Smoothing splines on Riemannian manifolds, with applications to 3D shape space

There has been increasing interest in statistical analysis of data lying in manifolds. This paper generalizes a smoothing spline fitting method to Riemannian manifold data based on the technique of unrolling, unwrapping and wrapping originally proposed by Jupp and Kent for spherical data. In particular, we develop such a fitting procedure for shapes of configurations in general m‐dimensional Euclidean space, extending our previous work for two‐dimensional shapes. We show that parallel transport along a geodesic on Kendall shape space is linked to the solution of a homogeneous first‐order differential equation, some of whose coefficients are implicitly defined functions. This finding enables us to approximate the procedure of unrolling and unwrapping by simultaneously solving such equations numerically, and so to find numerical solutions for smoothing splines fitted to higher dimensional shape data. This fitting method is applied to the analysis of some dynamic 3D peptide data.

[1]  Katie E. Severn,et al.  Manifold valued data analysis of samples of networks, with applications in corpus linguistics , 2019, The Annals of Applied Statistics.

[2]  I. Dryden,et al.  Principal nested shape space analysis of molecular dynamics data , 2019, The Annals of Applied Statistics.

[3]  H. Muller,et al.  Fréchet regression for random objects with Euclidean predictors , 2016, The Annals of Statistics.

[4]  Hongtu Zhu,et al.  Regression models on Riemannian symmetric spaces , 2017, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[5]  David B. Dunson,et al.  Extrinsic Local Regression on Manifold-Valued Data , 2015, Journal of the American Statistical Association.

[6]  Anuj Srivastava,et al.  Functional and Shape Data Analysis , 2016 .

[7]  Leif Ellingson,et al.  Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis , 2015 .

[8]  P. Thomas Fletcher,et al.  Intrinsic Polynomials for Regression on Riemannian Manifolds , 2014, Journal of Mathematical Imaging and Vision.

[9]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[10]  Ross C. Walker,et al.  An overview of the Amber biomolecular simulation package , 2013 .

[11]  Hongtu Zhu,et al.  VARYING COEFFICIENT MODEL FOR MODELING DIFFUSION TENSORS ALONG WHITE MATTER TRACTS. , 2013, The annals of applied statistics.

[12]  P. Thomas Fletcher,et al.  Geodesic Regression and the Theory of Least Squares on Riemannian Manifolds , 2012, International Journal of Computer Vision.

[13]  Anuj Srivastava,et al.  Fitting smoothing splines to time-indexed, noisy points on nonlinear manifolds , 2012, Image Vis. Comput..

[14]  Michael Pauley.  Jupp and Kent's cubics in Lie groups , 2011 .

[15]  I. Dryden,et al.  Shape curves and geodesic modelling , 2010 .

[16]  A. Munk,et al.  Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions. Discussion paper with rejoinder. , 2010 .

[17]  K. Mardia,et al.  Functional models of growth for landmark data , 2010 .

[18]  Hongtu Zhu,et al.  Intrinsic Regression Models for Positive-Definite Matrices With Applications to Diffusion Tensor Imaging , 2009, Journal of the American Statistical Association.

[19]  Charles A Laughton,et al.  COCO: A simple tool to enrich the representation of conformational variability in NMR structures , 2009, Proteins.

[20]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[21]  I. Dryden,et al.  Shape-space smoothing splines for planar landmark data , 2007 .

[22]  A. Munk,et al.  INTRINSIC SHAPE ANALYSIS: GEODESIC PCA FOR RIEMANNIAN MANIFOLDS MODULO ISOMETRIC LIE GROUP ACTIONS , 2007 .

[23]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

[24]  Stephan Huckemann,et al.  Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces , 2006, Advances in Applied Probability.

[25]  M. Karplus,et al.  Molecular dynamics and protein function. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[26]  P. Thomas Fletcher,et al.  Principal geodesic analysis for the study of nonlinear statistics of shape , 2004, IEEE Transactions on Medical Imaging.

[27]  R. Koenker,et al.  Penalized triograms: total variation regularization for bivariate smoothing , 2004 .

[28]  Huiling Le,et al.  Unrolling Shape Curves , 2003 .

[29]  B. Berne,et al.  Helix unfolding and intramolecular hydrogen bond dynamics in small α-helices in explicit solvent , 2002 .

[30]  K. Mardia,et al.  ‘Shape, Procrustes tangent projections and bilateral symmetry’ , 2001 .

[31]  A. Pressley.  Elementary Differential Geometry , 2000 .

[32]  H Le,et al.  Detection of shape changes in biological features * , 2000, Journal of microscopy.

[33]  T. K. Carne,et al.  Shape and Shape Theory , 1999 .

[34]  David G. Kendall,et al.  Shape & Shape Theory , 1999 .

[35]  B. Silverman,et al.  Nonparametric regression and generalized linear models , 1994 .

[36]  B. Silverman,et al.  Nonparametric Regression and Generalized Linear Models: A roughness penalty approach , 1993 .

[37]  P. Jupp,et al.  Fitting Smooth Paths to Spherical Data , 1987 .

[38]  D. Kendall.  SHAPE MANIFOLDS, PROCRUSTEAN METRICS, AND COMPLEX PROJECTIVE SPACES , 1984 .

[39]  M Karplus,et al.  Dynamics of proteins: elements and function. , 1983, Annual review of biochemistry.

[40]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[41]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[42]  H. Akaike.  A new look at the statistical model identification , 1974 .

[43]  I. J. Schoenberg.  SPLINE INTERPOLATION AND THE HIGHER DERIVATIVES. , 1964, Proceedings of the National Academy of Sciences of the United States of America.