Acknowledgements

We thank Prof. David E. Vaillancourt and Dr. Edward Ofori for providing the diffusion MRI scans of movement disorder patients. We also thank Dr. Xavier Pennec and the anonymous reviewer for their invaluable comments in improving this manuscript. This research was in part supported by the NSF grants IIS-1525431 and IIS-1724174.

7.1 Introduction

Manifold-valued data have gained much importance in recent times due to their expressiveness and ready availability of machines with powerful CPUs and large storage. For example, these data arise as diffusion tensors (manifold of symmetric positive definite matrices) [1,2], linear subspaces (the Grassmann manifold) [3–8], column orthogonal matrices (the Stiefel manifold) [3,9–11], directional data and probability densities (the hypersphere) [12–15], and others. A useful method of analyzing manifold-valued data is to compute statistics on the underlying manifold. The most popular statistic is a summary of the data, that is, the Riemannian barycenter (Fréchet mean, FM) [16–18], Fréchet median [19,20], and so on.

FM computation on Riemannian manifolds has been an active area of research for the past few decades. Several researchers have addressed this problem, and we refer the reader to [21,18,22,23,19,24–27]. In most of these works, the authors relied on the standard gradient descent-based iterative computation of the FM, which suffers from two major drawbacks in an online computation setting: (1) for each new sample, it has to compute the new FM from scratch, and (2) it requires the entire input data to be stored to estimate the new FM. Instead, an incremental that is, a recursive technique can address this problem more efficiently with respect to time/space utility. In this age of massive and continuous streaming data, samples are often acquired incrementally. Hence, also from the applications perspective, the desired algorithm should be recursive/inductive to maximize computational efficiency and account for availability of data, requirements that are seldom addressed in more theoretically oriented fields.

Recently, several incremental mean estimators for manifold-valued data have been reported [28–31,5,32]. Sturm [29] presented an incremental mean, the so-called inductive mean, and proved its convergence to the true FM for all nonpositively curved (NPC) spaces. In [33] the authors showed several algorithms (including a recursive algorithm) for FM computation for data residing in CAT(0) spaces, which are NPC. They also demonstrated several applications of the same to computer vision and medical imaging. Further, in [31] an incremental FM computation algorithm along with its convergence and applications was presented for a population of symmetric positive definite (SPD) matrices. Recently, Lim [30] presented an inductive FM to estimate the weighted FM of SPD matrices. The convergence analysis in all of these works is applicable only to the samples belonging to NPC spaces, and hence their convergence analysis does not apply to the case of the manifolds with positive sectional curvature, which are our “objects” of interest in this chapter. Arnaudon et al. [34] present a stochastic gradient descent algorithm for barycenter computation of probability measures on Riemannian manifolds under some conditions. Their algorithm is quite general as it is applicable both to nonpositively and positively curved Riemannian manifolds.

In this work we present a novel incremental FM estimator (iFME) of a set of samples on two Riemannian manifolds of positive sectional curvature, that is, the hypersphere and the special orthogonal group. Data samples from either of these aforementioned manifolds are very commonly encountered in medical imaging, computer vision, and computer graphics. To mention a few, the directional data that are often encountered in image processing and computer vision are points on the unit 2-sphere $\mathbb{S}(2)$ [12]. Further, $3\times 3$ rotation matrices can be parameterized by unit quaternions, which can be represented by points on the projective space $\mathcal{P}(3)$, i.e. couples of points and their antipodal point on the three-dimensional unit sphere $\mathbb{S}(3)$ [15]. Also, any probability density function, for example, orientation distribution function (ODF) in diffusion magnetic resonance imaging (MRI) [14], can be represented as points on the positive quadrant of a unit Hilbert sphere [35,13].

7.2 Riemannian geometry of the hypersphere

The hypersphere is a constant positive curvature Riemannian manifold that is commonly encountered in numerous application problems. Its geometry is well known, and here we will simply present (without derivations) the closed-form expressions for the Riemannian Exponential and Log maps as well as the geodesic between two points on it. Further, we also present the well-known square root parameterization of probability density functions (PDFs), which allows us to identify them with points on the unit Hilbert sphere. This will be needed in representing the probability density functions, namely, the ensemble average propagators (EAPs) derived from diffusion MRI, as points on the unit Hilbert sphere.

Without loss of generality we restrict the analysis to PDFs defined on the interval $[0,T]$ for simplicity: $\mathcal{P}=\{p:[0,T]\to \mathbb{R}|\forall s,p(s)⩾0,{\int }_0^{T}p(s)ds=1\}$. In [26], the Fisher–Rao metric was introduced to study the Riemannian structure of a statistical manifold (the manifold of probability densities). For a PDF $p\in \mathcal{P}$, the Fisher–Rao metric is defined as $\langle v,w\rangle ={\int }_0^{T}v(s)w(s)p(s)ds$ for $v,w\in T_p\mathcal{P}$. The Fisher–Rao metric is invariant to reparameterizations of the functions. To facilitate easy computations when using Riemannian operations, the square root density representation $\psi =\sqrt{p}$ was used in [13], which was originally proposed in [36,37] and further developed from geometric statistics view point in [38]. The space of square root density functions is defined as $\mathrm{\Psi }=\{\psi :[0,T]\to \mathbb{R}|\forall s,\psi (s)⩾0,{\int }_0^T{\psi }^2(s)ds=1\}$. As we can see, Ψ forms a convex subset of the unit sphere in a Hilbert space. Then, the Fisher–Rao metric can be written as $\langle v,w\rangle ={\int }_0^{T}v(s)w(s)ds$ for tangent vectors $v,w\in T_{\psi }\mathrm{\Psi }$. Given any two functions ${\psi }_i,{\psi }_j\in \mathrm{\Psi }$, the geodesic distance between these two points is given in closed form by $d({\psi }_i,{\psi }_j)={\cos }^{-1}(\langle {\psi }_i,{\psi }_j\rangle )$. The geodesic at ψ with a direction $\mathbf{v}\in T_{\psi }\mathrm{\Psi }$ is defined as $\mathrm{\Gamma }(t)=\cos (t)\psi +\sin (t)\frac{\mathbf{v}}{\Vert \mathbf{v}\Vert }$. The Riemannian exponential map can then be expressed by ${\mathsf{\text{Exp}}}_{\psi }(\mathbf{v})=\cos (\Vert \mathbf{v}\Vert )\psi +\sin (\Vert \mathbf{v}\Vert )\frac{\mathbf{v}}{\Vert \mathbf{v}\Vert }$, where $\Vert \mathbf{v}\Vert \in [0,\pi )$. The Riemannian inverse exponential map is then given by ${\mathsf{\text{Log}}}_{{\psi }_i}({\psi }_j)=u{\cos }^{-1}(\langle {\psi }_i,{\psi }_j\rangle )/\sqrt{\langle u,u\rangle }$, where $u={\psi }_j-\langle {\psi }_i,{\psi }_j\rangle {\psi }_i$. Note that, for the rest of this chapter, we will assume that the data points are within a geodesic ball of radius less than the injectivity radius so that the Riemannian exponential is unique, and hence there always exists the corresponding inverse exponential map. We can define the geodesic between ${\psi }_i$ and ${\psi }_j$ by ${\mathrm{\Gamma }}_{{\psi }_i}^{{\psi }_j}\left(t\right)=\cos (t)\psi +\sin (t)\frac{\mathbf{v}}{\Vert \mathbf{v}\Vert }$, where $\mathbf{v}={\mathsf{\text{Log}}}_{{\psi }_i}({\psi }_j)$ and $t\in [0,1]$. We will use the term geodesic to denote the shortest geodesic between two points.

Using the geodesic distance provided previously, we can define the Fréchet mean (FM) [16,17] of a set of points on the hypersphere as the minimizer of the sum of squared geodesic distances (the so-called Fréchet functional). Let $\mathcal{B}(\mathbf{x},\rho )$ be the geodesic ball centered at x with radius ρ, that is, $\mathcal{B}(\mathbf{x},\rho )=\{\mathbf{y}\in \mathbb{S}(k)|d(\mathbf{x},\mathbf{y})<\rho \}$. Authors in [39,18] showed that for any $\mathbf{x}\in S_k$ and for data samples in $\mathcal{B}(\mathbf{x},\frac{\pi }{2})$, the minimizer of the Fréchet functional exists and is unique. For the rest of this chapter, we will assume that this condition is satisfied for any given set of points $\left\{{\mathbf{x}}_i\right\}\subset \mathbb{S}(k)$. For more details on Riemannian geometry of the sphere, we refer the reader to Chapter 2 of [40] and references therein.

7.3 Weak consistency of iFME on the sphere

In this section we present a detailed proof of convergence of our recursive estimator on the $\mathbb{S}(k)$. The proposed method is similar in “spirit” to the incremental arithmetic mean update in the Euclidean space; given the old mean ${\mathbf{m}}_{n-1}$ and the new sample point ${\mathbf{x}}_n$, we define the new mean ${\mathbf{m}}_n$ as the weighted mean of ${\mathbf{m}}_{n-1}$ and ${\mathbf{x}}_n$ with the weights $\frac{n-1}{n}$ and $\frac{1}{n}$, respectively. From a geometric viewpoint, this corresponds to the choice of the point on the geodesic between ${\mathbf{m}}_{n-1}$ and ${\mathbf{x}}_n$ with the parameter $t=\frac{1}{n}$, that is, ${\mathbf{m}}_n={\mathrm{\Gamma }}_{{\mathbf{m}}_{n-1}}^{{\mathbf{x}}_n}(1/n)$.

Formally, let ${\mathbf{x}}_1,{\mathbf{x}}_2,\cdots ,{\mathbf{x}}_N$ be a set of N samples on the hypersphere $\mathbb{S}(k)$, all of which lie inside a geodesic ball of radius $\frac{\pi }{2}$. The incremental FM estimator (denoted by iFME) ${\mathbf{m}}_n$ with the given nth sample point ${\mathbf{x}}_n$ is defined by

${\mathbf{m}}_1={\mathbf{x}}_1,$

${\mathbf{m}}_n={\mathrm{\Gamma }}_{{\mathbf{m}}_{n-1}}^{{\mathbf{x}}_n}\left(\frac{1}{n}\right),$

where ${\mathrm{\Gamma }}_{{\mathbf{m}}_{n-1}}^{{\mathbf{x}}_n}$ is the shortest geodesic path from ${\mathbf{m}}_{n-1}$ to ${\mathbf{x}}_n$ ($\in S_k$), and $\frac{1}{n}$ is the weight assigned to the new sample point (in this case the nth sample), which is henceforth called the Euclidean weight. In the rest of this section we will show that if the number of given samples N tends to infinity, then the iFME estimates will converge to the expectation of the distribution from which the samples are drawn. Note that the geometric construction steps in the proof given further are not needed to compute the iFME; these steps are only needed to prove the weak consistency of iFME.

Our proof is based on the idea of projecting the samples ${\mathbf{x}}_i$ on the sphere to the tangent plane using the gnomonic projection [15] and performing the convergence analysis on the projected samples ${\tilde{\mathbf{x}}}_i$ in this linear space, instead of performing the analysis on the hypersphere. We take advantage of the fact that the geodesic curve between any pair of points on the hemisphere is projected to a straight line in the tangent space at the anchor point (in this case, without loss of generality, assumed to be the north pole) via the gnomonic projection. A figure depicting the gnomonic projection is shown in Fig. 7.1. Despite the simplifications used in the statistical analysis of the iFME estimates on the hypersphere using the gnomonic projection, there is one important obstacle that must be considered. Without loss of generality, suppose the true FM of the input samples $\left\{{\mathbf{x}}_i\right\}$ is the north pole. Then it can be shown through counter examples that:

This fact can be illustrated using two sample points on a unit circle ($S_1$), ${\mathbf{x}}_1={(\cos (\pi /6),\sin (\pi /6))}^t$ and ${\mathbf{x}}_2={(\cos (\pi /3),\sin (\pi /3))}^t$, whose intrinsic mean is $\mathbf{m}={(\cos (\pi /4),\sin (\pi /4))}^t$. Then the midpoint of the gnomonic projections of ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$, which are denoted by ${\tilde{\mathbf{x}}}_1$ and ${\tilde{\mathbf{x}}}_2$, is $\hat{\mathbf{m}}=\frac{\tan (\pi /3)+\tan (\pi /6)}{2}=1.1547\ne \tan (\pi /4)=\tilde{\mathbf{m}}$ (see Fig. 7.2).

For the rest of this section, without loss of generality, we assume that the FM of N given samples is located at the north pole. Since the gnomonic projection space is anchored at the north pole, this assumption leads to significant simplifications in our convergence analysis. However, a similar convergence proof can be developed for any arbitrary location of the FM, with the tangent (projection) space anchored at the location of this mean.

In what follows we prove that the use of Euclidean weights $w_n=\frac{1}{n}$ to update the incremental FM on the hypersphere corresponds to a set of weights in the projection space, denoted henceforth by $t_n$, for which the weighted incremental mean in the tangent plane converges to the true FM on the hypersphere, which in this case is the point of tangency.

Then we bound $t_n$ with respect to the radius ρ.

Note that as ρ tends to zero, $\cos (\rho )$ converges to one, and this lower bound tends to $\frac{1}{n}$, which is the case for the Euclidean space.

Thus far we have shown analytical bounds for the sequence of weights $t_n$ in the projection space corresponding to Euclidean weights on the sphere (Eq. (7.4)). We now prove the convergence of iFME estimates to the expectation of distribution from which the samples are drawn as the number of samples tends to infinity.

First note that $\text{Var}[{\tilde{\mathbf{m}}}_n]=t_n^2\text{Var}[\tilde{\mathbf{x}}]+{(1-t_n)}^2\text{Var}[{\tilde{\mathbf{m}}}_{n-1}]$. Since, $0⩽t_n⩽1$, we can see that $\text{Var}[{\tilde{\mathbf{m}}}_n]⩽\text{Var}[\tilde{\mathbf{x}}]$ for all n. Besides, for each n, the maximum of the right-hand side is achieved when $t_n$ attains either its minimum or maximum value. Therefore we need to prove the theorem for the following two values of $t_n$: (i) $t_n=\frac{\cos (\rho )}{n}$ and (ii) $t_n=\frac{1}{n{\cos }^3(\rho )}$. These two cases will be proved in Lemmas 7.3 and 7.4, respectively.

The inequality in Theorem 7.3 implies that as $n\to \infty$, for any $\rho \in [0,\pi /2)$, the variance of iFME estimates in the projection space tends to zero. Besides, when ρ approaches $\pi /2$, the corresponding power of n and $\cos (\rho )$ become very small, hence the rate of convergence gets slower. Note that instead of the weighting scheme used here (i.e., in the spirit of incremental mean in Euclidean space), we can choose a different weighting scheme that is intrinsic to the manifold (i.e., as a function of curvature) to speed up the convergence rate.

Now we will show that our proposed recursive FM estimator has a linear convergence rate.

7.4 Experimental results

We now evaluate the effectiveness of the iFME algorithm, compared to the nonincremental counterpart FME, for computing the FM of a finite set of samples on the sphere (northern hemisphere not including the equator). For FME, we used a gradient descent technique to minimize the sum of squared geodesic distances cost function. We report the results for samples drawn from a Log-Normal distribution (with mean at the north pole and the variance set to 0.2) on the upper hemisphere. A set of random samples are drawn from the distribution and incrementally input to both iFME and FME algorithms. The computation time needed for each method to compute the sample FM and the error were recorded for each new sample incrementally introduced. In all the experiments the computation time is reported for execution on a 3.3 GHz desktop with a quadcore Intel i7 processor and 24 GB RAM. The error is defined by the geodesic distance between the estimated mean (using either iFME or FME) and the expectation of the input distribution. Because of the randomness in generating the samples, we repeated this experiment 100 times for each case, and the mean execution time and the error for each method are shown.

The performances of iFME and FME are evaluated with respect to the execution time and error and are illustrated in Fig. 7.3. From these plots we can clearly see that iFME performs almost equally good in terms of error/accuracy but takes comparatively very less execution time.

7.5 Application to the classification of movement disorders

In this section we use an exact-PGA (exact principal geodesic analysis) algorithm presented in [44], which is applicable to data residing on a hypersphere. We will call this the ePGA algorithm. We will also use the tangent PCA (tPCA) algorithm presented in [45] for diffusion tensor fields as a comparison. This algorithm consists of (1) computing the FM of the input data, (2) projecting each data point to the tangent space at the FM using the Riemannian log-map, (3) performing standard PCA in the tangent plane, and (4) projecting the result (principal vectors) back to the manifold using the Riemannian exponential map.

In the ePGA algorithm we will use iFME to compute the FM and make use of the parallel transport operation on the hypersphere. For completeness, we now present the ePGA algorithm from [44] and used here.

The projection algorithm/step in the above algorithm has an analytic expression for manifolds with constant sectional curvature, that is, for hypersphere and hyperbolic space. Here we will present the projection algorithm.

Note that the parallel transport operation on the hypersphere can be expressed in analytic form. The formula for parallel transporting $\mathbf{u}\in T_{\mathbf{n}}\mathbb{S}(k)$ from n to m is given by

$\mathbf{w}={\mathrm{\Gamma }}_{\mathbf{n}\to \mathbf{m}}(\mathbf{p})=(\mathbf{p}-\mathbf{v}\left(\frac{{\mathbf{v}}^t\mathbf{p}}{{\Vert \mathbf{v}\Vert }^2}\right))+\frac{{\mathbf{v}}^t\mathbf{p}}{{\Vert \mathbf{v}\Vert }^2}(\mathbf{n}(-\sin (\Vert \mathbf{v}\Vert )\Vert \mathbf{v}\Vert )+\mathbf{v}\cos (\Vert \mathbf{v}\Vert )),$

where $\mathbf{v}=Lo{g}_{\mathbf{n}}\mathbf{m}$. We refer the reader to [44] for details of the ePGA algorithm.

The dataset for classification consists of high angular resolution diffusion image (HARDI) scans from (1) healthy controls, patients with (2) Parkinson's disease (PD), and (3) essential tremor (ET). We aim to automatically discriminate between these three classes using features derived from the HARDI data. This dataset consists of 25 controls, 24 PD, and 15 ET images. The HARDI data were acquired using a 3T Phillips MR scanner with the following parameters: $TR=7748$ ms, $TE=86$ ms, b-values: 0, $1000\frac{\mathrm{s}}{\mathrm{m}{\mathrm{m}}^{\mathrm{2}}}$, 64 gradient directions, and voxel size = $2\times 2\times 2\mathrm{m}{\mathrm{m}}^{\mathrm{3}}$.

Authors in [46] employed DTI-based analysis using scalar-valued features to address the problem of movement disorder classification. Later in [47] a PGA-based classification algorithm was proposed using Cauchy deformation tensors (computed from a nonrigid registration of patient scans to a HARDI control atlas), which are SPD matrices. In the next subsection we develop classification method based on (1) Ensemble Average Propagators (EAPs) derived from HARDI data within an ROI and (2) shapes of the ROI derived from the input population. Using a square root density parameterization [48], both features can be mapped to points on an unit Hilbert sphere, where the proposed iFME in conjunction with the ePGA method is applicable.

Classification results using the ensemble average propagator as features: To capture the full diffusional information, we chose to use the ensemble average propagator (EAP) at each voxel as our feature in the classification. We compute the EAPs using the method described in [49] and use the square root density parameterization of each EAP. This way the full diffusion information at each voxel is represented as a point on the unit Hilbert sphere.

We now present the classification algorithm, which is a combination of ePGA-based reduced representation and a nearest-neighbor classifier. The input to the ePGA algorithm are EAP features in this case. The input HARDI data are first rigidly aligned to the atlas computed from the control (normal) group, then EAPs are computed in a 3-D region of interest (ROI) in the midbrain. Finally, the EAP field extracted from each ROI image is identified with a point on the product manifold (the number of elements in the product is equal to the number of voxels in the ROI) of unit Hilbert spheres. This is in spirit similar to the case of the product manifold formalism in [45,47].

A set of 10 control, 10 PD, and 5 ET images are randomly picked as the test set, and the rest of the images are used for training. Also, classification is performed using ePGA, tPCA, and the standard PCA and is repeated 300 times to report the average accuracy. The results using EAP features are summarized in Table 7.2. It is evident that the accuracy of ePGA is better than that of the tPCA, whereas both methods are considerably more accurate than the standard PCA, as they account for the nonlinear geometry of the sphere.

Classification results using the shape features: In this section we evaluated the ePGA algorithm based on shape of the Substantia Nigra region in the given brain images for the task of movement disorder classification. We first collected random samples (point) on the boundary of each 3-D shape and applied the Schrödinger distance transform (SDT) technique in [50] to represent each shape as a point on the unit hypersphere. The size of the ROI for the 3-D shape of interest was set to $28\times 28\times 15$, and the resulting samples lie on an $S_{11759}$ manifold. Then we used ePGA for classification. The results reported in Table 7.1 depict the accuracy gained in classification when using ePGA compared to tPCA. Evidently, ePGA yields a better classification accuracy.

7.6 Riemannian geometry of the special orthogonal group

The set of all $n\times n$ orthogonal matrices is denoted by $\mathrm{O}(n)$, that is, $\mathrm{O}(n)=\{X\in {\mathbf{R}}^{n\times n}|X^{T}X=I_n\}$. The set of orthogonal matrices with determinant 1, denoted by the special orthogonal group $\mathfrak{so}(n)$, forms a compact subset of $\mathrm{O}(n)$. As $\mathfrak{so}(n)$ is a compact Riemannian manifold, by the Hopf–Rinow theorem it is also a geodesically complete manifold [51]. Its geometry is well understood, and we recall a few relevant concepts here and refer the reader to [51] for details. The manifold $\mathfrak{so}(n)$ has a Lie group structure, and the corresponding Lie algebra is defined as $\mathfrak{so}(n)=\{W\in {\mathbf{R}}^{n\times n}|W^T=-W\}$. In other words, $\mathfrak{so}(n)$ (the set of Left invariant vector fields with associated Lie bracket) is the set of $n\times n$ antisymmetric matrices. The Lie bracket $[,]$, operator on $\mathfrak{so}(n)$, is defined as the commutator, that is, $[U,V]=UV-VU$ for $U,V\in \mathfrak{so}(n)$. Now we can define a Riemannian metric on $\mathfrak{so}(n)$ as follows: ${\langle U,V\rangle }_X=trace\left(U^{T}V\right)$ for $U,V\in T_X\mathfrak{so}(n)$, $X\in \mathfrak{so}(n)$. Note that it can be shown that this is a biinvariant Riemannian metric. Under this biinvariant metric, now we define the Riemannian exponential and inverse exponential maps as follows. Let $X,Y\in \mathfrak{so}(n)$, $U\in T_X\mathfrak{so}(n)$. Then the Riemannian inverse exponential map is defined as

${\mathsf{\text{Log}}}_X(Y)=X\log (X^{T}Y),$

and the Riemannian exponential map is defined as

${\mathsf{\text{Exp}}}_X(U)=X\exp (X^{T}U),$

where exp and log are the matrix exponential and logarithm, respectively. Due of the computational complexity of matrix exponential, we may instead choose to use the Riemannian retraction map as follows. Given $W\in \mathfrak{so}(n)$, the Cayley map is a conformal mapping $\text{Cay}:\mathfrak{so}(n)$ $\to \mathfrak{so}(n)$ defined by $\text{Cay}(W)=(I_n+W){(I_n-W)}^{-1}$. Using the Cayley map, we can define the Riemannian retraction map as, ${\mathsf{\text{Ret}}}_X$ $:T_X\mathfrak{so}(n)\to \mathfrak{so}(n)$ by ${\text{Ret}}_X(W)=\text{Cay}(W)X$. Using the Riemannian exponential (retraction) and inverse exponential map, we can define the geodesic on $\mathfrak{so}(n)$ as ${\mathrm{\Gamma }}_X^Y(y)=\mathsf{\text{Exp}}\left(t{\mathsf{\text{Log}}}_X\left(Y\right)\right)$.

7.7 Weak consistency of iFME on $\mathfrak{so}(n)$

In this section we present a detailed proof of convergence of our recursive estimator on $\mathfrak{so}(n)$. Analogous to our FM estimator on $\mathbb{S}(k)$, we first define the FM estimator on $\mathfrak{so}(n)$ and then prove its consistency.

Formally, let $X_1,\dots ,X_N$ be a set of N samples on $\mathfrak{so}(n)$, all of which lie inside a geodesic ball of an appropriate radius such that FM exists and is unique. The iFME estimate $M_n$ of the FM with the nth given sample $X_n$ is defined by

$M_1=X_1,$

$M_n={\mathrm{\Gamma }}_{M_{n-1}}^{X_n}\left(\frac{1}{n}\right),$

where ${\mathrm{\Gamma }}_{M_{n-1}}^{X_n}$ is the shortest geodesic path from $M_{n-1}$ to $X_n$ ($\in \mathfrak{so}(n)$). In what follows we will show that as the number of given samples N tends to infinity, the iFME estimates will converge to the expectation of the distribution from which the samples are drawn.

By Proposition 7.1 we can express X as a product of $n(n-1)/2$ planar rotation matrices. Each planar rotation matrix can be mapped onto $\mathbb{S}(n-1)$, hence there exists a mapping $F:\mathsf{\text{SO}}(n)$ $\to \underset{n(n-1)/2\text{ times}}{\underbrace{\mathbb{S}(n-1)\times \cdots \times {\mathbb{S}}_{n-1}}}$. Let us denote this product space of hyperspheres by $\mathfrak{O}(n-1,\frac{n(n-1)}{2})$. Then F is a embedding of $\mathsf{\text{SO}}(n)$ in $\mathfrak{O}(n-1,\frac{n(n-1)}{2})$. Notice that as this product space has $n(n-1)/2$ components, we will use two indices $(i,j)$ to denote a component. Hence, given $X\in \mathsf{\text{SO}}(n)$, the $(i,j)$th component of $F(X)$ has ith and jth entries $\cos ({\theta }_{ij})$ and $\sin ({\theta }_{ij})$, respectively, and the rest of the $n-2$ entries are zero. Here ${\theta }_{ij}$ is the planar rotation angle in the plane spanned by ${\mathbf{e}}_i$ and ${\mathbf{e}}_j$. The following propositions allow us to prove that $\mathsf{\text{SO}}(n)$ is a submanifold of $\mathfrak{O}(n-1,\frac{n(n-1)}{2})$.

To do that, we will first use the Log-Euclidean metric on $\mathsf{\text{SO}}(n)$ defined as follows: Given $X,Y\in \mathsf{\text{SO}}(n)$, the metric $d(X,Y)=\frac{1}{2}{\Vert \log (X)-\log (Y)\Vert }_F$. Notice that this metric is adjoint invariant, as ${\Vert Z\log (X)Z^T-Z\log (Y)Z^T\Vert }_F$ $={\Vert \log (X)-\log (Y)\Vert }_F$ for some $Z\in \mathsf{\text{SO}}(n)$.