Geodesic Regression and the Theory of Least Squares on Riemannian Manifolds
TL;DR: Specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age, which can be generally applied to data on any manifold.
Abstract: This paper develops the theory of geodesic regression and least-squares estimation on Riemannian manifolds. Geodesic regression is a method for finding the relationship between a real-valued independent variable and a manifold-valued dependent random variable, where this relationship is modeled as a geodesic curve on the manifold. Least-squares estimation is formulated intrinsically as a minimization of the sum-of-squared geodesic distances of the data to the estimated model. Geodesic regression is a direct generalization of linear regression to the manifold setting, and it provides a simple parameterization of the estimated relationship as an initial point and velocity, analogous to the intercept and slope. A nonparametric permutation test for determining the significance of the trend is also given. For the case of symmetric spaces, two main theoretical results are established. First, conditions for existence and uniqueness of the least-squares problem are provided. Second, a maximum likelihood criteria is developed for a suitable definition of Gaussian errors on the manifold. While the method can be generally applied to data on any manifold, specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age.
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07 Jun 2015
TL;DR: It is shown that for spaces with conditionally negative definite distances the geodesic Laplacian kernel can be generalized while retaining positive definiteness, which implies that geodesIC La placian kernelsCan be generalized to some curved spaces, including spheres and hyperbolic spaces.
Abstract: We consider kernel methods on general geodesic metric spaces and provide both negative and positive results. First we show that the common Gaussian kernel can only be generalized to a positive definite kernel on a geodesic metric space if the space is flat. As a result, for data on a Riemannian manifold, the geodesic Gaussian kernel is only positive definite if the Riemannian manifold is Euclidean. This implies that any attempt to design geodesic Gaussian kernels on curved Riemannian manifolds is futile. However, we show that for spaces with conditionally negative definite distances the geodesic Laplacian kernel can be generalized while retaining positive definiteness. This implies that geodesic Laplacian kernels can be generalized to some curved spaces, including spheres and hyperbolic spaces. Our theoretical results are verified empirically.
94 citations
Cites background from "Geodesic Regression and the Theory ..."
...Emerging generalizations of learning tools such as regression [24, 33, 57] or transfer learning [27, 65] to nonlinear data spaces are encouraging....
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TL;DR: The results show that Riemannian polynomials provide a practical model for parametric curve regression, while offering increased flexibility over geodesics.
Abstract: We develop a framework for polynomial regression on Riemannian manifolds. Unlike recently developed spline models on Riemannian manifolds, Riemannian polynomials offer the ability to model parametric polynomials of all integer orders, odd and even. An intrinsic adjoint method is employed to compute variations of the matching functional, and polynomial regression is accomplished using a gradient-based optimization scheme. We apply our polynomial regression framework in the context of shape analysis in Kendall shape space as well as in diffeomorphic landmark space. Our algorithm is shown to be particularly convenient in Riemannian manifolds with additional symmetry, such as Lie groups and homogeneous spaces with right or left invariant metrics. As a particularly important example, we also apply polynomial regression to time-series imaging data using a right invariant Sobolev metric on the diffeomorphism group. The results show that Riemannian polynomials provide a practical model for parametric curve regression, while offering increased flexibility over geodesics.
92 citations
Cites background or methods from "Geodesic Regression and the Theory ..."
...So this approach subsumes geodesic regression as presented by Fletcher [12]....
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...In order to characterize how well our model fits a given set of data, we define the coefficient of determination of our regression curve γ (t), denoted R2 [12]....
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...Fletcher showed [12] that more nuanced modes of shape change are observed using geodesic regression....
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...However, for parametric curve regression, curve models are preferred that don’t depend on the data, such as the initial conditions of a geodesic [12]....
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...As Fletcher [12, Sect....
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TL;DR: Several submanifold mapping problems are discussed as they apply to metamorphosis, multiple shape spaces, and longitudinal time series studies of growth and atrophy via shape splines.
Abstract: The Computational Anatomy project is the morphome-scale study of shape and form, which we model as an orbit under diffeomorphic group action. Metric comparison calculates the geodesic length of the diffeomorphic flow connecting one form to another. Geodesic connection provides a positioning system for coordinatizing the forms and positioning their associated functional information. This article reviews progress since the Euler-Lagrange characterization of the geodesics a decade ago. Geodesic positioning is posed as a series of problems in Hamiltonian control, which emphasize the key reduction from the Eulerian momentum with dimension of the flow of the group, to the parametric coordinates appropriate to the dimension of the submanifolds being positioned. The Hamiltonian viewpoint provides important extensions of the core setting to new, object-informed positioning systems. Several submanifold mapping problems are discussed as they apply to metamorphosis, multiple shape spaces, and longitudinal time series...
82 citations
Proceedings Article•
05 Dec 2013TL;DR: This work presents a latent variable model for PGA that provides a probabilistic framework for factor analysis on manifolds, and develops a Monte Carlo Expectation Maximization algorithm, where the expectation is approximated by Hamiltonian Monte Carlo sampling of the latent variables.
Abstract: Principal geodesic analysis (PGA) is a generalization of principal component analysis (PCA) for dimensionality reduction of data on a Riemannian manifold. Currently PGA is defined as a geometric fit to the data, rather than as a probabilistic model. Inspired by probabilistic PCA, we present a latent variable model for PGA that provides a probabilistic framework for factor analysis on manifolds. To compute maximum likelihood estimates of the parameters in our model, we develop a Monte Carlo Expectation Maximization algorithm, where the expectation is approximated by Hamiltonian Monte Carlo sampling of the latent variables. We demonstrate the ability of our method to recover the ground truth parameters in simulated sphere data, as well as its effectiveness in analyzing shape variability of a corpus callosum data set from human brain images.
78 citations
Cites background or methods from "Geodesic Regression and the Theory ..."
...Following [8, 17], we use a generalization of the normal distribution for a Riemannian manifold as our noise model....
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...This distribution has the advantages that (1) it is applicable to any Riemannian manifold, (2) it reduces to a multivariate normal distribution (with isotropic covariance) whenM = R, and (3) much like the Euclidean normal distribution, maximum-likelihood estimation of parameters gives rise to least-squares methods (see [8] for details)....
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...WhenM is a symmetric space, this constant does not depend on the mean parameter, μ, because the distribution is invariant to isometrics (see [8] for details)....
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TL;DR: For the class of Cartan--Hadamard manifolds (which includes the data space in diffusion tensor imaging), the convergence of the proposed TV minimizing algorithms to a global minimizer is shown.
Abstract: We consider total variation minimization for manifold valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with $\ell^p$-type data terms in the manifold case. These algorithms are based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds. As an application, we consider denoising images which take their values in a manifold. We apply our algorithms to diffusion tensor images, interferometric SAR images as well as sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds (which includes the data space in diffusion tensor imaging) we show the convergence of the proposed TV minimizing algorithms to a global minimizer.
66 citations
Cites background from "Geodesic Regression and the Theory ..."
...Furthermore, statistical issues on Riemannian manifolds are topic of [30, 31, 32]....
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References
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Book•
01 Jan 1978TL;DR: In this article, the structure of semisimplepleasure Lie groups and Lie algebras is studied. But the classification of simple Lie algesbras and of symmetric spaces is left open.
Abstract: Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.
6,321 citations
"Geodesic Regression and the Theory ..." refers background in this paper
...More details can be found in standard references (Boothby 1986), including a complete classification of symmetric spaces (Helgason 1978)....
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TL;DR: This work presents tools for hierarchical clustering of imaged objects according to the shapes of their boundaries, learning of probability models for clusters of shapes, and testing of newly observed shapes under competing probability models.
Abstract: Using a differential-geometric treatment of planar shapes, we present tools for: 1) hierarchical clustering of imaged objects according to the shapes of their boundaries, 2) learning of probability models for clusters of shapes, and 3) testing of newly observed shapes under competing probability models. Clustering at any level of hierarchy is performed using a minimum variance type criterion and a Markov process. Statistical means of clusters provide shapes to be clustered at the next higher level, thus building a hierarchy of shapes. Using finite-dimensional approximations of spaces tangent to the shape space at sample means, we (implicitly) impose probability models on the shape space, and results are illustrated via random sampling and classification (hypothesis testing). Together, hierarchical clustering and hypothesis testing provide an efficient framework for shape retrieval. Examples are presented using shapes and images from ETH, Surrey, and AMCOM databases.
2,858 citations
Book•
06 Sep 2016
TL;DR: In this article, the authors proposed a planar procrustes analysis for two-dimensional data and showed that it is possible to estimate the size and shape of a shape in images.
Abstract: Preliminaries: Size Measures and Shape Coordinates. Preliminaries: Planar Procrustes Analysis. Shape Space and Distance. General Procrustes Methods. Shape Models for Two Dimensional Data. Tangent Space Inference. Size--and--Shape. Distributions for Higher Dimensions. Deformations and Describing Shape Change. Shape in Images. Additional Topics. References and Author Index. Index.
2,410 citations
"Geodesic Regression and the Theory ..." refers background or methods in this paper
...Related work includes statistical analysis of directional data (e.g., spheres) (Mardia and Jupp 2000) and analysis on shape manifolds (Dryden and Mardia 1998), where statistics are derived from probability distributions on specific manifolds (for example, the Fisher-von Mises distribution on…...
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..., spheres) (Mardia and Jupp 2000) and analysis on shape manifolds (Dryden and Mardia 1998), where statistics are derived from probability distributions on specific manifolds (for example, the Fisher-von Mises distribution on spheres)....
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Book•
01 Jan 1975
TL;DR: In this article, the authors present a revised edition of one of the classic mathematics texts published in the last 25 years, which includes updated references and indexes and error corrections and will continue to serve as the standard text for students and professionals in the field.
Abstract: This is a revised printing of one of the classic mathematics texts published in the last 25 years. This revised edition includes updated references and indexes and error corrections and will continue to serve as the standard text for students and professionals in the field.Differential manifolds are the underlying objects of study in much of advanced calculus and analysis. Topics such as line and surface integrals, divergence and curl of vector fields, and Stoke's and Green's theorems find their most natural setting in manifold theory. Riemannian plane geometry can be visualized as the geometry on the surface of a sphere in which "lines" are taken to be great circle arcs.
1,929 citations
"Geodesic Regression and the Theory ..." refers background in this paper
...More details can be found in standard references (Boothby 1986), including a complete classification of symmetric spaces (Helgason 1978)....
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TL;DR: In this article, the shape-space l. k m whose points represent the shapes of not totally degenerate /c-ads in IR m is introduced as a quotient space carrying the quotient metric.
Abstract: The shape-space l. k m whose points a represent the shapes of not totally degenerate /c-ads in IR m is introduced as a quotient space carrying the quotient metric. When m = 1, we find that Y\ = S k ~ 2 ; when m ^ 3, the shape-space contains singularities. This paper deals mainly with the case m = 2, when the shape-space I* ca n be identified with a version of CP*~ 2 . Of special importance are the shape-measures induced on CP k ~ 2 by any assigned diffuse law of distribution for the k vertices. We determine several such shape-measures, we resolve some of the technical problems associated with the graphic presentation and statistical analysis of empirical shape distributions, and among applications we discuss the relevance of these ideas to testing for the presence of non-accidental multiple alignments in collections of (i) neolithic stone monuments and (ii) quasars. Finally the recently introduced Ambartzumian density is examined from the present point of view, its norming constant is found, and its connexion with random Crofton polygons is established.
1,468 citations
"Geodesic Regression and the Theory ..." refers background in this paper
...Dating back to the groundbreaking work of Kendall (1984) and Bookstein (1986), modern shape analysis is concerned with the geometry of objects that is invariant to rotation, translation, and scale....
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