Relevant Differential Topology in Statistical Manifolds

2021 ◽  
pp. 154-178
Author(s):  
Michel Nguiffo-Boyom
Keyword(s):  
Differential Topology ◽  
Statistical Manifolds

scholarly journals Hooke’s law in statistical manifolds and divergences

2000 ◽  
Vol 159 ◽  
pp. 1-24 ◽  
Author(s):  
Masayuki Henmi ◽  
Ryoichi Kobayashi
Keyword(s):  
Classical Mechanics ◽  
Riemannian Metric ◽  
Legendre Transform ◽  
Point Function ◽  
Hooke’S Law ◽  
Affine Connections ◽  
Statistical Manifolds ◽  
Dual Affine ◽  
Hooke's Law

The concept of the canonical divergence is defined for dually flat statistical manifolds in terms of the Legendre transform between dual affine coordinates. In this article, we introduce a new two point function defined for any triple (g,∇, ∇*) of a Riemannian metric g and two affine connections ∇ and ∇*. We show that this interprets the canonical divergence without refering to the existence of special coordinates (dual affine coordinates) but in terms of only classical mechanics concerning ∇- and ∇*-geodesics. We also discuss the properties of the two point function and show that this shares some important properties with the canonical divergence defined on dually flat statistical manifolds.

scholarly journals Manifolds of differentiable densities

2018 ◽  
Vol 22 ◽  
pp. 19-34 ◽  
Author(s):  
Nigel J. Newton
Keyword(s):  
Continuous Function ◽  
Likelihood Function ◽  
Probability Measures ◽  
Fréchet Spaces ◽  
Finite Sample ◽  
Infinite Dimensional ◽  
Statistical Manifolds ◽  
Covariant Derivatives ◽  
Finite Measures ◽  
Non Parametric

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 Cbk with respect to appropriate reference measures. The case k = ∞, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and, unusually for the non-parametric setting, Amari’s α-covariant derivatives for all α ∈ ℝ. By construction, they are C∞-embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually (α = ±1) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the α-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the α-divergences are of class C∞.

scholarly journals Sasakian statistical manifolds

2017 ◽  
Vol 117 ◽  
pp. 179-186 ◽  
Author(s):  
Hitoshi Furuhata ◽  
Izumi Hasegawa ◽  
Yukihiko Okuyama ◽  
Kimitake Sato ◽  
Mohammad Hasan Shahid
Keyword(s):  
Statistical Manifolds

scholarly journals On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature

2020 ◽  
Vol 5 (4) ◽  
pp. 3495-3509
Author(s):  
Aliya Naaz Siddiqui ◽  
◽  
Mohammad Hasan Shahid ◽  
Jae Won Lee ◽  
Keyword(s):  
Ricci Curvature ◽  
Constant Curvature ◽  
Statistical Manifolds

scholarly journals Detection of the homotopy type of an object using differential invariants of an approximating map

2019 ◽  
Vol 43 (4) ◽  
pp. 611-617
Author(s):  
S.V. Kurochkin
Keyword(s):  
Neural Network ◽  
Data Analysis ◽  
Homotopy Type ◽  
Morse Theory ◽  
Object Representation ◽  
Differential Invariants ◽  
Topological Data Analysis ◽  
Higher Dimension ◽  
Differential Topology ◽  
Open Questions

A method of topological data analysis is proposed that allows one to find out the homotopy type of the object under study. Unlike mature and widely used methods based on persistent homologies, our method is based on computing differential invariants of some map associated with an approximating map. Differential topology tools and the analogy with the main result in Morse theory are used. The approximating map can be constructed in the usual way using a neural network or otherwise. The method allows one to identify the homotopy type of an object in the plane because the number of circles in the homotopy equivalent object representation as a wedge is expressed through the degree of some map associated with the approximating map. The performance of the algorithm is illustrated by examples from the MNIST database and transforms thereof. Generalizations and open questions relating to a higher-dimension case are discussed.

scholarly journals Lectures on Differential Topology

2021 ◽  
Author(s):  
Riccardo Benedetti
Keyword(s):  
Differential Topology
Sign in / Sign up

Export Citation Format

Share Document