Article Contents
Article Contents

Principal symmetric space analysis

• Principal Geodesic Analysis is a statistical technique that constructs low-dimensional approximations to data on Riemannian manifolds. It provides a generalization of principal components analysis to non-Euclidean spaces. The approximating submanifolds are geodesic at a reference point such as the intrinsic mean of the data. However, they are local methods as the approximation depends on the reference point and does not take into account the curvature of the manifold. Therefore, in this paper we develop a specialization of principal geodesic analysis, Principal Symmetric Space Analysis, based on nested sequences of totally geodesic submanifolds of symmetric spaces. The examples of spheres, Grassmannians, tori, and products of two-dimensional spheres are worked out in detail. The approximating submanifolds are geometrically the simplest possible, with zero exterior curvature at all points. They can deal with significant curvature and diverse topology. We show that in many cases the distance between a point and the submanifold can be computed analytically and there is a related metric that reduces the computation of principal symmetric space approximations to linear algebra.

Mathematics Subject Classification: Primary: 62H25, 53C35, 53C42.

 Citation:

• Figure 1.  In linearized Principal Geodesic Analysis, the data (here 20 points on a sphere) is pulled back to the tangent space (shown here as a disk) of the intrinsic mean using geodesics. Data points near the mean are well represented, but data points far from the mean (here, near the south pole) become far apart in the linear approximation

Figure 2.  Twenty data points on $S^2$ (dark blue) together with the point (pink) and the great circle (blue) that best approximate the data in the Riemannian metric, computed using nonlinear optimization. The best point (the intrinsic mean of the data) does not lie on the best great circle

Figure 3.  Results for datasets 1–3 of Example 2. In each case, the best subspheres that approximate a set of 20 points on $S^3$ is shown. Data points further from the best $S^2$ are shown smaller. The best $S^1$ is shown in blue, lying on the best $S^2$ in teal. In datasets 2 and 3, the axis of the best $S^0$ (which consists of two antipodal points) is shown in black. In dataset 3, this also coincides with a standard mean of the data

Figure 4.  The geodesic in $\mathbb{T} ^3$ through the origin in direction $d = [1, 2, 3]^\mathsf{T}$, seen from two different viewpoints. Viewing in direction $d$ (right) shows the lattice formed in $\mathbb{R}^3$ by the intersection of the lifted geodesic with an orthogonal plane

Figure 6.  Fitting data on a torus (see Example 4). The best 1-torus and 2-torus approximating 50 data points on $\mathbb{T} ^3$, are shown from two different viewing directions. The subtori have been chosen from those with fixed resonance relations, i.e., only their translations fitted

Figure 5.  Fitting data on a torus. Here the closed geodesic of best fit is computed to a set of 50 data points on $S^1\times S^1$. The data set is synthetic and has been chosen to lie near the geodesic with resonance relation $2 x_1 + 5 x_2 =$ const.; each data point has normal random noise of standard deviation $0.1/(2\pi)$ in each angle

Figure 7.  Fitting data on a polysphere (see Example 6). The rotations of points $x_i$ are plotted as cubes, whilst the points $y_i$ are plotted as circles; a different colour is chosen for each $i$. The great circle shows the best subspace $S^1\subset S^2\subset S^2\times S^2$

Figure 8.  Fitting data on a polysphere (see Example 6). The best approximating torus $S^1\times S^1\subset S^2\times S^2$ is shown with two great circles inside the two spheres. Due to the difficulty of plotting the nested approximation $S^1\subset S^2\times S^2$ we have plotted (right) the projection of the points in $S^2\times S^2$ to the approximating torus $S^1\times S^1$ and shown the best approximating $S^1$ as a subset of this

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