Intrinsic Analysis of Statistical Estimation

Oller, Josep M.; Corcuera, José Manuel

doi:10.1214/aos/1176324312

Cited by 65 publications

(68 citation statements)

References 15 publications

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“…[15] defined the mean values in manifolds as the exponential barycenters. To relate them with the Riemannian centers of mass, they determined the gradient of the variance.…”

Section: Corollary 1 (Characterization Of Riemannian Centers Of Mass)mentioning

confidence: 99%

“…Its goal is to study statistically geometric object living in differential manifolds. It is linked to the theory of statistical manifolds [14,15], which aims at providing a Riemannian structure to the the space of parameters of statistical distribution. However, the targeted geometrical objects are usually different.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For instance, [36] and [15] aimed at characterizing the performances of statistical parametric estimators, like the bias and the mean square error. [36] considered extrinsic statistics, based on the Euclidean distance of the embedding space, while [15] considered the intrinsic Riemannian distance, and refined the Cramer-Rao lower bound using bounds on the sectional curvature of the manifold. In [37,38], the authors focused on the asymptotic consistency properties of the extrinsic and intrinsic means and variances for large sample sizes, and were able to propose a central limit theorem for flat manifolds.…”

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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“…[15] defined the mean values in manifolds as the exponential barycenters. To relate them with the Riemannian centers of mass, they determined the gradient of the variance.…”

Section: Corollary 1 (Characterization Of Riemannian Centers Of Mass)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

2006

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“…Classically, in a manifold endowed with a Riemannian metric, the natural choice of mean is called the Riemannian center of mass or Fréchet mean. The Riemannian structure proves to be a powerful and consistent framework for computing simple statistics [72,50,53,[11][12][13]55] and can be extended to an effective computing framework on manifold-valued images [57]. On a Lie group, this Riemannian approach is consistent with the group operations if a biinvariant metric exists, which is for example the case for compact groups such as rotations [54,49].…”

Section: Introductionmentioning

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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References and Author Index

Mardia

Jupp

1999

Directional Statistics

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Intrinsic Analysis of Statistical Estimation

Cited by 65 publications

References 15 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

References and Author Index

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