scholarly journals Affine Differential Geometric Control Tools for Statistical Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1654
Author(s):  
Iulia-Elena Hirica ◽  
Cristina-Liliana Pripoae ◽  
Gabriel-Teodor Pripoae ◽  
Vasile Preda
Keyword(s):  
Lie Groups ◽  
Riemannian Manifolds ◽  
Information Geometry ◽  
Geometric Control ◽  
Statistical Manifold ◽  
Statistical Manifolds ◽  
Classical Setting ◽  
Theory Of Information

The paper generalizes and extends the notions of dual connections and of statistical manifold, with and without torsion. Links with the deformation algebras and with the Riemannian Rinehart algebras are established. The semi-Riemannian manifolds admitting flat dual connections with torsion are characterized, thus solving a problem suggested in 2000 by S. Amari and H. Nagaoka. New examples of statistical manifolds are constructed, within and beyond the classical setting. The invariant statistical structures on Lie groups are characterized and the dimension of their set is determined. Examples for the new defined geometrical objects are found in the theory of Information Geometry.

scholarly journals Entropic Dynamics on Gibbs Statistical Manifolds

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 494
Author(s):  
Pedro Pessoa ◽  
Felipe Xavier Costa ◽  
Ariel Caticha
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Exponential Family ◽  
Information Geometry ◽  
Arrow Of Time ◽  
Quantum Field ◽  
Statistical Manifold ◽  
Statistical Manifolds ◽  
Gibbs Distributions ◽  
Entropic Dynamics

Entropic dynamics is a framework in which the laws of dynamics are derived as an application of entropic methods of inference. Its successes include the derivation of quantum mechanics and quantum field theory from probabilistic principles. Here, we develop the entropic dynamics of a system, the state of which is described by a probability distribution. Thus, the dynamics unfolds on a statistical manifold that is automatically endowed by a metric structure provided by information geometry. The curvature of the manifold has a significant influence. We focus our dynamics on the statistical manifold of Gibbs distributions (also known as canonical distributions or the exponential family). The model includes an “entropic” notion of time that is tailored to the system under study; the system is its own clock. As one might expect that entropic time is intrinsically directional; there is a natural arrow of time that is led by entropic considerations. As illustrative examples, we discuss dynamics on a space of Gaussians and the discrete three-state system.

Bézier Curves on Riemannian Manifolds and Lie Groups With Kinematics Applications

1994 ◽  
Author(s):  
Frank C. Park ◽  
Bahram Ravani
Keyword(s):  
Riemannian Manifold ◽  
Rigid Body ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Group Structure ◽  
Recursive Algorithm ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Spatial Displacements ◽  
Natural Way

Abstract In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their algebraic group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body.

scholarly journals Isometric Signal Processing under Information Geometric Framework

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 332 ◽  
Author(s):  
Hao Wu ◽  
Yongqiang Cheng ◽  
Hongqiang Wang
Keyword(s):  
Signal Processing ◽  
Probability Distribution ◽  
Geometric Structure ◽  
Information Geometry ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Statistical Manifold ◽  
Intrinsic Parameter ◽  
Geometric Framework ◽  
Necessary And Sufficient

Information geometry is the study of the intrinsic geometric properties of manifolds consisting of a probability distribution and provides a deeper understanding of statistical inference. Based on this discipline, this letter reports on the influence of the signal processing on the geometric structure of the statistical manifold in terms of estimation issues. This letter defines the intrinsic parameter submanifold, which reflects the essential geometric characteristics of the estimation issues. Moreover, the intrinsic parameter submanifold is proven to be a tighter one after signal processing. In addition, the necessary and sufficient condition of invariant signal processing of the geometric structure, i.e., isometric signal processing, is given. Specifically, considering the processing with the linear form, the construction method of linear isometric signal processing is proposed, and its properties are presented in this letter.

Information geometry and statistical manifold

2003 ◽  
Vol 15 (1) ◽  
pp. 161-172 ◽  
Author(s):  
Nassar H. Abdel-All ◽  
H.N. Abd-Ellah ◽  
H.M. Moustafa
Keyword(s):  
Information Geometry ◽  
Statistical Manifold

scholarly journals r-Harmonic and Complex Isoparametric Functions on the Lie Groups $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$

2020 ◽  
Vol 58 (4) ◽  
pp. 477-496
Author(s):  
Sigmundur Gudmundsson ◽  
Marko Sobak
Keyword(s):  
Heisenberg Group ◽  
Lie Groups ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Semidirect Products ◽  
Simply Connected ◽  
Isoparametric Functions ◽  
General Method

Abstract In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ R m ⋉ R n and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$ R m ⋉ H 2 n + 1 , where $$\mathrm {H}^{2n+1}$$ H 2 n + 1 denotes the classical $$(2n+1)$$ ( 2 n + 1 ) -dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.

scholarly journals Riemannian Means as Solutions of Variational Problems

2006 ◽  
Vol 9 ◽  
pp. 86-103 ◽  
Author(s):  
Luís Machado ◽  
F. Silva Leite ◽  
Knut Hüper
Keyword(s):  
Riemannian Manifold ◽  
Variational Problem ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Single Point ◽  
Variational Problems ◽  
Compact Lie Groups ◽  
Data Set ◽  
The Given ◽  
Best Fit

We formulate a variational problem on a Riemannian manifoldMwhose solutions are piecewise smooth geodesies that best fit a given data set of time labelled points inM. By a limiting process, these solutions converge to a single point inM. which we prove to be the Riemannian mean of the given points for some particular Riemannian manifolds such as Euclidean spaces, connected and compact Lie groups, and spheres.

scholarly journals Attainable sets and one-parameter semigroups of sets

1991 ◽  
Vol 33 (2) ◽  
pp. 187-201 ◽  
Author(s):  
I. Chon ◽  
J. D. Lawson
Keyword(s):  
Control Systems ◽  
Lie Groups ◽  
Vector Fields ◽  
Geometric Control ◽  
Lie Theory ◽  
Attainable Sets ◽  
Widespread Application ◽  
Close Relationship ◽  
Basic Facts ◽  
Systems On Lie Groups

The methods of Lie theory have found widespread application in the study of the Lie algebras of vector fields on manifolds that arise naturally in geometric control theory (for some such applications, see [1]). Control systems on Lie groups themselves also have received considerable attention (see, for example, [9]). After reviewing basic facts about control systems on Lie groups, we derive the close relationship between attainable sets and Rådström's theory [12] of one-parameter semigroups of sets (Section 2). These ideas are then linked to the recently emerging Lie theory of semigroups [5]. The authors are indebted to the referee for pointing out some of the pertinent literature and analogous results from the area of geometric control.

scholarly journals Aspects of Geodesical Motion with Fisher-Rao Metric: Classical and Quantum

2018 ◽  
Vol 25 (01) ◽  
pp. 1850005 ◽  
Author(s):  
Florio M. Ciaglia ◽  
Fabio Di Cosmo ◽  
Domenico Felice ◽  
Stefano Mancini ◽  
Giuseppe Marmo ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Geometric Structure ◽  
Initial Conditions ◽  
Probability Distributions ◽  
End Points ◽  
Quantum Properties ◽  
Statistical Manifolds ◽  
Classical Setting ◽  
Qualitative Effect

The purpose of this paper is to exploit the geometric structure of quantum mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that minimise Shannon’s entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.

scholarly journals CONNECTIONS ON STATISTICAL MANIFOLDS OF DENSITY OPERATORS BY GEOMETRY OF NONCOMMUTATIVE Lp-SPACES

1999 ◽  
Vol 02 (01) ◽  
pp. 169-178 ◽  
Author(s):  
PAOLO GIBILISCO ◽  
TOMMASO ISOLA
Keyword(s):  
Unit Sphere ◽  
Von Neumann Algebra ◽  
Noncommutative Space ◽  
Statistical Manifold ◽  
Commutative Case ◽  
Von Neumann ◽  
Lp Spaces ◽  
Density Operators ◽  
Statistical Manifolds ◽  
Made In

Let [Formula: see text] be a statistical manifold of density operators, with respect to an n.s.f. trace τ on a semifinite von Neumann algebra M. If Sp is the unit sphere of the noncommutative space Lp(M, τ), using the noncommutative Amari embedding [Formula: see text], we define a noncommutative α-bundle-connection pair (ℱα, ∇α), by the pullback technique. In the commutative case we show that it coincides with the construction of nonparametric Amari–Čentsov α-connection made in Ref. 8 by Gibilisco and Pistone.

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