# Mean Estimation on the Diagonal of Product Manifolds
## 2 Background
### Riemannian Geometry  
A Riemannian metric \(g\) on a \(d\)-dimensional differentiable manifold \(M\) is a family of inner products \((g_{p})_{p\in M}\) on each tangent space \(T_{p}M\) varying smoothly in \(p\). The Riemannian metric allows for geometric definitions of, e.g., length of curves, angles of intersections, and volumes on manifolds. A differentiable curve on \(M\) is a map \(\gamma\colon[0,1]\to M\) for which the time derivative \(\gamma^{\prime}(t)\) belongs to \(T_{\gamma_{t}}M\), for each \(t\in(0,1)\). The length of the differentiable curve can then be determined from the Riemannian metric by \(L(\gamma):=\int_{0}^{1}\sqrt{g_{T^{\prime}}(\gamma^{\prime}(t),\gamma^{\prime }(t))}dt=\int_{0}^{1}\left\lVert\gamma^{\prime}(t)\right\rVert_{\gamma_{t}}dt\). Let \(p,q\in M\) and let \(\Gamma\) be the set of differentiable curves joining \(p\) and \(q\), i.e., \(\Gamma=\{\gamma\colon[0,1]\to M|\gamma(0)=p\text{ and }\gamma(1)=q\}\). The (Riemannian) distance between \(p\) and \(q\) is defined as \(d(p,q)=\min_{\gamma\in\Gamma}L(\gamma)\). Minimizing curves are called geodesics.  
A manifold can be parameterized using coordinate charts. The charts consist of open subsets of \(M\) providing a global cover of \(M\) such that each subset is diffeomorphic to  
Figure 1: **(left)** The mean estimator viewed as a projection onto the diagonal of a product manifold. Given a set \(x_{1},\dots,x_{n}\in M\), the tuple \((x_{1},\dots,x_{n})\) (blue dot) belongs to the product manifold \(M\times\dots\times M\). The mean estimator \(\hat{\mu}\) can be identified with the projection of \((x_{1},\dots,x_{n})\) onto the diagonal \(N\) (red dot). **(right)** Diffusion mean estimator in \(\mathbb{R}^{2}\) using Brownian bridges conditioned on the diagonal. Here a Brownian bridge \(X_{t}=(X_{1,t},\dots,X_{4,t})\) in \(\mathbb{R}^{8}\) is conditioned on hitting the diagonal \(N\subseteq\mathbb{R}^{8}\) at time \(T>0\). The components \(X_{j}\) each being two-dimensional processes are shown in the plot.  
an open subset of \(\mathbb{R}^{d}\), or, equivalently, \(\mathbb{R}^{d}\) itself. The exponential normal chart is often a convenient choice to parameterize a manifold for computational purposes. The exponential chart is related to the exponential map \(\exp_{p}\colon T_{p}M\to M\) that for each \(p\in M\) is given by \(\exp_{p}(v)=\gamma_{v}(1)\), where \(\gamma_{v}\) is the unique geodesic satisfying \(\gamma_{v}(0)=p\) and \(\gamma_{v}^{\prime}(0)=v\). For each \(p\in M\), the exponential map is a diffeomorphism from a star-shaped subset \(V\) centered at the origin of \(T_{p}M\) to its image \(\exp_{p}(V)\subseteq M\), covering all \(M\) except for a subset of (Riemannian) volume measure zero, \(\text{Cut}(p)\), the cut-locus of \(p\). The inverse map \(\log_{p}\colon M\backslash\text{Cut}(p)\to T_{p}M\) provides a local parameterization of \(M\) due to the isomorphism between \(T_{p}M\) and \(\mathbb{R}^{d}\). For submanifolds \(N\subseteq M\), the cut-locus \(\text{Cut}(N)\) is defined in a fashion similar to \(\text{Cut}(p)\), see e.g., [8].  
Stochastic differential equations on manifolds are often conveniently expressed using the frame bundle \(FM\), the fiber bundle which for each point \(p\in M\) assigns a frame or basis for the tangent space \(T_{p}M\), i.e., \(FM\) consists of a collection of pairs \((p,u)\), where \(u\colon\mathbb{R}^{d}\to T_{p}M\) is a linear isomorphism. We let \(\pi\) denote the projection \(\pi:FM\to M\). There exist a subbundle of \(FM\) consisting of orthonormal frames called the orthonormal frame bundle \(OM\). In this case, the map \(u\colon\mathbb{R}^{d}\to T_{p}M\) is a linear isometry.