We formulate a generalized version of the classical linear regression problem on Riemannian manifolds and derive the counterpart to the normal equations for the manifold of symmetric and positive definite matrices, equipped with the only metric that is invariant under the natural action of the general linear group. 1. Introduction The geometry of the set of symmetric and positive definite (SPD) matrices is in the focus of intensive research activity involving tensor analysis. The importance of SPD matrices lies in the fact that they encode image information. As a consequence, they appear in several contexts of computer vision, such as, for instance, in [1, 2], in medical image analysis to interpolate scattered diffusion tensor magnetic resonance imaging, [3, 4], but also in the area of continuum physics related to averaging methods for the case of the elasticity tensor of the generalized Hooke’s law [5]. Since the set of SPD matrices has a natural structure of Riemannian manifold, the rich theory of differential geometry can be used to solve real problems that may be formulated on this manifold. One particular problem of interest, that as far as we know has not been studied before, is that of approximating a set of data points in the SPD manifold by a geodesic. In the present paper, we first formulate the problem of finding the geodesic that best fits a given set of time-labelled points on a general Riemannian manifold. This corresponds to the natural generalization of the classical linear regression problem in . Solving this problem on a Riemannian manifold requires knowing explicit formulas for geodesics and for the geodesic distance between two points. Such is the case of connected and compact Lie groups, where geodesics are one-parameter subgroups or their translations, or of Euclidean spheres, where geodesics are the great circles. The geodesic regression problem has been studied for these two cases in [6], and the numerical implementation of the spherical case will appear soon in [7]. Our main objective is to derive the counterpart of the normal equations when the given data lies in the SPD manifold, equipped with a particular Riemannian metric that is affine-invariant. The paper is organized as follows. In Section 2, we start with the formulation of the geodesic fitting problem for the general case of a geodesically complete Riemannian manifold. In Section 3, we specialize to the case of the SPD manifold endowed with its natural affine-invariant Riemannian metric and gather all the necessary background to achieve our goal. The main result appears
References
[1]
F. Porikli, O. Tuzel, and P. Meer, “Covariance tracking using model update based on Lie algebra,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR '06), pp. 728–735, June 2006.
[2]
O. Tuzel, F. Porikli, and P. Meer, “Region covariance: a fast descriptor for detection and classification,” in Proceedings of the 9th European Conference on Computer Vision, 2006.
[3]
P. Filiard, X. Pennec, V. Arsigny, and N. Ayache, “Clinical DT-MRI estimation, smoothing, and fiber tracking with log-euclidean metrics,” IEEE Transactions on Medical Imaging, vol. 26, no. 11, pp. 1472–1482, 2007.
[4]
M. Moakher and P. Batchelor, “Symmetric positive-definite matrices: from geometry to applications and visualization,” in Visualization and Image Processing of Tensor Fields, J. Weickert and H. Hagen, Eds., Chapter 17, Springer, Berlin, Germany, 2005.
[5]
M. Moakher, “On the averaging of symmetric positive-definite tensors,” Journal of Elasticity, vol. 82, no. 3, pp. 273–296, 2006.
[6]
L. Machado and F. Silva Leite, “Fitting smooth paths on riemannian manifolds,” International Journal of Applied Mathematics & Statistics, no. J06, pp. 25–53, 2006.
[7]
L. Machado and T. Monteiro, “Geodesic regression on spheres: a numerical optimization approach,” in Proceedings of the 13th International Conference on Mathematical Methods in Science and Engineering (CMMSE '13), Almería, Spain.
[8]
M. P. do Carmo, Riemannian Geometry, Mathematics: Theory and Applications, Birk？user, 1992.
[9]
J. Jost, Riemannian Geometry and Geometric Analysis, Universitext; Springer, 6 edition, 2011.
[10]
J. W. Milnor, Morse Theory, Princeton University Press, Princeton, NJ, USA, 1963.
[11]
H. Karcher, “Riemannian center of mass and mollifier smoothing,” Communications on Pure and Applied Mathematics, vol. 30, pp. 509–541, 1977.
[12]
E. Batzies, L. Machado, and F. Silva Leite, The Geometric Mean and the Geodesic Fitting Problem on the Grassmann Manifold, Department of Mathematics; University of Coimbra, 2013.
[13]
L. Machado and F. Silva Leite, “Interpolation and polynomial fitting on the SPD manifold,” in Proceedings of the 52nd IEEE Conference on Decision and Control (CDC '13), Florence, Italy, December 2013.
[14]
F. Hiai and D. Petz, “Riemannian metrics on positive definite matrices related to means,” Linear Algebra and Its Applications, vol. 430, no. 11-12, pp. 3105–3130, 2009.
[15]
M. Moakher, “A differential geometric approach to the geometric mean of symmetric positive-definite matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 3, pp. 735–747, 2005.
[16]
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, NY, USA, 1991.
[17]
D. H. Sattinger and O. L. Weaver, Lie Groups and Algebras With Applications to Physics, Geometry, and Mechanics, Springer, 1980.
[18]
V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 29, no. 1, pp. 328–347, 2007.
[19]
P. E. Jupp and J. T. Kent, “Fitting smooth paths to spherical data,” Applied Statistics, vol. 36, no. 1, pp. 34–46, 1987.
[20]
P. Crouch and F. S. Leite, “The dynamic interpolation problem: on Riemannian manifolds, Lie groups, and symmetric spaces,” Journal of Dynamical and Control Systems, vol. 1, no. 2, pp. 177–202, 1995.
[21]
L. MacHado, F. Silva Leite, and K. Krakowski, “Higher-order smoothing splines versus least squares problems on Riemannian manifolds,” Journal of Dynamical and Control Systems, vol. 16, no. 1, pp. 121–148, 2010.
[22]
L. Noakes, G. Heinzinger, and B. Paden, “Cubic splines on curved spaces,” IMA Journal of Mathematical Control and Information, vol. 6, no. 4, pp. 465–473, 1989.
[23]
R. Giambò, F. Giannoni, and P. Piccione, “An analytical theory for Riemannian cubic polynomials,” IMA Journal of Mathematical Control and Information, vol. 19, no. 4, pp. 445–460, 2002.
