On the Limiting Behaviour of the Fundamental Geodesics of Information Geometry

The Information Geometry of extended exponential families has received much recent attention in a variety of important applications, notably categorical data analysis, graphical modelling and, more specifically, log-linear modelling. The essential geometry here comes from the closure of an exponential family in a high-dimensional simplex. In parallel, there has been a great deal of interest in the purely Fisher Riemannian structure of (extended) exponential families, most especially in the Markov chain Monte Carlo literature. These parallel developments raise challenges, addressed here, at a variety of levels: both theoretical and practical—relatedly, conceptual and methodological. Centrally to this endeavour, this paper makes explicit the underlying geometry of these two areas via an analysis of the limiting behaviour of the fundamental geodesics of Information Geometry, these being Amari’s (+1) and (0)-geodesics, respectively. Overall, a substantially more complete account of the Information Geometry of extended exponential families is provided than has hitherto been the case. We illustrate the importance and benefits of this novel formulation through applications

Tits Geometry and Positive Curvature

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three. We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but two that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a compact spherical building in the sense of Burns and Spatzier. Using this structure we classify up to equivariant diffeomorphism all polar actions on (simply connected) positively curved manifolds of cohomogeneity at least two.Comment: 43 pages, to appear in Acta Mathematic

3-D Face Recognition Using Geodesic-Map Representation and Statistical Shape Modelling

3-D face recognition research has received significant attention in the past two decades because of the rapid development in imaging technology and ever increasing security demand of modern society. One of its challenges is to cope with non-rigid deformation among faces, which is often caused by the changes of appearance and facial expression. Popular solutions to deal with this problem are to detect the deformable parts of the face and exclude them, or to represent a face in terms of sparse signature points, curves or patterns that are invariant to deformation. Such approaches, however, may lead to loss of information which is important for classification. In this paper, we propose a new geodesic-map representation with statistical shape modelling for handling the non-rigid deformation challenge in face recognition. The proposed representation captures all geometrical information from the entire 3-D face and provides a compact and expression-free map that preserves intrinsic geometrical information. As a result, the search for dense points correspondence in the face recognition task can be speeded up by using a simple image-based method instead of time-consuming, recursive closest distance search in 3-D space. An experimental investigation was conducted on 3-D face scans using publicly available databases and compared with the benchmark approaches. The experimental results demonstrate that the proposed scheme provides a highly competitive new solution for 3-D face recognition

Rank three geometry and positive curvature

An axiomatic characterization of buildings of type \CC_3 due to Tits is used to prove that any cohomogeneity two polar action of type \CC_3 on a positively curved simply connected manifold is equivariantly diffeomorphic to a polar action on a rank one symmetric space. This includes two actions on the Cayley plane whose associated \CC_3 type geometry is not covered by a building

Variational Methods for Biomolecular Modeling

Structure, function and dynamics of many biomolecular systems can be characterized by the energetic variational principle and the corresponding systems of partial differential equations (PDEs). This principle allows us to focus on the identification of essential energetic components, the optimal parametrization of energies, and the efficient computational implementation of energy variation or minimization. Given the fact that complex biomolecular systems are structurally non-uniform and their interactions occur through contact interfaces, their free energies are associated with various interfaces as well, such as solute-solvent interface, molecular binding interface, lipid domain interface, and membrane surfaces. This fact motivates the inclusion of interface geometry, particular its curvatures, to the parametrization of free energies. Applications of such interface geometry based energetic variational principles are illustrated through three concrete topics: the multiscale modeling of biomolecular electrostatics and solvation that includes the curvature energy of the molecular surface, the formation of microdomains on lipid membrane due to the geometric and molecular mechanics at the lipid interface, and the mean curvature driven protein localization on membrane surfaces. By further implicitly representing the interface using a phase field function over the entire domain, one can simulate the dynamics of the interface and the corresponding energy variation by evolving the phase field function, achieving significant reduction of the number of degrees of freedom and computational complexity. Strategies for improving the efficiency of computational implementations and for extending applications to coarse-graining or multiscale molecular simulations are outlined.Comment: 36 page

3-D Shape Matching for Face Analysis and Recognition

The aims of this paper are to introduce a 3-D shape matching scheme for automatic face recognition and to demonstrate its invariance to pose and facial expressions. The core of this scheme lies on the combination of non-rigid deformation registration and statistical shape modelling. While the former matches 3-D faces regardless of facial expression variations, the latter provides a low-dimensional feature vector that describes the deformation after the shape matching process, thereby enabling robust identification of 3-D faces. In order to assist establishment of accurate dense point correspondences, an isometric embedding shape representation is introduced, which is able to transform 3-D faces to a canonical form that retains the intrinsic geometric structure and achieve shape alignment of 3-D faces independent from individual’s facial expression. The feasibility and effectiveness of the proposed method was investigated using standard publicly available Gavab and BU-3DFE databases, which contain faces expressions and pose variations. The performance of the system was compared with the existing benchmark approaches and it demonstrates that the proposed scheme provides a competitive solution for the face recognition task with real-world practicality