Geometric Structures of Information

This book focuses on information geometry manifolds of structured data/information and their advanced applications featuring new and fruitful interactions between several branches of science: information science, mathematics and physics. It addresses interrelations between different mathematical domains like shape spaces, probability/optimization & algorithms on manifolds, relational and discrete metric spaces, computational and Hessian information geometry, algebraic/infinite dimensional/Banach information manifolds, divergence geometry, tensor-valued morphology, optimal transport theory, manifold & topology learning, and applications like geometries of audio-processing, inverse problems and signal processing.The book collects the most important contributions to the conference GSI'2017 - Geometric Science of Information.

Auteur

Frank Nielsen is Professor at the Laboratoire d'informatique de l'École polytechnique, Paris, France. His research aims at understanding the nature and structure of information and randomness in data, and exploiting algorithmically this knowledge in innovative imaging applications. For that purpose, he coined the field of computational information geometry (computational differential geometry) to extract information as regular structures whilst taking into account variability in datasets by grounding them in geometric spaces. Geometry beyond Euclidean spaces has a long history of revolutionizing the way we perceived reality. Curved spacetime geometry, sustained relativity theory and fractal geometry unveiled the scale-free properties of Nature. In the digital world, geometry is data-driven and allows intrinsic data analytics by capturing the very essence of data through invariance principles without being biased by such or such particular data representation.

Contenu
Rho-Tau Embedding of Statistical Models.- A class of non-parametric deformed exponentialstatistical models.- Statistical Manifolds Admitting Torsion and Partially Flat Spaces.- Conformal attening on the probability simplex and its applications to Voronoi partitions and centroids Atsumi Ohara.- Monte Carlo Information-Geometric Structures.- Information geometry in portfolio theory.- Generalising Frailty Assumptions in Survival Analysis: a Geometric Approach.- Some Universal Insights on Divergences forStatistics, Machine Learning and Articial Intelligence.- Information-Theoretic MatrixInequalities and Diusion Processes on Unimodular Lie Groups.