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

International audienceWhen 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

The Semiotic Fractures of Vulnerable Bodies: Resistance to the Gendering of Legal Subjects

While the turn to vulnerability in law responds to a recurrent critique by feminist scholars on the disembodiment of legal personhood, this article suggests that the mobilization of vulnerability in the criminal courts does not necessarily offer female drug mules a direct path to justice. Through an analysis of sentencing appeals of female drug mules in England and Wales, this article presents a feminist critique of the dispositif of the person and its relation to vulnerability. Discourses on drug mules’ vulnerability mobilize the trope of the colonial victim in need of protection, which is often translated into legal mercy. But mercy is rather an expression of biopower which inscribes not only fragility onto the bodies of drug mules by figuring them as exemplar paradigms of colonial subjectivity, but also reinvigorates the dispositif of gender implicit in the legal person. In this set-up, it would appear as if law and politics totalize the registers of life, in this case the contours of vulnerable body. The article suggests we must revisit the image of the wounded body in order to carve out a space for resistance. Drawing on Elaine Scarry and Judith Butler, it suggests vulnerable bodies are marked by a semiotic openness, which renders them subject to appropriation but also able to signify the precarity produced by the law through their resistance to representation

Bi-invariant Means on Lie Groups with Cartan-Schouten Connections

International audienceThe statistical Riemannian framework was pretty well developped for finite-dimensional manifolds. For Lie groups, left or right invariant metric provide a nice setting as the Lie group becomes a geodesically complete Riemannian manifold, thus also metrically complete. However, this Riemannian approach is fully consistent with the group operations only if a bi-invariant metric exists. Unfortunately, bi-invariant Riemannian metrics do not exist on most non compact and non-commutative Lie groups. In particular, such metrics do not exist in any dimension for {\it rigid-body transformations}, which form the most simple Lie group involved in biomedical image registration. The log-Euclidean framework, initially developed for symmetric positive definite matrices, was proposed as an alternative for affine transformations based on the log of matrices and for (some) diffeomorphisms based on Stationary Velocity Fields (SVFs). The idea is to rely on one-parameter subgroups, for which efficient algorithms exists to compute the deformation from the initial tangent vector (e.g. scaling and squaring). Previously, we showed that this framework allows to define bi-invariant means on Lie groups provided that the square-root (thus the log) of the transformations do exist. The goal of this note is to summarize the mathematical roots of these algorithms and to set the bases for comparing their properties with the left and right invariant metrics. The basis of our developments is the structure of affine connection instead of Riemannian metric. The connection defines the parallel transport, and thus a notion of geodesics (auto-parallel curves). Many local properties of Riemannian manifolds remains valid with affine connection spaces. In particular, there is still a local diffeomorphisms between the manifold and the tangent space using the exp and log maps. We explore invariant connections and show that there is a unique bi-invariant torsion-free Cartan-Schouten connection for which the geodesics are left and right translations of one-parameter subgroups. These group geodesics correspond to the ones of a left-invariant metric for the normal elements of the Lie algebra only. When a bi-invariant metric exists (we show that this is not always the case), then all elements are normal and Riemannian and group geodesics coincide. Finally we summarize the properties of the bi-invariant mean defined as the exponential barycenters of the canonical Cartan connection