A discrete framework to find the optimal matching between manifold-valued curves

The aim of this paper is to find an optimal matching between manifold-valued curves, and thereby adequately compare their shapes, seen as equivalent classes with respect to the action of reparameterization. Using a canonical decomposition of a path in a principal bundle, we introduce a simple algorithm that finds an optimal matching between two curves by computing the geodesic of the infinite-dimensional manifold of curves that is at all time horizontal to the fibers of the shape bundle. We focus on the elastic metric studied in the so-called square root velocity framework. The quotient structure of the shape bundle is examined, and in particular horizontality with respect to the fibers. These results are more generally given for any elastic metric. We then introduce a comprehensive discrete framework which correctly approximates the smooth setting when the base manifold has constant sectional curvature. It is itself a Riemannian structure on the product manifold of "discrete curves" given by a finite number of points, and we show its convergence to the continuous model as the size of the discretization goes to infinity. Illustrations of optimal matching between discrete curves are given in the hyperbolic plane, the plane and the sphere, for synthetic and real data, and comparison with dynamic programming is established

Computing distances and geodesics between manifold-valued curves in the SRV framework

This paper focuses on the study of open curves in a Riemannian manifold M, and proposes a reparametrization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. to define a Riemannian metric on the space of immersions M'=Imm([0,1],M) by pullback of a natural metric on the tangent bundle TM'. This induces a first-order Sobolev metric on M' and leads to a distance which takes into account the distance between the origins in M and the L2-distance between the SRV representations of the curves. The geodesic equations for this metric are given and exploited to define an exponential map on M'. The optimal deformation of one curve into another can then be constructed using geodesic shooting, which requires to characterize the Jacobi fields of M'. The particular case of curves lying in the hyperbolic half-plane is considered as an example, in the setting of radar signal processing

Suède et ses "Belles étrangères" : la littérature étrangère dans les bibliothèques suédoises/projet de recherche (La)

Ce projet de recherche se donne pour objet l\u27analyse du rapport des lecteurs suédois à la littérature étrangère à partir de 1\u27observation de la place qu\u27elle occupe dans les pratiques des professionnels et des usagers des bibliothèques publiques en Suède

Self-stabilizing binary search tree maintenance algorithm

Binary search tree is one of the most studied data structures. The main application of the binary search tree is in implementing efficient search operations. A binary search tree is a special binary tree which satisfies the property that for every processor p in the binary tree, the values of all the keys in the left subtree of p are smaller than that of p, and the values of all the keys in the right subtree of p are larger than that of p; We present a self-stabilizing [Dij74] algorithm to maintain a binary search tree given a binary tree structure and a sequence of integers as input. This protocol uses neither the processors identifiers nor the size of the tree but assumes the existence of a distinguished processor (the root). The algorithm is self-stabilizing, meaning that starting from an arbitrary state, it is guaranteed to reach a legitimate state in a finite number of steps. The proposed algorithm assures that the set of integers eventually sent to the output environment is a permutation of the integers received from the input environment. The algorithm stabilizes in 0(hn) time units, where h and n represent the height and size, respectively, of the tree. The proposed algorithm is aimed at the hardwired binary tree structures where the topology of the trees cannot be adaptive to the change of the input values, but the input values are organized within a predefined environment

The Fisher-Rao geometry of beta distributions applied to the study of canonical moments

This paper studies the Fisher-Rao geometry on the parameter space of beta distributions. We derive the geodesic equations and the sectional curvature, and prove that it is negative. This leads to uniqueness for the Riemannian centroid in that space. We use this Riemannian structure to study canonical moments, an intrinsic representation of the moments of a probability distribution. Drawing on the fact that a uniform distribution in the regular moment space corresponds to a product of beta distributions in the canonical moment space, we propose a mapping from the space of canonical moments to the product beta manifold, allowing us to use the Fisher-Rao geometry of beta distributions to compare and analyze canonical moments

Quantization and clustering on Riemannian manifolds with an application to air traffic analysis

International audienceThe goal of quantization is to find the best approximation of a probability distribution by a discrete measure with finite support. When dealing with empirical distributions, this boils down to finding the best summary of the data by a smaller number of points, and automatically yields a k-means-type clustering. In this paper, we introduce Competitive Learning Riemannian Quantization (CLRQ), an online quantization algorithm that applies when the data does not belong to a vector space, but rather a Riemannian manifold. It can be seen as a density approximation procedure as well as a clustering method. Compared to many clustering algorihtms, it requires few distance computations, which is particularly computationally advantageous in the manifold setting. We prove its convergence and show simulated examples on the sphere and the hyperbolic plane. We also provide an application to real data by using CLRQ to create summaries of images of covariance matrices estimated from air traffic images. These summaries are representative of the air traffic complexity and yield clusterings of the airspaces into zones that are homogeneous with respect to that criterion. They can then be compared using discrete optimal transport and be further used as inputs of a machine learning algorithm or as indexes in a traffic database