Tangent Space Methods for Approximation on Compact Homogeneous Manifolds

2001 ◽  
pp. 211-224
Author(s):  
Jeremy Levesley
Keyword(s):  
Tangent Space ◽  
Homogeneous Manifolds

Eigenvalues of the Curvature Operator for Certain Homogeneous Manifolds

1990 ◽  
Vol 42 (6) ◽  
pp. 981-999
Author(s):  
J. E. D'Atri ◽  
I. Dotti Miatello
Keyword(s):  
Riemannian Manifold ◽  
Tangent Space ◽  
Orthonormal Basis ◽  
Riemann Tensor ◽  
Inner Product ◽  
Curvature Operator ◽  
Curvature Function ◽  
Homogeneous Manifolds ◽  
Exterior Power ◽  
Image Position

Given a Riemannian manifold M, the Riemann tensor R induces the curvature operator on the exterior power of the tangent space, defined by the formula where the inner product is defined by From the symmetries of R, it follows that ρ is self-adjoint and so has only real eigenvalues. R also induces the sectional curvature function K on 2-planes in is an orthonormal basis of the 2-plane π.

Four-dimensional homogeneous manifolds satisfying some Einstein-like conditions

2020 ◽  
Vol 43 (3) ◽  
pp. 465-488
Author(s):  
Eduardo García-Río ◽  
Ali Haji-Badali ◽  
Rodrigo Mariño-Villar ◽  
M. Elena Vázquez-Abal
Keyword(s):  
Homogeneous Manifolds

scholarly journals Quantum Orlicz Spaces in Information Geometry

2004 ◽  
Vol 11 (04) ◽  
pp. 359-375 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Quantum Information ◽  
Tangent Space ◽  
Selfadjoint Operator ◽  
Affine Connection ◽  
Orlicz Spaces ◽  
Trace Class ◽  
Information Geometry ◽  
Young Function ◽  
Affine Structure ◽  
Cotangent Space

Let H0 be a selfadjoint operator such that Tr e−βH0 is of trace class for some β < 1, and let χɛ denote the set of ɛ-bounded forms, i.e., ∥(H0+C)−1/2−ɛX(H0+C)−1/2+ɛ∥ < C for some C > 0. Let χ := Span ∪ɛ∈(0,1/2]χɛ. Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρx = e−H0−X−ψx, X ∈ χ. We show that if Tr e−H0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ0, consisting of p-nearby states (those [Formula: see text] obeying C−1ρ1+p ≤ σ ≤ Cρ1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.

Manifold learning in local tangent space via extreme learning machine

2016 ◽  
Vol 174 ◽  
pp. 18-30 ◽  
Author(s):  
Qian Wang ◽  
Weiguo Wang ◽  
Rui Nian ◽  
Bo He ◽  
Yue Shen ◽  
...  
Keyword(s):  
Extreme Learning Machine ◽  
Manifold Learning ◽  
Tangent Space ◽  
Learning Machine ◽  
Local Tangent

scholarly journals Plant Leaf Recognition through Local Discriminative Tangent Space Alignment

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Chuanlei Zhang ◽  
Shanwen Zhang ◽  
Weidong Fang
Keyword(s):  
Dimensionality Reduction ◽  
Manifold Learning ◽  
Tangent Space ◽  
Reduction Method ◽  
Plant Leaf ◽  
Local Patch ◽  
Dimensionality Reduction Method

Manifold learning based dimensionality reduction algorithms have been payed much attention in plant leaf recognition as the algorithms can select a subset of effective and efficient discriminative features in the leaf images. In this paper, a dimensionality reduction method based on local discriminative tangent space alignment (LDTSA) is introduced for plant leaf recognition based on leaf images. The proposed method can embrace part optimization and whole alignment and encapsulate the geometric and discriminative information into a local patch. The experiments on two plant leaf databases, ICL and Swedish plant leaf datasets, demonstrate the effectiveness and feasibility of the proposed method.

The Kobayashi-pseudodistance on homogeneous manifolds

1990 ◽  
Vol 68 (1) ◽  
pp. 117-134 ◽  
Author(s):  
Jörg Winkelmann
Keyword(s):  
Homogeneous Manifolds ◽  
Kobayashi Pseudodistance

scholarly journals Two-Point Homogeneous Manifolds

1965 ◽  
Vol 68 ◽  
pp. 746-753 ◽  
Author(s):  
H.O. Singh Varma
Keyword(s):  
Homogeneous Manifolds
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