Recursive Computation of the Fréchet Mean on Non-positively Curved Riemannian Manifolds with Applications

Abstract

Computing the Riemannian center of mass or the finite sample Fréchet mean has attracted enormous attention lately due to the easy availability of data that are manifold valued. Manifold-valued data are encountered in numerous domains including but not limited to medical image computing, mechanics, statistics, machine learning. It is common practice to estimate the finite sample Fréchet mean by using a gradient descent technique to find the minimum of the Fréchet function when it exists. The convergence rate of this gradient descent method depends on many factors including the step size and the variance of the given manifold-valued data. As an alternative to the gradient descent technique, we propose a recursive (incremental) algorithm for estimating the Fréchet mean/expectation (iFEE) of the distribution from which the sample data are drawn. The proposed algorithm can be regarded as a geometric generalization of the well-known incremental algorithm for computing arithmetic mean, since it reinterprets this algebraic formula in terms of geometric operations on geodesics in the more general manifold setting. In particular, given known formulas for geodesics, iFEE does not require any optimization in contrast to the non-incremental counterparts and offers significant improvement in efficiency and flexibility. For the case of simply connected, complete and nonpositively curved Riemannian manifolds, we prove that iFEE converges to the true expectation in the limit. We present several experiments demonstrating the efficiency and accuracy of iFEE in comparison to the non-incremental counterpart for computing the finite sample Fréchet mean of symmetric positive definite matrices as well as applications of iFEE to K-means clustering and diffusion tensor image segmentation.

Appendix

In this appendix, we prove Proposition 2.1 presented in Sect. 2.2. The proof is entirely elementary if we assume the general property of CAT(0)-metric spaces [12] and Toponogov’s comparison theorem (specifically, the easier half of the theorem on manifolds of nonpositive sectional curvature) [7].

Complete Riemannian manifolds of nonpositive sectional curvature form an important subclass of CAT(0)-metric spaces [12]. For our purpose, the detailed definition of CAT(0)-metric spaces is not necessary; instead, we will recall only the features that are used in the proof. A geodesic triangle Γ on a complete Riemannian manifold $\mathcal{M}$ is the union of three geodesic segments joining three points $\mathbf{p},\mathbf{q},\mathbf{r} \in \mathcal{M}$: $\gamma _{1}(0) =\gamma _{3}(1) = \mathbf{p},\gamma _{1}(1) =\gamma _{2}(0) = \mathbf{q},\gamma _{2}(1) =\gamma _{3}(0) = \mathbf{r}$. Its comparison triangle Δ is a triangle in $\mathbb{R}^{2}$ with vertices p, q, r such that the lengths of the sides $\overline{pq},\overline{qr},\overline{rp}$ are equal to the lengths of the geodesic segments $\gamma _{1},\gamma _{2},\gamma _{3}$, respectively. Such comparison triangle always exists for any geodesic triangle in $\mathcal{M}$, and it is unique up to a rigid transform in $\mathbb{R}^{2}$. The correspondence between the three sides and segments extends naturally to points on the triangles as well: a point x ∈ Γ corresponds to a point x ∈ Δ if their associated sides correspond and their distances to the corresponding two endpoints are the same. For example, if x ∈ γ ₁ and $x \in \overline{pq}$, x corresponds to x if $\mathbf{d}_{\mathcal{M}}(\mathbf{x},\mathbf{p}) = \mathbf{d}_{\mathbb{R}^{2}}(x,p),$ and hence $\mathbf{d}_{\mathcal{M}}(\mathbf{x},\mathbf{q}) = \mathbf{d}_{\mathbb{R}^{2}}(x,q)$ as well. An important property enjoyed by any CAT(0)-metric space is that for any pair of points x, y on Γ and their corresponding points x, y on Δ, we have (see Fig. 2.5) $\mathbf{d}_{\mathcal{M}}(\mathbf{x},\mathbf{y}) \leq \mathbf{d}_{\mathbb{R}^{2}}(x,y).$ The importance of this inequality is the upper bound given by the Euclidean distance in $\mathbb{R}^{2}$, and it allows us to bound the integral of the squared distance function on $\mathcal{M}$ by an integral involving squared Euclidean distance that is considerably easier to manage. Finally, for the pair of triangles Γ, Δ, Toponogov’s comparison theorem asserts the angle $\angle (rpq)$ on Δ no smaller than $\angle (\mathbf{r}\mathbf{p}\mathbf{q})$ on Γ.

Fig. 2.5

A geodesic triangle in $\mathcal{M}$ and its comparison triangle in $\mathbb{R}^{2}$. Corresponding sides on the triangles have the same length. By Toponogov’s comparison theorem, the angle $\angle (qpr)$ is not less than the angle $\angle (\mathbf{q}\mathbf{p}\mathbf{r})$ due the nonpositive sectional curvature of $\mathcal{M}$. Furthermore, if $\gamma (t),\overline{\gamma }(t)$ denote the geodesic and straight line joining p, q and p, q, respectively, then the geodesic distance d _t between p and γ(t) is not greater than the Euclidean distance $\overline{d_{t}}$ between p and $\overline{\gamma }(t)$, i.e., $d_{t} \leq \overline{d_{t}}$

Armed with these results, the proof of Proposition 2.1 is straightforward and it involves comparing two triangles in the tangent space T _m . See Fig. 2.1. We restate Proposition 2.1 for convenience below and now present its proof.

Proposition 2.1.

Let x , y , z be two points on a complete Riemannian manifold $\mathcal{M}$ with nonpositive sectional curvature and γ(t) the unique geodesic path joining x , y such that $\gamma (0) = \mathbf{x},\gamma (1) = \mathbf{y}$ . Furthermore, let $x = \mathbf{Log}_{\mathbf{z}}(\mathbf{x}),y = \mathbf{Log}_{\mathbf{z}}(\mathbf{y})$ , and $\overline{\gamma }(t)$ denote the straight line joining x,y such that $\overline{\gamma (0)} = x,\overline{\gamma (1)} = y$ . Then, $\mathbf{d}_{\mathcal{M}}(\gamma (t),\,\mathbf{z}) \leq \|\overline{\gamma (t)}\|.$

Proof.

Given m _k , x _k+1 in $\mathcal{M}$, and m _k+1 determined according to the iFEE algorithm, we will denote $m_{k},x_{k+1}$ and m _k+1, their corresponding points in T _m under the Riemannian logarithm map Log _m . Without loss of generality, we will prove the proposition using $\mathbf{z} = \mathbf{m},\mathbf{x} = \mathbf{x}_{k+1}$, y = m _k . Let $a = \mathbf{d}_{\mathcal{M}}(\mathbf{x}_{k+1},\mathbf{m})$ and $b = \mathbf{d}_{\mathcal{M}}(\mathbf{m}_{k},\mathbf{m}).$ On T _m , we have the first triangle Σ formed by the three vertices: $x_{k+1},m_{k}$, and o the origin with the side lengths $\vert \overline{x_{k+1}\mathbf{o}}\vert = a,\vert \overline{m_{k}\mathbf{o}}\vert = b$. The geodesic triangle Γ on $\mathcal{M}$ spanned by $\mathbf{m},\mathbf{x}_{k+1},\mathbf{m}_{k}$ has a comparison triangle Δ in T _m spanned by o, p, q with $\vert \overline{p\mathbf{o}}\vert = a,\vert \overline{q\mathbf{o}}\vert = b$, and by Toponogov’s comparison theorem,

$$\displaystyle{\theta _{\sigma } \equiv \angle (x_{k+1}\mathbf{o}m_{k}) \leq \angle (p\mathbf{o}q) \equiv \theta _{\delta },}$$

since, by definition, $\angle (x_{k+1}\mathbf{o}m_{k}) = \angle (\mathbf{x}_{k+1}\mathbf{m}\mathbf{m}_{k})$.

For completing the proof, we need to show that the distance between any point on the side $\overline{pq}$ of Δ and the origin is not greater than the distance between its corresponding point on the side $\overline{x_{k+1}m_{k}}$ of Σ and the origin. Specifically, a point $u \in \overline{pq}$ can be written as $u = tp + (1 - t)q,$ for some 0 ≤ t ≤ 1 and its corresponding point v on $\overline{x_{k+1}m_{k}}$ is the point $v = tx_{k+1} + (1 - t)m_{k}.$ Since the triangle Δ is unique up to a rigid transform, we can, without loss of generality, assume that the two triangles Δ, Σ are contained in a two-dimensional subspace of T _m such that (using the obvious coordinates) they are spanned by the following two sets of three points:

$$\displaystyle\begin{array}{rcl} \varDelta:& & \qquad (0,0),(a,0),\,(b\cos \theta _{\delta },\,b\sin \theta _{\delta }), {}\\ \varSigma:& & \qquad (0,0),(a,0),\,(b\cos \theta _{\sigma },\,b\sin \theta _{\sigma }), {}\\ \end{array}$$

with θ _σ  ≤ θ _δ . Consequently, $u = (ta + (1 - t)b\cos \theta _{\delta },(1 - t)b\sin \theta _{\delta })$ and $v = (ta + (1 - t)b\cos \theta _{\sigma },(1 - t)b\sin \theta _{\sigma }),$ and their lengths are, respectively,

$$\displaystyle\begin{array}{rcl} \|u\|& =& \sqrt{t^{2 } a^{2 } + 2t(1 - t)ab\cos \theta _{\delta } + (1 - t)^{2}}, {}\\ \|v\|& =& \sqrt{t^{2 } a^{2 } + 2t(1 - t)ab\cos \theta _{\sigma } + (1 - t)^{2}}. {}\\ \end{array}$$

Since $\theta _{\sigma } \leq \theta _{\delta }$, it then follows that $\|u\| \leq \| v\|$.