Related papers: Divergence functions in Information Geometry
Information geometrical structure $(g^{(D_\alpha)}, \nabla^{(D_\alpha)},\nabla^{(D_\alpha)*})$ induced from the sandwiched R\'enyi $\alpha$-divergence $D_\alpha(\rho\|\sigma):=\frac{1}{\alpha (\alpha-1)}\log\,{\rm Tr}…
The principles of classical mechanics have shown that the inertial quality of mass is characterized by the kinetic energy. This, in turn, establishes the connection between geometry and mechanics. We aim to exploit such a fundamental…
In Riemannian geometry geodesics are integral curves of the Riemannian distance gradient. We extend this classical result to the framework of Information Geometry. In particular, we prove that the rays of level-sets defined by a…
A recent canonical divergence, which is introduced on a smooth manifold $\mathrm{M}$ endowed with a general dualistic structure $(\mathrm{g},\nabla,\nabla^*)$, is considered for flat $\alpha$-connections. In the classical setting, we…
We study a differential geometric construction, the warped product, on the background geometry for information theory. Divergences, dual structures and symmetric 3-tensor are studied under this construction, and we show that warped product…
Information geometry is a study of statistical manifolds, that is, spaces of probability distributions from a geometric perspective. Its classical information-theoretic applications relate to statistical concepts such as Fisher information,…
In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information…
Information geometry provides differential geometric concepts like a Riemannian metric, connections and covariant derivatives on spaces of probability distributions. We discuss here how these concepts apply to quantum field theories in the…
In the field of statistics, many kind of divergence functions have been studied as an amount which measures the discrepancy between two probability distributions. In the differential geometrical approach in statistics (information…
Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…
A general divergence measure for monotonic functions is introduced. Its connections with the f-divergence for convex functions are explored. The main properties are pointed out.
Given a real, finite-dimensional, smooth parallelizable Riemannian manifold $(\mathcal{N},G)$ endowed with a teleparallel connection $\nabla$ determined by a choice of a global basis of vector fields on $\mathcal{N}$, we show that the…
We describe topological gauge theories for which duality properties are encoded by construction. We study them for compact manifolds of dimensions four, eight and two. The fields and their duals are treated symmetrically, within the context…
This paper is concerned with the algebraic dual D*(\Omega) of the space of test functions D(\Omega). The emphasis is on failures and successes of D*(\Omega) as compared to the continuous dual D'(\Omega), the space of distributions.…
A new type of sectional curvature is introduced. The notion is purely algebraic and can be located in linear algebra as well as in differential geometry.
We survey some algebraic geometric aspects of mirror symmetry and duality in string theory. Some applications of computer algebra to algebraic geometry and string theory are shortly reviewed.
In this work, the dual flatness, which is connected with Statistics and Information geometry, of general $(\alpha,\beta)$-metrics (a new class of Finsler metrics) is studied. A nice characterization for such metrics to be dually flat under…
We describe the duality between different geometries which arises by considering the classical and quantum harmonic map problem. To appear in ``Essays on Mirror Manifolds II''.
We exhibit differential geometric structures that arise in numerical methods, based on the construction of Cauchy sequences, that are currently used to prove explicitly the existence of weak solutions to functional equations. We describe…
We examine the role of information geometry in the context of classical Cram\'er-Rao (CR) type inequalities. In particular, we focus on Eguchi's theory of obtaining dualistic geometric structures from a divergence function and then applying…