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 [START_REF] Allassonière | Toward a coherent statistical framework for dense deformable template estimation[END_REF][START_REF] Allassonnière | Construction of Bayesian deformable models via a stochastic approximation algorithm: a convergence study[END_REF][START_REF] Bigot | A deconvolution approach to estimation of a common shape in a shifted curves model[END_REF][START_REF] Bigot | Statistical M-estimation and consistency in large deformable models for image warping[END_REF][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, semi-simple, 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, the W m 's are independent copies of a standard Brownian sheet W on the topological space G with reference measure dg, and > 0 is the level of noise in the measurements. 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 the 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).

Some concrete examples of model (1.1) include the analysis of translated two-dimensional images, which corresponds to the case G = R 2 /Z 2 (the torus in dimension two), and which founds its applications in biomedical imaging or satellite remote sensing (see e.g. [START_REF] Li | A contour-based approach to multisensor image registration[END_REF][START_REF] Glasbey | A penalized likelihood approach to image warping[END_REF]). Other examples are rotation models for two-dimensional or three-dimensional images for which either G = SO [START_REF] Allassonnière | Construction of Bayesian deformable models via a stochastic approximation algorithm: a convergence study[END_REF] or G = SO(3) (the special orthogonal group in dimension 2 or 3) when the images at hand are observed through the action of random rotations (see e.g. [START_REF] Makadia | Rotation recovery from spherical images without correspondences[END_REF][START_REF] Grenander | Hilbert-Schmidt lower bounds for estimators on matrix lie groups for atr[END_REF][START_REF] Park | Deblurring of class-averaged images in single-particle electron microscopy[END_REF]). In particular, it is shown in [START_REF] Park | Deblurring of class-averaged images in single-particle electron microscopy[END_REF] that adopting a deconvolution approach over Lie groups (similar to the one developed in this paper) yields very satisfactory estimators of a mean pattern for applications in single-particle electron microscopy imaging. Therefore, the general methodology of this paper can be used in various practical problems as, in many applications, shape variability in a set of images can be seen as the random action of elements in a Lie group.

We will show that the density h of the random elements τ m ∈ G 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. Note that in the case of Abelian groups, the Fourier coefficients (associated to the irreducible representations of G) are one dimensional. However, 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 (unlike standard Fourier deconvolution on the torus).

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 → 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 [START_REF] Kim | Deconvolution density estimation on SO(N )[END_REF][START_REF] Kim | Optimal spherical deconvolution[END_REF][START_REF] Koo | Asymptotic minimax bounds for stochastic deconvolution over groups[END_REF][START_REF] Kim | Deconvolution density estimation on compact Lie groups[END_REF][START_REF] Luo | Deconvolution on the Euclidean motion group SE(3)[END_REF][START_REF] Park | Deblurring of class-averaged images in single-particle electron microscopy[END_REF][START_REF] Yazici | Stochastic deconvolution over groups[END_REF].

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 [START_REF] Koo | Asymptotic minimax bounds for stochastic deconvolution over groups[END_REF]: 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 straightforward to use because of 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). Note that by the compactness of G and thanks to the Peter-Weyl theorem, the set G is countable. 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 i2π g for some ∈ 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 (f ) = 1 0 f (g)e -i2π g dg. For G = R/Z, one thus retrieves the classical Fourier decomposition of a periodic function f :

[0, 1] → R as f (g) = ∈Z c (f )e i2π
= ||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 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 (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(A) R( f , f ),
where the above infimum is taken over the set all estimators.

The main goal of this paper is then to compute an upper bound on R( f , f ) and an asymptotic lower bound 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. The Wiener measure dW (g) (associated to the Brownian sheet W ) is defined through the Fourier basis of L 2 (G) (which diagonalizes the Laplace-Beltrami operator on G, see the appendix for a precise definition) as the random measure

dW (g) := π∈ Ĝ dπ k,l=1 X π kl d π (π(g)) k,l dg, (2.2)
where (π(g)) k,l is the (k, l) entry of the d π × d π matrix π(g), and the X π kl are independent and identically distributed normal variables with zero mean and variance 1. Let f ∈ L 2 (G). By definition, the integral of f with respect to the random measure dW (g) is

G f (g)dW (g) := π∈ Ĝ dπ k,l=1 X π kl d π G f (g)(π(g)) k,l dg.
For further details on the pointwise correlation structure and the regularity properties of Brownian sheets indexed by a manifold (compact Lie groups being a particular case), we refer to [START_REF] Istas | Karhunen-Loève expansion of spherical fractional Brownian motions[END_REF][START_REF] Istas | Manifold indexed fractional fields[END_REF][START_REF] Istas | On locally self-similar fractional random fields indexed by a manifold[END_REF]. Therefore, conditionally to

τ m each integral G f (g) dY m (g) of the "data" dY m (g) is a ran- dom 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]].
Hence, 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]], (2.3) 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 . (2.4)
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 , (2.5)
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.

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 and 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 (2.5)

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(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 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 (2.4). 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 (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(A) n 2s 2s+2ν+dimG R( f , f ) ≥ K 2 .
In the assumptions of Theorem 3.1 and Theorem 3.2, it is supposed that s > 2ν + dimG. For the upper bound, it is possible to weaken this assumption since a similar result can be obtained using the condition s > dim(G)/2. However, for the proof of the lower bound, the condition s > 2ν + dimG is a key technical assumption in our proof to obtain Theorem 3.2.

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 . Then, by Theorem B.2, one has

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 )]
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 ) ki ((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 (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.

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 (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 (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 (A)

inf f sup f ∈Hs(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 (A). Note that c π (f w ) = √ µ D w π which implies

f w ∈ H s (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 (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 . [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

A.2.3 A lower bound for C π,kl Let π ∈ G D , k, l ∈ [[1, d π ]] fixed. Denote by X = (c π (Y m )) (π,m)∈ G×[
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 = 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 ), (
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)
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. [START_REF] Kim | Deconvolution density estimation on SO(N )[END_REF]) 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 [START_REF] Bump | Lie groups[END_REF][START_REF] Duistermaat | Lie groups. Universitext[END_REF][START_REF] Sepanski | Compact Lie groups[END_REF]. 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.

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.

Definition 2 . 5 .

 25 Let k ∈ N * and |.| 2 be the standard Euclidean norm on C k . The operator norm of

2

 2

1 .

 1 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

( 2 )

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

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 .B.2 Peter-Weyl theoremLet 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 (π(g)) k,l = e k , π(g)e l the coordinates of π in the basisB π for k, l ∈ [[1, d π ]]. Theorem B.1. If G is a compact group then √ d π (π(•)) k,l π∈ G, k,l∈[[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 mappingc 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).

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.

  g .

	Definition 2.1. Let k ∈ N * . Let A ∈ M k×k (C) (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

Acknowledgements

The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-JCJC-SIMI1 DEMOS. We are very much indebted to the referees and the Associate Editor for their constructive comments and remarks that resulted in a significant improvement of the original manuscript.

, Y ] = 0 (Jacobi identity).

Definition C.3. The Killing form is the bilinear form B defined by

where ad(X) :

Since the group G is semi-simple, the Killing form B induces a norm on the Lie algebra g.

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

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 non vanishing roots.

Each root space g α with α ∈ Φ 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 the dual of the maximal torus (see the technical appendix in [START_REF] Koo | Asymptotic minimax bounds for stochastic deconvolution over groups[END_REF] for further details on this identification).