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arXiv:1608.03703v2 [] 26 Apr 2017 Template estimation in computational anatomy: Fréchet means in top and quotient spaces are not consistent Loïc Devilliers∗, Stéphanie Allassonnière†, Alain Trouvé‡, and Xavier Pennec§ April 27, 2017 Abstract In this article, we study the consistency of the template estimation with the Fréchet mean in quotient spaces. The Fréchet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases. Keyword : Template, Fréchet mean, group action, quotient space, inconsistency, consistency bias, empirical Fréchet mean, Hilbert space, manifold ∗ Université Côte d’Azur, Inria, France, [email protected] Ecole polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau, France ‡ CMLA, ENS Cachan, CNRS, Université Paris-Saclay, 94235 Cachan, France § Université Côte d’Azur, Inria, France † CMAP, 1 Contents 1 Introduction 3 2 Definitions, notations and generative model 5 3 Inconsistency for finite group when the template is point 3.1 Presence of inconsistency . . . . . . . . . . . . . . . . 3.2 Upper bound of the consistency bias . . . . . . . . . . 3.3 Study of the consistency bias in a simple example . . . . . . . . . . . . . . . . . . . . . 4 Inconsistency for any group when the template is point 4.1 Presence of an inconsistency . . . . . . . . . . . . . . 4.2 Analysis of the condition in theorem 4.1 . . . . . . . 4.3 Lower bound of the consistency bias . . . . . . . . . 4.4 Upper bound of the consistency bias . . . . . . . . . 4.5 Empirical Fréchet mean . . . . . . . . . . . . . . . . 4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Action of translation on L2 (R/Z) . . . . . . . 4.6.2 Action of discrete translation on RZ/NZ . . . . 4.6.3 Action of rotations on Rn . . . . . . . . . . . . . . . . . . . . 5 Fréchet means top and quotient spaces the template is a fixed point 5.1 Result . . . . . . . . . . . . . . . . . . 5.2 Proofs of these theorems . . . . . . . . 5.2.1 Proof of theorem 5.1 . . . . . . 5.2.2 Proof of theorem 5.2 . . . . . . 6 Conclusion and discussion a regular 8 9 12 13 not a fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 14 15 18 20 22 22 23 23 23 are not consistent when . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 25 26 26 28 28 A Proof of theorems for finite groups’ setting 29 A.1 Proof of theorem 3.2: differentiation of the variance in the quotient space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A.2 Proof of theorem 3.1: the gradient is not zero at the template . . 32 A.3 Proof of theorem 3.3: upper bound of the consistency bias . . . . 32 A.4 Proof of proposition 3.2: inconsistency in R2 for the action of translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 B Proof of lemma 5.1: differentiation of the variance in the top space 35 2 1 Introduction In Kendall’s shape space theory [Ken89], in computational anatomy [GM98], in statistics on signals, or in image analysis, one often aims at estimating a template. A template stands for a prototype of the data. The data can be the shape of an organ studied in a population [DPC+ 14] or an aircraft [LAJ+ 12], an electrical signal of the human body, a MR image etc. To analyse the observations, one assumes that these data follow a statistical model. One often models observations as random deformations of the template with additional noise. This deformable template model proposed in [GM98] is commonly used in computational anatomy. The concept of deformation introduces the notion of group action: the deformations we consider are elements of a group which acts on the space of observations, called here the top space. Since the deformations are unknown, one usually considers equivalent classes of observations under the group action. In other words, one considers the quotient space of the top space (or ambient space) by the group. In this particular setting, the template estimation is most of the time based on the minimisation of the empirical variance in the quotient space (for instance [KSW11, JDJG04, SBG08] among many others). The points that minimise the empirical variance are called the empirical Fréchet mean. The Fréchet means introduced in [Fré48] is comprised of the elements minimising the variance. This generalises the notion of expected value in non linear spaces. Note that the existence or uniqueness of Fréchet mean is not ensured. But sufficient conditions may be given in order to reach existence and uniqueness (for instance [Kar77] and [Ken90]). Several group actions are used in practice: some signals can be shifted in time compared to other signals (action of translations [HCG+ 13]), landmarks can be transformed rigidly [Ken89], shapes can be deformed by diffeomorphisms [DPC+ 14], etc. In this paper we restrict to transformation which leads the norm unchanged. Rotations for instance leave the norm unchanged, but it may seem restrictive. In fact, the square root trick detailed in section 5, allows to build norms which are unchanged, for instance by reparametrization of curves with a diffeomorphism, where our work can be applied. We raise several issues concerning the estimation of the template. 1. Is the Fréchet mean in the quotient space equal to the original template projected in the quotient space? In other words, is the template estimation with the Fréchet mean in quotient space consistent? 2. If there is an inconsistency, how large is the consistency bias? Indeed, we may expect the consistency bias to be negligible in many practicable cases. 3. If one gets only a finite sample, one can only estimate the empirical Fréchet mean. How far is the empirical Fréchet mean from the original template? These issues originated from an example exhibited by Allassonnière, Amit and Trouvé [AAT07]: they took a step function as a template and they added some 3 noise and shifted in time this function. By repeating this process they created a data sample from this template. With this data sample, they tried to estimate the template with the empirical Fréchet mean in the quotient space. In this example, minimising the empirical variance did not succeed in estimating well the template when the noise added to the template increases, even with a large sample size. One solution to ensure convergence to the template is to replace this estimation method with a Bayesian paradigm ([AKT10, BG14] or [ZSF13]). But there is a need to have a better understanding of the failure of the template estimation with the Fréchet mean. One can studied the inconsistency of the template estimation. Bigot and Charlier [BC11] first studied the question of the template estimation with a finite sample in the case of translated signals or images by providing a lower bound of the consistency bias. This lower bound was unfortunately not so informative as it is converging to zero asymptotically when the dimension of the space tends to infinity. Miolane et al. [MP15, MHP16] later provided a more general explanation of why the template is badly estimated for a general group action thanks to a geometric interpretation. They showed that the external curvature of the orbits is responsible for the inconsistency. This result was further quantified with Gaussian noise. In this article, we provide sufficient conditions on the noise for which inconsistency appears and we quantify the consistency bias in the general (non necessarily Gaussian) case. Moreover, we mostly consider a vector space (possibly infinite dimensional) as the top space while the article of Miolane et al. is restricted to finite dimensional manifolds. In a preliminary unpublished version of this work [ADP15], we proved the inconsistency when the transformations come from a finite group acting by translation. The current article extends these results by generalizing to any isometric action of finite and non-finite groups. This article is organised as follows. Section 2 details the mathematical terms that we use and the generative model. In sections 3 and 4, we exhibit sufficient condition that lead to an inconsistency when the template is not a fixed point under the group action. This sufficient condition can be roughly understand as follows: with a non zero probability, the projection of the random variable on the orbit of the template is different from the template itself. This condition is actually quite general. In particular, this condition it is always fulfilled with the Gaussian noise or with any noise whose support is the whole space. Moreover we quantify the consistency bias with lower and upper bounds. We restrict our study to Hilbert spaces and isometric actions. This means that the space is linear, the group acts linearly and leaves the norm (or the dot product) unchanged. Section 3 is dedicated to finite groups. Then we generalise our result in section 4 to non-finite groups. To complete this study, we extend in section 5 the result when the template is a fixed point under the group action and when the top space is a manifold. As a result we show that the inconsistency exists for almost all noises. Although the bias can be neglected when the noise level is sufficiently small, its linear asymptotic behaviour with respect to the noise level show that it becomes unavoidable for large noises. 4 2 Definitions, notations and generative model We denote by M the top space, which is the image/shape space, and G the group acting on M . The action is a map: G×M (g, m) → M 7→ g · m satisfying the following properties: for all g, g 0 ∈ G, m ∈ M (gg 0 )·m = g ·(g 0 ·m) and eG · m = m where eG is the neutral element of G. For m ∈ M we note by [m] the orbit of m (or the class of m). This is the set of points reachable from m under the group action: [m] = {g · m, g ∈ G}. Note that if we take two orbits [m] and [n] there are two possibilities: 1. The orbits are equal: [m] = [n] i.e. ∃g ∈ G s.t. n = g · m. 2. The orbits have an empty intersection: [m] ∩ [n] = ∅. We call quotient of M by the group G the set all orbits. This quotient is noted by: Q = M/G = {[m], m ∈ M }. The orbit of an element m ∈ M can be seen as the subset of M of all elements g · m for g ∈ G or as a point in the quotient space. In this article we use these two ways. We project an element m of the top space M into the quotient by taking [m]. Now we are interested in adding a structure on the quotient from an existing structure in the top space: take M a metric space, with dM its distance. Suppose that dM is invariant under the group action which means that ∀g ∈ G, ∀a, b ∈ M dM (a, b) = dM (g · a, g · b). Then we obtain a pseudo-distance on Q defined by: dQ ([a], [b]) = inf dM (g · a, b). (1) g∈G We remind that a distance on M is a map dM : M × M 7→ R+ such that for all m, n, p ∈ M : 1. dM (m, n) = dM (n, m) (symmetry). 2. dM (m, n) ≤ dM (m, p) + dM (p, n) (triangular inequality). 3. dM (m, m) = 0. 4. dM (m, n) = 0 ⇐⇒ m = n. A pseudo-distance satisfies only the first three conditions. If we suppose that all the orbits are closed sets of M , then one can show that dQ is a distance. In this article, we assume that dQ is always a distance, even if a pseudo-distance would be sufficient. dQ ([a], [b]) can be interpreted as the distance between the shapes a and b, once one has removed the parametrisation by the group G. In other words, a and b have been registered. In this article, except in section 5, we 5 suppose that the the group acts isometrically on an Hilbert space, this means that the map x 7→ g ·x is linear, and that the norm associated to the dot product is conserved: kg · xk = kxk. Then dM (a, b) = ka − bk is a particular case of invariant distance. We now introduce the generative model used in this article for M a vector space. Let us take a template t0 ∈ M to which we add a unbiased noise : X = t0 + . Finally we transform X with a random shift S of G. We assume that this variable S is independent of X and the only observed variable is: Y = S · X = S · (t0 + ), with E() = 0, (2) while S, X and  are hidden variables. Note that it is not the generative model defined by Grenander and often used in computational anatomy. Where the observed variable is rather Y 0 = S · t0 + 0 . But when the noise is isotropic and the action is isometric, one can show that the two models have the same law, since S ·  and  have the same probability distribution. As a consequence, the inconsistency of the template estimation with the Fréchet mean in quotient space with one model implies the inconsistency with the other model. Because the former model (2) leads to simpler computation we consider only this model. We can now set the inverse problem: given the observation Y , how to estimate the template t0 in M ? This is an ill-posed problem. Indeed for some element group g ∈ G, the template t0 can be replaced by the translated g ·t0 , the shift S by Sg −1 and the noise  by g, which leads to the same observation Y . So instead of estimating the template t0 , we estimate its orbit [t0 ]. By projecting the observation Y in the quotient space we obtain [Y ]. Although the observation Y = S · X and the noisy template X are different random variables in the top space, their projections on the quotient space lead to the same random orbit [Y ] = [X]. That is why we consider the generative model (2): the projection in the quotient space remove the transformation of the group G. From now on, we use the random orbit [X] in lieu of the random orbit of the observation [Y ]. The variance of the random orbit [X] (sometimes called the Fréchet functional or the energy function) at the quotient point [m] ∈ Q is the expected value of the square distance between [m] and the random orbit [X], namely: Q 3 [m] 7→ E(dQ ([m], [X])2 ) (3) An orbit [m] ∈ Q which minimises this map is called a Fréchet mean of [X]. If we have an i.i.d sample of observations Y1 , . . . , Yn we can write the empirical quotient variance: n Q 3 [m] 7→ n 1X 1X dQ ([m], [Yi ])2 = inf km − gi · Yi k2 . n i=1 n i=1 gi ∈G (4) Thanks to the equality of the quotient variables [X] and [Y ], an element which minimises this map is an empirical Fréchet mean of [X]. 6 In order to minimise the empirical quotient variance (4), the max-max algon P rithm1 alternatively minimises the function J(m, (gi )i ) = n1 km−gi ·Yi k2 over i=1 a point m of the orbit [m] and over the hidden transformation (gi )1≤i≤n ∈ Gn . With these notations we can reformulate our questions as: 1. Is the orbit of the template [t0 ] a minimiser of the quotient variance defined in (3)? If not, the Fréchet mean in quotient space is an inconsistent estimator of [t0 ]. 2. In this last case, can we quantify the quotient distance between [t0 ] and a Fréchet mean of [X]? 3. Can we quantify the distance between [t0 ] and an empirical Fréchet mean of a n-sample? This article shows that the answer to the first question is usually "no" in the framework of an Hilbert space M on which a group G acts linearly and isometrically. The only exception is theorem 5.1 where the top space M is a manifold. In order to prove inconsistency, an important notion in this framework is the isotropy group of a point m in the top space. This is the subgroup which leaves this point unchanged: Iso(m) = {g ∈ G, g · m = m}. We start in section 3 with the simple example where the group is finite and the isotropy group of the template is reduced to the identity element (Iso(t0 ) = {eG }, in this case t0 is called a regular point). We turn in section 4 to the case of a general group and an isotropy group of the template which does not cover the whole group (Iso(t0 ) 6= G) i.e t0 is not a fixed point under the group action. To complete the analysis, we assume in section 5 that the template t0 is a fixed point which means that Iso(t0 ) = G. In sections 3 and 4 we show lower and upper bounds of the consistency bias which we define as the quotient distance between the template orbit and the Fréchet mean in quotient space. These results give an answer to the second question. In section 4, we show a lower bound for the case of the empirical Fréchet mean which answers to the third question. As we deal with different notions whose name or definition may seem similar, we use the following vocabulary: 1. The variance of the noisy template X in the top space is the function E : m ∈ M 7→ E(km − Xk2 ). The unique element which minimises this function is the Fréchet mean of X in the top space. With our assumptions it is the template t0 itself. 2. We call variability (or noise level) of the template the value of the variance at this minimum: σ 2 = E(kt0 − Xk2 ) = E(t0 ). 1 The term max-max algorithm is used for instance in [AAT07], and we prefer to keep the same name, even if it is a minimisation. 7 3. The variance of the random orbit [X] in the quotient space is the function F : m 7→ E(dQ ([m], [X])2 ). Notice that we define this function from the top space and not from the quotient space. With this definition, an orbit [m? ] is a Fréchet mean of [X] if the point m? is a global minimiser of F . In sections 3 and 4, we exhibit a sufficient condition for the inconsistency, which is: the noisy template X takes value with a non zero probability in the set of points which are strictly closer to g · t0 for some g ∈ G than the template t0 itself. This is linked to the folding of the distribution of the noisy template when it is projected to the quotient space. The points for which the distance to the template orbit in the quotient space is equal to the distance to the template in the top space are projected without being folded. If the support of the distribution of the noisy template contains folded points (we only assume that the probability measure of X, noted P, is a regular measure), then there is inconsistency. The support of the noisy template X is defined by the set of points x such that P(X ∈ B(x, r)) > 0 for all r > 0. For different geometries of the orbit of the template, we show that this condition is fulfilled as soon as the support of the noise is large enough. The recent article of Cleveland et al. [CWS16] may seem contradictory with our current work. Indeed the consistency of the template estimation with the Fréchet mean in quotient space is proved under hypotheses which seem to satisfy our framework: the norm is unchanged under their group action (isometric action) and a noise is present in their generative model. However we believe that the noise they consider might actually not be measurable. Indeed, their top space is:   Z 1 L2 ([0, 1]) = f : [0, 1] → R such that f is measurable and f 2 (t)dt < +∞ . 0 The noise e is supposed to be in L2 ([0, 1]) such that for all t, s ∈ [0, 1], E(e(t)) = 0 and E(e(t)e(s)) = σ 2 1s=t , for σ > 0. This means that e(t) and e(s) are chosen without correlation as soon as s 6= t. In this case, it is not clear for us that the resulting function e is measurable, and thus that its Lebesgue integration makes sense. Thus, the existence of such a random process should be established before we can fairly compare the results of both works. 3 Inconsistency for finite group when the template is a regular point In this Section, we consider a finite group G acting isometrically and effectively on M = Rn a finite dimensional space equipped with the euclidean norm k k, associated to the dot product h , i. We say that the action is effective if x 7→ g · x is the identity map if and only if g = eG . Note that if the action is not effective, we can define a new effective action by simply quotienting G by the subgroup of the element g ∈ G such that x 7→ g · x is the identity map. 8 The template is assumed to be a regular point which means that the isotropy group of the template is reduced to the neutral element of G. Note that the measure of singular points (the points which are not regular) is a null set for the Lebesgue measure (see item 1 in appendix A.1). Example 3.1. The action of translation on coordinates: this action is a simplified setting for image registration, where images can be obtained by the translation of one scan to another due to different poses. More precisely, we take the vector space M = RT where G = T = (Z/N Z)D is the finite torus in Ddimension. An element of RT is seen as a function m : T → R, where m(τ ) is the grey value at pixel τ . When D = 1, m can be seen like a discretised signal with N points, when D = 2, we can see m like an image with N × N pixels etc. We then define the group action of T on RT by: τ ∈ T, m ∈ RT τ · m : σ 7→ m(σ + τ ). This group acts isometrically and effectively on M = RT . In this setting, if E(kXk2 ) < +∞ then the variance of [X] is well defined: F : m ∈ M 7→ E(dQ ([X], [m])2 ). In this framework, F is non-negative and continuous. Schwarz inequality we have: lim F (m) ≥ kmk→∞ (5) Thanks to Cauchy- lim kmk2 − 2kmkE(kXk) + E(kXk2 ) = +∞. kmk→∞ Thus for some R > 0 we have: for all m ∈ M if kmk > R then F (m) ≥ F (0) + 1. The closed ball B(0, R) is a compact set (because M is a finite vector space) then F restricted to this ball reached its minimum m? . Then for all m ∈ M , if m ∈ B(0, R), F (m? ) ≤ F (m), if kmk > R then F (m) ≥ F (0) + 1 > F (0) ≥ F (m? ). Therefore [m? ] is a Fréchet mean of [X] in the quotient Q = M/G. Note that this ensure the existence but not the uniqueness. In this Section, we show that as soon as the support of the distribution of X is big enough, the orbit of the template is not a Fréchet mean of [X]. We provide a upper bound of the consistency bias depending on the variability of X and an example of computation of this consistency bias. 3.1 Presence of inconsistency The following theorem gives a sufficient condition on the random variable X for an inconsistency: Theorem 3.1. Let G be a finite group acting on M = Rn isometrically and effectively. Assume that the random variable X is absolutely continuous with respect to the Lebesgue’s measure, with E(kXk2 ) < +∞. We assume that t0 = E(X) is a regular point. 9 g · t0 0 Cone(t0 ) t0 g 0 · t0 Figure 1: Planar representation of a part of the orbit of the template t0 . The lines are the hyperplanes whose points are equally distant of two distinct elements of the orbit of t0 , Cone(t0 ) represented in points is the set of points closer from t0 than any other points in the orbit of t0 . Theorem 3.1 states that if the support (the dotted disk) of the random variable X is not included in this cone, then there is an inconsistency. We define Cone(t0 ) as the set of points closer from t0 than any other points of the orbit [t0 ], see fig. 1 or item 6 in appendix A.1 for a formal definition. In other words, Cone(t0 ) is defined as the set of points already registered with t0 . Suppose that: P (X ∈ / Cone(t0 )) > 0, (6) then [t0 ] is not a Fréchet mean of [X]. The proof of theorem 3.1 is based on two steps: first, differentiating the variance F of [X]. Second, showing that the gradient at the template is not zero, therefore the template can not be a minimum of F . Theorem 3.2 makes the first step. Theorem 3.2. The variance F of [X] is differentiable at any regular points. For m0 a regular point, we define g(x, m0 ) as the almost unique g ∈ G minimising km0 − g · xk (in other words, g(x, m0 ) · x ∈ Cone(m0 )). This allows us to compute the gradient of F at m0 : ∇F (m0 ) = 2(m0 − E(g(X, m0 ) · X)). (7) This Theorem is proved in appendix A.1. Then we show that the gradient of F at t0 is not zero. To ensure that F is differentiable at t0 we suppose in the assumptions of theorem 3.1 that t0 = E(X) is a regular point. Thanks to theorem 3.2 we have: ∇F (t0 ) = 2(t0 − E(g(X, t0 ) · X)). Therefore ∇F (t0 )/2 is the difference between two terms, which are represented on fig. 2: on fig. 2a there is a mass under the two hyperplanes outside 10 g · t0 0 g · t0 Cone(t0 ) t0 0 g 0 · t0 Cone(t0 ) t0 Z g 0 · t0 (a) Graphic representation of the template t0 = E(X) mean of points of the support of X. (b) Graphic representation of Z = E(g(X, t0 ) · X). The points X which were outside Cone(t0 ) are now in Cone(t0 ) (thanks to g(X, t0 )). This part, in grid-line, represents the points which have been folded. Figure 2: Z is the mean of points in Cone(t0 ) where Cone(t0 ) is the set of points closer from t0 than g · t0 for g ∈ G \ eG . Therefore it seems that Z is higher that t0 , therefore ∇F (t0 ) = 2(t0 − Z) 6= 0. Cone(t0 ), so this mass is nearer from gt0 for some g ∈ G than from t0 . In the following expression Z = E(g(X, t0 ) · X), for X ∈ / Cone(t0 ), g(X, t0 )X ∈ Cone(t0 ) such points are represented in grid-line on fig. 2. This suggests that the point Z = E(g(X, t0 ) · X) which is the mean of points in Cone(t0 ) is further away from 0 than t0 . Then ∇F (t0 )/2 = t0 − Z should be not zero, and t0 = E(X) is not a critical point of the variance of [X]. As a conclusion [t0 ] is not a Fréchet mean of [X]. This is turned into a rigorous proof in appendix A.2. In the proof of theorem 3.1, we took M an Euclidean space and we work with the Lebesgue’s measure in order to have P(X ∈ H) = 0 for every hyperplane H. Therefore the proof of theorem 3.1 can be extended immediately to any Hilbert space M , if we make now the assumption that P(X ∈ H) = 0 for every hyperplane H, as long as we keep a finite group acting isometrically and effectively on M . Figure 2 illustrates the condition of theorem 3.1: if there is no mass beyond the hyperplanes, then the two terms in ∇F (t0 ) are equal (because almost surely g(X, t0 ) · X = X). Therefore in this case we have ∇F (t0 ) = 0. This do not prove necessarily that there is no inconsistency, just that the template t0 is a critical point of F . Moreover this figure can give us an intuition on what the consistency bias (the distance between [t0 ] and the set of all Fréchet mean in the quotient space) depends: for t0 a fixed regular point, when the variability of X (defined by E(kX − t0 k2 )) increases the mass beyond the hyperplanes on fig. 2 also increases, the distance between E(g(X, t0 ) · X) and t0 (i.e. the norm of ∇F (t0 )) augments. Therefore q the Fréchet mean should be further from t0 , (because at this point one should have ∇F (q) = 0 or q is a singular 11 point). Therefore the consistency bias appears to increase with the variability of X. By establishing a lower and upper bound of the consistency bias and by computing the consistency bias in a very simple case, sections 3.2, 3.3, 4.3 and 4.4 investigate how far this hypothesis is true. We can also wonder if the converse of theorem 3.1 is true: if the support is included in Cone(t0 ), is there consistency? We do not have a general answer to that. In the simple example section 3.3 it happens that condition (6) is necessary and sufficient. More generally the following proposition provides a partial converse: Cone(y) g · t0 y t0 O Cone(t0 ) g 0 · t0 Figure 3: y 7→ Cone(y) is continuous. When the support of the X is bounded and included in the interior of Cone(t0 ) the hatched cone. For y sufficiently close to the template t0 , the support of the X (the ball in red) is still included in Cone(y) (in grey), then F (y) = (E(kX − yk2 ). Therefore in this case, [t0 ] is at least a Karcher mean of [X]. Proposition 3.1. If the support of X is a compact set included in the interior of Cone(t0 ), then the orbit of the template [t0 ] is at least a Karcher mean of [X] (a Karcher mean is a local minimum of the variance). Proof. If the support of X is a compact set included in the interior of Cone(t0 ) then we know that X-almost surely: dQ ([X], [t0 ]) = kX −t0 k. Thus the variance at t0 in the quotient space is equal to the variance at t0 in the top space. Now by continuity of the distance map (see fig. 3) for y in a small neighbourhood of t0 , the support of X is still included in the interior of Cone(y). We still have dQ ([X], [y]) = kX − yk X-almost surely. In other words, locally around t0 , the variance in the quotient space is equal to the variance in the top space. Moreover we know that t0 = E(X) is the only global minimiser of the variance of X: m 7→ E(km − Xk2 ) = E(m). Therefore t0 is a local minimum of F the variance in the quotient space (since the two variances are locally equal). Therefore [t0 ] is at least a Karcher mean of [X] in this case. 3.2 Upper bound of the consistency bias In this Subsection we show an explicit upper bound of the consistency bias. 12 Theorem 3.3. When G is a finite group acting isometrically on M = Rn , we denote |G| the cardinal of the group G. If X is Gaussian vector: X ∼ N (t0 , s2 IdRn ), and m? ∈ argmin F , then we have the upper bound of the consistency bias: p (8) dQ ([t0 ], [m? ]) ≤ s 8 log(|G|). The proof is postponed in appendix A.3. When X ∼ N (t0 , s2 Idn ) the variability of X is σ 2 = E(||X − t0 ||2 )p= ns2 and we can write the upper bound of the bias: dQ ([t0 ], [m? ]) ≤ √σn 8 log |G|. This Theorem shows that the consistency bias is low when the variability of X is small, which tends to confirm our hypothesis in section 3.1. It is important to notice that this upper bound explodes when the cardinal of the group tends to infinity. 3.3 Study of the consistency bias in a simple example In this Subsection, we take a particular case of example 3.1: the action of translation with T = Z/2Z. We identify RT with R2 and we note by (u, v)T an element of RT . In this setting, one can completely describe the action of T on RT : 0 · (u, v)T = (u, v)T and 1 · (u, v)T = (v, u)T . The set of singularities is the line L = {(u, u)T , u ∈ R}. We note HPA = {(u, v)T , v > u} the half-plane above L and HPB the half-plane below L. This simple example will allow us to provide necessary and sufficient condition for an inconsistency at regular and singular points. Moreover we can compute exactly the consistency bias, and exhibit which parameters govern the bias. We can then find an equivalent of the consistency bias when the noise tends to zero or infinity. More precisely, we have the following theorem proved in appendix A.4: Proposition 3.2. Let X be a random variable such that E(kXk2 ) < +∞ and t0 = E(X). 1. If t0 ∈ L, there is no inconsistency if and only if the support of X is included in the line L = {(u, u), u ∈ R}. If t0 ∈ HPA (respectively in HPB ), there is no inconsistency if and only if the support of X is included in HPA ∪ L (respectively in HPB ∪ L). 2. If X is Gaussian: X ∼ N (t0 , s2 Id2 ), then the Fréchet mean of [X] exists and is unique. This Fréchet mean [m? ] is on the line passing through E(X) and perpendicular to L and the consistency bias ρ̃ = dQ ([t0 ], [m? ]) is the function of s and d = dist(t0 , L) given by:  2   Z 2 +∞ 2 r d ρ̃(d, s) = s r exp − g dr, (9) π ds 2 rs where g is a non-negative function on [0, 1] defined by g(x) = sin(arccos(x))− x arccos(x). (a) If d > 0 then s 7→ ρ̃(d, s) has an asymptotic linear expansion:  2 Z 2 +∞ 2 r ρ̃(d, s) ∼ s r exp − dr. s→∞ π 0 2 13 (10) (b) If d > 0, then ρ̃(d, s) = o(sk ) when s → 0, for all k ∈ N. (c) s → 7 ρ̃(0, s) is linear with respect to s (for d = 0 the template is a fixed point). Remark 3.1. Here, contrarily to the case of the action of rotation in [MHP16], it is not the ratio kE(X)k over the noise which matters to estimate the consistency bias. Rather the ratio dist(E(X), L) over the noise. However in both cases we measure the distance between the signal and the singularities which was {0} in [MHP16] for the action of rotations, L in this case. 4 Inconsistency for any group when the template is not a fixed point In section 3 we exhibited sufficient condition to have an inconsistency, restricted to the case of finite group acting on an Euclidean space. We now generalize this analysis to Hilbert spaces of any dimension included infinite. Let M be such an Hilbert space with its dot product noted by h , i and its associated norm k k. In this section, we do not anymore suppose that the group G is finite. In the following, we prove that there is an inconsistency in a large number of situations, and we quantify the consistency bias with lower and upper bounds. Example 4.1. The action of continuous translation: We take G = (R/Z)D acting on M = L2 ((R/Z)D , R) with: ∀τ ∈ G ∀f ∈ M (τ · f ) : t 7→ f (t + τ ) This isometric action is the continuous version of the example 3.1: the elements of M are now continuous images in dimension D. 4.1 Presence of an inconsistency We state here a generalization of theorem 3.1: Theorem 4.1. Let G be a group acting isometrically on M an Hilbert space, and X a random variable in M , E(kXk2 ) < +∞ and E(X) = t0 6= 0. If: P (dQ ([t0 ], [X]) < kt0 − Xk) > 0, (11)   P sup hg · X, t0 i > hX, t0 i > 0. (12) or equivalently: g∈G Then [t0 ] is not a Fréchet mean of [X] in Q = M/G. The condition of this Theorem is the same condition of theorem 3.1: the support of the law of X contains points closer from gt0 for some g than t0 . Thus the condition (12) is equivalent to E(dQ ([X], [t0 ])2 ) < E(kX − t0 k2 ). In other words, the variance in the quotient space at t0 is strictly smaller than the variance in the top space at t0 . 14 Proof. First the two conditions are equivalent by definition of the quotient distance and by expansion of the square norm of kt0 − Xk and of kt0 − gXk for g ∈ G. As above, we define the variance of [X] by:   2 F (m) = E inf kg · X − mk . g∈G In order to prove this Theorem, we find a point m such that F (m) < F (t0 ), which directly implies that [t0 ] is not be a Fréchet mean of [X]. In the proof of theorem 3.1, we showed that under condition (6) we had h∇F (t0 ), t0 i < 0. This leads us to study F restricted to R+ t0 : we define for a ∈ R+ f (a) = F (at0 ) = E(inf g∈G kg · X − ak2 ). Thanks to the isometric action we can expand f (a) by:   f (a) = a2 kt0 k2 − 2aE sup hg · X, t0 i + E(kXk2 ), (13) g∈G and explicit the unique element of R+ which minimises f :   E sup hg · X, t0 i g∈G . a? = kt0 k2 (14) For all x ∈ M , we have sup hg · x, t0 i ≥ hx, t0 i and thanks to condition (12) we g∈G get: E(sup hg · X, t0 i) > E(hX, t0 i) = hE(X), t0 i = kt0 k2 , (15) g∈G which implies a? > 1. Then F (a? t0 ) < F (t0 ).
Note that kt0 k2 (a? − 1) = E supg∈G hg · X, t0 i − E(hX, t0 i) (which is positive) is exactly − h∇F (t0 ), t0 i /2 in the case of finite group, see Equation (44). Here we find the same expression without having to differentiate the variance F , which may be not possible in the current setting. 4.2 Analysis of the condition in theorem 4.1 We now look for general cases when we are sure that Equation (12) holds which implies the presence of inconsistency. We saw in section 3 that when the group was finite, it is possible to have no inconsistency only if the support of the law is included in a cone delimited by some hyperplanes. The hyperplanes were defined as the set of points equally distant of the template t0 and g ·t0 for g ∈ G. Therefore if the cardinal of the group becomes more and more important, one could think that in order to have no inconsistency the space where X should takes value becomes smaller and smaller. At the limit it leaves only at most an hyperplane. In the following, we formalise this idea to make it rigorous. We show that the cases where theorem 4.1 cannot be applied are not generic cases. 15 First we can notice that it is not possible to have the condition (12) if t0 is a fixed point under the action of G. Indeed in this case hg · X, t0 i = X, g −1 t0 = hX, t0 i). So from now, we suppose that t0 is not a fixed point. Now let us see some settings when we have the condition (11) and thus condition (12). Proposition 4.1. Let G be a group acting isometrically on an Hilbert space M , and X a random variable in M , with E(kXk2 ) < +∞ and E(X) = t0 6= 0. If: 1. [t0 ] \ {t0 } is a dense set in [t0 ]. 2. There exists η > 0 such that the support of X contains a ball B(t0 , η). Then condition (12) holds, and the estimator is inconsistent according to theorem 4.1. B(t0 , η) O t0 g · t0 [t0 ] Figure 4: The smallest disk is included in the support of X and the points in that disk is closer from g · t0 than from t0 . According to theorem 4.1 there is an inconsistency. Proof. By density, one takes g · t0 ∈ B(t0 , η) \ {t0 } for some g ∈ G, now if we take r < min(kg ·t0 −t0 k/2, η −kg ·t0 −t0 k) then B(g ·t0 , r) ⊂ B(t0 , ). Therefore by the assumption we made on the support one has P(X ∈ B(g · t0 , r)) > 0. For y ∈ B(g · t0 , r) we have that kgt0 − yk < kt0 − yk (see fig. 4). Then we have: P (dQ ([X], [t0 ]) < kX − t0 k) ≥ P(X ∈ B(g · t0 , r)) > 0. Then we verify condition (12), and we can apply theorem 4.1. Proposition 4.1 proves that there is a large number of cases where we can ensure the presence of an inconsistency. For instance when M is a finite dimensional vector space and the random variable X has a continuous positive density (for the Lebesgue’s measure) at t0 , condition 2 of Proposition 4.1 is fulfilled. Unfortunately this proposition do not cover the case where there is no mass at the expected value t0 = E(X). This situation could appear if X has two modes for instance. The following proposition deals with this situation: 16 Proposition 4.2. Let G be a group acting isometrically on M . Let X be a random variable in M , such that E(kXk2 ) < +∞ and E(X) = t0 6= 0. If: 1. ∃ϕ s.t. ϕ : (−a, a) → [t0 ] is C 1 with ϕ(0) = t0 , ϕ0 (0) = v 6= 0. 2. The support of X is not included in the hyperplane v ⊥ : P(X ∈ / v ⊥ ) > 0. Then condition (12) is fulfilled, which leads to an inconsistency thanks to Theorem 4.1. Proof. Thanks to the isometric action: ht0 , vi = 0. We choose y ∈ / v ⊥ in the support of X and make a Taylor expansion of the following square distance (see also Figure 5) at 0: kϕ(x) − yk2 = kt0 + xv + o(x) − yk2 = kt0 − yk2 − 2x hy, vi + o(x). Then: ∃x? ∈ (−a, a) s.t. kx? k < a, x hy, vi > 0 and kϕ(x? ) − yk < kt0 − yk. For some g ∈ G, ϕ(x? ) = g · t0 . By continuity of the norm we have: ∃r > 0 s.t. ∀z ∈ B(y, r) kg · t0 − zk < kt0 − zk. Then P(kg·t0 −Xk < kt0 −Xk) ≥ P(X ∈ B(y, r)) > 0. Theorem 4.1 applies. Proposition 4.2 was a sufficient condition on inconsistency in the case of an orbit which contains a curve. This brings us to extend this result for orbits which are manifolds: Proposition 4.3. Let G be a group acting isometrically on an Hilbert space M , X a random variable in M , with E(kXk2 ) < +∞. Assume X = t0 + σ, where t0 6= 0 and E() = 0, and E(kk) = 1. We suppose that [t0 ] is a sub-manifold of M and write Tt0 [t0 ] the linear tangent space of [t0 ] at t0 . If: P(X ∈ / Tt0 [t0 ]⊥ ) > 0, (16) P( ∈ / Tt0 [t0 ]⊥ ) > 0, (17) which is equivalent to: then there is an inconsistency. Proof. First t0 ⊥ Tt0 [t0 ] (because the action is isometric), Tt0 [t0 ]⊥ = t0 + Tt0 [t0 ]⊥ , then the event {X ∈ Tt0 [t0 ]⊥ } is equal to { ∈ Tt0 [t0 ]⊥ }. This proves that equations (16) and (17) are equivalent. Thanks to assumption (16), we can choose y in the support of X such that y ∈ / Tt0 [t0 ]⊥ . Let us take v ∈ Tt0 [t0 ] 1 such that hy, vi = 6 0 and choose ϕ a C curve in [t0 ], such that ϕ(0) = t0 and ϕ0 (0) = v. Applying proposition 4.2 we get the inconsistency. Note that Condition (16) is very weak, because Tt0 [t0 ] is a strict linear subspace of M . 17 [t0 ] Tt0 [t0 ] y g · t0 O t0 Tt0 [t0 ]⊥ Figure 5: y ∈ / Tt0 [t0 ]⊥ therefore y is closer from g · t0 for some g ∈ G than t0 itself. In conclusion, if y is in the support of X, there is an inconsistency. 4.3 Lower bound of the consistency bias Under the assumption of Theorem 4.1, we have an element a? t0 such that F (a? t0 ) < F (t0 ) where F is the variance of [X]. From this element, we deduce lower bounds of the consistency bias: Theorem 4.2. Let δ be the unique positive solution of the following equation: δ 2 + 2δ (kt0 k + EkXk) − kt0 k2 (a? − 1)2 = 0. Let δ? be the unique positive solution of the following equation:   p δ 2 + 2δkt0 k 1 + 1 + σ 2 /kt0 k2 − kt0 k2 (a? − 1)2 = 0, (18) (19) where σ 2 = E(kX − t0 k2 ) is the variability of X. Then δ and δ? are two lower bounds of the consistency bias. Proof. In order to prove this Theorem, we exhibit a ball around t0 such that the points on this ball have a variance bigger than the variance at the point a? t0 , where a? was defined in Equation (14): thanks to the expansion of the function f we did in (13) we get : F (t0 ) − F (a? t0 ) = kt0 k2 (a? − 1)2 > 0, (20) Moreover we can show (exactly like equation (43)) that for all x ∈ M :   2 2 |F (t0 ) − F (x)| ≤ E inf kg · X − t0 k − inf kg · X − xk g∈G g∈G ≤ kx − t0 k (2kt0 k + kx − t0 k + E(k2Xk)) . (21) With Equations (20) and (21), for all x ∈ B(t0 , δ) we have F (x) > F (a? t0 ). No point in that ball mapped in the quotient space is a Fréchet mean of [X]. So 18 δ is a lower bound of the consistency bias. Now by usingthe fact that E(kXk) ≤ p p kt0 k2 + σ 2 , we get: 2|F (t0 )−F (x)| ≤ 2kx−t0 k×kt0 k 1 + 1 + σ 2 /kt0 k2 + kx − t0 k2 . This proves that δ? is also a lower bound of the consistency bias. δ? is smaller than δ, but the variability of X intervenes in δ? . Therefore we propose to study the asymptotic behaviour of δ? when the variability tends to infinity. We have the following proposition: Proposition 4.4. Under the hypotheses of Theorem 4.2, we write X = t0 + σ, with E() = 0, and E(kk2 ) = 1 and note ν = E(supg∈G hg, t0 /kt0 ki) ∈ (0, 1], we have that: p δ? ∼ σ( 1 + ν 2 − 1), σ→+∞ In particular, the consistency bias explodes when the variability of X tends to infinity. Proof. First, let us prove that that ν ∈ (0, 1] under the condition (12). We have ν ≥ E(h, t0 /kt0 ki = 0. By a reductio ad absurdum: if ν = 0, then sup hg, t0 i = h, t0 i almost surely. We have then almost surely: hX, t0 i ≤ g∈G supg∈G hgX, t0 i ≤ kt0 k2 + supg∈G σ hg, t0 i = kt0 k2 + σ h, t0 i ≤ hX, t0 i , which p is in contradiction with (12). Besides ν ≤ E(kk) ≤ Ekk2 = 1 Second, we exhibit equivalent of the terms in equation (19) when σ → +∞:   p 2kt0 k 1 + 1 + σ 2 /kt0 k2 ∼ 2σ. (22) Now by definition of a? in Equation (14) and the decomposition of X = t0 + σ we get:   1 E sup (hg · t0 , t0 i + hg · σ, t0 i) − kt0 k kt0 k(a? − 1) = kt0 k g∈G   1 kt0 k(a? − 1) ≤ E sup hg · σ, t0 i = σν (23) kt0 k g∈G   1 kt0 k(a? − 1) ≥ E sup hg · σ, t0 i − 2kt0 k = σν − 2kt0 k, (24) kt0 k g∈G The lower bound and the upper bound of kt0 k(a? −1) found in (23) and (24) are both equivalent to σν, when σ → +∞. Then the constant term of the quadratic Equation (19) has an equivalent: − kt0 k2 (a? − 1)2 ∼ −σ 2 ν 2 . (25) Finallye if we solve the quadratic Equation (19), we write δ? as a function of the coefficients of the quadratic equation (19). We use the equivalent of each of these terms thanks to equation (22) and (25), this proves proposition 4.4. 19 Remark 4.1. Thanks to inequality (24), if ktσ0 k < ν2 , then kt0 k2 (1 − a? )2 ≥ (σν −2kt0 k)2 , then we write δ? as a function of the coefficients of Equation (19), we obtain a lower bound of the inconsistency bias as a function of kt0 k, σ and ν for σ > 2kt0 k/ν: q p p δ? 2 2 ≥ −(1 + 1 + σ /kt0 k ) + (1 + 1 + σ 2 /kt0 k2 )2 + (σν/kt0 k − 2)2 . kt0 k Although the constant ν intervenes in this lower bound, it is not an explicit term. We now explicit its behaviour depending on t0 . We remind that:   1 ν= E sup hg, t0 i . kt0 k g∈G To this end, we first note that the set of fixed points under the action of G is a closed linear space, (because we can write it as an intersection of the kernel of the continuous and linear functions: x 7→ g · x − x for all g ∈ G). We denote by p the orthogonal projection on the set of fixed points Fix(M ). Then for x ∈ M , we have: dist(x, Fix(M )) = kx − p(x)k. Which yields: hg, t0 i = hg, t0 − p(t0 )i + h, p(t0 )i . (26) The right hand side of Equation (26) does not depend on g as p(t0 ) ∈ Fix(M ). Then:   kt0 kν = E sup hg, t0 − p(t0 )i + hE(), p(t0 )i . g∈G Applying the Cauchy-Schwarz inequality and using E() = 0, we can conclude that: ν≤ 1 dist(t0 , Fix(M ))E(kk) = dist(t0 /kt0 k, Fix(M ))E(kk). kt0 k (27) This leads to the following comment: our lower bound of the consistency bias is smaller when our normalized template t0 /kt0 k is closer to the set of fixed points. 4.4 Upper bound of the consistency bias In this Section, we find a upper bound of the consistency bias. More precisely we have the following Theorem: Proposition 4.5. Let X be a random variable in M , such that X = t0 + σ where σ > 0, E() = 0 and E(||||2 ) = 1. We suppose that [m? ] is a Fréchet mean of [X]. Then we have the following upper bound of the quotient distance between the orbit of the template t0 and the Fréchet mean of [X]: p dQ ([m? ], [t0 ]) ≤ σν(m∗ −m0 )+ σ 2 ν(m∗ − m0 )2 + 2dist(t0 , Fix(M ))σν(m∗ − m0 ), (28) where we have noted ν(m) = E(supg hg, m/kmki) ∈ [0, 1] if m 6= 0 and ν(0) = 0, and m0 the orthogonal projection of t0 on F ix(M ). 20 Note that we made no hypothesis on the template pin this proposition. We deduce from Equation (28) that √ dQ ([m? ], [t0 ]) ≤ σ + σ 2 + 2σdist(t0 , Fix(M )) is a O(σ) when σ → ∞, but a O( σ) when σ → 0, in particular the consistency bias can be neglected when σ is small. Proof. First we have: F (m? ) ≤ F (t0 ) = E(inf ||t0 − g(t0 + σ)||2 ) ≤ E(||σ||2 ) = σ 2 . g (29) Secondly we have for all m ∈ M , (in particular for m? ): F (m) = E(inf (km − gt0 k2 + σ 2 kk2 − 2hgσ, m − gt0 i)) ≥ dQ ([m], [t0 ])2 + σ 2 − 2E(suphσ, gmi). g (30) g With Inequalities (29) and (30) one gets: dQ ([m∗ ], [t0 ])2 ≤ 2E(sup hσ, gm? i) = 2σν(m? )||m? ||, g note that at this point, if m? = 0 then E(supg hσ, gm? i) = 0 and ν(m? ) = 0 although Equation (4.4) is still true even if m? = 0. Moreover with the triangular inequality applied at [m? ], [0] and [t0 ], one gets: km? k ≤ kt0 k + dQ ([m? ], [t0 ]) and then: dQ ([m∗ ], [t0 ])2 ≤ 2σν(m? )(dQ ([m∗ ], [t0 ]) + kt0 k). (31) We can solve inequality (31) and we get: p dQ ([m? ], [t0 ]) ≤ σν(m? ) + σ 2 ν(m? )2 + 2kt0 kσν(m? ), (32) We note by FX instead of F the variance in the quotient space of [X], and we want to apply inequality (32) to X − m0 . As m0 is a fixed point:   2 FX (m) = E inf kX − m0 − g · (m − m0 )k = FX−m0 (m − m0 ) g∈G Then m? minimises FX if and only if m? − m0 minimises FX−m0 . We apply Equation (32) to X − m0 , with E(X − m0 ) = t0 − m0 and [m? − m0 ] a Fréchet mean of [X − m0 ]. We get: p dQ ([m? −m0 ], [t0 −m0 ]) ≤ σν(m∗ −m0 )+ σ 2 ν(m∗ − m0 )2 + 2kt0 − m0 kσν(m∗ − m0 ). Moreover dQ ([m? ], [t0 ]) = dQ ([m? − m0 ], [t0 − m0 ]), which concludes the proof. 21 4.5 Empirical Fréchet mean In practice, we never compute the Fréchet mean in quotient space, only the empirical Fréchet mean in quotient space when the size of a sample is supposed to be large enough. If the empirical Fréchet in the quotient space means converges to the Fréchet mean in the quotient space then we can not use these empirical Fréchet mean in order to estimate the template. In [BB08], it has been proved that the empirical Fréchet mean converges to the Fréchet mean with a √1n convergence speed, however the law of the random variable is supposed to be included in a ball whose radius depends on the geometry on the manifold. Here we are not in a manifold, indeed the quotient space contains singularities, moreover we do not suppose that the law is necessarily bounded. However in [Zie77] the empirical Fréchet means is proved to converge to the Fréchet means but no convergence rate is provided. We propose now to prove that the quotient distance between the template and the empirical Fréchet mean in quotient space have an lower bound which is the asymptotic of the one lower bound of the consistency bias found in (18). Take X, X1 , . . . , Xn independent and identically distributed (with t0 = E(X) not a fixed point). We define the empirical variance of [X] by: n m ∈ M 7→ Fn (m) = n 1X 1X dQ ([m], [Xi ])2 = inf km − g · Xi k2 , n i=1 n i=1 g∈G and we say that [mn? ] is a empirical Fréchet mean of [X] if mn? is a global minimiser of Fn . Proposition 4.6. Let X, X1 , . . . , Xn independent and identically distributed random variables, with t0 = E(X). Let be [mn? ] be an empirical Fréchet mean of [X]. Then δn is a lower bound of the quotient distance between the orbit of the template and [mn? ], where δn is the unique positive solution of: ! n 1X 2 kXi k δ − kt0 k2 (an? − 1)2 = 0. δ + 2 ||t0 || + n i=1 an? is defined like a? in section 4.1 by: n P 1 sup hg · Xi , t0 i n i=1g∈G an? = . kt0 k2 We have that δn → δ by the law of large numbers. The proof is a direct application of theorem 4.2, but applied to the empirical law of X given by the realization of X1 , . . . , Xn . 4.6 Examples In this Subsection, we discuss, in some examples, the application of theorem 4.1 and see the behaviour of the constant ν. This constant intervened in lower bound of the consistency bias. 22 4.6.1 Action of translation on L2 (R/Z) We take an orbit O = [f0 ], where f0 ∈ C 2 (R/Z), non constant. We show easily that O is a manifold of dimension 1 and the tangent space at f0 is2 Rf00 . Therefore a sufficient condition on X such that E(X) = f0 to have an inconsistency is: P(X ∈ / f00⊥ ) > 0 according to proposition 4.3. Now if we denote by 1 the constant function on R/Z equal to 1. We have in this setting: that the set of fixed points under the action of G is the set of constant functions: Fix(M ) = R1 and: s 2 Z 1 Z 1 f0 (t) − f0 (s)ds dt. dist(f0 , Fix(M )) = kf0 − hf0 , 1i 1k = 0 0 This distance to the fixed points is used in the upper bound of the constant ν in Equation (27). Note that if f0 is not differentiable, then [f0 ] is not necessarily a manifold, and (4.3) does not apply. However proposition 4.1 does: if f0 is not a constant function, then [f0 ] \ {f0 } is dense in [f0 ]. Therefore as soon as the support of X contains a ball around f0 , there is an inconsistency. 4.6.2 Action of discrete translation on RZ/NZ We come back on example 3.1, with D = 1 (discretised signals). For some signal t0 , ν previously defined is:   1 ν= E max h, τ · t0 i . kt0 k τ ∈Z/NZ Therefore if we have a sample of size I of  iid, then: ν= I 1X 1 lim max hi , τi · t0 i , kt0 k I→+∞ I i=1 τi ∈Z/N Z By an exhaustive research, we can find the τi ’s which maximise the dot product, then with this sample and t0 we can approximate ν. We have done this approximation for several signals t0 on fig. 6. According the previous results, the bigger ν is, the more important the lower bound of the consistency bias is. We remark that the ν estimated is small, ν  1 for different signals. 4.6.3 Action of rotations on Rn Now we consider the action of rotations on Rn with a Gaussian noise. Take X ∼ N (t0 , s2 Idn ) then the variability of X is ns2 , then X has a decomposition: ] − 21 , 12 [ → O is a local parametrisation of O: f0 = ϕ(0), and we t 7→ f0 (. − t) 0 check that: lim kϕ(x) − ϕ(0) − xf0 kL2 = 0 with Taylor-Lagrange inequality at the order 2 Indeed ϕ : x→0 2. As a conclusion ϕ is differentiable at 0, and it is an immersion (since f00 6= 0), and D0 ϕ : x 7→ xf00 , then O is a manifold of dimension 1 and the tangent space of O at f0 is: Tf0 O = D0 ϕ(R) = Rf00 . 23 nu value for each signal 0.4 0.14456 0.082143 0.24981 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0 0.2 0.4 0.6 0.8 1 Figure 6: Different signals and their ν approximated with a sample of size 103 in RZ/100Z .  is here a Gaussian noise in RZ/100Z , such that E() = 0 and E(kk2 ) = 1. For instance the blue signal is a signal defined randomly, and when we approximate the ν which corresponds to that t0 we find ' 0.25. √ X = t0 + ns with E() = 0 and E(kk2 ) = 1. According to proposition 4.4 we have by noting δ? the lower bound of the consistency bias when s → ∞: p √ δ? → n(−1 + 1 + ν 2 ). s Now ν = E(supg∈G hg, t0 )i /kt0 k = E(kk) → 1 when n tends to infinity (expected value of the Chi distribution) we have that for n large enough: √ √ δ? ' n( 2 − 1). s→∞ s We compare this result with the exact computation of the consistency bias (noted here CB) made by Miolane et al. [MHP16], which writes with our current notations: CB √ Γ((n + 1)/2) lim = 2 . s→∞ s Γ(n/2) lim Using a standard Taylor expansion on the Gamma function, we have that for n large enough: CB √ lim ' n. s→∞ s As a conclusion, when the dimension of the space is large enough our lower bound and the exact computation of the √ bias have the same asymptotic behaviour. It differs only by the constant 2 − 1 ' 0.4 in our lower bound, 1 in the work of Miolane et al. [MP15]. 24 5 Fréchet means top and quotient spaces are not consistent when the template is a fixed point In this Section, we do not assume that the top space M is a vector space, but rather a manifold. We need then to rewrite the generative model likewise: let t0 ∈ M , and X any random variable of M such as t0 is a Fréchet mean of X. Then Y = S · X is the observed variable where S is a random variable whose value are in G. In this Section we make the assumption that the template t0 is a fixed point under the action of G. 5.1 Result Let X be a random variable on M and define the variance of X as: E(m) = E(dM (m, X)2 ). We say that t0 is a Fréchet mean of X if t0 is a global minimiser of the variance E. We prove the following result: Theorem 5.1. Assume that M is a complete finite dimensional Riemannian manifold and that dM is the geodesic distance on M . Let X be a random variable on M , with E(d(x, X)2 ) < +∞ for some x ∈ M . We assume that t0 is a fixed point and a Fréchet mean of X and that P(X ∈ C(t0 )) = 0 where C(t0 ) is the cut locus of t0 . Suppose that there exists a point in the support of X which is not a fixed point nor in the cut locus of t0 . Then [t0 ] is not a Fréchet mean of [X]. The previous result is finite dimensional and does not cover interesting infinite dimensional setting concerning curves for instance. However, a simple extension to the previous result can be stated when M is a Hilbert vector space since then the space is flat and some technical problems like the presence of cut locus point do not occur. Theorem 5.2. Assume that M is a Hilbert space and that dM is given by the Hilbert norm on M . Let X be a random variable on M , with E(kXk2 ) < +∞. We assume that t0 = E(X). Suppose that there exists a point in the support of the law of X that is not a fixed point for the action of G. Then [t0 ] is not a Fréchet mean of [X]. Note that the reciprocal is true: if all the points in the support of the law of X are fixed points, then almost surely, for all m ∈ M and for all g ∈ G we have: dM (X, m) = dM (g · X, m) = dQ ([X], [m]). Up to the projection on the quotient, we have that the variance of X is equal to the variance of [X] in M/G, therefore [t0 ] is a Fréchet mean of [X] if and only if t0 is a Fréchet mean of X. There is no inconsistency in that case. 25 Example 5.1. Theorem 5.2 covers the interesting case of the Fisher Rao metric on functions: F = {f : [0, 1] → R | f is absolutely continuous}. Then considering for G the group of smooth diffeomorphisms γ on [0, 1] such that γ(0) = 0 and γ(1) = 1, we have a right group action G × F → F given by γ · f = f ◦ γ. The Fisher Rao metric is built as a pull back metric q of the 2 2 ˙ L ([0, 1], R) space through the map Q : F → L given by: Q(f ) = f / |f˙|. This square root trick is often used, see for instance [KSW11]. Note that in this case, Rt Q is a bijective mapping with inverse given by q 7→ f with √ f (t) = 0 q(s)|q(s)|ds. We can define a group action on M = L2 as: γ · q = q ◦ γ γ̇, for which one can check easily by a change of variable that: p p kγ · q − γ · q 0 k2 = kq ◦ γ γ̇ − q 0 ◦ γ γ̇k2 = kq − q 0 k2 . So up to the mapping Q, the Fisher Rao metric on curve corresponds to the situation M where theorem 5.2 applies. Note that in this case the set of fixed points under the action of G corresponds in the space F to constant functions. We can also provide an computation of the consistency bias in this setting: Proposition 5.1. Under the assumptions of theorem 5.2, we write X = t0 + σ where t0 is a fixed point, σ > 0, E() = 0 and E(kk2 ) = 1, if there is a Fréchet mean of [X], then the consistency bias is linear with respect to σ and it is equal to: σ sup E(sup hv, g · i). kvk=1 g∈G Proof. For λ > 0 and kvk = 1, we compute the variance F in the quotient space of [X] at the point t0 + λv. Since t0 is a fixed point we get: F (t0 +λv) = E( inf kt0 +λv−gXk2 ) = E(kXk2 )−kt0 k2 −2λE(sup hv, g(X − t0 )i)+λ2 . g∈G g Then we minimise F with respect to λ, and after we minimise with respect to v (with kvk = 1). Which concludes. 5.2 5.2.1 Proofs of these theorems Proof of theorem 5.1 We start with the following simple result, which aims to differentiate the variance of X. This classical result (see [Pen06] for instance) is proved in appendix B in order to be the more self-contained as possible: Lemma 5.1. Let X a random variable on M such that E(d(x, X)2 ) < +∞ for some x ∈ M . Then the variance m 7→ E(m) = E(dM (m, X)2 ) is a continuous 26 function which is differentiable at any point m ∈ M such that P(X ∈ C(m)) = 0 where C(m) is the cut locus of m. Moreover at such point one has: ∇E(m) = −2E(logm (X)), where logm : M \ C(m) → Tm M is defined for any x ∈ M \ C(m) as the unique u ∈ Tm M such that expm (u) = x and kukm = dM (x, m). We are now ready to prove theorem 5.1. Proof. (of theorem 5.1) Let m0 be a point in the support of M which is not a fixed point and not in the cut locus of t0 . Then there exists g0 ∈ G such that m1 = g0 m0 6= m0 . Note that since x 7→ g0 x is a symmetry (the distance is equivariant under the action of G) have that m1 = g0 m0 ∈ / C(g0 t0 ) = C(t0 ) (t0 is a fixed point under the action of G). Let v0 = logt0 (m0 ) and v1 = logt0 (m1 ). We have v0 6= v1 and since C(t0 ) is closed and the logt0 is continuous application on M \ C(t0 ) we have: lim →0 P(X 1 E(1X∈B(m0 ,) logt0 (X)) = v0 . ∈ B(m0 , )) (we use here the fact that since m0 is in the support of the law of X, P(X ∈ B(m0 , )) > 0 for any  > 0 so that the denominator does not vanish and the fact that since M is a complete manifold, it is a locally compact space (the closed balls are compacts) and logt0 is locally bounded). Similarly: lim →0 P(X 1 E(1X∈B(m0 ,) logt0 (g0 X)) = v1 . ∈ B(m0 , )) Thus for sufficiently small  > 0 we have (since v0 6= v1 ): E(logt0 (X)1X∈B(m0 ,) ) 6= E(logt0 (g0 X)1X∈B(m0 ,) ). (33) By using using a reductio ad absurdum, we suppose that [t0 ] is a Fréchet mean of [X] and we want to find a contradiction with (33). In order to do that we introduce simple functions as the function x 7→ 1x∈B(m0 ,) which intervenes in Equation (33). Let s : M → G be a simple function (i.e. a measurable function with finite number of values in G). Then x 7→ h(x) = s(x)x is a measurable function3 . Now, let Es (x) = E(d(x, s(X)X)2 ) be the variance of the variable s(X)X. Note that (and this is the main point): ∀g ∈ G 3 Indeed if: s = dM (t0 , x) = dM (gt0 , gx) = dM (t0 , gx) = dQ ([t0 ], [x]), n P gi 1Ai where (Ai )1≤i≤n is a partition of M (such that the sum is always i=1 defined). Then for any Borel set B ⊂ M we have: h−1 (B) = n S gi−1 (B) ∩ Ai is a measurable i=1 set since x 7→ gi x is a measurable function. 27 we have: Es (t0 ) = E(t0 ). Assume now that [t0 ] a Fréchet mean for [X] on the quotient space and let us show that Es has a global minimum at t0 . Indeed for any m, we have: Es (m) = E(dM (m, s(X)X)2 ) ≥ E(dQ ([m], [X])2 ) ≥ E(dQ ([t0 ], [X])2 ) = Es (t0 ). Now, we want to apply lemma 5.1 to the random variables s(X)X and X at the point t0 . Since we assume that X ∈ / C(t0 ) almost surely and X ∈ / C(t0 ) implies s(X)X ∈ / C(t0 ) we get P(s(X)X ∈ C(t0 )) = 0 and the lemma 5.1 applies. As t0 is a minimum, we already know that the differential of Es (respectively E) at t0 should be zero. We get: E(logt0 (X)) = E(logt0 (s(X)X)) = 0. (34) Now we apply Equation (34) to a particular simple function defined by s(x) = g0 1x∈B(m0 ,) + eG 1x∈B(m . We split the two expected values in (34) into two / 0 ,) parts: E(logt0 (X)1X∈B(m0 ,) ) + E(logt0 (X)1X ∈B(m ) = 0, (35) / 0 ,) ) = 0. E(logt0 (g0 X)1X∈B(m0 ,) ) + E(logt0 (X)1X ∈B(m / 0 ,) (36) By substrating (35) from (36), one gets: E(logt0 (X)1X∈B(m0 ,) ) = E(logt0 (g0 X)1X∈B(m0 ,) ), which is a contradiction with (33). Which concludes. 5.2.2 Proof of theorem 5.2 Proof. The extension to theorem 5.2 is quite straightforward. In this setting many things are now explicit since d(x, y) = kx − yk , ∇x d(x, y)2 = 2(x − y), logx (y) = y − x and the cut locus is always empty. It is then sufficient to go along the previous proof and to change the quantity accordingly. Note that the local compactness of the space is not true in infinite dimension. However this was only used to prove that the log was locally bounded but this last result is trivial in this setting. 6 Conclusion and discussion In this article, we exhibit conditions which imply that the template estimation with the Fréchet mean in quotient space is inconsistent. These conditions are rather generic. As a result, without any more information, a priori there is inconsistency. The behaviour of the consistency bias is summarized in table 1. Surely future works could improve these lower and upper bounds. In a more general case: when we take an infinite-dimensional vector space quotiented by a non isometric group action, is there always an inconsistency? An important example of such action is the action of diffeomorphisms. Can we estimate the consistency bias? In this setting, one estimates the template (or 28 Table 1: Behaviour of the consistency bias with respect to σ 2 the variability of X = t0 + σ. The constants Ki ’s depend on the kind of noise, on the template t0 and on the group action. Consistency bias : CB G is any group Supplementary properties for G a finite group √ Upper bound of CB CB ≤ σ + 2 σ 2 + K1 σ CB ≤ K2 σ (theorem 3.3) (proposition 4.5) Lower bound of CB for σ → ∞ CB ≥ L ∼ K3 σ (proposition 4.4) σ→∞ when the template is not a fixed point √ Behavior of CB for σ → 0 when CB ≤ U ∼ K4 σ CB = o(σ k ), ∀k ∈ N in the σ→0 0 the template is not a fixed point section 3.3, can we extend this result for finite group? CB = σ sup E(supg∈G hv, gi) (proposition 5.1) CB when the template is a fixed point kvk=1 an atlas), but does not exactly compute the Fréchet mean in quotient space, because a regularization term is added. In this setting, can we ensure that the consistency bias will be small enough to estimate the original template? Otherwise, one has to reconsider the template estimation with stochastic algorithms as in [AKT10] or develop new methods. A Proof of theorems for finite groups’ setting A.1 Proof of theorem 3.2: differentiation of the variance in the quotient space In order to show theorem 3.2 we proceed in three steps. First we see some following properties and definitions which will be used. Most of these properties are the consequences of the fact that the group G is finite. Then we show that the integrand of F is differentiable. Finally we show that we can permute gradient and integral signs. 1. The set of singular points in Rn , is a null set (for the Lebesgue’s measure), since it is equal to: [ ker(x 7→ g · x − x), g6=eG a finite union of strict linear subspaces of Rn thanks to the linearity and effectively of the action and to the finite group. 2. If m is regular, then for g, g 0 two different elements of G, we pose: H(g · m, g 0 · m) = {x ∈ Rn , kx − g · mk = kx − g 0 · mk}. Moreover H(g · m, g 0 · m) = (g · m − g 0 · m)⊥ is an hyperplane. 29 3. For m a regular point we define the set of points which are equally distant from two different points of the orbit of m: [ H(g · m, g 0 · m). Am = g6=g 0 Then Am is a null set. For m regular and x ∈ / Am the minimum in the definition of the quotient distance : dQ ([m], [x]) = minkm − g · xk, g∈G (37) is reached at a unique g ∈ G, we call g(x, m) this unique element. 4. By expansion of the squared norm: g minimises km − g · xk if and only if g maximises hm, g · xi. 5. If m is regular and x ∈ / Am then: ∀g ∈ G \ {g(x, m)}, km − g(x, m) · xk < km − g · xk, by continuity of the norm and by the fact that G is a finite group, we can find α > 0, such that for µ ∈ B(m, α) and y ∈ B(x, α): ∀g ∈ G \ {g(x, m)} kµ − g(x, m) · yk < kµ − g · yk. (38) Therefore for such y and µ we have: g(x, m) = g(y, µ). 6. For m a regular point, we define Cone(m) the convex cone of Rn : Cone(m) = {x ∈ Rn / ∀g ∈ G kx − mk ≤ kx − g · mk} (39) n = {x ∈ R / ∀g ∈ G hm, xi ≥ hgm, xi}. This is the intersection of |G| − 1 half-spaces: each half space is delimited by H(m, gm) for g 6= eG (see fig. 1). Cone(m) is the set of points whose projection on [m] is m, (where the projection of one point p on [m] is one point g · m which minimises the set {kp − g · mk, g ∈ G}). 7. Taking a regular T point m allows us to see the T quotient. For every point x ∈ Rn we have: [x] Cone(m) 6= ∅, card([x] Cone(m)) ≥ 2 if and only if x ∈ Am . The borders of the cone is Cone(m)\Int(Cone(m)) = Cone(m)∩ Am (we denote by Int(A) the interior of a part A). Therefore Q = Rn /G can be seen like Cone(m) whose border have been glued together. The proof of theorem 3.2 is the consequence of the following lemmas. The first lemma studies the differentiability of the integrand, and the second allows us to permute gradient and integral sign. Let us denote by f the integrand of F: 30 ∀ m, x ∈ M f (x, m) = minkm − g · xk2 . (40) g∈G Thus we have: F (m) = E(f (X, m)). The min of differentiable functions is not necessarily differentiable, however we prove the following result: Lemma A.1. Let m0 be a regular point, if x ∈ / Am0 then m 7→ f (x, m) is differentiable at m0 , besides we have: ∂f (x, m0 ) = 2(m0 − g(x, m0 ) · x) ∂m (41) Proof. If m0 is regular and x ∈ / Am0 then we know from the item 5 of the appendix A.1 that g(x, m0 ) is locally constant. Therefore around m0 , we have: f (x, m) = km − g(x, m0 ) · xk2 , which can differentiate with respect to m at m0 . This proves the lemma A.1. Now we want to prove that we can permute the integral and the gradient sign. The following lemma provides us a sufficient condition to permute integral and differentiation signs thanks to the dominated convergence theorem: Lemma A.2. For every m0 ∈ M we have the existence of an integrable function Φ : M → R+ such that: ∀m ∈ B(m0 , 1), ∀x ∈ M |f (x, m0 ) − f (x, m)| ≤ km − m0 kΦ(x). (42) Proof. For all g ∈ G, m ∈ M we have: kg · x − m0 k2 − kg · x − mk2 = hm − m0 , 2g · x − (m0 + m)i ≤ km − m0 k × (km0 + mk + k2xk) 2 minkg · x − m0 k ≤ km − m0 k (km0 + mk + k2xk) + kg · x − mk2 g∈G minkg · x − m0 k2 ≤ km − m0 k (km0 + mk + k2xk) + minkg · x − mk2 g∈G 2 g∈G 2 minkg · x − m0 k − minkg · x − mk ≤ km − m0 k (2km0 k + km − m0 k + k2xk) g∈G g∈G By symmetry we get also the same control of f (x, m) − f (x, m0 ), then: |f (x, m0 ) − f (x, m)| ≤ km0 − mk (2km0 k + km − m0 k + k2xk) (43) The function Φ should depend on x or m0 , but not on m. That is why we take only m ∈ B(m0 , 1), then we replace km−m0 k by 1 in (43), which concludes. 31 A.2 Proof of theorem 3.1: the gradient is not zero at the template To prove it, we suppose that ∇F (t0 ) = 0, and we take the dot product with t0 : h∇F (t0 ), t0 i = 2E(hX, t0 i − hg(X, t0 ) · X, t0 i) = 0. (44) The item 4 of (x, m) 7→ g(x, m) seen at appendix A.1 leads to: hX, t0 i − hg(X, t0 ) · X, t0 i ≤ 0 almost surely. So the expected value of a non-positive random variable is null. Then hX, t0 i − hg(X, t0 ) · X, t0 i = 0 almost surely hX, t0 i = hg(X, t0 ) · X, t0 i almost surely. Then g = eG maximizes the dot product almost surely. Therefore (as we know that g(X, t0 ) is unique almost surely, since t0 is regular): g(X, t0 ) = eG almost surely, which is a contradiction with Equation (6). A.3 Proof of theorem 3.3: upper bound of the consistency bias In order to show this Theorem, we use the following lemma: Lemma A.3. We write X = t0 + where E() = 0 and we make the assumption that the noise  is a subgaussian random variable. This means that it exists c > 0 such that:   2 s kmk2 . (45) ∀m ∈ M = Rn , E(exp(h, mi)) ≤ c exp 2 If for m ∈ M we have: p ρ̃ := dQ ([m], [t0 ]) ≥ s 2 log(c|G|), (46) p ρ̃2 − ρ̃s 8 log(c|G|) ≤ F (m) − E(kk2 ). (47) then we have: Proof. (of lemma A.3) First we expand the right member of the inequality (47):   E(kk2 ) − F (m) = E max(kX − t0 k2 − kX − gmk2 ) g∈G We use the formula kAk2 − kA + Bk2 = −2 hA, Bi − kBk2 with A = X − t0 and B = t0 − gm:  
E(kk2 ) − F (m) = E max −2 hX − t0 , t0 − gmi − kt0 − gmk2 = E(max ηg ), g∈G g∈G (48) 32 with ηg = −kt0 − gmk2 + 2 h, gm − t0 i. Our goal is to find a lower bound of F (m) − E(kk2 ), that is why we search an upper bound of E(maxηg ) with the g∈G Jensen’s inequality. We take x > 0 and we get by using the assumption (45):   X exp(xE(max ηg )) ≤ E(exp(max xηg )) ≤ E  exp(xηg ) g∈G g∈G ≤ X g∈G 2 exp(−xkt0 − gmk )E(exp(h, 2x(gm − t0 )i) g X ≤c exp(−xkt0 − gmk2 ) exp(2s2 x2 kgm − t0 k2 ) g X ≤c exp(kgm − t0 k2 (−x + 2x2 s2 )) (49) g Now if (−x + 2t2 x2 ) < 0, we can take an upper bound of the sum sign in (49) by taking the smallest value in the sum sign, which is reached when g minimizes kg · m − t0 k multiplied by the number of elements summed. Moreover (−x + 2x2 s) < 0 ⇐⇒ 0 < x < 2s12 . Then we have: exp(xE(max ηg )) ≤ c|G| exp(ρ̃2 (−x + 2x2 s2 )) as soon as 0 < x < g∈G 1 . 2s2 Then by taking the log: E(maxηg ) ≤ g∈G log c|G| + (2xs2 − 1)ρ̃2 . x (50) Now we find the x which optimizes inequality (50).p By differentiation, the right member of inequality (50) is minimal for x? = log c|G|/2/(sρ̃) which is a valid choice because x? ∈ (0, 2s12 ) by using the assumption (46). With the equations (48) and (50) and x? we get the result. Proof. (of theorem 3.3) We take m? ∈ argmin F , ρ̃ = dQ ([m? ], [t0 ]), and  = 2 2 X − tp 0 . We have: F (m? ) ≤ F (t0 ) ≤ E(kk ) then F (m? ) − E(kk ) ≤ 0. If ρ̃ > s 2 log(|G|) then we can apply lemma A.3 with c = 1. Thus: p ρ̃2 − ρ̃s 8 log(|G|) ≤ 2F (m? ) − E(kk2 ) ≤ 0, p p which yields to ρ̃ ≤ s 8 log(|G|). If ρ̃ ≤ s 2 log(|G|), we have nothing to prove. Note that the proof of this upper bound does not use the fact that the action is isometric, therefore this upper bound is true for every finite group action. 33 A.4 Proof of proposition 3.2: inconsistency in R2 for the action of translation Proof. We suppose that E(X) ∈ HPA ∪ L. In this setting we call τ (x, m) one of element of the group G = T which minimises kτ · x − mk see (37) instead of g(x, m). The variance in the quotient space at the point m is:   F (m) = E min kτ · X − mk2 = E(kτ (X, m) · X − mk2 ). τ ∈Z/2Z As we want to minimize F and F (1 · m) = F (m), we can suppose that m ∈ HPA ∪ L. We can completely write what take τ (x, m) for x ∈ M : • If x ∈ HPA ∪ L we can set τ (x, m) = 0 (because in this case x, m are on the same half plane delimited by L the perpendicular bisector of m and −m). • If x ∈ HPB then we can set τ (x, m) = 1 (because in this case x, m are not on the same half plane delimited by L the perpendicular bisector of m and −m). This allows use to write the variance at the point m ∈ HPA :


F (m) = E kX − mk2 1{X∈HPA ∪L} + E k1 · X − mk2 1{X∈HPB } Then we define the random variable Z by: Z = X1X∈HPA ∪L + 1 · X1X∈HPB , such that for m ∈ HPA we have: F (m) = E(kZ − mk2 ) and F (m) = F (1 · m). Thus if m? is a global minimiser of F , then m? = E(Z) or m? = 1 · E(Z). So the Fréchet mean of [X] is [E(Z)]. Here instead of using theorem 3.1, we can work explicitly: Indeed there is no inconsistency if and only if E(Z) = E(X), (E(Z) = 1 · E(X) would be another possibility, but by assumption E(Z), E(X) ∈ HPA ), by writing X = X1X∈HPA + X1X∈HPB ∪L , we have: E(Z) = E(X) ⇐⇒ E(1 · X1X∈HPB ∪L ) = E(X1X∈HPB ∪L ) ⇐⇒ 1 · E(X1X∈HPB ∪L ) = E(X1X∈HPB ∪L ) ⇐⇒ E(X1X∈HPB ∪L ) ∈ L ⇐⇒ P(X ∈ HPB ) = 0, Therefore there is an inconsistency if and only if P(X ∈ HPB ) > 0 (we remind that we made the assumption that E(X) ∈ HPA ∪ L). If E(X) is regular (i.e. E(X) ∈ / L), then there is an inconsistency if and only if X takes values in HPB , (this is exactly the condition of theorem 3.1, but in this particular case, this is a necessarily and sufficient condition). This proves point 1. Now we make the assumption that X follows a Gaussian noise in order compute E(Z) (note that we could take another noise, as long as we are able to compute E(Z)). For that we convert to polar coordinates: (u, v)T = E(X) + (r cos θ, r sin θ)T where r > 0 et θ ∈ [0, 2π]. We also define: d = dist(E(X), L), E(X) is a regular point if 34 and only if d > 0. We still suppose that E(X) = (α, β)T ∈ HPA ∪ L. First we parametrise in function of (r, θ) the points which are in HPB : v < u ⇐⇒ β + r sin θ < α + r cos θ ⇐⇒ β−α √ π < 2 cos(θ + ) r 4 d π < cos(θ + ) r h 4 i π π ⇐⇒ θ ∈ − − arccos(d/r), − + arccos(d/r) and d < r 4 4 ⇐⇒ Then we compute E(Z): E(Z) =E(X1X∈HPA ) + E(1 · X1X∈HPB )    exp − r2 Z d Z 2π  2s2 α + r cos θ rdθdr E(Z) = 2 β + r sin θ 2πs 0 0    exp − r2 Z +∞ Z 2π− π4 −arccos( dr )  2 2s α + r cos θ rdrdθ + 2 β + r sin θ d π 2πs arccos( r )− 4 d  2 r  Z +∞ Z − π4 +arccos( dr )  β + r sin θ exp − 2s2 + rdrdθ α + r cos θ d 2πs2 d −π 4 −arccos( r )   Z +∞ 2 r2 √ r exp(− 2s d 2) =E(X) + 2g dr × (−1, 1)T , 2 πs r d We compute ρ̃ = dQ ([E(X)], [E(Z)]) where dQ is the distance in the quotient space defined in (1). As we know that E(X), E(Z) are in the same half-plane delimited by L, we have: ρ̃ = dQ ([E(Z)], [E(X)]) = kE(Z) − E(X)k. This proves eq. (9), note that items 2a to 2c are the direct consequence of eq. (9) and basic analysis. B Proof of lemma 5.1: differentiation of the variance in the top space Proof. By triangle inequality it is easy to show that E is finite and continuous everywhere. Moreover, it is a well known fact that x 7→ dM (x, z)2 is differentiable at any m ∈ M \ C(z) (i.e. z ∈ / C(m)) with derivative −2 logm (z). Now since: |dM (x, z)2 − dM (y, z)2 | = |dM (x, z) − dM (y, z)kdM (x, z) + dM (y, z)| ≤ dM (x, y)(2dM (x, z) + dM (y, x)), we get in a local chart φ : U → V ⊂ Rn at t = φ(m) we have locally around t that: h 7→ dM (φ−1 (t), φ−1 (t + h)), 35 is smooth and |dM (φ−1 (t), φ−1 (t+h))| ≤ C|h| for a C > 0. Hence for sufficiently small h, |dM (φ−1 (t), z)2 − dM (φ−1 (t + h), z)2 | ≤ C|h|(2dM (m, z) + 1). We get the result from dominated convergence Lebesgue theorem with E(dM (m, X)) ≤ E(dM (m, X)2 + 1) < +∞. References [AAT07] Stéphanie Allassonnière, Yali Amit, and Alain Trouvé. Towards a coherent statistical framework for dense deformable template estimation. 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Tiny SSD: A Tiny Single-shot Detection Deep Convolutional Neural Network for Real-time Embedded Object Detection arXiv:1802.06488v1 [] 19 Feb 2018 Alexander Wong, Mohammad Javad Shafiee, Francis Li, Brendan Chwyl Dept. of Systems Design Engineering University of Waterloo, DarwinAI {a28wong, mjshafiee}@uwaterloo.ca, {francis, brendan}@darwinai.ca Abstract—Object detection is a major challenge in computer vision, involving both object classification and object localization within a scene. While deep neural networks have been shown in recent years to yield very powerful techniques for tackling the challenge of object detection, one of the biggest challenges with enabling such object detection networks for widespread deployment on embedded devices is high computational and memory requirements. Recently, there has been an increasing focus in exploring small deep neural network architectures for object detection that are more suitable for embedded devices, such as Tiny YOLO and SqueezeDet. Inspired by the efficiency of the Fire microarchitecture introduced in SqueezeNet and the object detection performance of the singleshot detection macroarchitecture introduced in SSD, this paper introduces Tiny SSD, a single-shot detection deep convolutional neural network for real-time embedded object detection that is composed of a highly optimized, non-uniform Fire subnetwork stack and a non-uniform sub-network stack of highly optimized SSD-based auxiliary convolutional feature layers designed specifically to minimize model size while maintaining object detection performance. The resulting Tiny SSD possess a model size of 2.3MB (∼26X smaller than Tiny YOLO) while still achieving an mAP of 61.3% on VOC 2007 (∼4.2% higher than Tiny YOLO). These experimental results show that very small deep neural network architectures can be designed for real-time object detection that are well-suited for embedded scenarios. Keywords-object detection; deep neural network; embedded; real-time; single-shot I. I NTRODUCTION Object detection can be considered a major challenge in computer vision, as it involves a combination of object classification and object localization within a scene (see Figure 1). The advent of modern advances in deep learning [7], [6] has led to significant advances in object detection, with the majority of research focuses on designing increasingly more complex object detection networks for improved accuracy such as SSD [9], R-CNN [1], Mask R-CNN [2], and other extended variants of these networks [4], [8], [15]. Despite the fact that such object detection networks have showed stateof-the-art object detection accuracies beyond what can be achieved by previous state-of-the-art methods, such networks are often intractable for use for embedded applications due to computational and memory constraints. In fact, even faster variants of these networks such as Faster R-CNN [13] are only Figure 1. Tiny SSD results on the VOC test set. The bounding boxes, categories, and confidences are shown. capable of single-digit frame rates on a high-end graphics processing unit (GPU). As such, more efficient deep neural networks for real-time embedded object detection is highly desired given the large number of operational scenarios that such networks would enable, ranging from smartphones to aerial drones. Recently, there has been an increasing focus in exploring small deep neural network architectures for object detection that are more suitable for embedded devices. For example, Redmon et al. introduced YOLO [11] and YOLOv2 [12], which were designed with speed in mind and was able to achieve real-time object detection performance on a high-end Nvidia Titan X desktop GPU. However, the model size of YOLO and YOLOv2 remains very large in size (753 MB and 193 MB, respectively), making them too large from a memory perspective for most embedded devices. Furthermore, their object detection speed drops considerably when running on embedded chips [14]. To address this issue, Tiny YOLO [10] was introduced where the network architecture was reduced considerably to greatly reduce model size (60.5 MB) as well as greatly reduce the number of floating point operations required (just 6.97 billion operations) at a cost of object detection accuracy (57.1% on the twenty-category VOC 2017 test set). Similarly, Wu et al. introduced SqueezeDet [16], a fully convolutional neural network that leveraged the efficient Fire microarchitecture introduced in SqueezeNet [5] within an end-to-end object detection network architecture. Given that the Fire microarchitecture is highly efficient, the resulting SqueezeDet had a reduced model size specifically for the purpose of autonomous driving. However, SqueezeDet has only been demonstrated for objection detection with limited object categories (only three) and thus its ability to handle larger number of categories have not been demonstrated. As such, the design of highly efficient deep neural network architectures that are well-suited for real-time embedded object detection while achieving improved object detection accuracy on a variety of object categories is still a challenge worth tackling. In an effort to achieve a fine balance between object detection accuracy and real-time embedded requirements (i.e., small model size and real-time embedded inference speed), we take inspiration by both the incredible efficiency of the Fire microarchitecture introduced in SqueezeNet [5] and the powerful object detection performance demonstrated by the single-shot detection macroarchitecture introduced in SSD [9]. The resulting network architecture achieved in this paper is Tiny SSD, a single-shot detection deep convolutional neural network designed specifically for realtime embedded object detection. Tiny SSD is composed of a non-uniform highly optimized Fire sub-network stack, which feeds into a non-uniform sub-network stack of highly optimized SSD-based auxiliary convolutional feature layers, designed specifically to minimize model size while retaining object detection performance. This paper is organized as follows. Section 2 describes the highly optimized Fire sub-network stack leveraged in the Tiny SSD network architecture. Section 3 describes the highly optimized sub-network stack of SSD-based convolutional feature layers used in the Tiny SSD network architecture. Section 4 presents experimental results that evaluate the efficacy of Tiny SSD for real-time embedded object detection. Finally, conclusions are drawn in Section 5. II. O PTIMIZED F IRE S UB - NETWORK S TACK The overall network architecture of the Tiny SSD network for real-time embedded object detection is composed of two main sub-network stacks: i) a non-uniform Fire sub-network stack, and ii) a non-uniform sub-network stack of highly optimized SSD-based auxiliary convolutional feature layers, with the first sub-network stack feeding into the second subnetwork stack. In this section, let us first discuss in detail the design philosophy behind the first sub-network stack of the Tiny SSD network architecture: the optimized fire sub-network stack. A powerful approach to designing smaller deep neural network architectures for embedded inference is to take a more principled approach and leverage architectural design strategies to achieve more efficient deep neural network microarchitectures [3], [5]. A very illustrative example of such a principled approach is the SqueezeNet [5] network architecture, where three key design strategies were leveraged: 1) reduce the number of 3 × 3 filters as much as possible, Figure 2. An illustration of the Fire microarchitecture. The output of previous layer is squeezed by a squeeze convolutional layer of 1 × 1 filters, which reduces the number of input channels to 3 × 3 filters. The result of the squeeze convolutional layers is passed into the expand convolutional layer which consists of both 1 × 1 and 3 × 3 filters. 2) reduce the number of input channels to 3 × 3 filters where possible, and 3) perform downsampling at a later stage in the network. This principled designed strategy led to the design of what the authors referred to as the Fire module, which consists of a squeeze convolutional layer of 1x1 filters (which realizes the second design strategy of effectively reduces the number of input channels to 3 × 3 filters) that feeds into an expand convolutional layer comprised of both 1 × 1 filters and 3 × 3 filters (which realizes the first design strategy of effectively reducing the number of 3 × 3 filters). An illustration of the Fire microarchitecture is shown in Figure 2. Inspired by the elegance and simplicity of the Fire microarchitecture design, we design the first sub-network stack of the Tiny SSD network architecture as a standard convolutional layer followed by a set of highly optimized Fire modules. One of the key challenges to designing this sub-network stack is to determine the ideal number of Fire modules as well as the ideal microarchitecture of each of the Fire modules to achieve a fine balance between object detection performance and model size as well as inference speed. First, it was determined empirically that 10 Fire modules in the optimized Fire sub-network stack provided strong object detection performance. In terms of the ideal microarchitecture, the key design parameters of the Fire microarchitecture are the number of filters of each size (1 × 1 or 3 × 3) that form this microarchitecture. In the SqueezeNet network architecture that first introduced the Fire microarchitecture [5], the microarchitectures of the Fire modules are largely uniform, with many of the modules sharing the same microarchitecture configuration. In an effort to achieve more optimized Fire microarchitectures on a permodule basis, the number of filters of each size in each Fire Table I T HE OPTIMIZED F IRE SUB - NETWORK STACK OF THE T INY SSD NETWORK ARCHITECTURE . T HE NUMBER OF FILTERS AND INPUT SIZE TO EACH LAYER ARE REPORTED FOR THE CONVOLUTIONAL LAYERS AND F IRE MODULES . E ACH F IRE MODULE IS REPORTED IN ONE ROW FOR A BETTER REPRESENTATION . ”x@S – y@E1 – z@E3" STANDS FOR x NUMBERS OF 1 × 1 FILTERS IN THE SQUEEZE CONVOLUTIONAL LAYER , y NUMBERS OF 1 × 1 FILTERS AND z NUMBERS OF 3 × 3 FILTERS IN THE EXPAND CONVOLUTIONAL LAYER . Type / Stride Conv1 / s2 Pool1 / s2 Fire1 Fire2 Figure 3. An illustration of the network architecture of the second sub-network stack of Tiny SSD. The output of three Fire modules and two auxiliary convolutional feature layers, all with highly optimized microarchitecture configurations, are combined together for object detection. module is optimized to have as few parameters as possible while still maintaining the overall object detection accuracy. As a result, the optimized Fire sub-network stack in the Tiny SSD network architecture is highly non-uniform in nature for an optimal sub-network architecture configuration. Table I shows the overall architecture of the highly optimized Fire sub-network stack in Tiny SSD, and the number of parameters in each layer of the sub-network stack. III. O PTIMIZED S UB - NETWORK S TACK OF SSD- BASED C ONVOLUTIONAL F EATURE L AYERS In this section, let us first discuss in detail the design philosophy behind the second sub-network stack of the Tiny SSD network architecture: the sub-network stack of highly optimized SSD-based auxiliary convolutional feature layers. One of the most widely-used and effective object detection network macroarchitectures in recent years has been the single-shot multibox detection (SSD) macroarchitecture [9]. The SSD macroarchitecture augments a base feature extraction network architecture with a set of auxiliary convolutional feature layers and convolutional predictors. The auxiliary convolutional feature layers are designed such that they decrease in size in a progressive manner, thus enabling the flexibility of detecting objects within a scene across different scales. Each of the auxiliary convolutional feature layers can then be leveraged to obtain either: i) a confidence score for a object category, or ii) a shape offset relative to default bounding box coordinates [9]. As a result, a number of object detections can be obtained per object category in this manner in a powerful, end-to-end single-shot manner. Inspired by the powerful object detection performance and multi-scale flexibility of the SSD macroarchitecture [9], the second sub-network stack of Tiny SSD is comprised of a set of auxiliary convolutional feature layers and convo- Pool3 / s2 Fire3 Fire4 Pool5 / s2 Fire5 Fire6 Fire7 Fire8 Pool9 / s2 Fire 9 Pool10 / s2 Fire10 Filter Shapes 3 × 3 × 57 3×3 15@S – 49@E1 – 53@E3 Concat1 15@S – 54@E1 – 52@E3 Concat2 3×3 29@S – 92@E1 – 94@E3 Concat3 29@S – 90@E1 – 83@E3 Concat4 3×3 44@S – 166@E1 – 161@E3 Concat5 45@S – 155@E1 – 146@E3 Concat6 49@S – 163@E1 – 171@E3 Concat7 25@S – 29@E1 – 54@E3 Concat8 3×3 37@S – 45@E1 – 56@E3 Concat9 3×3 38@S – 41@E1 – 44@E3 Concat10 Input Size 300 × 300 149 × 149 74 × 74 74 × 74 74 × 74 37 × 37 37 × 37 37 × 37 18 × 18 18 × 18 18 × 18 18 × 18 18 × 18 9×9 4×4 lutional predictors with highly optimized microarchitecture configurations (see Figure 3). As with the Fire microarchitecture, a key challenge to designing this sub-network stack is to determine the ideal microarchitecture of each of the auxiliary convolutional feature layers and convolutional predictors to achieve a fine balance between object detection performance and model size as well as inference speed. The key design parameters of the auxiliary convolutional feature layer microarchitecture are the number of filters that form this microarchitecture. As such, similar to the strategy taken for constructing the highly optimized Fire sub-network stack, the number of filters in each auxiliary convolutional feature layer is optimized to minimize the number of parameters while preserving overall object detection accuracy of the full Tiny SSD network. As a result, the optimized sub-network stack of auxiliary convolutional feature layers in the Tiny SSD network architecture is highly non-uniform in nature for an optimal sub-network architecture configuration. Table II shows the overall architecture of the optimized sub-network stack of the auxiliary convolutional feature layers within the Tiny SSD network architecture, along with the number of Table II T HE OPTIMIZED SUB - NETWORK STACK OF THE AUXILIARY CONVOLUTIONAL FEATURE LAYERS WITHIN THE T INY SSD NETWORK ARCHITECTURE . T HE INPUT SIZES TO EACH CONVOLUTIONAL LAYER AND KERNEL SIZES ARE REPORTED . Type / Stride Conv12-1 / s2 Conv12-2 Conv13-1 Conv13-2 Fire5-mbox-loc Fire5-mbox-conf Fire9-mbox-loc Fire9-mbox-conf Fire10-mbox-loc Fire10-mbox-conf Fire11-mbox-loc Fire11-mbox-conf Conv12-2-mbox-loc Conv12-2-mbox-conf Conv13-2-mbox-loc Conv13-2-mbox-conf Filter Shape 3 × 3 × 51 3 × 3 × 46 3 × 3 × 55 3 × 3 × 85 3 × 3 × 16 3 × 3 × 84 3 × 3 × 24 3 × 3 × 126 3 × 3 × 24 3 × 3 × 126 3 × 3 × 24 3 × 3 × 126 3 × 3 × 24 3 × 3 × 126 3 × 3 × 16 3 × 3 × 84 Input Size 4×4 4×4 2×2 2×2 37 × 37 37 × 37 18 × 18 18 × 18 9×9 9×9 4×4 4×4 2×2 2×2 1×1 1×1 parameters in each layer. Model size 60.5MB 2.3MB mAP (VOC 2007) 57.1% 61.3% Table IV R ESOURCE USAGE OF T INY SSD. Model Name Tiny SSD V. E XPERIMENTAL R ESULTS AND D ISCUSSION To study the utility of Tiny SSD for real-time embedded object detection, we examine the model size, object detection accuracies, and computational operations on the VOC2007/2012 datasets. For evaluation purposes, the Tiny YOLO network [10] was used as a baseline reference comparison given its popularity for embedded object detection, and was also demonstrated to possess one of the smallest model sizes in literature for object detection on the VOC 2007/2012 datasets (only 60.5MB in size and requiring just 6.97 billion operations). The VOC2007/2012 datasets consist of natural images that have been annotated with 20 different types of objects, with illustrative examples shown in Figure 4. The tested deep neural networks were trained using the VOC2007/2012 training datasets, and the mean average precision (mAP) was computed on the VOC2007 test dataset to evaluate the object detection accuracy of the deep neural networks. A. Training Setup Table III O BJECT DETECTION ACCURACY RESULTS OF T INY SSD ON VOC 2007 TEST SET. T INY YOLO RESULTS ARE PROVIDED AS A BASELINE COMPARISON . Model Name Tiny YOLO [10] Tiny SSD reductions while having a negligible effect on object detection accuracy. Total number of Parameters 1.13M Total number of MACs 571.09M IV. PARAMETER P RECISION O PTIMIZATION In this section, let us discuss the parameter precision optimization strategy for Tiny SSD. For embedded scenarios where the computational requirements and memory requirements are more strict, an effective strategy for reducing computational and memory footprint of deep neural networks is reducing the data precision of parameters in a deep neural network. In particular, modern CPUs and GPUs have moved towards accelerated mixed precision operations as well as better handling of reduced parameter precision, and thus the ability to take advantage of these factors can yield noticeable improvements for embedded scenarios. For Tiny SSD, the parameters are represented in half precision floating-point, thus leading to further deep neural network model size The proposed Tiny SSD network was trained for 220,000 iterations in the Caffe framework with training batch size of 24. RMSProp was utilized as the training policy with base learning rate set to 0.00001 and γ = 0.5. B. Discussion Table III shows the model size and the object detection accuracy of the proposed Tiny SSD network on the VOC 2007 test dataset, along with the model size and the object detection accuracy of Tiny YOLO. A number of interesting observations can be made. First, the resulting Tiny SSD possesses a model size of 2.3MB, which is ∼26X smaller than Tiny YOLO. The significantly smaller model size of Tiny SSD compared to Tiny YOLO illustrates its efficacy for greatly reducing the memory requirements for leveraging Tiny SSD for real-time embedded object detection purposes. Second, it can be observed that the resulting Tiny SSD was still able to achieve an mAP of 61.3% on the VOC 2007 test dataset, which is ∼4.2% higher than that achieved using Tiny YOLO. Figure 5 demonstrates several example object detection results produced by the proposed Tiny SSD compared to Tiny YOLO. It can be observed that Tiny SSD has comparable object detection results as Tiny YOLO in some cases, while in some cases outperforms Tiny YOLO in assigning more accurate category labels to detected objects. For example, in the first image case, Tiny SSD is able to detect the chair in the scene, while Tiny YOLO misses the chair. In the third image case, Tiny SSD is able to identify the dog in the scene while Tiny YOLO detects two bounding boxes around the dog, with one of the bounding boxes incorrectly labeling it as cat. This significant improvement Figure 4. Example images from the Pascal VOC dataset. The ground-truth bounding boxes and object categories are shown for each image. in object detection accuracy when compared to Tiny YOLO illustrates the efficacy of Tiny SSD for providing more reliable embedded object detection performance. Furthermore, as seen in Table IV, Tiny SSD requires just 571.09 million MAC operations to perform inference, making it well-suited for real-time embedded object detection. These experimental results show that very small deep neural network architectures can be designed for real-time object detection that are wellsuited for embedded scenarios. VI. C ONCLUSIONS In this paper, a single-shot detection deep convolutional neural network called Tiny SSD is introduced for real-time embedded object detection. Composed of a highly optimized, non-uniform Fire sub-network stack and a non-uniform subnetwork stack of highly optimized SSD-based auxiliary convolutional feature layers designed specifically to minimize model size while maintaining object detection performance, Tiny SSD possesses a model size that is ∼26X smaller than Tiny YOLO, requires just 571.09 million MAC operations, while still achieving an mAP of that is ∼4.2% higher than Tiny YOLO on the VOC 2007 test dataset. These results demonstrates the efficacy of designing very small deep neural network architectures such as Tiny SSD for real-time object detection in embedded scenarios. ACKNOWLEDGMENT The authors thank Natural Sciences and Engineering Research Council of Canada, Canada Research Chairs Program, DarwinAI, and Nvidia for hardware support. 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In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 779–788, 2016. [12] Joseph Redmon and Ali Farhadi. YOLO9000: better, faster, stronger. arXiv preprint, 1612, 2016. [13] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster R-CNN: Towards real-time object detection with region proposal networks. In Advances in neural information processing systems, pages 91–99, 2015. [14] Mohammad Javad Shafiee, Brendan Chywl, Francis Li, and Alexander Wong. Fast YOLO: A fast you only look once system for real-time embedded object detection in video. arXiv preprint arXiv:1709.05943, 2017. Input Image Tiny YOLO Tiny SSD Figure 5. Example object detection results produced by the proposed Tiny SSD compared to Tiny YOLO. It can be observed that Tiny SSD has comparable object detection results as Tiny YOLO in some cases, while in some cases outperforms Tiny YOLO in assigning more accurate category labels to detected objects. This significant improvement in object detection accuracy when compared to Tiny YOLO illustrates the efficacy of Tiny SSD for providing more reliable embedded object detection performance. [15] Abhinav Shrivastava, Abhinav Gupta, and Ross Girshick. Training region-based object detectors with online hard example mining. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 761–769, 2016. [16] Bichen Wu, Forrest Iandola, Peter H Jin, and Kurt Keutzer. Squeezedet: Unified, small, low power fully convolutional neural networks for real-time object detection for autonomous driving. arXiv preprint arXiv:1612.01051, 2016. 

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