x) dx + 1 2 R σ ′′ (x) (∂ x r) 2 (t, x) dx and - R ∂ x (σ∂ x r)(t, x) ψ ′ (x) ∂ x r(t, x) dx = - 1 2 R σ(x) ψ ′ (x) ∂ x ((∂ x r) 2 )(t, x) dx - R σ ′ (x) ψ ′ (x) (∂ x r) 2 (t, x) dx = 1 2 R (σψ ′ ) ′ (x) (∂ x r) 2 (t, x) dx - R σ ′ (x) ψ ′ (x) (∂ x r) 2 (t, x) dx.
Notice that all above integrals exist and all integrations by parts are valid from the density of C ∞ c (R) in H 2 σ (R) and from Lemma 1 and (3.29) (in particular, σ ′ /σ and σ ′′ /σ are bounded). Moreover, it follows from (3.29) that 1 2

(σψ ′ ) ′ (x) -σ ′ (x)ψ ′ (x) + 1 2 σ ′′ (x) = σ(x)ψ ′′ (x) ≤ σ(x)|ψ ′′ | ∞
for all x ∈ R. Thus, the last integral in (3.32) is bounded from above by

R ∂ x (σ∂ x r)(t, x) Lr(t, x) dx ≤ - R σ(x) (∂ 2 x r) 2 (t, x) dx + |ψ ′′ | ∞ R σ(x) (∂ x r) 2 (t, x) dx.
Then, for all t > 0,

Z ′ (t) ≤ - R σ(x) (∂ 2 x r) 2 (t, x) dx -(K -|ψ ′′ | ∞ ) R σ(x) (∂ x r) 2 (t, x) dx, ≤ - R σ(x) (∂ 2 x r) 2 (t, x) dx - R σ(x) (∂ x r) 2 (t, x) dx,
from the choice of K. The proof of Lemma 2 is thereby complete.

Proof of property (2.15). We are now ready to state some convergence results for the function υ. Note first that W (t) ≥ 0 and W ′ (t) ≤ 0 for all t > 0. Thus W (t) converges to W ∞ ≥ 0 as t → +∞. Similarly, Z(t) converges to Z ∞ ≥ 0 as t → +∞, which implies that ∂ x r(t, •) 2 also converges as t → +∞. From (3.30) and the convergence of r(t, •) 2 and ∂ x r(t, •) 2 to finite limits, it also follows that

∂ x r(t, •) 2 → 0 as t → +∞.
On the other hand, for all t > 0, there holds

r 2 (t, •) ∞ ≤ 1 2 R |∂ x (σr 2 )(t, x)| dx ≤ R σ(x) |r(t, x)| |∂ x r(t, x)| dx + 1 2 R |σ ′ (x)| r 2 (t, x) dx. (3.33)
Using Cauchy-Schwarz inequality, we can see that the first term goes to 0 as t → +∞ since ∂ x r(t, •) 2 goes to 0 and r(t, •) 2 is bounded. If σ ′ were nonnegative on R, we could drop the modulus in the second term and then, integrating by parts, we would get as above that r 2 (t,

•) ∞ ≤ 2 r(t, •) 2 ∂ x r(t, •) 2 for all t > 0.
However, the function σ ′ may not be nonnegative on R. Let us now prove that r 2 (t, •) ∞ still goes to 0 as t → +∞ in the general case. First, since ψ ′ is bounded from Lemma 1 and σ(x) ∼ e cx as x → -∞, there holds σ

′ (x) = ψ ′ (x)σ(x) → 0 as x → -∞. Moreover, if c = c * = 2 f ′ (0), it follows from the asymptotic property (2.8) that σ(x) = (Ax + B) 2 + o(1) as x → +∞ if U (x) = (Ax + B)e -c * x/2 + O e -(c * /2+δ)x as x → +∞ with δ > 0, A > 0 and B ∈ R, or σ(x) → B 2 > 0 as x → +∞ if U (x) = Be -c * x/2 + O e -(c * /2+δ)x as x → +∞,
where δ > 0 and B > 0. On the other hand, if c > c * , it follows from (2.10), (3.29) and the proof of Lemma 1 that σ ′ (x) → +∞ as x → +∞.

Finally, in all cases it is possible to construct a constant S > 0 and a function ρ ∈ C 1 (R) such that ρ ′ ≥ 0 on R and Sσ(x) ≤ ρ(x) ≤ σ(x) for all x ∈ R.

The norms associated to ρ are equivalent to those defined by σ. Then, denoting

w ρ,2 = R ρ(x)w 2 (x)dx 1/2 and w ρ,∞ = sup x∈R |ρ(x)w(x)|,
and applying equation (3.33) to these norms, one infers that

S r 2 (t, •) ∞ ≤ r 2 (t, •) ρ,∞ ≤ 2 r(t, •) ρ,2 ∂ x r(t, •) ρ,2 ≤ 2 r(t, •) 2 ∂ x r(t,

Extinction in (α √ t, +∞) and in R: proofs of Theorem 1 and Proposition 1

Before completing the proof of Theorem 1 and Proposition 1, let us first state two auxiliary lemmas. They provide some uniform estimates of υ in intervals of the type (α √ t, A + ct) or the whole real line R when bounds for υ(t, •) are known at the positions A + ct, in the intervals (A + ct, +∞) and/or at -∞. These two lemmas will be used in all cases (A), (B) and (C).

Lemma 3 Let f be of type (A), (B) or (C), let (c, U ) be a front satisfying (1.3) and let υ solve (1.5) with υ 0 satisfying (1.4). Let µ ∈ [0, 1] and A 0 ∈ R be such that lim sup t→+∞ υ(t, A + ct) ≤ µ (resp. lim inf t→+∞ υ(t, A + ct) ≥ µ) for all A < A 0 . Then, for all ε > 0, there exist α 0 > 0 and A < A 0 such that The proofs of Lemmas 3 and 4 are postponed at the end of this section.

lim sup t→+∞ max α √ t≤x≤A+ct υ(t, x) ≤ µ + ε for all α ≥ α 0 , ( 3
End of the proof of Theorem 1. Let υ be the solution of (1.5) with υ 0 satisfying (1.4) and (2.13). To get (2.14), pick any ε > 0 and observe that property (2.15) and Lemma 3 with µ = 0 (and an arbitrary A 0 ) yield the existence of α 0 > 0 and A ∈ R such that lim sup

t→+∞ max α √ t≤x≤A+ct υ(t, x) ≤ ε for all α ≥ α 0 .
Property (2.14) follows then from (3.35). This proves Theorem 1.

End of the proof of Proposition 1. We make the additional assumption (2.16). Notice that the assumptions of Lemma 4 are fulfilled with λ = µ = 0, from (2.16) and (3.35). It follows that the inequality (3.37) holds with λ = µ = 0, which implies that υ(t, x) → 0 uniformly on R as t → +∞. The proof of Proposition 1 is thereby complete.

The proofs of Lemmas 3 and 4 are based on the construction of explicit sub-or super-solutions of (1.5) in suitable domains in the (t, x) coordinates, and on the fact that the coefficient g(u(t, x)) = g(U (x -ct)) in (1.5) is exponentially small when x -ct → -∞.

Proof of Lemma 3. Let us first consider the case of the upper bounds. Let µ ∈ [0, 1] and A 0 ∈ R be as in the statement and pick any ε ∈ (0, 1). Let j ε be the positive function defined on (-∞, 0) by

j ε (y) = ε 1 - 1 1 -y for all y < 0. (3.38)
The functions j ′ ε and j ′′ ε are negative on (-∞, 0) and -j ′′ ε (y) -cj ′ ε (y) ∼ εc/y 2 as y → -∞. We recall the asymptotic behavior of U at -∞:

U (y) = 1 -Be νy + O e (ν+δ)y as y -∞, (3.39)
where δ > 0, B > 0 and ν = (-c + c 2 -4f ′ (1))/2 > 0. This property (3.39) and the negativity of f ′ (1) yield the existence of a real number A < min(A 0 , 0) such that

0 ≤ g(U (y)) ≤ -j ′′ ε (y) -cj ′ ε (y) 2 + ε for all y ∈ (-∞, A]. (3.40)
From the assumptions made in Lemma 3, there is t 0 > 0 such that υ(t, A + ct) ≤ µ + ε for all t ≥ t 0 .

Now, let us define the function υ by

υ(t, x) = h(t -t 0 , x -A -ct 0 ) + j ε (x -ct) for all t ≥ t 0 and x ∈ [A + ct 0 , A + ct],
where h solves the heat equation

∂ t h(t, x) = ∂ 2 x h(t, x), t > 0, x ∈ R, h(0, x) = 2 1 (-∞,0) (x) + (µ + ε)1 (0,+∞) (x), x ∈ R.
Let us check that υ is a supersolution of (1.5) in the domain t ≥ t 0 and x

∈ [A + ct 0 , A + ct]. Firstly, observe that µ + ε ≤ υ(t, x) ≤ 2 + ε (3.41) for all t ≥ t 0 and x ∈ [A + ct 0 , A + ct] since µ + ε < 2 and 0 < j ε < ε on (-∞, 0). It follows from (3.26) that υ(t, A + ct 0 ) < 1 ≤ 2 + µ + ε 2 = h(t -t 0 , 0) ≤ υ(t, A + ct 0 ) for all t > t 0 .
On the other hand,

υ(t, A + ct) ≤ µ + ε ≤ h(t -t 0 , c(t -t 0 )) ≤ υ(t, A + ct) for all t ≥ t 0 .
Lastly, from (3.40) and (3.41), the function υ satisfies, for all t > t 0 and x ∈ (A + ct 0 , A + ct),

∂ t υ(t, x) -∂ 2 x υ(t, x) -g(U (x -ct))υ(t, x) = -j ′′ ε (x -ct) -cj ′ ε (x -ct) -g(U (x -ct))υ(t, x) ≥ -g U (x -ct) (2 + ε) -j ′′ ε (x -ct) -cj ′ ε (x -ct) ≥ 0. (3.42)
The maximum principle applied to (1.5) implies that

υ(t, x) ≤ υ(t, x) for all t ≥ t 0 and x ∈ [A + ct 0 , A + ct]. Fix α 0 > 0 so that 2 -µ -ε √ π -α0/2 -∞ e -z 2 dz ≤ ε.
Let α ≥ α 0 and t 1 > t 0 be such that A + ct 0 < α √ t < A + ct for all t ≥ t 1 . Since h(t, •) is decreasing for all t > 0, there holds, for all t ≥ t 1 ,

max α √ t≤x≤A+ct υ(t, x) ≤ max α √ t≤x≤A+ct υ(t, x) = max α √ t≤x≤A+ct h(t -t 0 , x -A -ct 0 ) + j ε (x -ct) ≤ h(t -t 0 , α √ t -A -ct 0 ) + ε ≤ µ + 2ε + 2 -µ -ε √ π (-α √ t+A+ct0)/ √ 4(t-t0)
-∞ e -y 2 dy.

Hence, for all α ≥ α 0 , lim sup

t→+∞ max α √ t≤x≤A+ct υ(t, x) ≤ µ + 2ε + 2 -µ -ε √ π -α/2
-∞ e -y 2 dy ≤ µ + 3ε.

Since ε > 0 is arbitrarily small, the conclusion (3.36) follows.

As far as the lower bounds are concerned, with the same type of arguments as above and since g(U (y)) is nonnegative near -∞, one can show that for any ε ∈ (0, 1), there exist A < A 0 and t 0 > 0 such that

υ(t, x) ≥ h(t -t 0 , x -A -ct 0 ) for all t ≥ t 0 and x ∈ [A + ct 0 , A + ct],
where h solves the heat equation in R with initial condition h(0,

•) = -1 (-∞,0) + (µ -ε)1 (0,+∞) . It follows that lim inf t→+∞ min α √ t≤x≤A+ct υ(t, x) ≥ µ -2ε for α > 0 large enough.
The proof of Lemma 3 is thereby complete.

Proof of Lemma 4. Let λ ∈ [0, 1] and µ ∈ [0, 1] be given as in the statement and pick any ε > 0. Let j ε be the function defined as in (3.38) and let A < 0 be such that

0 ≤ g(U (y)) ≤ -j ′′ ε (y) -cj ′ ε (y) max(max(λ, µ) + ε, 1) + ε for all y ∈ (-∞, A]. (3.43)
The assumptions made in Lemma 4 yield the existence of t 0 > 0 such that

υ(t, x) ≤ µ + ε for all t ≥ t 0 and x ∈ [A + ct, +∞). (3.44)
If υ 0 satisfies lim sup x→-∞ υ 0 (x) ≤ λ, then the comparison with the heat equation and the equality g(1) = 0 imply that lim sup x→-∞ υ(t, x) ≤ λ for all t > 0. On the other hand, if w solves (1.5) with an initial condition w 0 in L p (R) for some p ∈ [1, +∞), then heat kernel estimates and the boundedness of g(U (x -ct)) imply that w(t, •) is also in L p (R) for all t > 0, while w(t, •) is uniformly continuous on R from standard parabolic estimates. Finally, w(t, x) → 0 as x → -∞ for all t > 0. Now, if max (υ 0 -λ, 0) ∈ L p (R) for some p ∈ [1, +∞), then by writing υ 0 ≤ λ + max (υ 0 -λ, 0), the previous arguments and the linearity of (1.5) imply that lim sup x→-∞ υ(t, x) ≤ λ for all t > 0. In any case, at time t = t 0 , there exists

B ≤ A such that υ(t 0 , x) ≤ λ + ε for all x ∈ (-∞, B + ct 0 ]. (3.45)
Now, let us define the function υ by

υ(t, x) = h(t -t 0 , x) + j ε (x -ct) for all t ≥ t 0 and x ∈ (-∞, A + ct],
where h solves the heat equation

∂ t h(t, x) = ∂ 2 x h(t, x), t > 0, x ∈ R, h(0, x) = max(max(λ, µ) + ε, υ(t 0 , x)), x ∈ R. (3.46)
Let us check that υ is a supersolution of (1.5) in the domain t ≥ t 0 and x ≤ A + ct. Firstly, observe that max (λ, µ)

+ ε ≤ υ(t, x) ≤ max(max (λ, µ) + ε, 1) + ε (3.47)
for all t ≥ t 0 and x ≤ A + ct, from (3.26), (3.38) and the maximum principle applied to the heat equation (3.46). Then, from equation (3.44), we get that

υ(t, A + ct) ≤ µ + ε ≤ υ(t, A + ct) for all t ≥ t 0 .
Moreover, by definition of υ(t 0 , •), there holds

υ(t 0 , x) ≤ max(max (λ, µ) + ε, υ(t 0 , x)) ≤ υ(t 0 , x) for all x ∈ (-∞, A + ct 0 ].
Finally, from (3.43) and (3.47), υ satisfies the following inequality, for all t > t 0 and x ∈ (-∞, A + ct),

∂ t υ(t, x) -∂ 2 x υ(t, x) -g(U (x -ct))υ(t, x) = -j ′′ ε (x -ct) -cj ′ ε (x -ct) -g(U (x -ct))υ(t, x) ≥ -g(U (x -ct)) max(max(λ, µ) + ε, 1) + ε -j ′′ ε (x -ct) -cj ′ ε (x -ct) ≥ 0.
The maximum principle applied to (1.5) implies that υ(t, x) ≤ υ(t, x) for all t ≥ t 0 and x ∈ (-∞, A + ct].

(3.48)

Next, we claim that

υ(t, x) -j ε (x -ct) → max (λ, µ) + ε uniformly on (-∞, A + ct] as t → +∞.
Indeed, since υ(t 0 , •) satisfies (3.44) and (3.45), the initial condition h(0, •) of h is the sum of the constant max (λ, µ) + ε and a nonnegative compactly supported continuous function. By linearity and standard properties of the heat equation on R, it follows that h(t, •) → max (λ, µ) + ε uniformly on R as t → +∞. Hence, υ(t, x) -j ε (x -ct) → max (λ, µ) + ε uniformly on (-∞, A + ct] as t → +∞. From (3.48), we get that lim sup

t→+∞ sup x∈(-∞,A+ct] υ(t, x) ≤ max (λ, µ) + ε + sup y∈(-∞,A] j ε (y) = max (λ, µ) + 2ε.
Then, from equation (3.44) we get that lim sup

t→+∞ sup x∈R υ(t, x) ≤ max (λ, µ) + 2ε.
Since ε > 0 is arbitrary, the conclusion (3.37) follows and the proof of Lemma 4 is complete.

Remark 2 If the initial condition υ 0 of (1.5) is such that υ 0 (x) → λ ∈ [0, 1] as x → -∞, then, as already noticed, υ(t, x) → λ as x → -∞ for all t > 0, since g(1) = 0. In particular, sup R υ(t, •) ≥ λ for all t > 0. Therefore, if υ 0 satisfies (1.4), (2.13) and lim x→-∞ υ 0 (x) = λ ∈ [0, 1], then the proof of Lemma 4 shows that sup R υ(t, •) → λ as t → +∞.

The description of pushed fronts

This section is devoted to the proofs of Theorem 2 and Proposition 2. We begin by proving formula (2.21). The proof of this formula draws its inspiration from the front stability analysis [START_REF] Sattinger | On the stability of waves of nonlinear parabolic systems[END_REF][START_REF] Sattinger | Weighted norms for the stability of traveling waves[END_REF] and especially from the lecture notes of Gallay [START_REF] Gallay | Ondes progressives dans les sytèmes de reaction-diffusion[END_REF]. It is based on some properties of the self-adjoint Schrödinger operator L defined by

L = -∂ 2 x + c 2 4 -g(U (x)) (4.49)
∂ t υ * (t, x) + Lυ * (t, x) = 0, t > 0, x ∈ R, υ * (0, x) = υ 0 (x) e cx/2 , x ∈ R. (4.51)
The main spectral properties of L are stated in Section 4.1. Then, Section 4.2 is devoted to the proof of formula (2.21). The proofs of Theorem 2 and Proposition 2 are given in Section 4.3.

Preliminary lemmas

Let X c/2 be the weighted space defined by i) The essential spectrum σ e (L) of L is equal to c 2 /4 -max (f ′ (0), 0), +∞ .

X c/2 = w ∈ L 2 loc (R) | R w 2 (x) e cx dx < ∞ . ( 4
ii) The point spectrum of L is included in 0, c 2 /4 -max (f ′ (0), 0) . Moreover, λ = 0 is the smallest eigenvalue of L and the function x → φ(x) = U (x) e cx/2 spans the kernel of L.

iii) The following spectral decomposition of L 2 (R) holds:

L 2 (R) = Im(P ) ⊕ ker(P ), (4.53)

where the operator P : L 2 (R) → L 2 (R) is the spectral projection onto the kernel of L, that is

P (w) = R ϕw ϕ for all w ∈ L 2 (R) with ϕ = φ/ φ L 2 (R) .
Proof. It uses standard results and it is just sketched here for the sake of completeness.

i) The coefficients of the operator L are not constant but converge exponentially to two limits as x → ±∞. It follows that L is a relatively compact perturbation of the operator L 0 defined by L 0 = -∂ 2

x + (c 2 /4 -g ∞ (x)), where

g ∞ (x) = g(0) = f ′ (0) if x < 0, g(1) = 0 if x > 0.
Then, Theorem A.2 in [START_REF] Henry | Geometric Theory of Semilinear Parabolic Equations[END_REF] implies that the essential spectrum σ e (L) of the operator L is equal to the spectrum σ(L 0 ) of the operator L 0 . Since L 0 is self-adjoint in L 2 (R), we get

σ e (L) = σ(L 0 ) = c 2 /4 -max (f ′ (0), 0), +∞ .
ii) The operator L is a self-adjoint operator in L 2 (R), so the eigenvalues of L are in R. Moreover, since (c, U ) satisfies equation (1.3) and U belongs to X c/2 , one has necessarily that c 2 > 4f ′ (0) (whatever the sign of f ′ (0) be) and the function x → φ(x) = U (x)e cx/2 is in L 2 (R) (and then in H 2 (R) by adapting the arguments used the proof of Lemma 1). Furthermore, φ is an eigenvector of L associated to the eigenvalue λ = 0, that is Lφ = 0, and the eigenvalue is simple, from elementary arguments based on the exponential behavior at ±∞. On the other hand, since φ is positive, Sturm-Liouville theory implies that λ = 0 is the lowest value of the spectrum of L. Together with i), we finally get that the point spectrum of L is a discrete subset of the interval 0, c 2 /4 -max(f ′ (0), 0) .

iii) The function ϕ = φ/ φ L 2 (R) is a normalized eigenvector of L associated to the eigenvalue 0. Since L is self-adjoint, the operator P : L 2 (R) → L 2 (R) defined as in Lemma 5 is the spectral projection onto the kernel of L. Then, the spectral decomposition (4.53) holds, where Im(P ) = βϕ, β ∈ R and ker

(P ) = w ∈ L 2 (R) | R ϕw = 0 .
Let us come back to the Cauchy problem ∂ t w + Lw = 0. From Lemma 5, the semigroup e -tL t≥0 generated by -L satisfies the following properties: Lemma 6 If the front U solving (1.3) with c > 0 belongs to X c/2 , then there exist two constants C > 0 and η > 0 such that |e -tL w| ∞ ≤ Ce -ηt |w| ∞ for all t ≥ 0 and w ∈ ker(P ) ∩ L ∞ (R).

(4.54)

Proof. From Lemma 5, the decomposition (4.53) is stable by L. Moreover, the restriction of L to the space ker(P ) is a sectorial operator whose spectrum is included in {z ∈ C | ℜe(z) > η} for some small η > 0. The conclusion (4.54) follows from [START_REF] Henry | Geometric Theory of Semilinear Parabolic Equations[END_REF][START_REF] Pazy | Semigroup of Linear Operators and Applications to Partial Differential Equations[END_REF].

Proof of formula (2.21)

Let f satisfy the assumptions of Theorem 2 and let (c, U ) be either the pushed critical front when c = c * > 2 f ′ (0) in case (A) or the unique front satisfying (1.3) in cases (B) and (C). Let υ be the solution of the Cauchy problem (1.4)-(1.5) and let υ be defined by υ(t, x) = υ(t, x + ct). First of all, υ solves (2.7) and from the maximum principle the comparison (3.26) still holds. Moreover, since U satisfies (2.9), (2.11) or (2.12), U and υ(t, •) -for all t ≥ 0 -belong to the weighted space X c/2 defined by (4.52). Next, let υ * be defined by (4.50). Since υ(t, •) is in X c/2 for all t ≥ 0, the function υ * (t, •) belongs to L 2 (R), for all t ≥ 0. Furthermore, ϕ and υ * (0, •) belong to L ∞ (R) from (2.9), (2.11) and (2.12). From (4.51) and Lemma 5, the initial condition υ * (0, •) can be split in L 2 (R) as follows:

υ * (0, •) = P (υ * (0, •)) + w, where        P (υ * (0, •)) = R ϕ(s) υ (0, s) ds ϕ and ϕ(x) = e cx/2 U (x) R U 2 (s) e cs ds 1/2 for all x ∈ R, w = υ * (0, •) -P (υ * (0, •)) ∈ ker(P ) ∩ L ∞ (R).
Since Lϕ = 0, it follows that υ * (t, •) = P (υ * (0, •)) + e -tL w for all t ≥ 0.

(x) → λ ∈ [0, 1] as x → -∞, then sup R υ(t, •) → max (λ, p(υ 0 )) as t → +∞.

Spreading properties inside the pushed fronts: proofs of Theorem 2 and Proposition 2

The previous section 4.2 shows that, in the pushed case, the right spreading speed of υ in the reference frame is equal to c, in the sense that

c = inf γ > 0 | υ(t, • + γt) → 0 uniformly in (0, +∞) as t → +∞ .
In this section, we prove that in the pushed case the left spreading speed of υ is actually at least equal to 0. More precisely, we prove that the solution υ moves to the left in the reference frame, at least at a sublinear rate proportional to √ t, in the sense that lim inf t→+∞ υ(t, α √ t) > 0 for all α ≤ 0 (and, in fact, for all α ∈ R). This corresponds to formula (2.20) in Theorem 2. We also obtain some estimates, which are more precise than (2.21), on the asymptotic profile of the solution υ in sets of the type (α √ t, +∞) with α > 0 large enough. Lastly, we prove Proposition 2, which shows that the solution υ cannot spread to the left with a positive speed if υ 0 is small near -∞, in the sense of (2.16). The proofs of the pointwise estimates stated in Theorem 2 are based on formula (2.21) and on the construction of explicit sub-and super-solutions of (1.5) in the reference frame. Let υ be the solution of problem (1.5) with initial condition υ 0 satisfying (1.4). Theorem 2 implies that υ(t, A + ct) → p(υ 0 )U (A) > 0 as t → +∞. Choose any real number ν such that 0 < ν < p(υ 0 )U (A) and let t 0 > 0 be such that υ(t, A + ct) ≥ ν for all t ≥ t 0 . Now, let us construct a subsolution υ of the problem (1.5) in the domain t ≥ t 0 and x ≤ A + ct. Let υ be defined by υ(t, x) = h(t -t 0 , x -A -ct 0 ) for all t ≥ t 0 and x ≤ A + ct, where h is the solution of the heat equation ∂ t h = ∂ 2 x h with initial condition h(0, •) = ν1 (0,+∞) in R. On the one hand, there holds

υ(t, A + ct) ≤ sup R h(t -t 0 , •) ≤ ν ≤ υ(t, A + ct) for all t ≥ t 0 and υ(t 0 , x) = ν1 (0,+∞) (x -A -ct 0 ) = 0 ≤ υ(t 0 , x) for all x ∈ (-∞, A + ct 0 ),
while, on the other hand, it follows from (4.59) that

∂ t υ(t, x) -∂ 2 x υ(t, x) -g(U (x -ct))υ(t, x) ≤ 0
for all t > t 0 and x ∈ (-∞, A + ct). Then the maximum principle applied to (1.5) implies that υ(t, x) ≥ υ(t, x) for all t ≥ t 0 and x ≤ A + ct.

Lastly, let α be any fixed real number. Let us now turn to the proof of property (2.18). Let ε be any positive real number less than 2p(υ 0 ). From (2.21) and U ≤ U (-∞) = 1, Lemma 3 applied with µ = p(υ 0 ) (upper bound) and with µ = p(υ 0 )-ε/2 (lower bound) yields the existence of α 0 > 0, A ∈ R and t 1 > 0 such that α √ t < A + ct for all t ≥ t 1 and p(υ 0 ) -ε ≤ υ(t, x) ≤ p(υ 0 ) + ε for all t ≥ t 1 , x ∈ [α √ t, A + ct] and α ≥ α 0 .

Even if it means decreasing A, one can assume without loss of generality that p(υ 0 )(1 -U (A)) ≤ ε. This implies that, for all t ≥ t 1 and α ≥ α 0 , max From property (2.21) (and the fact that υ(t, x + ct) -p(υ 0 )U (x) → 0 as t → +∞ uniformly in [A, +∞), as observed at the end of the proof of (2.21)), we also know that there exists t 2 ≥ t 1 such that for all t ≥ t 2 , max x≥A+ct |υ(t, x) -p(υ 0 )U (x -ct)| ≤ ε.

Remember that, in the above formula, the supremum is a maximum, from (3.26) and the continuity of U and υ(t, •) for all t > 0. We conclude that lim sup Since ε > 0 is arbitrarily small, this completes the proof of Theorem 2.

Description of the left spreading speed: proof of Proposition 2

In addition to (1.4), we assume that υ 0 satisfies (2.16). As already observed in the proof of Lemma 4, this implies that υ(t, x) → 0 as x → -∞ for all t > 0. Let ε be any real number in (0, 1) and let j ε be the function defined on (-∞, 0) by (3.38). Let A < 0 be such that (3.40) holds. Since υ(1, x) → 0 as x → -∞, one can assume without loss of generality that υ(1, x) ≤ ε for all x ≤ A.

As in the proof of Lemmas 3 and 4, let us now construct a super-solution υ of (1.5) in the domain t ≥ 1 and x ≤ A. More precisely, let us set υ(t, x) = h(t -1, x -A) + j ε (x -ct) for all t ≥ 1 and x ≤ A, where h solves the heat equation

∂ t h(t, x) = ∂ 2
x h(t, x), t > 0, x ∈ R, h(0, x) = ε 1 (-∞,0) (x) + 2 1 (0,+∞) (x), x ∈ R.

There holds υ(

Remark 4

The proof of the lower bound of υ given in Subsection 4.3.1 implies immediately that, under the assumptions of Theorem 2, lim inf t→+∞ υ(t, x) ≥ p(υ 0 )/2 locally uniformly in x ∈ R. Furthermore, for any ε > 0, an adaptation of the above proof given in the present subsection implies that, under the assumptions of Proposition 2, there are A < 0 negative enough, t 0 > 0 positive enough and B < min(A + ct 0 , 0) negative enough such that υ(t, x) ≤ h(t -t 0 , x) + j ε (x -ct) for all t > t 0 and x ≤ ct + A, where h solves the heat equation ∂ t h = ∂ 2 x h with initial condition h(0, •) = ε1 (-∞,B) + 1 (B,ct0+A) + (p(υ 0 ) + ε)1 (ct0+A,+∞) . Therefore, since ε > 0 can be arbitrarily small, it follows that, under the assumptions of Proposition 2, lim sup t→+∞ υ(t, x) ≤ p(υ 0 )/2 locally uniformly in x ∈ R, and finally υ(t, x) → p(υ 0 )/2 as t → +∞ locally uniformly in x ∈ R.

Theorem 2 (

 2 Pushed case) Let f be either of type (A) with the minimal speed c * satisfying c * > 2 f ′ (0), or of type (B) or (C). Let (c, U ) be either the critical front with c = c * > 2 f ′ (0) in case (A) or the unique front in case (B) or (C). Let υ be the solution of the Cauchy problem (1.5) with the initial condition υ 0 satisfying (1.4). Then

  and D is a linear operator of dispersion. The classical examples of D are the homogeneous diffusion operators such as the Laplacian, D(u) = D∂ 2x u with D > 0, the heterogeneous diffusion operators of the form D

Lemma 4

 4 , x) ≥ µ -ε for all α ≥ α 0 . Let f be of type (A), (B) or (C), let (c, U ) be a front satisfying (1.3) and let υ solve (1.5) with υ 0 satisfying (1.4). Let λ and µ ∈ [0, 1] be such that either lim sup x→-∞ υ 0 (x) ≤ λ or max (υ 0 -λ, 0) ∈ L p (R) for some p ∈ [1, +∞), and lim sup t→+∞ max x≥A+ct υ(t, x) ≤ µ for all A ∈ R. Then lim sup t→+∞ sup x∈R υ(t, x) ≤ max (λ, µ).(3.37)

with domain H 2 (

 2 R), where g(s) = f (s)/s, f satisfies the hypotheses of Theorem 2, and (c, U ) is either the pushed critical front in case (A) when c = c * > 2 f ′ (0), or the unique front satisfying (1.3) in case (B) or (C). The properties of the semigroup generated by -L play an essential role in the large time behavior of the solution υ of the Cauchy problem (2.7). Indeed, the function υ * defined by υ * (t, x) = e cx/2 υ(t, x) for all t ≥ 0 and x ∈ R (4.50) satisfies the Cauchy problem

.52) Lemma 5

 5 If the front U solving (1.3) with c > 0 belongs to X c/2 , then the operator L defined by (4.49) satisfies the following properties:

  the existence of C > 0 and η > 0 such that |υ * (t, •) -P (υ * (0, •))| ∞ = |e -tL w| ∞ ≤ Ce -ηt |w| ∞ for all t ≥ 0. (4.56) Equation (4.55) and the definition (4.50) of υ * imply then that, for all t > 0 and x ∈ R, υ(t, x) = e -cx/2 P (υ * (0, •))(x) + e -tL w (x) = e -cx/2 ϕ(x) R ϕ(s) υ * (0, s) ds + e -cx/2 e -tL w (x) = p(υ 0 )U (x) + e -cx/2

4. 3 . 1 2

 312 Description of the right spreading speed and the asymptotic profile of solutions: proof of TheoremLet f fulfill the assumptions of Theorem 2 and let (c, U ) be either the pushed critical front in the monostable case (A) with speed c = c * > 2 f ′ (0), or the unique front in the bistable (B) and ignition (C) cases. First, as in the first inequality of (3.40), since g(U (y)) → 0 + as y → -∞, there is A ∈ R such that g(U (y)) ≥ 0 for all y ∈ (-∞, A].(4.59)

  x) -p(υ 0 )U (x -ct)| ≤ max α √ t≤x≤A+ct |υ(t, x) -p(υ 0 )| + p(υ 0 ) 1 -U (A) ≤ 2ε.

  t→+∞ max x≥α √ t |υ(t, x) -p(υ 0 )U (x -ct)| ≤ 2ε for all α ≥ α 0 .

  and let υ be the solution of the Cauchy problem (1.5) with the initial condition υ 0 satisfying (1.4) and

		+∞
		0	e cx υ 2 0 (x) dx < +∞.	(2.13)
	Then
	lim sup t→+∞	max x≥α √	t	υ(t, x) → 0 as α → +∞. 1	(2.14)

  +∞ and even uniformly in any interval of the type [A, +∞) with A ∈ R. This proves (2.21).Proof. The proof of (4.58) is a consequence of (2.21) and Lemma 4. More precisely, let ε be any positive real number in (0, 1) and let A be any real number. From the previous paragraph there is t 0 > 0 such that υ(t, x) ≤ p(υ 0 )U (x -ct) + ε ≤ p(υ 0 ) + ε for all t ≥ t 0 and x ≥ A + ct.On the other hand, since lim inf t→+∞ sup R υ(t, •) ≥ p(υ 0 ) from (2.21) and U (-∞) = 1, we get that sup R υ(t, •) → p(υ 0 ) as t → +∞. As in Remark 2, the above proof implies that if υ 0 satisfies (1.4) and υ 0

	Lemma 4 applied with λ = µ = p(υ 0 ) implies that
		lim sup t→+∞	sup x∈R	υ(t, x) ≤ p(υ 0 ).
	Remark 3
	Under an additional assumption on υ 0 , the following lemma holds:
	Lemma 7 Under the assumptions of Theorem 2, if υ 0 satisfies the additional assumption
	lim sup x→-∞	υ 0 (x) ≤ p(υ 0 ), or max (υ 0 -p(υ 0 ), 0) ∈ L p (R) for some p ∈ [1, +∞),	(4.57)
	then
		sup R	υ(t, •) → p(υ 0 ) as t → +∞.	(4.58)

e -tL w (x), where p(υ 0 ) ∈ (0, 1] is given in

(2.19)

. It follows from (4.56) that υ(t, x + ct) -p(υ 0 )U (x) → 0 uniformly on compacts as t →

  There exists t 1 > t 0 such that, for all t ≥ t

		ν √ π	+∞ (-α √ t+A+ct0)/	√	4(t-t0)
	Therefore, which together with (2.21) yields (2.20). lim inf t→+∞ min α √ t≤x≤ct+A	υ(t, x) ≥	ν √ π	+∞ -α/2	e -y 2 dy > 0,

1 , one has α √ t < ct+A and υ(t, x) ≥ υ(t, x) = h(t -t 0 , x -A -ct 0 ) for all x ∈ [α √ t, ct + A].

Since h(t, •) is increasing in R for all t > 0, we get, that for all t ≥ t 1 and

x ∈ [α √ t, ct + A], υ(t, x) ≥ h(t -t 0 , α √ t -A -ct 0 ) =

e -y 2 dy.

  ≤ 3ε for all α ≤ α 0 < 0 with |α 0 | large enough. Since ε can be arbitrarily small, the proof of Proposition 2 is thereby complete.

							√ t] is
	reached since υ ≥ 0 and υ(t, -∞) = 0, and there holds
	max x≤α √	t	υ(t, x) ≤ sup x≤α √ t
			max x≤α √ t	υ(t, x) ≤ 2ε +	2 -ε √ π	+∞ -(A+α √ t)/	4(t-1) √	e -y 2 dy
	for t large enough, whence
			lim sup t→+∞	max x≤α √ t	υ(t, x)

1, x) ≤ ε ≤ υ(1, x) for all x < A, while υ(t, A) ≤ 1 ≤ (2 + ε)/2 = h(t -1, 0) ≤ υ(t, A) for all t > 1 from

(3.26)

and

(3.38)

. Furthermore,

∂ t υ(t, x) -∂ 2 x υ(t, x) -g U (x -ct) υ(t, x) ≥ 0 for all t > 1 and x < A,

as in (3.42). It follows from the maximum principle applied to (1.5) that υ(t, x) ≤ υ(t, x) for all t ≥ 1 and x ≤ A.

For any fixed α < 0, and t > 0 large enough so that α

√ t < A, the maximum of υ(t, •) on (-∞, α h(t -1, x -A) + j ε (x -ct) ≤ h(t -1, α √ t -A) + ε since h(t,

•) is increasing in R for all t > 0. Thus, for any fixed α < 0,

Notice that the max in (2.14) is reached from (1.6) and the continuity of υ(t, •) for all t > 0.

The authors are supported by the French "Agence Nationale de la Recherche" within the projects ColonSGS and PREF-ERED.


Émilie Morvant

Stéphane Ayache

Amaury Habrard

Adaptation de domaine parcimonieuse Adaptation de domaine parcimonieuse par pondération de bonnes fonctions de similarité

Keywords: Apprentissage Multi-Tâches et Transfert, Méthodes à Noyaux, Adaptation de Domaine

L'adaptation de domaine est une problématique importante dans laquelle les données sources d'apprentissage et les données cibles de test sont supposées suivre deux distributions différentes. Nous nous plaçons dans le cadre difficile où aucune information sur les étiquettes cibles n'est disponible. D'un point de vue théorique, Ben-David et al. ont montré qu'un classifieur a de meilleures garanties de généralisation lorsque les distributions marginales des données sources et cibles sont proches. Nous présentons une approche basée sur un cadre de Balcan et al. permettant l'apprentissage de classifieurs linéaires à partir de fonctions de similarité n'ayant besoin ni d'être symétriques ni d'être semi-définies positives. Nous exploitons cette propriété pour repondérer la fonction de similarité afin de construire itérativement un espace de projection dans lequel les deux distributions marginales sont proches. Notre approche, formulée sous la forme d'un programme linéaire en norme 1, infère des modèles très parcimonieux montrant de bonnes performances d'adaptation. Nous l'évaluons expérimentalement sur des données synthétiques et des corpus réels d'annotations d'images.

Introduction

Pour de nombreux problèmes d'apprentissage automatique, il est généralement admis que les données d'apprentissage sont représentatives des données test. Cependant, cette hypothèse forte n'est pas toujours vérifiée en pratique. Cet inconvénient peut être contourné par des méthodes de Transfer Learning [START_REF] Pan | A survey on transfer learning[END_REF] où le but est d'adapter un modèle depuis un domaine source vers un domaine cible. Ce papier s'intéresse au problème d'adaptation de domaine (AD) pour lequel les données test sont supposées tirées selon une distribution -le domaine cible -différente de celle génératrice des données d'apprentissage -le domaine source [START_REF] Quionero-Candela J | Dataset Shift in Machine Learning[END_REF]. Différentes approches d'AD ont été proposées dans la littérature pour améliorer les performances de certaines méthodes classiques. Alors qu'une majorité d'entre elles supposent l'utilisation d'étiquettes cibles (Ben-David et al., 2010a;Bergamo & Torresani, 2010;Daumé III, 2007;[START_REF] Schweikert | An empirical analysis of domain adaptation algorithms for genomic sequence analysis[END_REF], nous considérons le cadre plus difficile où aucune de ces étiquettes n'est disponible. Ben-David et al. (2010a); [START_REF] Mansour | Domain adaptation : Learning bounds and algorithms[END_REF] ont démontré qu'un classifieur appris uniquement sur des données sources étiquetées peut être performant sur les données cibles si les distributions maginales source et cible sont proches, sous l'hypothèse que les domaines soient corrélés. Intuitivement, une AD judicieuse équivaut à rapprocher les distributions tout en maintenant de bonnes performances sur le domaine source. De cette idée découle différentes méthodes de repondération des données sources en fonction de différentes hypothèses ou mesures de divergence [START_REF] Huang | Correcting sample selection bias by unlabeled data[END_REF][START_REF] Mansour | Domain adaptation : Learning bounds and algorithms[END_REF][START_REF] Sugiyama | Direct importance estimation with model selection and its application to covariate shift adaptation[END_REF]. Bruzzone & Marconcini (2010) ont quant à eux proposé un processus itératif basé sur les SVMs : à chaque étape, des exemples sources sont retirés de l'échantillon d'apprentissage et des exemples cibles étiquetés par le modèle courant sont ajoutés. Une autre approche vise à construire un nouvel espace de projection où les deux distributions sont proches (Ben-David et al., 2006). Cependant, ce type de méthodes repose essentiellement sur des heuristiques et reste spécifique à certaines tâches. Dans cet article, nous proposons une méthode d'AD pour la classification binaire, inspirée du cadre récent de Balcan et al. (2008a,b) autorisant l'apprentissage à partir d'une bonne fonction de similarité, n'ayant besoin d'être ni symétrique ni semi-définie positive (SDP) (i.e. qui n'est pas un noyau). Ils ont montré qu'un classifieur linéaire performant peut être appris dans un espace de projection défini par un ensemble de similarités vis-à-vis de points dits raisonnables. Notre idée est de modifier automatiquement cet espace, en rapprochant les points sources et cibles, pour aboutir à une bonne adaptation. Pour ce faire, nous proposons une régularisation centrée sur les points raisonnables à la fois proches des exemples sources et cibles. Notre approche est itérative est exploite les propriétés d'une fonction de similarité en la repondérant pour construire un nouvel espace de projection dans lequel les deux distributions sont proches et tout en la maintenant bonne. Cette méthode offre ainsi une meilleure flexibilité que celles basées sur les SVMs. Sa formulation en norme 1 permet d'obtenir des modèles très parcimonieux. Nous l'évaluons sur un problème jouet, ainsi que sur un corpus réel d'annotation d'images.

Ce papier est organisé comme suit. Les Sections 2 et 3 introduisent res-pectivement les cadres d'AD et de Balcan et al. (2008a). Notre méthode est présentée en Section 4, puis est évaluée expérimentalement en Section 5.

Adaptation de Domaine

Soient X l'espace d'entrée et Y = {-1, 1} l'ensemble d'étiquettes. Un domaine est défini comme une distribution de probabilité selon X × Y . Dans le cadre de l'AD, les distributions différentes P S et P T désignent respectivement le domaine source et le domaine cible. D S et D T sont les distributions maginales respectives selon X. Un algorithme d'AD prend en entrée un échantillon source étiqueté, LS = {(x i , y i )} m i=1 , i.i.d. selon P S et à un échantillon cible non-étiqueté, T S = {x j } m j=1 , i.i.d. selon D T . Soit h : X → Y une hypothèse. L'espérance de l'erreur de h sur le domaine source P S est la probabilité qu'elle fasse une erreur de classification : (2006, 2010a) définissant une borne majorant l'erreur sur le domaine cible.

err S (h) = E (x,y)∼P S L 01 h(x) -y où L 01 (h(x) -y) est la fonction perte 0-1 et vaut 1 si h(x) = y et 0 sinon. L'
Théorème 1 (Ben-David et al. (2006, 2010a)) Soit H une classe d'hypothèses symétrique, pour tout h ∈ H :

err T (h) ≤ err S (h) + 1 2 d H (D S , D T ) + ν, où ν = err S (h * ) + err T (h * ) avec h * = argmin h∈H err S (h) + err T (h).
Cette borne dépend de l'erreur sur le domaine source qui peut facilement se minimiser selon le principe ERM. Le terme ν correspond à l'erreur jointe de la meilleure hypothèse possible sur les deux domaines.

dH (U, U ) = 2 1 -min h∈H ê rr U,U (h) (1) avec ê rr U,U (h) = 1 m x:h(x)=-1 1 x∈U + x:h(x)=1

Apprentissage à partir d'une Bonne Fonction de Similarité

Cette section présente la classe H de classifieurs linéaires définis à partir d'une similarité. Toute fonction K : X × X → [-1, 1] peut être considérée comme fonction de similarité sur X. De nombreux algorithmes font appel à de telles fonctions. C'est le cas des SVMs qui utilisent une similarité symétrique et semi-définie positive (SDP) appelée noyau. L'apprentissage s'effectue alors implicitement dans l'espace de grande dimension définit par ce noyau. Cependant être SDP est une contrainte forte et la définition d'un bon noyau reste une tâche difficile en général. Afin de contourner certaines de ces limitations, Balcan et al. (2008a,b) ont récemment proposé un cadre d'apprentissage où la définition d'une bonne fonction de similarité est plus intuitive. Nous allons le présenter en commençant par cette définition.

Définition 1 (Balcan et al. (2008a))

Une fonction de similarité K est une ( , γ, τ )-bonne fonction de similarité pour un problème P s'il existe un fonction indicatrice (aléatoire) R(x) qui définit un ensemble de points raisonnables tel que les conditions suivantes soient vérifiées :

(i) une proportion 1des exemples (x, y) satisfont ) points raisonnables (non-étiquetés) tirés selon P . Considérons φ R : X → R d la projection définie par φ R i (x) = K(x, x i ), i ∈ {1, .., d} induite par les similarités avec les points de R. Alors, avec une probabilité d'au moins 1 -δ sur l'échantillon aléatoire R, la distribution induite φ R (P ) dans R d admet un séparateur d'erreur au plus + δ par rapport à une marge L 1 d'au moins γ/2.

E (x ,y )∼P yy K(x, x )|R(x ) ≥ γ, (ii) P r x [R(x )] ≥ τ .
Pour un problème donné, si K est une ( , γ, τ )-bonne fonction de similarité associée à une quantité pertinente de points raisonnables, alors il existe avec une grande probabilité un séparateur linéaire d'erreur faible dans l'espace φ R (espace induit par les similarités avec les d points raisonnables). Ainsi, avec du exemples non-étiquetés et dl exemples étiquetés, un séparateur défini par α ∈ R du peut être inféré efficacement en résolvant un problème d'optimisation linéaire. Ceci conduit à un algorithme d'apprentissage en deux étapes : sélectionner un ensemble de points raisonnables puis apprendre un classifieur linéaire dans l'espace de projection induit. L'apprentissage de α est basée sur la perte Hinge ([1 -z] + = max(0, 1 -z)) (Balcan et al., 2008a) :

min α dl i=1 1 - du j=1 α j y i K(x i , x j ) + , s.t. du j=1 |α j | ≤ 1/γ (2)
CAp 2011

Nous dénotons par "classifieur SF" un classifieur linéaire appris via ce principe. Une formulation en programmation linéaire du Pb.(2) sera proposée. (2) comme un programme linéaire classique en y introduisant des slack variables ξ gérant la perte Hinge et en approximant la contrainte L1 par une régularisation en norme 1 pondérée par un paramètre λ. Cette régularisation est reformulée en remplaçant les variables α par des ensembles de variables α + et α -, tels que α = α + -α -. Elle n'est pas équivalente à la contrainte L1 mais tend à la satisfaire.

Définissons maintenant notre terme de régularisation. Nous selectionnons deux sous-ensemble U S et U T de même taille issus respectivement de LS et T S. Ensuite, nous construisons un couplage biparti, C ST , entre les points de

U S et U T en minimisant : (xs,xt)∈C ST φ R (x s ) -φ R (x t ) 2 2
. Étant donné cet ensemble C ST qui contient les paires d'exemples (source,cible) de U S × U T proches, nous voulons selectionner les points raisonnables -donc α -qui minimisent la norme 1 : ( t φ R (x s ) -t φ R (x t )) diag(α) 1 pour chaque paire (x s , x t ) ∈ C ST (où t défini la transposée). Grâce à la présence de α, ce terme tend à selectionner un espace dans lequel des points sources et cibles sont difficiles à séparer, ce qui, d'après l'Eq.( 1), tend à une décroissance de dH . Notons que ces choix sont effectués arbitrairement puisque la distribution cible est inconnue. Nous intégrons cette régularisation dans un programme linéaire en introduisant des slack variables η, coefficientées par un paramètre C.

Notre problème d'optimisation global est défini dans l'Eq.( 3) : un programme linéaire classique et efficacement résoluble en temps polynomial. La régularisation en norme 1 infère naturellement des modèles parcimonieux.

min α + ,α -,ξ,η dl i=1 ξ i + du j=1   C (xs,xt)∈C ST η j st + λ α + j + α - j   s.t. ∀i = 1, . . . , dl : ξ i ≥ 0, ξ i ≥ 1 - dl i=1 (α + j -α - j )y i K(x i , x j ) ∀j = 1, . . . , du, ∀(x s , x t ) ∈ C ST : α + j ≥ 0, α - j ≥ 0 η j st ≥ (α + j -α - j )(K(x s , x j ) -K(x t , x j )) η j st ≥ -(α + j -α - j )(K(x s , x j ) -K(x t , x j )).
(3) Les points raisonnables sélectionnés par l'algorithme, i.e. ceux associés à un poids α j non nul, définissent une projection dans laquelle les deux distributions sont plus proches. En effet, ils contribuent à la minimisation de la norme 1 sur les C ST , ce que nous exploitons ensuite en proposant une approche itérative repondérant la fonction de similarité les α. Ce processus peut être vu comme une contraction de l'espace pour minimiser dH . Supposons qu'à l'itération l et avec la fonction de similarité K l (repondérée), la résolution de l'Eq.(3) nous renvoie de nouveaux poids α l . Nous proposons alors de définir K l+1 en pondérant K l vis-à-vis de chacun des points raisonnables x j ∈ R (en la renormalisant éventuellement pour assurer K l+1 ∈ [-1, 1]) telle que :

K l+1 (x, x j ) = α l j K l (x, x j ).
Une bonne fonction de similarité n'a pas besoin d'être symétrique ou SDP. Notre normalisation est valide, si et seulement si, la nouvelle similarité est encore suffisamment bonne sur le domaine source (notons que l'ensemble R est mis à jour pour ne contenir que les points de poids non nul). Ce processus de repondération permet en fait de relier la qualité de la similarité au classifieur h l appris à l'itération l. En effet, pour tout x, on a :

E x ∈R K l+1 (x, x ) = 1 du du j=1 K l+1 (x, x j ) = 1 du du j=1 α l j K l (x, x j ) = 1 du h l (x).
CAp 2011

Cette qualité dépend donc de celle du classifieur, contrôlée par le terme issu de l'Eq.(3) de minimisation de l'erreur sur le domaine source ( i ξ i ).

Classifieur Inverse et Validation

Étant donné un classifieur h l , son classifieur inverse h r l est le classifieur appris dans l'espace φ R l (considéré à l'itération l) à partir des exemples cibles étiquetés par h l : {(x, sign(h l (x)))} x∈T S . En s'inspirant de l'idée de Zhong et al. (2010), nous évaluons la qualité de notre classifieur h l à l'aide de celle du classifieur inverse sur le domaine source. Une idée similaire a été proposée en tant que validation circulaire par Bruzzone & Marconcini (2010). Étant donnés k sous-ensembles de l'échantillon étiqueté source (LS = ∪ k i=1 LS i ), nous en utilisons k-1 en tant qu'exemples étiquetés pour résoudre le Pb.(3) et évaluons h r l sur le dernier sous-ensemble. L'erreur finale correspond à la moyenne des erreurs sur les k sous-ensembles :

ê rr S (h r l ) = 1 k k i=1 ê rr LS i (h r l ).

Critère d'Arrêt

D'après le Th.1, l'erreur sur le domaine cible est bornée par la somme de trois termes : (i) l'erreur sur le domaine source, (ii) la distance entre les deux distributions et (iii) l'erreur jointe du meilleur classifieur sur les deux domaines. Résoudre le Pb.(3) produit une décroissance naturelle de (i) et (ii). L'erreur jointe (iii) peut être associée à la capacité d'adaptation de notre classe d'hypothèses H dans l'espace de projection courant. Contrôler ce terme et sa diminution au cours des itérations semblerait être un critère attractif pour l'arrêt de l'agorithme. Cependant, puisqu'aucune information sur le domaine cible n'est disponible, nous proposons d'évaluer l'erreur jointe, à une itération l donnée, comme étant l'erreur du classifieur inverse h r l sur les deux domaines. La justification de ce choix est liée au fait que si les domaines sont suffisamment proches et reliés, alors le classifieur inverse doit être performant sur la tâche source (Bruzzone & Marconcini, 2010). En d'autres termes, nous devons être capable de passer d'un problème à l'autre dans l'espace de projection. Si l'erreur jointe, estimée par h r l , augmente entre deux itérations, le nouvel espace construit n'est plus pertinent et l'espace précédent est à préférer. L'erreur de h r l sur le domaine source est estimée par validation inverse, tandis que l'erreur sur le domaine cible est évaluée par cross-validation sur l'échantillon cible automatiquement étiqueté. Notre algorithme stoppe à l'itération l lorsque l'erreur jointe empirique du classifieur inverse, ê rr S (h r l+1 )+ ê rr T (h r l+1 ), a atteint un point de convergence ou a augmenté significativement. L'erreur jointe étant positive et bornée par 0, la convergence est assurée. Notons que ce critère est lié à la qualité de la similarité : lorsqu'elle n'est plus suffisamment bonne, l'erreur empirique sur le domaine source augmente dramatiquement et l'algorithme s'arrête. Ce critère permet à la fois le contrôle de la qualité de l'espace de projection construit et de celle de la fonction de similarité.

L'Algorithme DASF

Notre approche globale, appelée DASF, est présentée dans l'Algo.1. Un point clé de la méthode réside dans le choix des ensembles U S ⊆ LS et U T ⊆ T S servant au calcul du couplage biparti C ST . Sachant que l'ensemble des points pertinents permettant une bonne adaptation dépend généralement du problème considéré, nous proposons de les sélectionner à l'aide du classifieur inverse h r l et à chaque itération. Ils correspondent aux exemples pour lesquels le classifieur admet une forte ou une faible confiance en ses étiquettes. Soient δ + S > 0, δ + T > 0, δ - S < 0, δ - T < 0, alors U S et U T sont définis par :

U S = h r l (x) > δ + S OU h r l (x) < δ - S | x ∈ LS U T = h r l (x) > δ + T OU h r l (x) < δ - T | x ∈ T S , tels que |U S | = |U T |.
À chaque itération, nous choisissons les paramètres δ +/-S/T et λ, C de l'Eq.(3) minimisant l'erreur jointe de h r l en fonction d'une recherche par grille.

Algorithme 1 DASF entrée Fonction de similarité K, ensemble R, échantillons LS et T S sortie Classifieur de la dernière itération

h DASF h 0 (•) ← du j=1 K(•, x j ) ; K 1 ← K ; l ← 1 tant que Le critère d'arrêt n'est pas vérifié faire Construire U S ⊆ LS, U T ⊆ T S avec h r l-1 Construire C ST α l ← Résoudre le Pb.(3) avec K l et C ST K l+1 ← MAJ de K l selon α l MAJ R ; l + + fin tant que retourner h DASF (•) = x j ∈R α l j K l (•, x j )
CAp 2011

Expérimentations

Dans cette partie, nous évaluons notre approche DASF sur un problème jouet puis sur une tâche réelle d'annotation d'images. Notre fonction de similarité est basée sur le noyau Gaussien. Pour qu'elle ne soit ni symétrique ni SDP, nous la normalisons telle que chaque similarité vis-à-vis d'un point raisonnable ait une une moyenne de 0 et une variance unitaire sur l'ensemble d'apprentissage. Cependant, cette normalisation dépend des échantillons et n'offre pas toujours la meilleure ( , γ, τ )-qualité par rapport au noyau Gaussien. La similarité avec les meilleures garanties sur le domaine source est choisie a priori. Par la suite, les expérimentations faisant appel à une similarité normalisée sont indiquées par * . Comme nous le verrons, elles correspondent généralement aux tâches d'AD plus difficiles. DASF est comparé à un SVM classique, appris uniquement sur le domaine source, et à un Transductive SVM semi-supervisé (Vapnik, 1998) (TSVM). Pour ces deux algorithmes, nous utilisons la librairie SVM-light [START_REF] Joachims | Transductive inference for text classification using support vector machines[END_REF] avec un noyau Gaussien (les paramètres sont choisis par cross-validation sur les données sources). De plus, nous nous comparons à un classifieur SF appris uniquement sur le domaine source. La divergence dH entre les deux distributions est estimée à l'aide d'un classifieur SF visant à séparer l'échantillon source du cible. D'après l'Eq.( 1), une valeur proche de 0 implique des distributions proches tandis qu'une grande valeur indique une tâche d'AD potentiellement difficile.

Problème Jouet Synthétique

Ici, notre domaine source correspond à un problème de classification binaire classique de deux lunes jumelles (une classe par lune, cf. Fig. 1). Nous considérons 7 domaines cibles différents, chacun produit par une rotation

Classification d'Images

Dans cette partie, nous expérimentons notre approche sur les corpus Pas-calVOC'07 [START_REF] Everingham | The PASCAL Visual Object Classes Challenge 2007[END_REF] et TrecVid'07 [START_REF] Smeaton | High-Level Feature Detection from Video in TRECVid : a 5-Year Retrospective of Achievements[END_REF] TAB. 3: F-mesure estimée sur les domaines cibles TrecVid. Dans la dernière expérience, nous choisissons 4 concepts partagés par Trec-Vid et PascalVOC. Pour chacun d'entre eux, nous gardons notre domaine source PascalVOC tel quel et prenons, comme domaine cible, un ensemble d'exemples TrecVid de même ratio +/que le domaine source. dH est de l'ordre de 1.4, les deux corpus sont très différents et la tâche d'AD est donc potentiellement difficile. Les résultats évalués par la F-mesure sont reportés dans la Tab. 3. TSVM et DASF obtiennent des performances similaires, excepté pour le concept PERSON où DASF est significativement meilleur. Nous notons encore une fois que les modèles construits sont à la fois parcimonieux et performants. Enfin, pour ces tâches difficiles la similarité normalisée est souvent préférée ( * ).

Conclusion

Cet article propose une approche d'AD qui tire avantage du cadre de Balcan et al. (2008a,b), permettant d'utiliser une fonction de similarité potentiellement non SDP et non symétrique et donc plus flexible. La méthode construit itérativement un espace de projection définie par repondération des similarités vis-à-vis de points raisonnables à la fois proches des exemples sources et cibles. Nous avons montré expérimentalement sur différentes tâches que notre méthode produit des modèles toujours plus parcimonieux et pourvus de bonnes capacités d'adaptation. C'est un avantage certain en perspective d'applications sur des problèmes en grande dimension.

Dans le futur, nous avons l'intention d'étendre notre approche en autorisant l'utilisation de quelques étiquettes cibles, afin d'induire un meilleur espace de projection tel que [START_REF] Daumé | Co-regularization based semi-supervised domain adaptation[END_REF]. Un autre objectif serait de considérer différents domaines sources [START_REF] Duan | Domain adaptation from multiple sources via auxiliary classifiers[END_REF].

  En d'autres termes, la majorité des exemples doivent être plus similaires aux points raisonnables de même classe qu'à ceux de classe opposée, avec une confiance γ. De plus, au moins une proportion τ des points doivent être raisonnables. La Def.1 inclut aussi bien des noyaux valides que des similarités non-SDP non-symétrique : c'est une généralisation des noyaux(Balcan et al., 2008a,b). Le Th. 2 pose les conditions suffisantes pour apprendre un classifieur linéaire performant dans l'espace induit par les points raisonnables. Théorème 2 (Balcan et al. (2008a)) Soit K une ( , γ, τ )-bonne fonction de similarité pour un problème d'apprentissage P . Soit S = {x 1 , .., x d } un échantillon de d = 2 τ (log 2 δ + 8 log(2/δ) γ 2

  4. Adaptation de Domaine par Pondération d'une Fonction de Similarité Nous présentons maintenant notre méthode d'AD basée sur la pondération d'une bonne fonction de similarité. Rappelons que dans le cadre de Balcan et al., l'apprentissage d'un classifieur linéaire performant est possible dans l'espace φ R des similarités à l'ensemble des points raisonnables. D'après le Th.1, nous cherchons un espace φ R tel que les distributions source et cible soient proches. Pour ce faire, nous proposons une régularisation aidant à la selection des points raisonnables pertinents pour les deux domaines. La repondérantion des similarités, basée sur le classifieur linéaire appris, induit un nouvel espace φ R . Le processus est ensuite itéré jusqu'à un critère d'arrêt défini à l'aide d'une approche par validation inverse. 4.1. La Méthode d'Apprentissage Proposée Considérons dl points sources étiquetés, LS = {(x i , y i )} dl i=1 , tirés selon P S , du points raisonnables, R = {x j } du j=1 , tirés selon D S et un échantillon d'exemples cibles non-étiquetés T S tirés selon D T . Tout d'abord, nous reformulons le Pb.

FIG. 1 :

 1 FIG.1: Gauche : un échantillon source. Droite : un échantillon cible (50 • ).

  erreur err T sur le domaine cible est définie de manière équivalente. ê rr

S et ê rr T sont les erreurs empiriques associées. On note H la classe d'hypothèses de X vers Y considérée.

Nous présentons maintenant le cadre théorique d'AD de Ben-David et al.

  Résultats obtenus pour le problème jouet des lunes jumelles.anti-horaire (selon 7 angles) du domaine source. Plus l'angle est grand, plus le problème devient difficile. Pour chaque domaine, nous générons 300 instances (150 de chaque classe). La capacité en généralisation des algorithmes est évaluée sur un échantillon de test composé de 1500 exemples tirés selon le domaine cible (et non utilisé par les algorithmes). Chacun des problèmes d'AD est répété 10 fois. La performance moyenne de chaque méthode est calculée et reportée dans la Tab.1. Nous donnons aussi le nombre moyen de Vecteurs Support (VS) calculés par TSVM, le nombre de points raisonnables (RAIS.) trouvés par DASF et une estimation de dH entre les domaines dans les espaces initial φ R 0 et final φ R f inal . Nous faisons les remarques suivantes. -DASF est en moyenne plus performant. Il est significativement meilleur pour tous les problèmes d'angle supérieur à 20 • . Dès 60 • , la difficulté augmente et la performance de TSVM chute alors que DASF reste compétitif. La similarité normalisée est alors préférée ( * ). -Les points RAIS. sont significativement moins nombreux que les VS. C'est la confirmation que DASF produit des modèles très parcimonieux avec de bonnes performances. Cette quantité est 3 à 12 fois plus faible. -À la dernière itération, la distance entre les domaines est plus basse.Notre approche rapproche donc bien les distributions. Cependant, l'algorithme tend à construire un "petit" espace pour les problèmes difficiles. Ceci est probablement dû à la nécessité d'avoir des distributions suffisamment proches, et peut induire une perte de performance. Deux exécutions de DASF. À gauche pour une rotation de 30 • , à droite de 50 • . En ordonnée, nous lisons à gauche le taux d'erreur, à droite le taux de divergence. Nous observons le taux d'erreur sur les échantillons test source et cible des classifieurs h i appris à chaque itération, la divergence dH entre les deux distributions, l'erreur jointe du classifieur inverse et l'erreur sur l'échantillon test cible d'un classifieur SF appris sans AD (comme baseline). Dans les deux cas, dH diminue significativement en comparaison à l'itération 1. L'algorithme stoppe quand l'erreur jointe atteint un minimum, après une diminution continue. Pour l'exemple de rotation 30 • , DASF construit un classifieur d'erreur nulle sur l'échantillon de test cible. Pour le problème 50 • , la difficulté est plus grande et DASF infère un classifieur performant, meilleur que le classifieur SF appris seulement sur les données sources. Notons que l'erreur sur l'échantillon source augmente, ce qui est attendu puisque le but est d'être performant sur le domaine cible.

	Taux d'erreur	Adaptation de domaine parcimonieuse 50 • 60 • * 70 • * 80 • * 90 • * 60.00 47.18 26.12 19.22 17.2 ±1.08 ±2.82 ±3.12 ±0.28 ±0.37 57.85 43.93 39.2 35.93 36.73 ±4.81 ±4.46 ±9.64 ±10.93 ±10.17 70.91 64.72 21.28 18.92 17.49 ±0.88 ±9.10 ±1.26 ±1.10 ±1.12 37 38 35 37 36 ±1.50 ±2.67 ±2.93 ±2.10 ±1.69 99.55 91.03 81.27 65.23 61.95 30 • 40 • 75.99 68.84 ±0.92 ±0.85 81.81 72.55 ±4.62 ±7.60 78.98 74.66 ±2.31 ±2.17 37 37 ±3.77 ±2.66 60.91 59.75 ±1.19 ±3.30 ±4.36 ±6.36 ±4.88 ±2.24 ±2.11 10 9 8 4 4 4 3 ±1.59 ±2.21 ±3.27 ±0.99 ±2.16 ±1.84 ±1.06 1.16 1.31 1.34 1.34 1.32 1.33 1.31 ±0.04 ±0.04 ±0.04 ±0.03 ±0.03 ±0.03 ±0.05 0.66 0.82 0.85 0.39 0.40 0.49 0.45 ±0.11 ±0.13 ±0.11 ±0.15 ±0.05 ±0.12 ±0.09 0.25 0 0.5 0.75 Divergence 0.25 0.5 0.75 1 2 3 4 5 6 0.25 0.5 0.75 Taux d'erreur Divergence Iteration erreur source erreur cible divergence erreur jointe du classifieur inverse erreur cible du classifieur SF FIG. 2: La Fig.2 montre l'exécution de DASF sur deux problèmes d'AD de rota-ANGLE ROTATION 20 • SVM 89.68 ±0.78 SF 92.4 ±3.13 TSVM 100 ±0.00 VS 28 ±1.92 DASF 99.80 ±0.40 RAIS. 10 ±2.32 dH DANS φ R 0 0.58 ±0.04 dH DANS φ R f inal 0.33 ±0.12 0 0.25 0.5 0.75 1 2 3 4 5 6 7 8 9 Iteration erreur source erreur cible divergence erreur jointe du classifieur inverse erreur cible du classifieur SF TAB. 1: CAp 2011
		tions différentes.

  . L'objectif est l'identification d'objets visuels classiques -de concepts visuels -dans des images. Le corpus TrecVid est constitué d'images extraites de vidéos et peut aussi être vu comme corpus d'images. Dans nos expériences, Nous sélectionnons 14 concepts issus de ce corpus et notre but est d'améliorer la performance (en classification) sur l'ensemble de test. Pour ce problème, la divergence entre les ensembles d'apprentissage et de test est faible ( 0.05), les deux domaines étant proches. En CAp 2011 général, pour chaque concept, le ratio d'images positives et négatives (ratio +/-) est inférieur à 10%. Nous proposons une évaluation de la capacité d'AD de DASF lorsque ce ratio +/diffère entre les domaines source et cible, définissant ainsi une tâche d'AD plus difficile. Nous générons un échantillon source par concept, constitué de tous les exemples d'apprentissage positifs et d'exemples négatifs indépendemment tirés tel que le ratio +/soit 1 3 / 2 3 . L'échantillon cible est l'échantillon test originel. Pour les 4 méthodes décrites précédemment, nous apprenons un classifieur binaire associé à chaque concept. Comme le ratio +/est faible sur l'échantillon de test, nous choisissons d'évaluer les performances au sens de la F-mesure et reportons les résultats dans la Tab.2. Remarquons premièrement que TSVM est moins performant, probablement à cause de l'absence supposée d'information sur le domaine cible et donc à l'impossibilité d'estimer le ratio +/-. Correctement paramétré, SVM obtient même de meilleures performances que TSVM sur de nombreuses tâches. DASF est, quant à lui, plus performant en moyenne. Il améliore à chaque fois la performance du classifieur SF et est le plus efficace pour 11 concepts. De plus, il construit toujours des modèles significativement plus parcimonieux. En illustration, la Fig.3 est l'ensemble des points RAIS. sélectionnés pour le le concept PERSON.

	CONC.	BIRD		BOTTLE *	BUS	CAR	CHAIR	CYCLE	COW
	SVM	0.18		0.01	0.16	0.28		0.24		0.10	0.15
	VS	867		587	476	1096		1195		392	681
	SF	0.17		0.11	0.12	0.33		0.21		0.14	0.11
	TSVM	0.14		0.11	0.16	0.37		0.22		0.13	012	MOY.
	SS	814		718	445	631		864		390	888	SVM	0.22
	DASF	0.19		0.12	0.17	0.38		0.26		0.16	0.16	VS	670
	RAIS.	134		78	94	51		229		192	203	SF	0.19
	CONC.	DOG *	MONITOR	PERSON *	PLANE	PLANT		SOFA	TRAIN	TSVM	0.19
	SVM	0.24		0.16	0.56	0.34		0.12		0.16	0.36	VS	710
	VS	436		698	951	428		428		631	510	DASF	0.25
	SF	0.18		0.12	0.48	0.26		0.13		0.13	0.20	RAIS.	127
	TSVM	0.22		0.12	0.44	0.18		0.10		0.15	0.19
	VS	704		861	1111	585		406		866	652
	DASF	0.23		0.14	0.58	0.33		0.14		0.18	0.42
	RAIS.	42		161	6	141		208		167	75
		CONCEPT	CAR *	MONITOR *	PERSON *	PLANE	MOY.
		SVM		0.43	0.19	0.52		0.32		0.36
		SF		0.52	0.34	0.45		0.54		0.46
		TSVM		0.52	0.37	0.46		0.61		0.49
		VS		631	741	1024		259		664
		DASF		0.53	0.37	0.57		0.59		0.52
		RAIS.		50	36	19		81		47

TAB. 2: F-mesure sur les domaines cibles PascalVOC. MOY. est le résultat moyen sur les 14 concepts. nous prenons, comme représentation des images, un descripteur visuel défini par les scores de prédictions sur 15 concepts "intermédiaires" tels que ciel, herbe, peau, etc. Chacun d'entre eux est détecté par un classifieur SVM à partir de moments couleurs et d'orientations de contours sur 260 blocs de 32 × 32 pixels (la dimension est de 3900). Cette représentation a été utilisée avec succès dans de précédentes évaluations TrecVid (Ayache et al., 2007). Nous considérons dans cette section deux expériences. Tout d'abord, le corpus PascalVOC est constitué d'un ensemble de 5000 images d'apprentissage et 5000 de test. FIG. 3: PascalVOC : Les 6 points raisonnables pour le concept PERSON, les 3 premiers sont positifs, les 3 derniers négatifs.

Les auteurs considèrent la divergence sur H∆H, l'espace de la différence symétrique des hypothèses. Voir Ben-David et al. (2010a) pour plus de détails.

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