Information geometry

About: Information geometry is a research topic. Over the lifetime, 1725 publications have been published within this topic receiving 44354 citations.

A Riemannian Framework for Tensor Computing

Abstract: Tensors are nowadays a common source of geometric information. In this paper, we propose to endow the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc. We have previously shown that the Riemannian metric provides a powerful framework for generalizing statistics to manifolds. In this paper, we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data. For instance, most interpolation and Gaussian filtering schemes can be tackled efficiently through a weighted mean computation. Linear and anisotropic diffusion schemes can be adapted to our Riemannian framework, through partial differential evolution equations, provided that the metric of the tensor space is taken into account. For that purpose, we provide intrinsic numerical schemes to compute the gradient and Laplace-Beltrami operators. Finally, to enforce the fidelity to the data (either sparsely distributed tensors or complete tensors fields) we propose least-squares criteria based on our invariant Riemannian distance which are particularly simple and efficient to solve.

Self-duality in four-dimensional Riemannian geometry

Abstract: We present a self-contained account of the ideas of R. Penrose connecting four-dimensional Riemannian geometry with three-dimensional complex analysis. In particular we apply this to the self-dual Yang-Mills equations in Euclidean 4-space and compute the number of moduli for any compact gauge group. Results previously announced are treated with full detail and extended in a number of directions.

Spectral Asymmetry and Riemannian Geometry

Abstract: This has an analytic continuation to the whole s-plane as a meromorphic function of 5 and s = 0 is not a pole: moreover CA(o) can be computed as an explicit integral over the manifold [9]. In this note we shall introduce a refinement of this invariant when A is no longer positive and we shall study its geometrical significance for an important class of operators (first order systems) arising from Riemannian geometry. A full exposition will be given elsewhere. Suppose therefore that A is self-adjoint and elliptic but no longer positive. The eigenvalues are now real but can be positive or negative. We define, for Re(s) large,

Pedestrian Detection via Classification on Riemannian Manifolds

Abstract: We present a new algorithm to detect pedestrian in still images utilizing covariance matrices as object descriptors. Since the descriptors do not form a vector space, well known machine learning techniques are not well suited to learn the classifiers. The space of d-dimensional nonsingular covariance matrices can be represented as a connected Riemannian manifold. The main contribution of the paper is a novel approach for classifying points lying on a connected Riemannian manifold using the geometry of the space. The algorithm is tested on INRIA and DaimlerChrysler pedestrian datasets where superior detection rates are observed over the previous approaches.