Jérémie Bigot
email: jeremie.bigot@math.univ-toulouse.fr

Claire Christophe
email: claire.christophe@math.univ-toulouse.fr

Sébastien Gadat
email: sebastien.gadat@math.univ-toulouse.fr

Random action of compact Lie groups and minimax estimation of a mean pattern

Keywords: Grenander's pattern theory, Shape variability, Lie groups, Harmonic analysis, Random action, Mean pattern estimation, Reference template, Deconvolution, Sobolev space, Minimax rate. AMS classifications: Primary 62G08, secondary 43A30

This paper considers the problem of estimating a mean pattern in the setting of Grenander's pattern theory. Shape variability in a data set of curves or images is modeled by the random action of elements in a compact Lie group on an infinite dimensional space. In the case of observations contaminated by an additive Gaussian white noise, it is shown that estimating a reference template in the setting of Grenander's pattern theory falls into the category of deconvolution problems over Lie groups. To obtain this result, we build an estimator of a mean pattern by using Fourier deconvolution and harmonic analysis on compact Lie groups. In an asymptotic setting where the number of observed curves or images tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Sobolev balls. This rate depends on the smoothness of the density of the random Lie group elements representing shape variability in the data, which makes a connection between estimating a mean pattern and standard deconvolution problems in nonparametric statistics.

Introduction

In signal and image processing, data are often in the form of a set of n curves or images Y 1 , . . . , Y n . In many applications, observed curves or images have a similar structure which may lead to the assumption that these observations are random elements which vary around the same mean pattern (also called reference template). However, due to additive noise and shape variability in the data, this mean pattern is typically unknown and has to be estimated. In this setting, a widely used approach is Grenander's pattern theory [START_REF] Grenander | General pattern theory -A mathematical study of regular structures[END_REF][START_REF] Grenander | Pattern Theory: From Representation to Inference[END_REF] which models shape variability by the action of a Lie group on an infinite dimensional space of curves or images. In the last decade, the study of transformation Lie groups to model shape variability of images has been an active research field, and we refer to [START_REF] Trouvé | Local geometry of deformable templates[END_REF][START_REF] Trouvé | Shape spaces[END_REF] for a recent overview of the theory of deformable templates. Currently, there is also a growing interest in statistics on the problem of estimating the mean pattern of a set of curves or images using deformable templates [AAT07, AKT10, BG10, BGL09, [START_REF] Bigot | Estimation of translation, rotation and scaling between noisy images using the fourier mellin transform[END_REF][START_REF] Ma | Bayesian template estimation in computational anatomy[END_REF]. In this paper, we focus on the problem of constructing asymptotically minimax estimators of a mean pattern using noncommutative Lie groups to model shape variability. The main goal of this paper is to show that estimating a reference template in the setting of Grenander's pattern theory falls into the category of deconvolution problems over Lie groups as formulated in [START_REF] Koo | Asymptotic minimax bounds for stochastic deconvolution over groups[END_REF].

To be more precise, let G be a connected and compact Lie group. Let L 2 (G) be the Hilbert space of complex valued, square integrable functions on the group G with respect to the Haar measure dg. We propose to study the nonparametric estimation of a complex valued function f ⋆ : G → C in the following deformable white noise model

dY m (g) = f m (g) dg + ε dW m (g), g ∈ G, m ∈ [[1, n]]
(1.1) where f m (g) = f ⋆ (τ -1 m g). The τ m 's are independent and identically distributed (i.i.d) random variables belonging to G and the W m 's are independent standard Brownian sheets on the topological space G with reference measure dg. For all m = 1, . . . , n, τ m is also supposed to be independent of W m .

In (1.1) the function f ⋆ is the unknown mean pattern to estimate in the asymptotic setting n → +∞, and L 2 (G) represents an infinite dimensional space of curves or images. The τ m 's are random variables acting on L 2 (G) and they model shape variability in he data. The W m model intensity variability in the observed curves or images. In what follows, the random variables τ m are also supposed to have a known density h ∈ L 2 (G). We will show that h plays the role of the kernel a convolution operator that has to be inverted to construct an optimal (in the minimax sense) estimator of f ⋆ . Indeed, since W m has zero expectation, it follows that the expectation of the m-th observation in (1.1) is equal to

Ef m (g) = G f ⋆ (τ -1 g)h(τ ) dτ for any m ∈ [[1, n]].
Therefore, Ef m (g) = f ⋆ * h is the convolution over the group G between the function f ⋆ and the density h. Hence, we propose to build an estimator of f ⋆ using a regularized deconvolution method over Lie groups. This class of inverse problems is based on the use of harmonic analysis and Fourier analysis on compact Lie groups to transform convolution in a product of Fourier coefficients. However, unlike standard Fourier deconvolution on the torus, when G is not a commutative group, the Fourier coefficients of a function in L 2 (G) are no longer complex coefficients but grow in dimension with increasing "frequency". This somewhat complicates both the inversion process and the study of the asymptotic minimax properties of the resulting estimators.

In [START_REF] Bigot | Semiparametric estimation of shifts on compact lie groups for image registration[END_REF], a model similar to (1.1) has been studied where n is held fixed, and the τ m 's are not random but deterministic parameters to be estimated in the asymptotic setting ǫ → 0 using semi-parametric statistics techniques. The potential of using noncommutative harmonic analysis for various applications in engineering is well described in [START_REF] Chirikjian | Engineering applications of noncommutative harmonic analysis[END_REF]. The contribution of this paper is thus part of the growing interest in nonparametric statistics and inverse problems on the use of harmonic analysis on Lie groups [Kim98, KK02, KK08, KR01, LKKK11, PMRC10, Yaz04].

Our construction of an estimator of the mean pattern in (1.1) is inspired by the following problem of stochastic deconvolution over Lie groups introduced in [KK08]: estimate f ⋆ ∈ L 2 (G) from the regression model

y j = G f ⋆ (τ -1 g j )h(τ ) dτ + η j , g j ∈ G, j ∈ [[1, n]] (1.2)
where h is a known convolution kernel, the g j 's are "design points" in G, and the η j 's are independent realizations of a random noise process with zero mean and finite variance. In [START_REF] Koo | Asymptotic minimax bounds for stochastic deconvolution over groups[END_REF] a notion of asymptotic minimaxity over L 2 (G) is introduced, and the authors derive upper and lower bounds for a minimax risk over Sobolev balls. In this paper we also introduce a notion of minimax risk in model (1.1). However, deriving upper and lower bounds of the minimax risk for the estimation of f ⋆ is significantly more difficult in (1.1) than in model (1.2). This is due to the fact that there are two sources of noise in model (1.1): a source of additive Gaussian noise W m which is a classical one for studying minimax properties of an estimator, and a source of shape variability due to the τ m 's which is much more difficult to treat. In particular, standard methods to derive lower bounds of the minimax risk in classical white noise models such as Fano's Lemma are not adapted to the source of shape variability in (1.1). We show that one may use the Assouad's cube technique (see e.g. [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF] and references therein), but it has to be carefully adapted to model (1.1). The paper is organized as follows. In Section 2, we describe the construction of our estimator using a deconvolution step and Fourier analysis on compact Lie groups. We also define a notion of asymptotic optimality in the minimax sense for estimators of the mean pattern. In Section 3, we derive an upper bound on the minimax risk that depends on smoothness assumptions on the density h. A lower bound on the minimax risk is also given. All proofs are gathered in a technical appendix. At the end of the paper, we have also included some technical materials about Fourier analysis on compact Lie groups, along with some formula for the rate of convergence of the eigenvalues of the Laplace-Beltrami operator which are needed to derive our asymptotic rates of convergence.

Mean pattern estimation via deconvolution on Lie groups

In this section, we use various concepts from harmonic analysis on Lie groups which are defined in Appendix B.

Sobolev space in L 2 (G)

Let G be the set of equivalence classes of irreducible representations of G that is identified to the set of unitary representations of each class. For π ∈ G and g ∈ G one has that π(g) ∈ GL dπ ×dπ (C) (the set of d π × d π nonsingular matrices with complex entries) where d π is the dimension of π. By the Peter-Weyl theorem (see Appendix B.2), any function f ∈ L 2 (G) can be decomposed as

f (g) = π∈ G d π Tr (π(g)c π (f )) , (2.1)
where Tr is the trace operator and c π (f ) = G f (g)π(g -1 ) dg is the π-th Fourier coefficient of f (a d π × d π matrix). The decomposition formula (2.1) is an analogue of the usual Fourier analysis in L 2 ([0, 1]) which corresponds to the situation G = R/Z (the torus in dimension 1) for which G = Z, the representations π are the usual trigonometric polynomials π(g) = e i2πℓg for some ℓ ∈ Z (with the bold symbol π denoting the number Pi). In this case, the matrices c π (f ) are one-dimensional (d π = 1) and they equal the standard Fourier coefficients

c π (f ) = c ℓ (f ) = 1 0 f (g)e -i2πℓg dg. For G = R/Z, one thus retrieves the classical Fourier decomposition of a periodic function f : [0, 1] → R as f (g) = ℓ∈Z c ℓ (f )e i2πℓg . Definition 2.1. Let k ∈ N * . Let A ∈ M k×k (C) (
= ||f || 2 L 2 (G) := G |f (g)| 2 dg = π∈ G d π c π (f ) 2 F for any f ∈ L 2 (G).
The following definitions of a Sobolev norm and Sobolev spaces have been proposed in [START_REF] Koo | Asymptotic minimax bounds for stochastic deconvolution over groups[END_REF].

Definition 2.2. Let f ∈ L 2 (G) and s > dim(G)/2. The Sobolev norm of order s of f is defined by

f 2 Hs = G |f (g)| 2 dg + π∈ G λ s π d π Tr c π (f )c π (f ) t = G |f (g)| 2 dg + π∈ G λ s π d π c π (f ) 2 F
, where λ π is the eigenvalue value of π associated to the Laplace-Beltrami operator induced by the Riemannian structure of the Lie group G.

Definition 2.3. Let s > dim(G)/2 and denote by C ∞ (G) the space of infinitely differentiable functions on G. The Sobolev space H s (G) of order s is the completion of C ∞ (G) with respect to the norm • Hs . Let A > 0. The Sobolev ball of radius A and order S in L 2 (G) is defined as

H s (G, A) = f ∈ H s (G) : f 2 Hs ≤ A 2 .
It can be checked that H s (G) corresponds to the usual notion of a Sobolev space in the case G = R/Z. Now, let f ∈ L 2 (G) be an estimator of f ⋆ i.e. a measurable mapping of the random processes Y m , m = 1, . . . , n taking its value in L 2 (G). The quadratic risk of an estimator f is defined as

R( f , f ⋆ ) = E f -f ⋆ 2 = E G | f (g) -f ⋆ (g)| 2 dg .
Definition 2.4. The minimax risk over Sobolev balls associated to model (1.1) is defined as

R n (A, s) = inf f ∈L 2 (G) sup f ⋆ ∈Hs(G,A) R( f , f ⋆ ),
where the above infimum is taken over the set all estimators.

The main goal of this paper is then to derive asymptotic upper and lower bounds on the minimax risk R n (A, s) as n → +∞.

Construction of the estimator

First, note that the white noise model (1.1) has to be interpreted in the following sense: let f ∈ L 2 (G), then conditionally to τ m each integral G f (g) dY m (g) of the "data" dY m (g) is a random variable normally distributed with mean G f (g)f ⋆ (τ -1 m g) dg and variance

ε 2 G |f (g)| 2 dg. Moreover, E G f 1 (g) dW m (g) G f 2 (g) dW m (g) = G f 1 (g)f 2 (g) dg for f 1 , f 2 ∈ L 2 (G) and any m ∈ [[1, n]].
Therefore, using Fourier analysis on compact Lie groups, one may re-write model (1.1) in the Fourier domain as

c π (Y m ) = G π(g -1 ) dY m (g) = c π (f m ) + εc π (W m ), for π ∈ G and m ∈ [[1, n]], (2.2) where c π (f m ) = G f m (g)π(g -1 ) dg and c π (W m ) = G π(g -1 ) dW m (g). Note that c π (f m ) = G f ⋆ (τ -1 m g)π(g -1 ) dg = G f ⋆ (g)π((τ m g) -1
) dg which implies that

c π (f m ) = c π (f ⋆ )π(τ -1 m ), m ∈ [[1, n]].
Remark also that the coefficients (c π (W m )) k,l of the matrix c π (W m ) ∈ M dπ ,dπ (C) are independent complex random variables that are normally distributed with zero expectation and variance d -1 π . Moreover, note that

E π(τ -1 m ) = c π (h) and E (c π (Y m )) = c π (f ⋆ )c π (h).
Therefore, if we assume that c π (h) is an invertible matrix, it follows that an unbiased estimator of the the π-th Fourier coefficient of f ⋆ is given by the following deconvolution step in the Fourier domain

c π (f ⋆ ) = 1 n n m=1 c π (Y m )c π (h) -1 . (2.3)
An estimator of f ⋆ can then be constructed by defining for g ∈ G

f ⋆ T (g) = π∈ G T d π Tr π(g) c π (f ⋆ ) = 1 n n m=1 π∈ G T d π Tr π(g)c π (Y m )c π (h) -1 , (2.4)
where G T = π ∈ G : λ π < T for some T > 0 whose choice has to be discussed (note that the cardinal of G T is finite).

Regularity assumptions on the density h

It is well-known that the difficulty of a deconvolution problem is quantified by the smoothness of the convolution kernel. The rate of convergence that can be expected from any estimator depends on such smoothness assumptions. This issue has been well studied in the nonparametric statistics literature on standard deconvolution problems (see e.g. [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF]). Following the approach proposed in [START_REF] Koo | Asymptotic minimax bounds for stochastic deconvolution over groups[END_REF], we now discuss a smoothness assumption on the convolution kernel h.

Definition 2.5. Let k ∈ N * and |.| 2 be the standard Euclidean norm on C k . The operator norm of A ∈ M k×k (C) is A op = sup u =0 |Au| 2 |u| 2 . Definition 2.6. A function f ∈ L 2 (G) is said to be smooth of order ν ≥ 0 if c π (f ) is an invertible matrix for any π ∈ G, and if there exists two constants C 1 , C 2 > 0 such that c π (f ) -1 2 op ≤ C 1 λ ν π et c π (f ) 2 op ≤ C 2 λ -ν π for all π ∈ G.
Assumption 2.1. The density h is smooth of order ν ≥ 0.

Note that Assumption 2.1 corresponds to the case where, in most applications, the convolution kernel h leads to an inverse problem that is ill-posed, meaning in particular that there is no bounded inverse deconvolution kernel. This can be seen in the assumption c π (f ) -1 2 op ≤ C 1 λ ν π which accounts for the setting where lim λπ→+∞ c π (f ) -1 op = +∞ meaning that the mapping f → f * h does not have a bounded inverse in L 2 (G). Example of such convolution kernels are discussed in [START_REF] Kim | Deconvolution density estimation on compact Lie groups[END_REF][START_REF] Koo | Asymptotic minimax bounds for stochastic deconvolution over groups[END_REF], and we refer to these papers and references therein for specific examples.

Upper and lower bounds

The following theorem gives the asymptotic behavior of the quadratic risk of fT over Sobolev balls using an appropriate choice for the regularization parameter T . Theorem 3.1. Suppose that Assumption 2.1 holds. Let fT be the estimator defined in (2.4) with

T = T n = ⌊n 2 2s+2ν+dim(G) ⌋. Let s > 2ν + dim(G). Then, there exists a constant K 1 > 0 such that lim sup n→∞ sup f ⋆ ∈Hs(G,A) n 2s 2s+2ν+dim(G) R( fTn , f ⋆ ) ≤ K 1 .
Therefore, under Assumption 2.1 on the density h, Theorem 3.1 shows that the quadratic risk R( fTn , f ⋆ ) is of polynomial order of the sample size n, and that this rate deteriorates as the smoothness ν of h increases. The fact that estimating f ⋆ becomes harder with larger of ν (the so-called degree of ill-posedness) is well known in standard deconvolution problems (see e.g. [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF] and references therein). Hence, Theorem 3.1 shows that a similar phenomenon holds in model (1.1) when using the deconvolution step (2.3). The rate of convergence n

- 2s 2s+2ν+dim(G)
corresponds to the minimax rate in model (1.2) for the problem of stochastic deconvolution over Lie groups as described in [START_REF] Koo | Asymptotic minimax bounds for stochastic deconvolution over groups[END_REF].

Then, thanks to the Theorem 3.2 below, there exists a connection between mean pattern estimation in the setting of Grenander's pattern theory [START_REF] Grenander | General pattern theory -A mathematical study of regular structures[END_REF][START_REF] Grenander | Pattern Theory: From Representation to Inference[END_REF] and the analysis of deconvolution problems in nonparametric statistics. Indeed, in the following theorem, we derive an asymptotic lower bound on H s (G, A) for the minimax risk R n (A, s) which shows that the rate of convergence n -2s 2s+2ν+dim(G) cannot be improved. Thus, fTn is an optimal estimator of f ⋆ in the minimax sense.

Theorem 3.2. Suppose that Assumption 2.1 holds. Let s > 2ν + dimG. Then, there exists a constant K 2 > 0 such that lim inf

n→∞ inf f ∈L 2 (G) sup f ⋆ ∈Hs(G,A) n 2s 2s+2ν+dimG R( f , f ⋆ ) ≥ K 2 . A Technical Appendix A.1 Proof of Theorem 3.1
By the classical bias/variance decomposition of the risk one has

R( f ⋆ T , f ⋆ ) = E f ⋆ T -E f ⋆ T 2 + E f ⋆ T -f ⋆ 2 .
Let us first give an upper bound for the bias

E f ⋆ T -f ⋆ 2
. By linearity of the trace operator and by inverting expectation and sum (since Card( G T ) is finite) one obtains that

E f ⋆ T -f ⋆ 2 = π∈ G T d π Tr π(g) 1 n n m=1 E (c π (Y m )) c π (h) -1 -c π (f ⋆ ) - π∈ G\ G T d π Tr [π(g)c π (f ⋆ )] 2 . Since the (c π (Y m )) m 's are i.i.d. random variables and E(c π (Y m )) = E(c π (f m )) = c π (f ⋆ )c π (h) we obtain that E f ⋆ T -f ⋆ 2 = π∈ G\ G T d π Tr [π(g)c π (f ⋆ )] 2 . Then, by Theorem B.2, one has that E f ⋆ T -f ⋆ 2 = π∈ G\ G T d π Tr c π (f ⋆ )c π (f ⋆ ) t .
Finally since π / ∈ G T and f 2 Hs ≤ A 2 we obtain the following upper bound for the bias

E f ⋆ T -f ⋆ 2 ≤ T -s A 2 . (A.1)
Let us now compute an upper bound for the variance term E

f ⋆ T -E f ⋆ T 2 . E f ⋆ T -E f ⋆ T 2 = E   π∈ G T d π Tr π(g) 1 n n m=1 c π (Y m )c π (h) -1 -c π (f ⋆ ) 2   ≤ 2 E   1 n n m=1 π∈ G T d π Tr π(g) c π (f m )c π (h) -1 -c π (f ⋆ ) 2   E 1 + 2 E   ε π∈ G T d π Tr π(g) 1 n n m=1 c π (W m )c π (h) -1 2   E 2 , using that c π (Y m ) = c π (f m ) + εc π (W m ).
Let us first consider the term E 2 . By Theorem B.2 and by decomposing the trace

E 2 = ε 2 E   π∈ G T d π Tr   1 n 2 n m,m ′ =1 c π (W m )c π (h) -1 c π (W ′ m )c π (h) -1 t     = ε 2 E   π∈ G T d π 1 n 2 n m,m ′ =1 dπ k,j=1 c π (W m )c π (h) -1 kj c π (W ′ m )c π (h) -1 kj   = ε 2 E   π∈ G T d π 1 n 2 n m,m ′ =1 dπ k,j=1 dπ i,i ′ =1 (c π (W m )) ki (c π (h) -1 ) ij (c π (W ′ m ) k,i ′ ((c π (h) -1 ) i ′ j   .
By the Fubini-Tonneli theorem, we can invert sum and integral, and since (((c π (W m )) kl ) k,l are i.i.d. Gaussian variables with zero expectation and variance d -1 π , it follows that

E 2 = ε 2 π∈ G T d π 1 n 2 n m=1 dπ k,j,i=1 (c π (h) -1 ) ij (c π (h) -1 ) ij E (c π (W m )) ki ((c π (W m )) k,i = ε 2 n π∈ G T d π dπ k,j,i=1 |(c π (h) -1 ) ij | 2 d -1 π = ε 2 n π∈ G T d π dπ j,i=1 |(c π (h) -1 ) ij | 2
Then thanks to the properties of the operator norm, one has

dπ j=1 |(c π (h) -1 ) ij | 2 ≤ c π (h) -1 2
op , and therefore

E 2 ≤ ε 2 n π∈ G T d 2 π c π (h) -1 2 op . (A.2)
Let us now compute an upper bound for

E 1 . Since c π (f m ) = c π (f ⋆ )π(τ -1 m )
and by Theorem B.2,

E 1 = E   π∈ G T d π Tr π(g) c π (f ⋆ ) 1 n n m=1 π(τ -1 m )c π (h) -1 -c π (f ⋆ ) 2   = E   π∈ G T d π (c π (f ⋆ ) 1 n n m=1 π(τ -1 m )c π (h) -1 -c π (f ⋆ ) 2 F   .
By Fubini-Tonelli theorem, we invert sum and integral, and since the random variables τ m are i.i.d.

E 1 = 1 n π∈ G T d π E c π (f ⋆ )π(τ -1 1 )c π (h) -1 -c π (f ⋆ ) 2 F = 1 n π∈ G T d π E c π (f ⋆ )π(τ -1 1 )c π (h) -1 2 F + c π (f ⋆ ) 2 F -2Tr c π (f ⋆ )π(τ -1 1 )c π (h) -1 c π (f ⋆ ) t ,
where the last equality follows by definition of the Frobenius norm. Now remark that,

E Tr c π (f ⋆ )π(τ -1 1 )c π (h) -1 c π (f ⋆ ) t = Tr c π (f ⋆ )E(π(τ -1 1 ))c π (h) -1 c π (f ⋆ ) t = Tr c π (f ⋆ )c π (h)c π (h) -1 c π (f ⋆ ) t = c π (f ⋆ ) 2 F ,
and let us compute

E c π (f ⋆ )π(τ -1 1 )c π (h) -1 2 F . Recall that P Q F ≤ P F Q op
for any P, Q ∈ M dπ×dπ (C) and that the operator norm is a multiplicative norm, which implies that

E c π (f ⋆ )π(τ -1 1 )c π (h) -1 2 F = E c π (f ⋆ ) 2 F π(τ -1 1 )c π (h) -1 2 op = c π (f ⋆ ) 2 F E π(τ -1 1 ) 2 op c π (h) -1 2 op ,
Since the operator norm is the smallest matrix norm one has that E π(τ -1 1 )

2 op ≤ E π(τ -1 1 ) 2 F . Now since π(τ -1 1 ) 2 F = Tr π(τ -1 1 )π(τ -1 1 ) t = Tr π(τ -1 1 )π(τ 1 ) = Tr [Id dπ ], it follows that E π(h -1 1 )
2 op ≤ d π , and therefore

E 1 ≤ 1 n π∈ G T d π c π (f ⋆ ) 2 F d π c π (h) -1 2 op -1 . (A.3)
Thus, combining the bounds (A.2) and (A.3)

E f ⋆ T -E f ⋆ T 2 ≤ 2 n π∈ G T d 2 π c π (f ⋆ ) 2 F c π (h) -1 2 op - 1 d π + ε 2 c π (h) -1 2 op ≤ 2 n π∈ G T d 2 π c π (h) -1 2 op c π (f ⋆ ) 2 F + ε 2 . Since f ⋆ ∈ H s (G, A), this implies that c π (f ⋆ ) 2 F ≤ M , for some constant M that is independent of π and f ⋆ . Hence c π (f ⋆ ) 2 F + ε 2 ≤ (M + ε 2 ). Assumption 2.1 on the smoothness of h thus implies E f ⋆ T -E f ⋆ T 2 ≤ 2C 1 (M + ε 2 ) n π∈ G T d 2 π λ ν π ≤ 2C 1 (M + ε 2 ) n T ν π∈ G T d 2 π ≤ C n T ν+(dim(G)/2) , (A.4)
where the last inequality follows by Proposition C.1, and C > 0 is some constant that is independent of f ⋆ ∈ H s (G, A). Therefore, combining the bounds (A.1) and (A.25) it follows that

R( f ⋆ T , f ⋆ ) ≤ L(T ) where L(T ) = T -s A 2 + C n T ν+(dim(G)/2) (note that L(T ) does not depend on f ⋆ ∈ H s (G, A)).
Let us now search among the estimators ( f ⋆ T ) T the ones which minimize the upper bound of the quadratic risk. It is clear that the function T → L(T ) has a mini- G) . which completes the proof of Theorem 3.1.

mum at T = ⌊n 2 2s+2ν+dim(G) ⌋ such that L(⌊n 2 2s+2ν+dim(G) ⌋) ≤ A 2 n -2s 2s+2ν+dim(G) + C n n 2ν+dim(G) 2s+2ν+dim(G) ≤ C ′′ n -2s 2s+2ν+dim (

A.2 Proof of Theorem 3.2

To obtain a lower bound, we use an adaptation of the Assouad's cube technique (see e.g. [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF] and references therein) to model (1.1) which differs from the standard white noise models classically studied in nonparametric statistics. Note that for any subset Ω ⊂ H s (G, A)

inf f sup f ⋆ ∈Hs(G,A) R( f , f ⋆ ) ≥ inf f sup f ⋆ ∈Ω R( f , f ⋆ ).
The main idea is to find an appropriate subset Ω of test functions that will allow us to compute an asymptotic lower bound for inf f sup f ⋆ ∈Ω R( f ⋆ , f ⋆ ) and thus the result of Theorem 3.2 will immediately follow by the above inequality.

A.2.1 Choice of a subset Ω of test functions

Let us consider a set Ω of the following form:

Ω = Ω D =    f * w : G → C : ∀g ∈ G, f * w (g) = √ µ D π∈ G D d π dπ k,l w π,kl (π(g)) kl , w π,kl ∈ {-d -1/2 π , d -1/2 π }    ,
where

G D = π ∈ G : D ≤ λ π < 2D and µ D ∈ R + .
To simplify the presentation of the proof, we will write

f w = f * w . Let Ω = π∈ G D {-d -1/2 π , d -1/2 π } d 2 π .
In what follows, the notation w = (w π,kl ) π∈ G D ,1≤k,l≤dπ ∈ Ω is used to denote the set of coefficients w π,kl taking their value in {-d

-1/2 π , d -1/2 π }.
The notation E w will be used to denote expectation with respect to the distribution P w of the random processes Y m , m ∈ [[1, n]] in model (1.1) under the hypothesis that f ⋆ = f w .

Note that any f w ∈ Ω can be written as

f w (g) = √ µ D π∈ G D d π Tr [π(g)w π ]
, where w π = (w π,kl ) 1≤k,l≤dπ . Let |Ω| = Card(Ω) and let us search for a condition on µ D such that Ω ⊂ H s (G, A). Note that c π (f w ) = √ µ D w π which implies

f w ∈ H s (G, A) ⇐⇒ f w 2 Hs ≤ A 2 ⇐⇒ π∈ G D d π Tr √ µ D w π √ µ D w π t + π∈ G D λ s π d π Tr √ µ D w π √ µ D w π t ≤ A 2 ⇐⇒ π∈ G D (1 + λ s π )µ D d 2 π ≤ A 2 ,
using the equality Tr w π w π t = dπ k,l=1 w 2 π,kl = d π which follows from the fact that |w π,kl

| = d -1/2 π . Since π ∈ G D , one has that λ π < 2D, and thus µ D π∈ G D d 2 π ≤ 2 -s D -s A 2 /2 =⇒ π∈ G D (1 + λ s π )µ D d 2 π ≤ A 2 .
Moreover by Proposition C.1 we have that for D sufficiently large, π∈ G D d 2 π ≤ CD dimG/2 , for some constant C > 0, and therefore for such a D, it follows that

µ D ≤ 2 -s D -s-dimG/2 (A 2 /2)C -1 =⇒ µ D π∈ G D d 2 π ≤ 2 -s D -s A 2 /2.
Hence, there exists a sufficiently large D 0 such that for all D ≥ D 0 the condition µ D ≤ KD -s-dimG/2 for some K > 0 (independent of D) implies that Ω ⊂ H s (G, A). In what follows, we thus assume that µ D = κD -s-dimG/2 for some 0 ≤ κ ≤ K and D ≥ D 0 .

A.2.2 Minoration of the quadratic risk over Ω Note that the supremum over Ω of the quadratic risk of any estimator f can be bounded from below as follows. First, remark that by Theorem B.2

sup fw∈Ω R( f , f w ) = sup fw∈Ω E w f -f w 2 ≥ sup w∈ Ω π∈ G D d π dπ k,l=1 E w (c π ( f )) kl - √ µ D w π,kl 2 ≥ 1 | Ω| w∈ Ω π∈ G D d π dπ k,l=1 E w (c π ( f )) kl - √ µ D w π,kl 2 = 1 | Ω| π∈ G D d π dπ k,l=1 w∈ Ω E w (c π ( f )) kl - √ µ D w π,kl 2 (A.5) with |Ω| = 2 π∈ G D d 2 π . Now, define for all π ∈ G D , k, l ∈ [[1, d π ]] the coefficients w * π,kl = argmin v∈ -d -1/2 π ,d -1/2 π (c π ( f )) kl - √ µ D v .
The inequalities

√ µ D w π,kl -w * π,kl ≤ √ µ D w π,kl -(c π ( f )) kl + √ µ D w * π,kl -(c π ( f )) kl ≤ 2 √ µ D w π,kl -(c π ( f )) kl , imply that 1 4 µ D w π,kl -w * π,kl 2 ≤ √ µ D w π,kl -(c π ( f )) kl 2
, and thus by inequality (A.5)

sup fw∈Ω R( f , f ) ≥ µ D 4| Ω| π∈ G D d π dπ k,l=1 w∈ Ω E w w π,kl -w * π,kl 2 ≥ µ D 4| Ω| π∈ G D d π dπ k,l=1 w∈ Ω w π,kl =d -1/2 π E w w π,kl -w * π,kl 2 +E w (π,kl) w (π,kl) π,kl -w * π,kl 2 , (A.6)
where for all

π ′ ∈ G D , k ′ , l ′ ∈ [[1, d π ]], we define w (π,kl) = (w (π,kl) π ′ ,k ′ l ′ ) is such that      w (π,kl) π ′ ,k ′ l ′ = w π ′ ,k ′ l ′ if π ′ = π or (k ′ , l ′ ) = (k, l) w (π,kl) π ′ ,k ′ l ′ = -w π,kl if π ′ = π and (k ′ , l ′ ) = (k, l)
.

Note that the above minoration depends on f . Let us introduce the notation

C π,kl := E w w π,kl -w * π,kl 2 + E w (π,kl) w (π,kl) π,kl -w * π,kl 2 .
In what follows, we show that C π,kl can be bounded from below independently of f .

A.2.3 A lower bound for C π,kl Let π ∈ G D , k, l ∈ [[1, d π ]] be fixed. Denote by X = (c π (Y m )) (π,m)∈ G×[[1,n]
] the data set in the Fourier domain. In what follows, the notation E w,τ is used to denote expectation with respect to the distribution P w,τ of the random processes Y m , m ∈ [[1, n]] in model (1.1) conditionally to τ = (τ 1 , . . . , τ n ) and under the hypothesis that f ⋆ = f w . The notation w = 0 is used to denote the hypothesis f ⋆ = 0 in model (1.1). Therefore, using these notations, one can write that

C π,kl = G n E w,τ w π,kl -w * π,kl 2 + E w,τ w (π,kl) π,kl -w * π,kl 2 h(τ 1 )...h(τ n ) dτ 1 ... dτ n , = G n E 0,τ w π,kl -w * π,kl 2 dP w,τ dP 0,τ (X) + w (π,kl) π,kl -w * π,kl 2 dP w (π,kl) ,τ dP 0,τ (X) h(τ 1 )...h(τ n ) dτ 1 ... dτ n = G n E 0 w π,kl -w * π,kl 2 dP w,τ dP 0 (X) + w (π,kl) π,kl -w * π,kl 2 dP w (π,kl) ,τ dP 0 (X) h(τ 1 )...h(τ n ) dτ 1 ... dτ n ,
where the last equality follows from the fact that, under the hypothesis f ⋆ = 0, the data X in model (1.1) do not depend on τ . By inverting sum and integral, and using Fubini-Tonneli theorem we obtain

C π,kl = E 0 G n w π,kl -w * π,kl 2 dP w,τ dP 0 (X) + w (π,kl) π,kl -w * π,kl 2 dP w (π,kl) ,τ dP 0 (X) h(τ 1 )...h(τ n ) dτ 1 ... dτ n = E 0 w π,kl -w * π,kl 2 G n dP w,τ dP 0 (X)h(τ 1 )...h(τ n ) dτ 1 ... dτ n + w (π,kl) π,kl -w * π,kl 2 G n dP w (π,kl) ,τ dP 0 (X)h(τ 1 )...h(τ n ) dτ 1 ... dτ n .
Introduce the notations

Q(X) = G n dP w,α dP 0 (X)h(α 1 )...h(α n ) dα 1 ... dα n and Q (π,kl) (X) = G n
dP w (π,kl) ,α dP 0 (X)h(α 1 )...h(α n ) dα 1 ... dα n .

Since w (π,kl)

π,kl -w * π,kl = -w π,kl -w * π,kl with w π,kl ∈ -d

-1/2 π , d -1/2 π and w * π,kl ∈ -d -1/2 π , d -1/2 π , it follows that C π,kl ≥ 4d -1 π E 0 min Q(X), Q (π,kl) (X) = 4d -1 π E 0 Q(X) min 1, Q (π,kl) (X) Q(X) = 4d -1 π E 0 G n dP w,τ dP 0 (X)h(τ 1 )...h(τ n ) dτ 1 ... dτ n min 1, Q (π,kl) (X) Q(X) = 4d -1 π G n E 0 dP w,τ dP 0 (X)h(τ 1 )...h(τ n ) dτ 1 ... dτ n min 1, Q (π,kl) (X) Q(X) = 4d -1 π G n E w,τ min 1, Q (π,kl) (X) Q(X) h(τ 1 )...h(τ n ) dτ 1 ... dτ n = 4d -1 π E w min 1, Q (π,kl) (X) Q(X) . (A.7)
Let us now compute a lower bound for E w min 1,

Q (π,kl) (X) Q(X)
. Note that for any 0 < δ < 1,

E w min 1, Q (π,kl) (X) Q(X) ≥ δP w Q (π,kl) (X) Q(X) > δ . (A.8) Proposition A.1. Let π ∈ G D , k, l ∈ [[1, d π ]] be fixed. Let µ D = κD -s-dimG/2 and D = n 2 2s+2ν+dimG
. Suppose that s > 2ν + dimG. Then, there exists 0 < δ < 1 and a constant C > 0 such that lim inf

n→∞ P w Q (π,kl) (X) Q(X) > δ > C.
Proof. Throughout the proof, we assume that µ D = κD -s-dimG/2 and D = n 2 2s+2ν+dimG . To simplify the presentation, we also write E = E w and P = P w . Then, thanks to Proposition C.1, it follows that d 2 π ∼ D (dimG)/2 for λ π ∈ G D , and therefore, under the assumption that s > 2ν + dimG, one obtains the following relations (needed later on in the proof) . In the proof, we also make repeated use of the fact that

nµ 3/2 D d 3 π → 0, nµ 2 D d 4 π → 0, nd 4 π µ 2 D D -ν → 0 as n → +∞, (A.
w π 2 F = d π and wπ 2 F = d π . (A.11) Since c π (Y m ) = √ µ D w π π(τ -1 m )+ c π (W m
) (under the hypothesis that f ⋆ = f w ) and using the fact that wπ 2 F = w π 2 F , simple calculations on the likelihood ratios dPw,α dP 0 (X) and

dP w (π,kl) ,α dP 0 (X) yield that Q (π,kl) (X) Q(X) = n m=1 G exp( Z(1) m + Z(2) m )h(α m ) dα m n m=1 G exp(Z (1) m + Z (2) m )h(α m ) dα m where Z(1) m = d π µ D w π π(τ -1 m ), wπ π(α -1 m ) F , Z(2) m = d π √ µ D c π (W m ), wπ π(α -1 m ) F , Z (1) m = d π µ D w π π(τ -1 m ), w π π(α -1 m ) F , Z (2) m = d π √ µ D c π (W m ), w π π(α -1 m ) F .
Note that by Cauchy-Schwarz's inequality

| Z(1) m | 2 ≤ d 2 π µ 2 D w π π(τ -1 m ) 2 F wπ π(α -1 m ) 2 F = d 2 π µ 2 D w π 2 F wπ 2 F , and
|Z (1) m | 2 ≤ d 2 π µ 2 D w π π(τ -1 m ) 2 F w π π(α -1 m ) 2 F = d 2 π µ 2 D w π 4 F .
Since the coefficients of the matrix c π (W m ) are independent complex Gaussian random variables with zero expectation and variance d -1

π , one has that Z(2) m (resp. Z

m ) is a Gaussian random variable with zero mean and variance

d π µ D wπ π(α -1 m ) 2 F = d π µ D wπ 2 F (resp. d π µ D w π π(α -1 m ) 2 F = d π µ D w π 2 F ). Thence, by (A.11), one obtains that E| Z(1) m | 2 ≤ µ 2 D d 4 π , E|Z (1) m | 2 ≤ µ 2 D d 4 π and E| Z(2) m | 2 = µ D d 2 π , E|Z (2) m | 2 = µ D d 2 π . (A.12)
Therefore, (A.9) and Markov's inequality imply that

| Z(1) m | 2 = o p n -1 , | Z(2) m | 3 = o p n -1 , | Z(1) m Z(2) m | = o p n -1 , (A.13) and |Z (1) m | 2 = o p n -1 , |Z (2) m | 3 = o p n -1 , |Z (1) m Z (2) m | = o p n -1 . (A.14)
Hence, using (A.13), (A.14) and the second order Taylor expansion exp(z

) = 1 + z + z 2 2 + O z 3 it follows that log Q (π,kl) (X) Q(X) = n m=1 log 1 + G Z(1) m + Z(2) m + 1 2 | Z(2) m | 2 h(α m ) dα m + o p n -1 - n m=1 log 1 + G Z (1) m + Z (2) m + 1 2 |Z (2) m | 2 h(α m ) dα m + o p n -1 .
Then, using (A.14) and the second order expansion log(1

+ z) = z -z 2 2 + O z 3 yield log Q (π,kl) (X) Q(X) = n m=1 G Z(1) m + Z(2) m + 1 2 | Z(2) m | 2 h(α m ) dα m - 1 2 G Z(1) m + Z(2) m + 1 2 | Z(2) m | 2 h(α m ) dα m 2 (A.15) - n m=1 G Z (1) m + Z (2) m + 1 2 |Z (2) m | 2 h(α m ) dα m + 1 2 G Z (1) m + Z (2) m + 1 2 |Z (2) m | 2 h(α m ) dα m 2 (A.16) + o p (1) .
Let us now study the expansion of the quadratic term (A.16). Since

c π (h) = G π(τ -1 m )h(α m ) dα m , it follows by Cauchy-Schwarz's inequality that n m=1 G Z (1) m h(α m ) dα m 2 = d 2 π µ 2 D n m=1 w π π(τ -1 m ), w π c π (h) 2 F ≤ nd 2 π µ 2 D w π 2 F w π c π (h) 2 F ≤ nd 2 π µ 2 D w π 4 F c π (h) 2 op ≤ C 2 nd 4 π µ 2 D D -ν = o(1) .
for some constant C 2 > 0, where the last inequality is a consequence of Assumption 2.1, the fact that λ -1 π ≤ D -1 for λ π ∈ G D and the third relation in (A.9). By Jensen's inequality and (A.9) and since the Z

(2) m 's are i.i.d. Gaussian random variables with zero mean and variance µ D d 2 π one obtains that

E n m=1 G |Z (2) m | 2 h(α m ) dα m 2 ≤ n m=1 G E|Z (2) m | 4 h(α m ) dα m ≤ 3nµ 2 D d 4 π = o(1) ,
and thus Markov's inequality implies that n m=1 G |Z

(2)

m | 2 h(α m ) dα m 2 = o p (1)
. Now, using (A.9) and (A.12) it follows that

E n m=1 G Z (1) m h(α m ) dα m G Z (2) m h(α m ) dα m ≤ nµ 3/2 D d 3 π = o(1) , which implies that n m=1 G Z (1) m h(α m ) dα m G Z (2) m h(α m ) dα m = o p (1). Finally, using (A.14), it follows that n m=1 G Z (2) m h(α m ) dα m G |Z (2) m | 2 h(α m ) dα m = o p (1)
. By applying the same arguments to the expansion of the quadratic term (A.15), one finally obtains that log

Q (π,kl) (X) Q(X) = n m=1 G Z(1) m + Z(2) m + 1 2 | Z(2) m | 2 h(α m ) dα m - 1 2 G Z(2) m h(α m ) dα m 2 - n m=1 G Z (1) m + Z (2) m + 1 2 |Z (2) m | 2 h(α m ) dα m + 1 2 G Z (2) m h(α m ) dα m 2 + o p (1) .
Using that wπ

2 F = w π 2
F and the equality

-w π c π (h), ( wπ -w π )c π (h) F - 1 2 w π c π (h) 2 F + 1 2 wπ c π (h) 2 F - 1 2 ( wπ -w π )c π (h) 2 F = 0 one obtains that log Q (π,kl) (X) Q(X) = n m=1 d π µ D w π π(τ -1 m ) -w π c π (h), ( wπ -w π )c π (h) F (A.17) + n m=1 d π √ µ D c π (W m ), ( wπ -w π )c π (h) F (A.18) + n m=1 1 2 G | Z(2) m | 2 h(α m ) dα m - n 2 d π µ D wπ 2 F (A.19) - n m=1 1 2 G |Z (2) m | 2 h(α m ) dα m + n 2 d π µ D w π 2 F (A.20) - n m=1 1 2 G Z(2) m h(α m ) dα m 2 + n 2 d π µ D wπ c π (h) 2 F (A.21) + n m=1 1 2 G Z (2) m h(α m ) dα m 2 - n 2 d π µ D w π c π (h) 2 F (A.22) - n 2 d π µ D ( wπ -w π )c π (h) 2 F + o p (1) . (A.23)
Control of the term (A.23). Thanks to Assumption 2.1 and the fact that λ -1 π ≤ D -1 for λ π ∈ G D and (A.9), it follows by (A.10) that

nd π µ D ( wπ -w π )c π (h) 2 F ≤ nd π µ D c π (h) 2 op wπ -w π 2 F ≤ 4nµ D D -ν = O(1) , (A.24)
and thus the term (A.23) is bounded in probability.

Control of the term (A.17). Remark that (A.9) can be used to prove that

Var n m=1 d π µ D w π π(τ -1 m ), ( wπ -w π )c π (h) F ≤ nd 2 π µ 2 D w π 2 F ( wπ -w π )c π (h) 2 F ≤ nd 2 π µ 2 D w π 2 F wπ -w π 2 F c π (h) 2 op ≤ 4nd 2 π µ 2 D D -ν = o(1) ,
and therefore by Chebyshev's inequality the term (A.17) converges to zero in probability.

Control of the term (A.18). First, since the coefficients of the matrix c π (W m ) are independent complex Gaussian random variables with zero expectation and variance d -1 π , one has that

T m = d π √ µ D c π (W m ), (
n m=1 G | Z(2) m | 2 h(α m ) dα m = n m=1 Var G | Z(2) m | 2 h(α m ) dα m ≤ n m=1 E G | Z(2) m | 2 h(α m ) dα m 2 ≤ n G E| Z(2) 1 | 4 h(α 1 ) dα 1 ≤ 3nµ 2 D d 4 π = o(1) ,
and thus the terms (A.19) and (A.20) converge to zero in probability by Chebyshev's inequality.

Control of the terms (A.21) and (A.22). Similarly, by Jensen's inequality and (A.9) one has that

Var n m=1 G Z(2) m h(α m ) dα m 2 ≤ n G E| Z(2) 1 | 4 h(α 1 ) dα 1 ≤ 3nµ 2 D d 4 π = o(1) ,
and thus the terms (A.21) and (A.22) converge to zero in probability by Chebyshev's inequality.

Combining the above controls of the terms (A.17) to (A.23), one obtains that log

Q (π,kl) (X) Q(X)
is bounded in probability which completes the proof of Proposition A.1. Now, recall that using (A.6) and (A.7)

sup fw∈Ω R( f , f ) ≥ µ D 4| Ω| π∈ G D d π dπ k,l=1 w∈ Ω w π,kl =d -1/2 π C π,kl ≥ µ D 4| Ω| π∈ G D d π dπ k,l=1 w∈ Ω w π,kl =d -1/2 π 4d -1 π E w min 1, Q (π,kl) (X) Q(X)
Combining inequality (A.8) and Proposition A.1 one obtains that there exists a constant C > 0 (not depending on n) such that with the choice D = n 2 2s+2ν+dimG and for all sufficiently large

n sup fw∈Ω R( f , f ) ≥ µ D | Ω| π∈ G D d π dπ k,l=1 w∈ Ω w π,kl =d -1/2 π d -1 π C ≥ C 2 µ D π∈ G D d 2 π ,
where we have the fact that for any π, k, l the cardinality of the set {w ∈ Ω with w π,kl = d

-1/2 π } is | Ω|/2. Now, let 0 < ρ < 1. Thanks to Proposition C.1, it follows that for η = ρW 2 dimG/2 -1 2 dimG/2 +1 , one has that (W + η)D dimG/2 ≥ π∈ G D d 2 π ≥ (W -η)D dimG/2
for all sufficiently large D, where W is the constant defined in (C.1). Hence,

π∈ G D d 2 π = π : λπ <2D d 2 π - π : λπ <D d 2 π ≥ (W -η)(2D) dimG/2 -(W + η)D dimG/2 = W ′ D dimG/2 , with W ′ = (1 -ρ)W (2 dimG/2 -1) > 0. Taking D = n 2 2s+2ν+dimG
and since µ D = κD -s-dimG/2 we finally obtain that

n 2s 2s+2ν+dimG sup fw∈Ω R( f , f ) ≥ n 2s 2s+2ν+dimG KD -s-dimG/2 D dimG/2 = K,
for some constant K > 0 not depending on n, which completes the proof of Theorem 3.2.

B Some background on noncommutative harmonic analysis

In this appendix, some aspects of the theory of the Fourier transform on compact Lie groups are summarized. For detailed introductions to Lie groups and noncommutative harmonic analysis we refer to the books [Bum04, DK00, Sep07]. Throughout the Appendix, it is assumed that G is a connected and compact Lie group.

B.1 Representations

Definition B.1. Let V be a finite-dimensional C-vector space. A representation of G in V is a continuous homomorphism π : G → GL(V ), where GL(V ) is the set of automorphisms of V . The representation π is said to be irreducible if, for any g ∈ G, the only invariant subspaces by the automorphism π(g) are {0} and V .

If G is a compact group and π is an irreducible representation in V , then the vector space V is finite dimensional, and we denote by d π the dimension of V . By choosing a basis for V , it is often convenient to identify π(g) with a matrix of size d π × d π with complex entries. Definition B.2. Two representations π, π ′ in V are called equivalent if there exists M ∈ GL(V ) such that π(g) = M π ′ (g)M -1 for all g ∈ G.

Definition B.3. A representation π is said to be unitary if π(g) is a unitary operator for every g ∈ G.

Let π be a representation in V . Then, there exists an inner product on V such that π is unitary. This means that any irreducible representation π in V is equivalent to an irreducible representation that is unitary.

B.2 Peter-Weyl theorem

Let π ∈ G be a representation in a Hilbert space V . Let B π = (e 1 , ..., e dπ ) a basis of V . For g ∈ G, denote by φ π ij (g) = e i , π(g)e j the coordinates of π in the basis

B π for i, j ∈ [[1, d π ]]. Theorem B.1. If G is a compact group then √ d π φ π ij (.) π∈ G, i,j∈[[1,dπ]]
is an orthonormal basis of the Hilbert space L 2 (G) endowed with the inner product f, h = G f (g)h(g)dg.

B.3 Fourier transform and convolution in L 2 (G)

Let π ∈ G and define for any f ∈ L 2 (G) the linear mapping

c π (f ) : V → V v → G f (g)π(g) T v dg = G f (g)π(g -1 )v dg.
Note that the matrix c π (f ) is the generalization to functions in L 2 (G) of the usual notion of Fourier coefficients.

Definition B.5. Let f ∈ L 2 (G) and π ∈ G. We call c π (f ) the π-th Fourier coefficient of f . Theorem B.2. Let f ∈ L 2 (G). Then f (g) = π∈ G d π Tr (π(g)c π (f )) , and ||f || 2 L 2 (G) = π∈ G d π Tr c π (f )c π (f ) t = π∈ G d π c π (f ) 2 F
, where • F denotes the Frobenius norm of a matrix. Definition B.6. Let f, h ∈ L 2 (G). The convolution of f and h is defined as the function

(f * h)(g) = G f (g ′-1 g)h(g ′ ) dg ′ for g ∈ G. Proposition B.2. Let f, h ∈ L 2 (G) then c π (f * h) = c π (f )c π (h).

C Laplace-Beltrami operator on a compact Lie group

For further details on the material presented in this section we refer to the technical appendix in [START_REF] Koo | Asymptotic minimax bounds for stochastic deconvolution over groups[END_REF] and to the book [START_REF] Faraut | Analysis on Lie groups[END_REF]. In this section, we still assume that G is a connected and compact Lie group. In what follows, with no loss of generality, we identify (through an isomorphism) G to a subgroup of GL r×r (C) (the set of r × r nonsingular matrices with complex entries) for some integer r > 0. where ad(X) : g → g : Y → [X, Y ] is an endomorphism of g.

C.1 Lie algebra

C.2 Roots of a Lie algebra

A torus in G is a connected Abelian subgroup of G. It is well known that in a compact Lie group G, there exists (up to an isomorphism) a maximal torus. Let us fix such a maximal torus that we denote by T. Denote by t the Lie algebra of T, which is a maximal Abelian subalgebra of g. Let h = t + it be the complexification of t. Then, h is a maximal Abelian subalgebra of g such that the linear transformations (ad(H)) H∈h are simultaneously diagonalizable. Denote by h * the dual space of h. Let α ∈ h * , and define

g α = {X ∈ g : ∀H ∈ h, [H, X] = α(H)X} .
Definition C.4. α ∈ h * is said to be a root of g with respect to h, if g α is nonzero, and in this case g α is called the corresponding root space. We also denote by Φ ⊂ h * the set of roots.

Each root space is of dimension 1. One has that g 0 = h (by the maximal property of h) and g can be decomposed as the following direct sum g = h α∈ Φ g α , called the root space decomposition of g. To each α ∈ Φ we associate the hyperplane H α ⊂ h * that is orthogonal to α. The set of all hyperplanes

C.3 Laplace-Beltrami operator

The Laplace-Beltrami operator is a generalization to Riemannian manifolds (such as Lie groups) of the usual Laplacian operator. We will denote this operator by ∆. To state the following proposition, note that one may identify the set G with a subset of Φ + (see the technical appendix in [START_REF] Koo | Asymptotic minimax bounds for stochastic deconvolution over groups[END_REF] for further details on this identification).

  9) and nµ D D -ν = O(1) as n → +∞. (A.10) Without loss of generality, we consider the case where w π,kl = -d . To simplify the presentation, we also introduce the notation wπ = w (π,kl) π

Definition B. 4 .

 4 We denote by G the set of equivalence classes of irreducible representations of G, and we identify G to the set of unitary representations of each class.Proposition B.1. Let g ∈ G and π ∈ G, then π(g -1 ) = π(g) t .

  Definition C.1. A one parameter subgroup of G is a group homomorphism c : R → G. Theorem C.1. Let c : R → GL r×r (C) one parameter subgroup of GL r×r (C). Then c is C ∞ and c(t) = exp(tA), with A = dc dt (0). Definition C.2. Let M r×r (C) be the set of r × r matrices with complex entries. The mapping [., .] : M r×r (C) 2 → M r×r (C) : X, Y → [X, Y ] = XY -Y X is called a Lie bracket. A Lie algebra is the C-vector space g = {X ∈ M r×r (C) : exp(tX) ∈ G ∀t ∈ R} endowed with the bilinear form [., .] : g × g → g : X, Y → [X, Y ], which satisfies [X, Y ] = -[Y, X] and [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0 (Jacobi identity). Definition C.3. The Killing form is the bilinear form B defined by B : g → C : X, Y → Tr [ad(X)ad(Y )] ,

  H α : α ∈ Φ partition h * into a finite number of open convex regions called the Weyl chambers of h * . In what follows, we choose and fix a fundamental Weyl chamber denoted by K. Definition C.5. Let Φ be the set of real roots and Φ + = {α ∈ Φ : ∀β ∈ K α, β } be the set of positive roots. Denote one-half of the sum of positive roots by ρ = 1 2 α∈Φ + α.

Proposition C. 1 .

 1 The elements of G are the eigenfunctions of ∆. Let π ∈ G. The eigenvalue of π is λ π = π + ρ 2 -ρ 2 , where • is the norm induced by the Killing form. For π ∈ G, one has the following relationship between d π and λ π π∈ G:λπ<Td 2 π = W T (dimG)/2 + o(T (dimG)/2) as T → ∞, the volume of G, the bold symbol π denoting the number Pi and Γ(.) being the classical gamma function.

  the set of k × k matrices with complex entries).

	The Frobenius norm of A is defined by A 2 F = Tr AA t . It is the norm induced by the
	inner product A, B F = Tr (AB	t ) of two matrices A, B ∈ M k×k (C).
	By Parseval's relation, it follows that ||f || 2

  wπ -w π )c π (h) F are i.i.d. Gaussian random variables with zero mean and variance d π µ D ( wπ -w π )c π (h) 2 F . Using inequality (A.24) it follows that

			n
		Var	m=1	T m = nd π µ D ( wπ -w π )c π (h) 2 F = O(1) ,		(A.25)
	and standard arguments in concentration of Gaussian variables imply that for any t > 0
	P	n m=1	T m ≥ t ≤ 2 exp -	t 2 2nd π µ D ( wπ -w π )c π (h) 2 F	.	(A.26)
	Therefore, combining (A.25) and (A.26) imply that the term (A.18) is bounded in probability.
	Control of the terms (A.19) and (A.20). Remark that Jensen's inequality, the fact that the
	Z(2) m 's are i.i.d. Gaussian random variables with zero mean and variance µ D d 2 π and (A.9) imply
	that
	Var

Acknowledgements

The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-JCJC-SIMI1 DEMOS.


Jérémie Bigot

Jean-Michel Loubes

Myriam Vimond

Semiparametric estimation of rigid transformations on compact Lie groups

We study a simple model for the estimation of rigid transformations between noisy images. The transformations are supposed to belong to a compact Lie group, and a new matching criteria based on the Fourier transform is proposed. Consistency and asymptotic normality of the resulting estimators are studied. Some simulations are used to illustrate the methodology, and we describe potential applications of this approach to various image registration problems.

Introduction

Originating in Grenander's pattern theory, transformation Lie groups are commonly used to model the deformations of images. The study of the properties and intrinsic geometries of such deformation groups is now an active field of research (see e.g. [START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF], [START_REF] Klassen | Analysis of planar shapes using geodesic paths on shape spaces[END_REF], [START_REF] Marsland | Constructing diffeomorphic representations for the groupwise analysis of non-rigid registrations of medical images[END_REF], [START_REF] Michor | Riemannian geometries on spaces of planar curves[END_REF], [START_REF] Pennec | Statistical computing on non-linear spaces for computational anatomy[END_REF], [START_REF] Trouvé | Metamorphoses through lie group action[END_REF]). In this setting, an important problem is the estimation of the deformations that may exist between similar images in the presence of additive noise. Rigid displacements such as translations, rotations or affine transformations can be modeled by finite dimensional Lie groups. Hence, the estimation of rigid deformation parameters can be formulated as a semi-parametric estimation problem which is an important field of research in mathematical statistics [START_REF] Bickel | Efficient and Adaptive Estimation for Semiparametric Models[END_REF]. Indeed, semiparametric modeling is concerned with statistical problems where the parameters of interest are finite dimensional but where their observation is blurred by an infinite dimensional parameter. Here, the finite-dimensional parameters are the Lie group elements, and the infinite-dimensional parameter is an unknown 2D or 3D image, which is warped to obtain the different deformations. This image, that has to be recovered, is often called a template. The semiparametric framework provides optimal recovery of the warping parameters, contrary to nonparametric methods, leading to a better estimation of the template obtained by aligning the observed images.

As Lie groups are typically nonlinear spaces, an important question is the development of information geometry tools to extend classical notions such as asymptotic normality and efficiency, or the Cramer-Rao bound originally proposed for parameters lying in an Euclidean space. In the context of parametric statistics, several generalizations of these concepts to arbitrary manifolds have been proposed (see e.g. [START_REF] Smith | Covariance, subspace, and intrinsic Cramér-Rao bounds[END_REF]). However, in the more general situation of semiparametric models, there is not so much work dealing with the estimation of parameters lying in a Lie group.

A first attempt in this direction has been proposed in [START_REF] Gamboa | Semi-parametric estimation of shifts[END_REF], [START_REF] Vimond | Efficient estimation in homothetic shifted in regression models[END_REF] for the simple problem of recovering shifts between one-dimensional noisy curves observed on an interval. The goal of this paper is extend such an approach to the more general case where the shift parameters belong to a compact Lie group. Thanks to a general shift property of the Fourier transform on compact Lie groups, a matching criterion based on the Fourier transform of the data can be defined, and we study some statistical properties of the resulting estimators.

In Section 2, a simple model for shifts on groups is introduced. Some properties on the Fourier transform are briefly reviewed (see [START_REF] Sepanski | Compact Lie groups[END_REF] for further details), and are then used to define a general matching criterion on compact Lie groups. In Section 3, the consistency and the asymptotic normality of the estimators are established. Some numerical simulations are presented in Section 4, and extensions of the model are given in Section 5 for the registration of spherical images. Finally, we give some perspectives for future work. Note that the proofs of our theorems are based on the theory of M-estimation (see e.g. [START_REF] Van Der Vaart | Asymptotic statistics[END_REF]), but are quite long and technical, and are thus not given in this manuscript but can be found in [START_REF] Bigot | Semiparametric Estimation of Shifts on Lie Groups for Image Registration[END_REF].

2 A shift model on Lie groups and a matching criteria based on the Fourier transform

The Fourier transform on compact Lie groups

Let G be a compact Lie group. Denote by e the identity element, and by hg the binary operation between two elements h, g ∈ G. Let L 2 (G) be the Hilbert space of complexed valued, square integrable functions on the group G with respect to the Haar measure dg.

To define a Fourier transform on L 2 (G), a fundamental tool is the theory of group representations, which aims at studying the properties of groups via their representations as linear transformation of vector spaces. More precisely, a representation is an homomorphism from the group G to the automorphism group of a vector space. So let V be a finite-dimensional vector space, a representation of G in V is thus defined as a continuous homomorphism π : G → GL(V ), where GL(V ) denotes the set of automorphisms of V. Hence it provides a linear transformation which depends on the vector space on which the group acts.

Every irreducible representation π of a compact group G in a vector space V is finite dimensional, so we denote by d π the dimension of V . By choosing a basis for V , it is often convenient to identify π(g) with a matrix of size d π × d π with complex entries. The function g → Trπ(g) is called the character of π. The characters carry the essential information about the group representation. Moreover, the fundamental theorem of Schur orthogonality states that the characters form an orthonormal system in L 2 (G) when π ranges over the dual set Ĝ of irreducible and equivalent representations of G. In the case of compact groups, the dual Ĝ is a countable set, and the Peter-Weyl Theorem states that the characters are dense in L 2 (G). Indeed, if π is a finite dimensional representation of G in the vector space V, then one can define, for every f ∈ L 2 (G), the linear mapping π(f

) : V → V by π(f )v = G f (g)π(g) T vdg, for v ∈ V.
The matrix π(f ) is the generalization to the case of compact group of the usual notion of Fourier coefficient. Then, Peter-Weyl Theorem implies that (for simplicity the same notation is used for π and its equivalence class in Ĝ)

f (g) = π∈ Ĝ d π Tr (π(g)π(f )) and f 2 L 2 (G) = π∈ Ĝ d π Tr π(f )π(f ) T (1)
In the sequel, we will also denote by A, B HS = Tr (A T B) the Hilbert-Schmidt inner product between two finite dimensional d π × d π matrices A and B.

A simple shift model on groups

Let X be a subset of R d (with d = 2, 3 in our applications) and G be a compact Lie group acting on X . A general deformation model for a set of J noisy signals would be

Y j (x) = f * (g j -1 • x) + W j (x) for x ∈ X , (2)
where f * : X → R is an unknown template, W j (x) is some additive noise, and g j are the deformations to estimate. Our interest is to provide a very general framework for image registration under warping effect given by model [START_REF] Bigot | Semiparametric Estimation of Shifts on Lie Groups for Image Registration[END_REF], and thus to deal with the general problem of estimation of the elements of a Lie Group G acting on a space X . In this paper, a simplest model for which X = G is studied to give the main ideas of our approach. An example for which X = G is given in Section 5 to show that our approach can be extended to more complex situations, and we discuss possible extensions to more complex situations in the concluding section. Now, consider the following white noise model : for j = 1, . . . , J and g ∈ G dY j (g) = f j (g)dg + ǫdW j (g),

where

f j (g) = f * (h * j -1 g).
The function f * : G → R is the unknown common shape of the observed images Y j . The parameter h * j , j = 1, . . . , J are the unknown shift parameters that we wish to estimate. W j , j = 1, . . . , J are independent standard Brownian sheets on the topological space G with measure dg, ǫ is an unknown noise level parameter which will tend to zero in our asymptotic considerations. Note that the white noise model ( 3) is a continuous model which is a very useful tool for the theoretical study of statistical problem in image analysis. In practice, the noisy images Y j are typically discretely sampled on a regular grid, but the model (3) can be shown to lead to comparable asymptotic theory in a sampled data model [START_REF] Brown | Asymptotic equivalence of nonparametric regression and white noise[END_REF].

Obviously, without any further restriction on the set of possible shifts, the model ( 3) is not identifiable. Indeed, if s is an element of G with s = e, then one can replace the h * j 's in equation ( 3) by hj = h * j s and f * by f (g) = f * (sg) without changing the formulation of the model. To ensure identification, we further assume that the set of parameters G J is reduced to the subset

A 0 ⊂ A such that A 0 = {(h 1 , . . . , h J ) ∈ A, h 1 = e}. (4)
This choice implies that the first image will be a reference and all the other images will be warped onto this particular choice. Now, remark that since π(g)

T = π(g -1 ), one has that π(f j ) = π(f )π(h * j -1
) for all j = 1, . . . , J. This equality is classically referred to as the shift property of the Fourier transform, and is at the heart of our estimation procedure to exhibit the shift parameters h * j .

A matching criterion based on the Fourier transform

For h = (h 1 , . . . , h J ) ∈ A 0 , we propose to minimize the following criterion inspired by recent results in [START_REF] Gamboa | Semi-parametric estimation of shifts[END_REF] and [START_REF] Vimond | Efficient estimation in homothetic shifted in regression models[END_REF] for the estimation of shifts between curves:

M (h 1 , . . . , h J ) = 1 J J j=1 f j • T hj - 1 J J j ′ =1 f j ′ • T h j ′ 2 L 2 (G) , (5)
where

T h : g ∈ G → hg ∈ G.
Using the Parseval-Plancherel formula, the criterion may be rewritten in the Fourier domain as:

M (h) = M (h 1 , . . . , h J ) = 1 J J j=1 π∈ Ĝ d π π(f j )π(h j ) - 1 J J j ′ =1 π(f j ′ )π(h j ′ ) 2 HS , (6)
for h = (h 1 , . . . , h J ) ∈ A 0 . Given that π(f j ) = π(f * )π(h * j -1 ), the criterion M has a minimum at h * = (h * 1 , . . . , h * J ) such that M (h * ) = 0.

The empirical criterion

Our estimation method is then based on the Fourier Transform of the noisy data given by model [START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF]. Let π an irreducible representation of G into V . We consider the following linear mappings from V to V which are defined from the model (3):

π(Y j ) = G π(g -1 )dY j (g) = π(f j ) + ǫπ(W j ), j = 1 . . . J,
where π(W j ) = G π(g -1 )dW j (g), j = 1 . . . J. Let us denote by (π kl (W j )) the matrix coefficients of π(W j ) : π kl (W j ) = G π kl (g -1 )dW j (g). Using the Schur orthogonality and the fact that W j is a standard Brownian sheet on G, one obtains that the complex variables π kl (W j ) are independent identically distributed Gaussian variables N C (0, d -1 π ). Let Ĝǫ be a finite subset of Ĝ such that the sequence Ĝǫ increases when ǫ tends to 0 and ∪ ǫ>0 Ĝǫ = Ĝ. Practical choices for the set Ĝǫ will be discussed later on for the case of Abelian groups and the non-commutative group SO [START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF]. Then, we consider the following criterion:

M ǫ (h 1 , . . . , h J ) = 1 J J j=1 π∈ Ĝǫ d π π(Y j )π(h j ) - 1 J J j ′ =1 π(Y j ′ )π(h j ′ ), 2 HS (7)
and our M-estimator is given by ĥǫ = arg min h∈A0 M ǫ (h).

3 Consistency and asymptotic normality of the M-estimator

Theorem 1. Assume that f * ∈ L 2 (G) is such that there does not exist a closed subgroup H (except H = {e} or H = G) such that f (gh) = f (g) for all g ∈ G and h ∈ H. Moreover, suppose that for all π ∈ Ĝ such that π(f * ) is not identically null, then π(f * ) is invertible.
Then, M has a unique minimum at h * , and if

lim ǫ→0 ǫ 2 π∈ Ĝǫ d 2 π = 0, then ĥǫ converges in probability to h * = (h * 1 , . . . , h * J ).
This Theorem shows the consistency of the estimators of the parameters h * when the noise level goes to zero (recall that in the discretized model, this is equivalent to the fact that the number of observations increase). We stress that this result is obtained by minimizing a quadratic contrast without prior knowledge of the main pattern, thanks to the empirical criterion [START_REF] Deguchi | 3D fundus pattern reconstruction and display from multiple fundusimages[END_REF] which enables to get rid of f * . Asymptotic normality of these estimators and thus rates of convergence will provided in Theorem 2. The first condition of Theorem 1 ensures identifiability of the model since a function which would not satisfy this condition would be shift invariant, preventing any estimation. The second condition can be viewed as a smoothness condition which ensures unicity of the minimization scheme. In the following, we will denote by ℜ(z) the real part of any complex number z.

As the estimator ĥǫ takes its values in a Lie group, it is not that obvious to define a notion of asymptotic normality as the space A 0 is typically not a linear space if the group G is not Abelian. To overcome this, a classical approach is to use the exponential map to "project" ĥǫ into the Lie algebra G of G. If we can write ĥǫ = exp(û ǫ ), then one can study the asymptotic normality of ûǫ which belongs to the linear space G supposed to be of finite dimension p. For this, we first re-express the criterion M ǫ defined on G J as a function Mǫ defined on G J . If G is a compact group, then the exponential map is surjective, but it is not necessarily injective. Hence, define U to be a compact neighborhood of 0 ∈ G J onto which the exponential map is a smooth diffeomorphism, and let A = exp(U). For u ∈ U J , we re-express our criterion as M (u 1 , . . . , u J ) = M (exp(u 1 ), . . . , exp(u J )), and Mǫ (u 1 , . . . , u J ) = M ǫ (exp(u 1 ), . . . , exp(u J )). Then, define ûǫ = (û 1 , . . . , ûJ ) = arg min u∈U1 Mǫ (u 1 , . . . , u J ),

where U 1 = (u 1 , . . . , u J ) ∈ U J , u 1 = 0 . Suppose that h * 1 = exp(u * 1 ), . . . , h * J = exp(u * J ), with u * = (u 1 , . . . , u J ) ∈ U J , then the following result holds:

Theorem 2. Assume that the conditions of Theorem 1 hold. Moreover, assume that for all j = 2, . . . , J and k = 1, . . . , p

lim ǫ→0 ǫ π∈ Ĝǫ d 2 π d e π d u * j exp(x k ) 2 HS = 0, (8)
and that

lim ǫ→0 sup u1∈U    π∈ Ĝ\ Ĝǫ d π π(f * ) 2 HS d e π d u1 exp(x k1 ) 2 HS    = 0 (9) lim ǫ→0 sup u1∈U    π∈ Ĝ\ Ĝǫ d π π(f * ) 2 HS ∇ x k 2 u1 d e π d u1 exp(x k1 ) HS    = 0 (10) lim ǫ→0 ǫ 2 sup u1∈U    π∈ Ĝǫ d 2 π d e π d u1 exp(x k1 ) 2 HS    = 0 (11) lim ǫ→0 ǫ 2 sup u1∈U    π∈ Ĝǫ d 2 π ∇ x k 2 u1 d e π d u1 exp(x k1 ) HS    = 0, ( 12
)
where x 1 , . . . , x p is an arbitrary basis of G. Then,

ǫ -1 (û ǫ -u * ) → N (0, H -1 ΣH -1 ), as ǫ → 0,
where Σ is a positive definite (J -1)p × (J -1)p matrix whose entries for 2 ≤ j 1 , j 2 ≤ J and 1 ≤ k 1 , k 2 ≤ p are given by

Σ (j1,k1)×(j1,k1) = 4 J 2 π∈ Ĝ d π 1 - 1 J π(f * )d e π d u * j 1 exp(x k1 ) 2 HS Σ (j1,k1)×(j1,k2) = 4 J 2 π∈ Ĝ d π 1 - 1 J ℜ π(f * )d e π d u * j 1 exp(x k1 ) , π(f * )d e π d u * j 1 exp(x k2 ) HS ,
and for j 1 = j 2 by

Σ (j1,k1)×(j2,k2) = - 4 J 2 π∈ Ĝ d π 1 J ℜ π(f * )d e π d u * j 1 exp(x k1 ) , π(f * )d e π d u * j 2 exp(x k2 ) HS ,
and where H is a positive definite (J -1)p × (J -1)p matrix whose entries for 2 ≤ j 1 , j 2 ≤ J and 1 ≤ k 1 , k 2 ≤ p are given by

H (j1,k1)×(j1,k2) = - 2 J π∈ Ĝ d π ℜ π(f * )d e π d u * j 1 exp(x k2 ) d e π d u * j 1 exp(x k1 ) + π(f * )∇ x k 2 u * j 1 d e π d u * j 1 exp(x k1 ) , π(f * ) HS + 1 J π(f * )d e π d u * j 1 exp(x k1 ) , π(f * )d e π d u * j 1 exp(x k2 ) HS ,
and for j 1 = j 2 by

H (j1,k1)×(j2,k2) = - 2 J 2 π∈ Ĝ d π ℜ π(f * )d e π d u * j 1 exp(x k1 ) , π(f * )d e π d u * j 2 exp(x k2 ) HS .
The convergence result in the above theorem must be understood for the vector ûǫ = (û 2 , . . . , ûJ ) ∈ G J-1 since the first component is fixed to û1 = 0. Moreover, the notation d e π stands for the differential of π at e, while d u1 exp(x k1 ) corresponds to the differential of the exponential at u 1 in the direction x k1 . We point out that the estimator converges at the parametric rate of convergence, and thus optimal rate of convergence, which would not have been the case if we had considered a preliminar estimate of f * . This is one of the main achievements of the semiparametric type methodology proposed in this paper. Proving the optimality up to the constants imply studying the semiparametric efficiency of the estimators and falls beyond the scope of this paper. Some intuitions about such a result is provided in Section 6.

Abelian groups : the special case of the torus

The assumptions of Theorem 2 are rather technical and difficult to state in the very general case. However, for Abelian groups their statement is much simpler, which is due to the fact that the mapping d exp u : G → G reduces to the identity on G i.e. d u exp(v) = v for all u ∈ G and v ∈ G. The assumptions can be rewritten as

lim ǫ→0 ǫ 2 #{ Ĝǫ } = 0, lim ǫ→0 ǫ π∈ Ĝǫ |d e π(x k )| 2 = 0, lim ǫ→0 π∈ Ĝ\ Ĝǫ |π(f * )| 2 d e π(x k ) 2 = 0, (13)
where x 1 , . . . , x p is an arbitrary basis of G. Thus, Condition (??) states that the common shape is differentiable and its derivatives are square integrable on G. Conditions [START_REF] Michor | Riemannian geometries on spaces of planar curves[END_REF] give some assumptions on the choice of Ĝǫ .

As an illustrative example, let us consider the case where G = (R/Z) p which corresponds to the classical multi-dimensional Fourier decomposition of a func-

tion f ∈ L 2 ([0, 1] p ) f (x) = ℓ∈Z p c ℓ (f )e ℓ (x), for x = (x 1 , . . . , x p ) ∈ [0, 1] d and ℓ = x = (ℓ 1 , . . . , ℓ p ) ∈ Z d , where e ℓ (x) = π(x) = e -i2π( P p k=1 ℓ k x k ) and c ℓ (x) = π(f ) = [0,1] d f (x)e ℓ (x)dx.
Note that also that d e π(x k ) = -i2πℓ k . Now, take Ĝǫ = {(ℓ 1 , . . . , ℓ p ) ∈ Z p , |ℓ k | ≤ ℓ ǫ for all k = 1, . . . , p}, for some positive integer ℓ ǫ . Then, the following corollary holds:

Corollary 1. Let G = (R/Z) p and f * ∈ L 2 ([0, 1] p ) be a periodic function satisfying the conditions of Theorem 1. Assume that h * ∈ G J or equivalently that u * ∈ ([0, 1] p ) J . If

ǫℓ p+2 ǫ = o(1) and (ℓ1,...,ℓp)∈Z p |ℓ 1 | 2 + . . . + |ℓ p | 2 |c ℓ (f * )| 2 < ∞, then ǫ -1 (û ǫ -u * ) → N (0, Γ -1
), as ǫ → 0, where the matrix Γ is given by

Γ (j1,k1)×(j1,k2) = ℓ∈Z 1 - 1 J |c ℓ (f * )| 2 (2π) 2 ℓ k1 ℓ k2 , Γ (j1,k1)×(j2,k2) = - 1 J ℓ∈Z |c ℓ (f * )| 2 (2π) 2 ℓ k1 ℓ k2 for j 1 = j 2 ,
4 Numerical simulations and some illustrative examples

A general gradient descent algorithm

To compute the estimator ĥǫ one has to minimize the function M ǫ (h). As this criterion is defined on a Lie group, a direct numerical optimization is generally not feasible. Finding minima of functions defined on a Lie group is generally done by reformulating the problem as an optimization problem on its Lie algebra. Since the expression of the gradient of Mǫ (u) is available in a closed form, the following gradient descent algorithm with an adaptive step can be easily implemented: Step 3 : if M (m) -M (m + 1) ≥ ρ(M (1) -M (m + 1)) then return to Step 2, else stop the iterations, and take ĥǫ = exp(u m+1 ).

Initialization : let u 0 = 0 ∈ G J , γ 0 = 1 ∇ u 0 Mǫ , M (0) = Mǫ (u 0 ),
In the above algorithm, ρ > 0 is a small stopping parameter and κ > 1 is a parameter to control the choice of the adaptive step γ m . Note that to satisfy the identifiability constraints the first p components of u m are held fixed to zero at each iteration m.

Registration of translated 2D images

As an illustrative example, we consider the registration of translated 2D images (see [START_REF] Glasbey | A penalized likelihood approach to image warping[END_REF] for a related work in a similar setting for 2D images). The chosen template f * is the Shepp-Logan phantom image [START_REF] Anil | Fundamentals of digital image processing[END_REF] of size N ×N with N = 100 (see Figure 1). Noisy images can be generated by translating this image and adding Gaussian noise to each pixel value:

Y j (i 1 , i 2 ) = f * ( i 1 N -h 1 j , i 2 N -h 2 j ) + σz j (i 1 , i 2 ), 1 ≤ i 1 , i 2 ≤ N, j = 1, . . . , J (14)
where i 1 , i 2 denotes a pixel position in the image, z j (i 1 , i 2 ) ∼ i.i.d. N (0, 1), σ is the level of noise, and h 1 j , h 2 j ∈ [0, 1] are the unknown translation parameters to estimate. One could argue that the sampled data model ( 14) does not truly correspond to the white noise model (3). However, as previously explained there exists a correspondence between these two models in the sense that they are asymptotically equivalent if ǫ = σ N (see [START_REF] Brown | Asymptotic equivalence of nonparametric regression and white noise[END_REF]). We have repeated M = 100 simulations with J = 6 noisy images simulated from the model [START_REF] Pennec | Statistical computing on non-linear spaces for computational anatomy[END_REF]. The various values taken for the translation parameters are the bold numbers given in Table 1. A typical example of a simulation run is shown in Figure 1. Here, G = [0, 1] × [0, 1] and its Lie algebra is G = R 2 . The criterion Mǫ (u) can be easily implemented via the use of the fast Fourier transform for 2D images:

Mǫ (u) = 1 J J j=1 |ℓ1|≤ℓǫ |ℓ2|≤ℓǫ y j ℓ1,ℓ2 e i2π(ℓ1u j 1 +ℓ2u j 2 ) - 1 J J j ′ =1 y j ′ ℓ1,ℓ2 e i2π(ℓ1u j ′ 1 +ℓ2u j ′ 2 )
2 for u = (u 1 1 , u 2 1 , . . . , u 1 J , u 2 J ), and where the y j ℓ1,ℓ2 's are the empirical Fourier coefficients of the image Y j . Moreover, if (x 1 1 , x 2 1 , . . . , x 1 J , x 2 J ) denotes the canonical basis of the product space (R 2 ) J , then the components of the gradient of Mǫ (u) are given by

∂ ∂x k j Mǫ (u) = - 2 J |ℓ1|≤ℓǫ |ℓ2|≤ℓǫ ℜ   (i2πℓ k )y j ℓ1,ℓ2 e i2π(ℓ1u j 1 +ℓ2u j 2 ) ( 1 J J j ′ =1 y j ′ ℓ1,ℓ2 e i2π(ℓ1u j ′ 1 +ℓ2u j ′ 2 ) )   .
According to Corollary 1, the smoothing parameter ℓ ǫ should be chosen such that ǫℓ 4 ǫ = o(1). Since the models ( 14) and ( 3) are asymptotically equivalent if

ǫ = σ N , this condition becomes ℓ ǫ = ℓ N = o(N 1/4
). With N = 100, this suggests to take ℓ N ≤ 100 1/4 ≈ 3.16. In Table 1 we give the empirical average of the estimated parameters over the M = 100 simulations, for the choice ℓ N = 3, together with their standard deviation. The results are extremely satisfactory as averages are very close to the true values and standard deviations are very small.

Registration of spherical images

In many applications, data can be organized as spherical images i.e. as functions defined on unit sphere 2 . For instance, spherical images are widely used in robotics since the sphere is a domain where perspective projection can be mapped, and an important question is the estimation of the camera rotation from such images (see [START_REF] Makadia | Rotation Recovery from Spherical Images without Correspondences[END_REF]). Obviously such data do not correspond exactly to the shift model on group (3) as spherical images are defined on S 2 while the shifts parameters belong the special orthogonal group SO [START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF]. However this setting corresponds to the general model ( 2) with X = S 2 and G = SO(3), and a matching criterion similar to the one defined in equation ( 6) can still be defined by combining the spherical harmonics on S 2 with the irreducible representations of SO(3). Indeed, let x ∈ S 2 be a point on the unit sphere parameterized with spherical coordinates θ ∈ [0, π] and φ ∈ [0, 2π[. Then any f ∈ L 2 (S 2 ) can be decomposed as (see [START_REF] Chirikjian | Engineering Applications of Noncommutative Harmonic Analysis[END_REF])

f (x) = +∞ ℓ=0 ℓ m=-ℓ c m ℓ (f )ψ m ℓ (x), with c m ℓ (f ) = π 0 2π 0 f (θ, φ)ψ m ℓ (θ, φ)dφ sin(θ)dθ,
where the functions (ψ m ℓ , ℓ ∈ N, m = -ℓ, . . . , ℓ) are the usual spherical harmonics which form an orthonormal basis of L 2 (S 2 ). Since these functions form a basis for the irreducible representations (π ℓ ) ℓ∈N of SO(3) which are matrices of size (2ℓ + 1) × (2ℓ + 1), it follows that the action of a rotation h ∈ SO(3) on a function f ∈ L 2 (S 2 ) is given by (see [START_REF] Chirikjian | Engineering Applications of Noncommutative Harmonic Analysis[END_REF])

f (h -1 x) = +∞ ℓ=0 c ℓ (f ) T π ℓ (h -1 )ψ ℓ (x) for all x ∈ S 2 , ( 15
)
where c ℓ (f ) = (c m ℓ (f )) m=-ℓ,...,ℓ and ψ ℓ (x) = (ψ m ℓ (x)) m=-ℓ,...,ℓ denotes vectors in C 2ℓ+1 . Now, suppose that one observes a set of noisy spherical images f j that satisfy the following shift model: for j = 1, . . . , J and x ∈ S 2 dZ j (x) = f j (x)dx + ǫdW j (x) with dx = dφ sin(θ)dθ, [START_REF] Sepanski | Compact Lie groups[END_REF] where f j (x) = f * (h * j -1 x), and h * j ∈ SO(3), j = 1, . . . , J are rotation parameters to estimate. For h = (h 1 , . . . , h J ) ∈ G J , the shift property [START_REF] Sepanski | Compact Lie groups[END_REF] and the orthonormality of the spherical harmonics imply that the following matching criterion

N (h) = 1 J J j=1 f j • T hj - 1 J J j ′ =1 f j ′ • T h j ′ 2 L 2 (S 2 ) , (17)
where T hj : x ∈ S 2 → h j x ∈ S 2 , can be written as

N (h) = 1 J J j=1 +∞ ℓ=0 c ℓ (f j ) T π ℓ (h j ) - 1 J J j ′ =1 c ℓ (f j ′ ) T π ℓ (h j ′ ) 2 C 2ℓ+1 , (18)
where

c ℓ (f j ) T = c ℓ (f * ) T π ℓ (h * -1 j
) and • 2 C 2ℓ+1 denotes the usual euclidean norm in C 2ℓ+1 . Then, remark that the spherical harmonic coefficients of the noisy images Z j are given by (in vector form) by c ℓ (Z j ) = S 2 ψ ℓ (x)dZ j (x) = c ℓ (f j ) + ǫc ℓ (W j ), j = 1 . . . J, where c ℓ (W j ) = S 2 ψ ℓ (x)dW j (x) is a complex random vector whose components are independent and identically distributed Gaussian variables N C (0, 1). Now, let ℓ ǫ be an appropriate frequency cut-off parameter. The following empirical criterion can thus be proposed for registering spherical images:

N ǫ (h 1 , . . . , h J ) = 1 J J j=1 ℓǫ ℓ=0 c ℓ (Z j ) T π ℓ (h j ) - 1 J J j ′ =1 c ℓ (Z j ′ ) T π ℓ (h j ′ ) 2 C 2ℓ+1 , (19)
and an M-estimator of the rotation parameters is given by ĥǫ = arg min h∈A0 N ǫ (h).

The criterion N ǫ is very similar to the criterion M ǫ , and the study of the consistency and the asymptotic normality of ĥǫ can be done by following exactly the arguments as those developed in the previous sections.

Some perspectives and future work

The results on the asymptotic normality of the estimators show that there exists a significant difference between semi-parametric estimation on a linear Euclidean space and semi-parametric estimation on a nonlinear manifold. If the group G is non-commutative, then the asymptotic covariance matrix of the estimator ûǫ depends on the point u * (and thus on h * ). Hence, this matrix can be interpreted as a Riemanian metric on G. This is a classical result in parametric statistics for random variables whose law is indexed by parameters belonging to a finite-dimensional manifold. In such setting, the Fisher information matrix is a Riemanian metric and lower bounds analogue to the classical Cramer-Rao bound for parameters in an Euclidean space can be derived (see e.g. [START_REF] Smith | Covariance, subspace, and intrinsic Cramér-Rao bounds[END_REF]). If G is supposed to be an Abelian group, then the asymptotic covariance matrix of the estimator does not depend on the point h * since the parameter space for the shifts is a flat manifold.

An important issue is then the question of optimality of our estimators. We are currently studying analogs of the Cramer-Rao bound for the semi-parametric model [START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF], and in particular we are investigating if the covariance matrix given in Theorem 2 corresponds to the Fisher information matrix of this model. This result would provide a proof of the optimality of our reconstruction, even in a non asymptotic framework.

Another important question is the implementation of our approach for noncommutative groups. The numerical computation of our method for the registration of spherical images is more involved that the one used for the alignment of 2D images. Indeed, one has to deal with both the problem of computing the Fourier transform for images defined on a sphere, and with the problem of computing the irreducible representations of the group SO(3) from its Lie algebra. We are currently working on the development of an efficient and fast numerical scheme to minimize the criterion N ǫ , and we believe that this approach could yield good results for the registration of spherical images.

Finally, it should be mentioned that this work is rather preliminary and practical applications are restricted to estimation of shifts in 2D images and rotation on a sphere for the registration of spherical images. Another application that would be of great interest is the analysis of images of the fundus of the eye as described in [START_REF] Deguchi | 3D fundus pattern reconstruction and display from multiple fundusimages[END_REF]. However, many other interesting applications in medical images involve the study of more sophisticated Lie groups of transformations that are generally non-compact. We believe that there is a good chance to obtain satisfactory results for the estimation of deformations on non-commutative and compact groups. However, an important challenge is to investigate the extension of this work to the case of non-compact groups and at least to identify the difficulties introduced by such a generalization.

  and set m = 0. Step 2 : let u new = u mγ m ∇ u m Mǫ and M (m + 1) = Mǫ (u new ) While M (m + 1) > M (m) do γ m = γ m /κ, and u new = u mγ m ∇ u m Mǫ , and M (m + 1) = Mǫ (u new ) End while Then, take u m+1 = u new and set m = m + 1.

Fig. 1 .

 1 Fig.1. A typical simulation run for J = 6 images generated from the model[START_REF] Pennec | Statistical computing on non-linear spaces for computational anatomy[END_REF].

Table 1 .

 1 Average and standard deviation (in brackets) of the estimators ĥj = ( ĥ1

	j , ĥ2 j )

Acknowledgments

We are very much indebted to the three anonymous referees for their constructive criticism, comments and remarks that resulted in a substantial revision of the original manuscript.