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Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics †

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Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics † Entropy 2015, 17, 1850-1881; doi:10.3390/e17041850 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics y Nina Miolane * and Xavier Pennec INRIA, Asclepios project-team, 2004 Route des Lucioles, BP93, Sophia Antipolis Cedex F-06902, France; E-Mail: [email protected] y This paper is an extended version of our paper published in MaxEnt 2014, Amboise, France, 21–26 September 2014. * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +33-492-387-182. Received: 31 January 2015 / Accepted: 20 March 2015 / Published: 31 March 2015 Abstract: In computational anatomy, organ’s shapes are often modeled as deformations of a reference shape, i.e., as elements of a Lie group. To analyze the variability of the human anatomy in this framework, we need to perform statistics on Lie groups. A Lie group is a manifold with a consistent group structure. Statistics on Riemannian manifolds have been well studied, but to use the statistical Riemannian framework on Lie groups, one needs to define a Riemannian metric compatible with the group structure: a bi-invariant metric. However, it is known that Lie groups, which are not a direct product of compact and abelian groups, have no bi-invariant metric. However, what about bi-invariant pseudo-metrics? In other words: could we remove the assumption of the positivity of the metric and obtain consistent statistics on Lie groups through the pseudo-Riemannian framework? Our contribution is two-fold. First, we present an algorithm that constructs bi-invariant pseudo-metrics on a given Lie group, in the case of existence. Then, by running the algorithm on commonly-used Lie groups, we show that most of them do not admit any bi-invariant (pseudo-) metric. We thus conclude that the (pseudo-) Riemannian setting is too limited for the definition of consistent statistics on general Lie groups. Keywords: Lie group; Lie algebra; statistics; pseudo-Riemannian Entropy 2015, 17 1851 1. Introduction 1.1. Modeling with Lie Groups Data can be modeled as elements of Lie groups in many different fields: computational anatomy, robotics, paleontology, etc. Indeed, Lie groups are continuous groups of transformations and, thus, appear naturally whenever one deals with articulated objects or shapes. Regarding articulated objects, one can take examples in robotics or in computational anatomy. In robotics, first, a spherical arm is obviously an articulated object. The positions of the arm can be modeled as the elements of the three-dimensional Lie group of rotations SO(3). In computational anatomy, then, the spine can be modeled as an articulated object. In this context, each vertebra is considered as an orthonormal frame that encodes the rigid body transformation from the previous vertebra. Thus, as the human spine has 24 vertebrae, a configuration of the spine can be modeled as an element of the Lie group SE(3)23, where SE(3) is the Lie group of rigid body transformations in 3D, i.e., the Lie group of rotations and translations in R3, also called the special Euclidean group. Regarding shapes, the general model of d’Arcy Thompson suggests representing shape data as the diffeomorphic deformations of a reference shape [1], thus as elements of an infinite dimensional Lie group of diffeomorphisms. This framework can be applied as well in paleontology compared to in computational medicine. In palaeontology, first, a monkey skull or a human skull can be modeled as the diffeomorphic deformation of a reference skull. In computational medicine, then, the shape of a patient’s heart can be modeled as the diffeomorphic deformation of a reference shape. Obviously, many more examples could be given, also in other fields. 1.2. Statistics on Lie Groups Once data are represented as elements of a Lie group, we may want to perform statistical analysis on them for prediction or quantitative modeling. Thus, we want to perform statistics on Lie groups. How can we define an intrinsic statistical framework that is efficient on all Lie groups? How do we compute the mean or the principal modes of variation for a sample of Lie group elements? In order to train our intuition, we consider finite dimensional Lie groups here. To define a statistical framework, it seems natural to start with the definition of a mean. The definition of mean on a Lie group exemplifies the issues one can encounter while defining the whole statistical framework. We know that the usual definition of the mean is the weighted sum of the data elements of the sample. However, this definition is linear, and Lie groups are not linear in general. Consequently, we cannot use this definition on Lie groups: we could get a mean of Lie group elements that is not a Lie group element. One can consider as an example the half sum of two rotation matrices that is not always a rotation matrix. In fact, the definition of the mean on a Lie group should be consistent with the group structure. This consistency leads to several requirements of the mean, or properties. First, the mean of Lie group elements should be in the Lie group. Then, it seems natural to require that a left or right translation of the dataset should translate its mean accordingly. Figure1 illustrates the case when this condition is Entropy 2015, 17 1852 fulfilled. Finally, the inversion of all data elements should lead to an inverted mean. A mean verifying all of these properties is said to be bi-invariant. Lh Rh N N N fh ∗ gigi=1 fgigi=1 fgi ∗ hgi=1 N Figure 1. Left and right translation of a dataset fgigi=1 on the Lie group G. The initial N N dataset fgigi=1 has a mean represented in red. The left translated dataset fh ∗ gigi=1 has a N mean represented in blue. The right translated dataset fgi ∗ hgi=1 has a mean represented in green. We require that the mean of the (right or left) translated dataset is the translation of the red mean, which is the case in this illustration: the blue mean is the left translation of the red mean, and the green mean is the right translation of the red mean. A naturally bi-invariant candidate for the mean on Lie groups is the group exponential barycenter [2] defined as follows. A group exponential barycenter m of the dataset fgigi=1;::;N is a solution, if there are some, of the following group barycenter equation: N X (−1) Log(m ∗ gi) = 0 (1) i=1 where Log is the group logarithm. As the group exponential barycenter is naturally bi-invariant, we call a group exponential barycenter a bi-invariant mean. The local existence and uniqueness of the bi-invariant mean have been proven if the dispersion of the data is small enough. “Local” means that the data are assumed to be in a sufficiently small normal convex neighborhood of some point of the Lie group. Now, we want to provide a computational framework for the bi-invariant mean that would set the foundations for computations on Lie groups statistics in general. For that, we are interested in characterizing the global domains of existence and uniqueness of the bi-invariant mean. By “global domain”, we mean, for example, a ball of maximal radius, such that any probability measure with support included in it would have a unique bi-invariant mean. Note that there is a priori no problem having several means, which can be called several “modes”, or no mean at all. Our aim is rather to characterize the different situations that may occur: no mean, one unique mean, several means. 1.3. Using Riemannian and Pseudo-Riemannian Structures for Statistics on Lie Groups To this aim, we are interested in additional geometric structures on Lie groups that could help, by providing computational tools. For example, we are interested in a distance on a Lie group, that could enable one to measure the radii of balls. Such a distance could obviously help with characterizing balls of maximal radius. However, a Lie group is a group that carries an additional manifold structure, and one can define a pseudo-metric on a manifold, making it a pseudo-Riemannian manifold. Thus, we can Entropy 2015, 17 1853 add a pseudo-metric on Lie groups, which then induces a pseudo-distance. Could this additional pseudo-Riemannian structure help to define the statistical framework on Lie groups in practice? We consider first the case of the Riemannian structure, i.e., when the pseudo-metric is in fact a metric (positive definite). Several definitions of the mean on Riemannian manifolds have been proposed in the literature: the Fréchet mean, the Karcher mean or the Riemannian exponential barycenter [3–8]. For example, the Riemannian exponential barycenters are defined as the critical points of the variance of the 2 1 PN 2 N data, defined as: σ (y) = N i=1 dist(xi; y) , where fxigi=1 are the data and dist the distance induced by the Riemannian metric. The Riemannian framework provides theorems for the global existence and uniqueness domains of this mean [7–11], ensuring the computability of statistics on Riemannian manifolds. These represent exactly the kind of results that we would like to have for the bi-invariant mean on Lie groups. Thus, one may wonder if we can apply this computational framework for statistics on Lie groups and, more particularly, for the bi-invariant mean, by adding a Riemannian metric on the Lie group. In fact, the notions of Riemannian mean and group exponential barycenter (or bi-invariant mean) coincide when the Riemannian metric is itself bi-invariant.
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