# American Institute of Mathematical Sciences

December  2019, 6(2): 251-276. doi: 10.3934/jcd.2019013

## Principal symmetric space analysis

 1 Department of Mathematical Sciences, NTNU, Gjøvik, Norway 2 School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand 3 School of Fundamental Sciences, Massey University, Palmerston North, New Zealand

Received  April 2019 Revised  October 2019 Published  November 2019

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.

Citation: Charles Curry, Stephen Marsland, Robert I McLachlan. Principal symmetric space analysis. Journal of Computational Dynamics, 2019, 6 (2) : 251-276. doi: 10.3934/jcd.2019013
##### References:
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##### References:
 [1] J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom., 104 (2016), 187-217.  doi: 10.4310/jdg/1476367055. [2] B. Y. Chen and T. Nagano, Totally geodesic submanifolds of symmetric spaces, Ⅱ, Duke Math J., 45 (1978), 405-425.  doi: 10.1215/S0012-7094-78-04521-0. [3] J. H. Conway, R. H. Hardin and N. J. Sloane, Packing lines, planes, etc.: Packings in Grassmannian spaces, Experiment. Math., 5 (1996), 139-159.  doi: 10.1080/10586458.1996.10504585. [4] J. Damon and J. S. Marron, Backwards principal component analysis and principal nested relations, J. Math. Imaging Vision, 50 (2014), 107-114.  doi: 10.1007/s10851-013-0463-2. [5] B. Eltzner, S. Jung and S. Huckermann, Dimension reduction on polyspheres with application to skeletal representations, in Geometric Science of Information, Lecture Notes in Comput. Sci., 9389, Springer, Cham, 2015, 22–29. doi: 10.1007/978-3-319-25040-3_3. [6] P. T. Fletcher, C. Lu, S. M. Pizer and S. C. Joshi, Principal geodesic analysis for the study of nonlinear statistics of shape, IEEE Transactions on Medical Imaging, 23 (2004), 995-1005.  doi: 10.1109/TMI.2004.831793. [7] P. T. Fletcher and S. C. Joshi, Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors, in Computer Vision and Mathematical Methods in Medical and Biomedical Image Analysis, LNCS, 3117, Springer, Berlin, 2004, 87–98. [8] T. Fletcher, Geodesic regression on Riemannian manifolds, in Proceedings of the Third International Workshop on Mathematical Foundations of Computational Anatomy-Geometrical and Statistical Methods for Modelling Biological Shape Variability, 2001, 75–86. [9] C. G. Gebhardt, M. C. Steinbach and R. Rolfes, Understanding the nonlinear dynamics of beam structures: A principal geodesic analysis approach, Thin-Walled Structures, 140 (2019), 357-372.  doi: 10.1016/j.tws.2019.03.009. [10] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Series in the Mathematical Sciences, 3, Johns Hopkins University Press, Baltimore, MD, 1989. [11] S. Huckemann, T. Hotz and A. Munk, Intrinsic shape analysis: Geodesic PCA for Riemannian manifolds modulo isometric Lie group actions, Statist. Sinica, 20 (2010), 1-58. [12] I. Jolliffe, Principal Component Analysis, Springer Series in Statistics, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4757-1904-8. [13] S. Jung, I. L. Dryden and J. S. Marron, Analysis of principal nested spheres, Biometrika, 99 (2012), 551-568.  doi: 10.1093/biomet/ass022. [14] K. Kenobi, I. L. Dryden and H. Le, Shape curves and geodesic modelling, Biometrika, 97 (2010), 567-584.  doi: 10.1093/biomet/asq027. [15] S. Klein, Totally geodesic submanifolds in Riemannian symmetric spaces, in Differential Geometry, World Sci. Publ., Hackensack, NJ, 2009, 136–145. doi: 10.1142/9789814261173_0013. [16] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, John Wiley & Sons, Inc., New York, 1996. [17] J. Lawrence, Enumeration in torus arrangements, European J. Combin., 32 (2011), 870-881.  doi: 10.1016/j.ejc.2011.02.003. [18] A. K. Lenstra, H. W. Lenstra Jr. and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann., 261 (1982), 515-534.  doi: 10.1007/BF01457454. [19] X. Pennec, Barycentric subspace analysis on manifolds, Ann. Statist., 46 (2018), 2711-2746.  doi: 10.1214/17-AOS1636. [20] Q. Rentmeesters, A gradient method for geodesic data fitting on some symmetric Riemannian manifolds, in 2011 50$^{th}$ IEEE Conference on Decision and Control and European Control Conference, 2011, 7141–7146. doi: 10.1109/CDC.2011.6161280. [21] H. J. S. Smith, On systems of linear indeterminate equations and congruences, Phil. Trans. Roy. Soc. London, 151 (1861), 293-326. [22] S. Sommer, F. Lauze and M. Nielsen, Optimization over geodesics for exact principal geodesic analysis, Adv. Comput. Math., 40 (2014), 283-313.  doi: 10.1007/s10444-013-9308-1. [23] W. Utschick, Tracking of signal subspace projectors, IEEE Trans. Signal Process., 50 (2002), 769-778.  doi: 10.1109/78.992119. [24] J. A. Wolf, Elliptic spaces in Grassmann manifolds, Illinois J. Math., 7 (1963), 447-462.  doi: 10.1215/ijm/1255644952. [25] D. Wubben, D. Seethaler, J. Jalden and G. Matz, Lattice reduction, IEEE Signal Processing Magazine, 28 (2011), 70-91.  doi: 10.1109/MSP.2010.938758.
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
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
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
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
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
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
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 $S^1\subset S^2\subset S^2\times S^2$
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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