Sur le barycentre d’une probabilité dans une variété

Émery, Michel; Mokobodzki, Gabriel

doi:10.1007/bfb0100858

Cited by 67 publications

(58 citation statements)

References 8 publications

(6 reference statements)

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“…A similar definition that uses convex functions on the manifold instead of metric properties proposed by Emery [27] and Arnaudon [52,28]. A function from M to R is convex if its restriction to all geodesic is convex (considered as a function from R to R).…”

Section: Other Possible Definitions Of the Mean Pointsmentioning

confidence: 99%

“…For instance, [24] derived central limit theorems on different families of groups and semi-groups with specific algebraic properties. Since then, several authors in the area of stochastic differential geometry and stochastic calculus on manifolds proposed results related to mean values [25,26,27,28,29,30]. On the applied mathematics and computer science side, people get interested in computing and optimizing in specific manifolds, like rotations and rigid body transformations [4,31,32,33,34], Stiefel and Grassmann manifolds [35], etc.…”

Section: Introductionmentioning

confidence: 99%

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Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

2006

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In medical image analysis and high level computer vision, there is an intensive use of geometric features like orientations, lines, and geometric transformations ranging from simple ones (orientations, lines, rigid body or affine transformations, etc.) to very complex ones like curves, surfaces, or general diffeomorphic transformations. The measurement of such geometric primitives is generally noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare observations, or to test hypotheses. Unfortunately, even simple geometric primitives often belong to manifolds that are not vector spaces. In previous works [1, 2], we investigated invariance requirements to build some statistical tools on transformation groups and homogeneous manifolds that avoids paradoxes. In this paper, we consider finite dimensional manifolds with a Riemannian metric as the basic structure. Based on this metric, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and χ 2 law. We provide a new proof of the characterization of Riemannian centers of mass and an original gradient descent algorithm to efficiently compute them. The notion of Normal law we propose is based on the maximization of the entropy knowing the mean and covariance of the distribution. The resulting family of pdfs spans the whole range from uniform (on compact manifolds) to the point mass distribution. Moreover, we were able to provide tractable approximations (with their limits) for small variances which show that we can effectively implement and work with these definitions.

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Section: Other Possible Definitions Of the Mean Pointsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

2006

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“…This 'critical condition' equation can also be taken as the definition of the mean, which leads to the notion of exponential barycenter [23,19].…”

Section: A Barycentric Definition Of the Mean?mentioning

confidence: 99%

“…An idea to investigate this link is the following. The bi-invariant mean defined in this work is a special instance of the exponential barycenters proposed in [23,5,6] for Riemannian manifolds. The existence and uniqueness of the exponential barycenters was recently established in affine connection manifolds which are convex with semi-local convex geometry (CSLCG) by Arnaudon and Li [4].…”

Section: Perspectivesmentioning

confidence: 99%

Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups

Pennec

Arsigny

2012

Matrix Information Geometry

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Abstract. When performing statistics on elements of sets that possess a particular geometric structure, it is desirable to respect this structure. For instance in a Lie group, it would be judicious to have a notion of a mean which is stable by the group operations (composition and inversion). Such a property is ensured for Riemannian center of mass in Lie groups endowed with a bi-invariant Riemannian metric, like compact Lie groups (e.g. rotations). However, bi-invariant Riemannian metrics do not exist for most non compact and non-commutative Lie groups. This is the case in particular for rigid-body transformations in any dimension greater than one, which form the most simple Lie group involved in biomedical image registration. In this paper, we propose to replace the Riemannian metric by an affine connection structure on the group. We show that the canonical Cartan connections of a connected Lie group provides group geodesics which are completely consistent with the composition and inversion. With such a non-metric structure, the mean cannot be defined by minimizing the variance as in Riemannian Manifolds. However, the characterization of the mean as an exponential barycenter gives us an implicit definition of the mean using a general barycentric equation. Thanks to the properties of the canonical Cartan connection, this mean is naturally bi-invariant. We show the local existence and uniqueness of the invariant mean when the dispersion of the data is small enough. We also propose an iterative fixed point algorithm and demonstrate that the convergence to the invariant mean is at least linear. In the case of rigid-body transformations, we give a simple criterion for the global existence and uniqueness of the bi-invariant mean, which happens to be the same as for rotations. We also give closed forms for the bi-invariant mean in a number of simple but instructive cases, including 2D rigid transformations. For general linear transformations, we show that the bi-invariant mean is a generalization of the (scalar) geometric mean, since the determinant of the bi-invariant mean is the geometric mean of the determinants of the data. Finally, we extend the theory to higher order moments, in particular with the covariance which can be used to define a local bi-invariant Mahalanobis distance.

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“…16]. Most applications of these concepts are related to random variables concentrated on domains which can be described geometrically in terms of convex geometry (see 25]).…”

Section: Introductionmentioning

confidence: 99%

Brownian motion and the formation of singularities in the heat flow for harmonic maps

Thalmaier¹

1996

Probability Theory and Related Fields

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Summary. We develop a general framework for a stochastic interpretation of certain nonlinear PDEs on manifolds. The linear operation of taking expectations is replaced by the concept of \martingale means", namely the notion of deterministic starting points of martingales (with respect to the Levi-Civita connection) ending up at a prescribed state. We formulate a monotonicity condition for the Riemannian quadratic variation of such martingales that allows us to turn smallness of the quadratic variation into a priori gradient bounds for solutions of the nonlinear heat equation. Such estimates lead to simple criteria for blow-ups in the nonlinear heat ow for harmonic maps with small initial energy.

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Sur le barycentre d’une probabilité dans une variété

Cited by 67 publications

References 8 publications

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups

Brownian motion and the formation of singularities in the heat flow for harmonic maps

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