Classifies toric dually flat manifolds into complex space forms.
problem Classifying 1D toric dually flat manifolds.
method Using complex space forms and exponential families.
result Toric dually flat manifolds are complex space forms.
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,051 papers · 148 categories
Trend · papers per month
48 results for “dually flat spaces”
Kahler toric manifolds linked to dually flat spaces via affine isometry.
problem Mapping Kähler toric manifolds to dually flat spaces.
method Affine isometric maps and equivariant Kähler immersions.
result Existence of lifting procedure between Kähler toric manifolds and dually flat spaces.
The paper studies a generalized Pythagorean theorem on dually flat spaces via toric geometry.
problem Understanding the geometry of dually flat spaces and their toric Kähler manifolds.
method Introducing a dually flat structure and Bregman divergence on the boundary of toric Kähler manifolds.
result A continuity and generalized Pythagorean theorem for the divergence on the boundary.
Minimization of Functions on Dually Flat Spaces Using Geodesic Descent Based on Dual Connectionsstat.CO
Geodesic descent optimizes likelihood in dually flat spaces.
problem Maximum likelihood estimation in exponential families.
method m-geodesic and e-geodesic updates on dually flat spaces.
result Geodesic updates can reach maximum likelihood estimator in one step.
New divergences introduced in dually flat spaces with properties.
problem Measuring discrepancy between probability distributions in dually flat spaces.
method Introducing two types of divergences based on affine coordinates and potentials, and deriving relational equations.
result Generalization of the law of cosines and new inequalities between divergences.
In this paper, we prove that a strongly convex complex Finsler metric on a domain is projectively flat (resp. dually flat) if and only if comes from a strongly convex complex Minkowski metric.
In this paper, we consider Randers change of some special metrics. First we find the fundamental metric tensor and Cartan tensor of these Randers changed metrics. Next, we establish a general formula for inverse of fundamental metric tensors of these metrics. Finally, we find the necessary and su…
A new geometric structure for singular models is introduced.
problem Degenerate metrics in dually flat structures.
method Introducing quasi-Hessian manifolds with degenerate metrics and symmetric cubic tensors.
result Established Amari-Nagaoka's extended Pythagorean and projection theorems for singular models.
Dually flat structure with escort probability and its application to alpha-Voronoi diagramscond-mat.stat-mech
This paper studies geometrical structure of the manifold of escort probability distributions and shows its new applicability to information science. In order to realize escort probabilities we use a conformal transformation that flattens so-called alpha-geometry of the space of discrete probability distributions, which…
On dually flat -metricsmath.DG
In this paper, I will show how to use -deformations to deal with dual flatness of -metrics. It is a natural continuation of the research on dually flat Randers metrics(see arxiv:1209.1150). -deformations is a new method in Riemann-Finsler geometry, it is introduced by the author(see arxiv:1209.0845).
In this work, the dual flatness, which is connected with Statistics and Information geometry, of general -metrics (a new class of Finsler metrics) is studied. A nice characterization for such metrics to be dually flat under some suitable conditions is provided and all the solutions are completely determined. By …
Investigate AR-Finsler metrics for local dual flatness and projective flatness.
problem Locally dually and projectively flat AR-Finsler metrics
method Derive necessary and sufficient conditions and a compatibility relation.
result Establish a rigidity result for AR-Finsler metrics.
Study invariant connections on multivariate Gaussian distributions.
problem Understanding statistical connections on multivariate Gaussian distributions.
method Investigate invariant connections on with the Fisher metric.
result Explicitly determined invariant connections and their moduli spaces.
Introduces Legendre bundle for dually flat manifolds and quantum field theories.
problem Understanding duality in geometric structures and quantum field theories.
method Introduces Legendre bundle and para-Kähler structure.
result Exponential families and Hessian QFTs are realizations of the Legendre bundle.
The class of spherically symmetric Finsler metrics is studied and locally dually flat and projectively flat spherically symmetric Finsler metrics is classified.
This paper studies the geometry of immersions into statistical manifolds. A necessary and sufficient condition is obtained for statistical manifold structures to be dual to each other for a non-degenerate equiaffine immersion. Then we obtain conditions for realizing an n-dimensional statistical manifold in an (n+1)-dim…
In this paper, we find a condition under which a Finsler space with Kropina change of mth-root metric is projectively related to a mth-root metric and also we find a condition under which this Kropina transformed mth-root metric is locally dually flat. Moreover we find the condition for its Projective flatness.
Exponential families and mixture families are parametric probability models that can be geometrically studied as smooth statistical manifolds with respect to any statistical divergence like the Kullback-Leibler (KL) divergence or the Hellinger divergence. When equipping a statistical manifold with the KL divergence, th…
Moment polytope of toric exponential families is a projection of a simplex.
problem Understanding the geometry of exponential families in finite sample spaces.
method Toric torification and projection of higher-dimensional simplices.
result Moment polytope is a projection of a higher-dimensional simplex.
New insights into Markov chain geometry via positive transition measures.
problem Lack of statistical meaning in the space of transition probabilities.
method Constructing an extension of the space of transition probabilities using Amari's theory of positive measures.
result Introduction of a new dually flat structure for the space of positive transition measures.
Introduces new info-geometric structure for dynamics on graphs and hypergraphs.
problem Modeling dynamics on discrete structures like graphs and hypergraphs.
method Introduces two dually flat structures: one on vertex space and another on edge space.
result Extends gradient flows to include nonequilibrium dynamics.
Study information geometry of warped product spaces, finding special connections.
problem Understanding information geometry in warped product spaces.
method Examined warped products with dually flat connections, characterized connections on base space.
result Characterized connections on base space as -connections with .
In this paper, we prove that every m-th root metric with isotropic mean Berwald curvature reduces to a weakly Berwald metric. Then we show that an m-th root metric with isotropic mean Landsberg curvature is a weakly Landsberg metric. We find necessary and sufficient condition under which conformal -change of an m-th…
In this paper, I will show how to use beta-deformations to deal with dual flatness of Randers metrics. beta-deformations is a new method in Riemann-Finsler geometry, it is introduced by the author(see arxiv:1209.0845). Later on I will provide more applications of the new kind of deformations in Finsler geometry.
On m-th root Finsler metricsmath.DG
In this paper, we characterize locally dually flat and Antonelli -th root Finsler metrics. Then, we show that every -th root Finsler metric of isotropic mean Berwald curvature reduces to a weakly Berwald metric.
Geodesic sprays on Finsler manifolds studied with covariant coefficients.
problem Understanding geometric properties of Finsler metrics through covariant coefficients.
method Introduced -covariant coefficients and studied their geometric consequences.
result Existence and uniqueness of spray scalar for projectively flat metrics.
In this paper, we consider Kropina change of -th root Finsler metrics. We find necessary and sufficient condition under which the Kropina change of an -th root Finsler metric be locally dually flat. Then we prove that the Kropina change of an -th root Finsler metric is locally projectively flat if and only if …
Study of pseudometric properties on domains in Nagano spaces.
problem Characterize pseudometrics on domains in real-type Nagano spaces.
method Analyze Kobayashi-type pseudometrics on domains, proving properties and computing specific cases.
result The pseudometric is a genuine metric under certain conditions and has specific properties in higher rank.
Study on spherical Finsler metrics with isotropic curvature rigidity.
problem Characterizing and understanding spherically symmetric Finsler metrics with isotropic -curvature.
method Provided the correct formula for mean Berwald curvature, established differential equations for projective and dual flatness, and derived a rigidity result.
result Rigidity result on spherically symmetric Finsler metrics with isotropic -curvature.
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 pseudo-distance are generated by the sum of two tangent vectors. By relying on these vector…
Paper studies statistical manifolds with logarithmic divergences.
problem Understanding statistical manifolds induced by logarithmic divergences.
method Constructs dual foliation of the statistical manifold.
result Extends dual foliation of a dually flat manifold.
In this paper, we show that the projection of a dualistic structure defined on a twisted product manifold induces dualistic structures on the base and the fiber manifolds, and conversely. Then under some conditions on the Ricci curvature and the Weyl conformal tensor we characterize dually flat structures on twisted pr…
The dual variety X* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X* is a hypersurface in the dual space (P^N)*. If dim X* < N - 1, then the variety X is called dually degenerate. The authors refine these definitions for a varie…
A new statistical model uses Orlicz-Sobolev spaces with Gaussian weight.
problem Statistical modeling of infinite-dimensional probability measures.
method Affine statistical bundle on Gaussian Orlicz-Sobolev space.
result Provides tools for solving infinite-dimensional evolution problems.
In this paper, we characterize locally dually flat generalized m-th root Finsler metrics. Then we find a condition under which a generalized m-th root metric is projectively related to a m-th root metric. Finally, we prove that if a generalized m-th root metric is conformal to a m-th root metric, then both of them redu…
A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper we showed that gradient maps of exponentially concave functions provide solutions to a Monge-Kantorovich optimal transport problem and give a better gradient approximat…
This paper studies moduli spaces of statistical structures on Lie groups.
problem Understanding statistical structures on Lie groups.
method Introduced and studied moduli spaces for left-invariant statistical structures on Lie groups.
result Moduli spaces of left-invariant Riemannian metrics are singletons for certain Lie groups.
Geometrically decomposes Kähler functions on toric manifolds.
problem Decomposing Kähler functions on Kähler toric manifolds.
method Defining spectrum of Kähler functions and proving spectral decomposition theorem.
result Geometric spectral theory for Kähler functions established.
We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class with respect to appropriate reference measures. The case , in which the manifolds are modelled on Fréchet spaces, is included.…
The paper extends gradient flow and relaxation studies to non-flat Riemannian manifolds.
problem Understanding gradient flows and relaxation in non-flat Riemannian manifolds.
method Developed a criterion for comparing relaxation along gradient descent curves using non-metricity tensor.
result Revealed a universal asymmetry: warming up is faster than cooling down.
On the product of two Finsler manifolds M1 M2, we consider the twisted metric F which is construct by using Finsler metrics F1 and F2 on the manifolds M1 and M2, respectively. We introduce horizontal and vertical distributions on twisted product Finsler manifold and study Creducible and semi-C-reducible properties of t…
Paper finds local normal forms for wavefronts in flat coordinates.
problem Understanding local diffeomorphic types of wavefronts.
method Using connections and the metric, criteria for wavefront types are derived in affine flat coordinates.
result Local normal forms of -wavefronts in affine flat coordinates are derived.
In this paper, we consider doubly warped product (DWP) Finsler manifolds with some non-Riemannian curvature properties. First, we study Berwald and isotropic mean Berwald DWP-Finsler manifolds. Then we prove that every proper Douglas DWP-Finsler manifold is Riemannian. We show that a proper DWP-manifold is Landsbergian…
The Yukawa term in statistical mechanics quantifies information generation.
problem Understanding information generation in statistical mechanics.
method Defining a statistical product and Yukawa term to quantify information generation.
result The Yukawa term diverges in the quantum case, indicating Bose-Einstein condensation.
New geometric perspective for optimal learning on hexagonal structures.
problem Optimal learning process on hexagonal structures.
method Local trivial fibrations and Ceva's theorem.
result Learning can be defined on hexagonal structures.
Introduces a new geometric structure for statistical manifolds with degenerate metrics.
problem Degenerate metrics in statistical manifolds affect geometric structures and applications.
method Introduces quasi-Codazzi structure for degenerate metrics and coherent tangent bundles.
result Generalizes geometric structures and relations for statistical models with degenerate metrics.
Formulates mechanics for probability distributions on statistical manifold.
problem Formulating mechanics for probability distributions on statistical manifold.
method Information-geometric formulation of Classical Mechanics on statistical manifold, using dually-flat connection and Hilbert bundle structure.
result Provides coherent formalism for Lagrangian and Hamiltonian mechanics on statistical bundle.
The report analyzes Legendre decomposition for tensor data.
problem Finding effective lower dimensional representations of tensors.
method Theoretical analysis of dual parameters and dually flat manifold properties, followed by experimental verification and clustering.
result Parameters on submanifold cannot be directly used as low-rank representations.