Optimal interpolants on Grassmann manifolds

Erchuan Zhang, Lyle Noakes

Research output: Contribution to journalArticle

Abstract

The Grassmann manifoldGrm(Rn) of all m-dimensional subspaces of the n-dimensional space Rn(m< n) is widely used in image analysis, statistics and optimization. Motivated by interpolation in the manifold Gr2(R4) , we first formulate the differential equation for desired interpolation curves called Riemannian cubics in symmetric spaces by the Pontryagin maximum principle (PMP) and then narrow down to it in Gr2(R4). Although computation on this low-dimensional manifold may not occur heavy burden for modern machines, theoretical analysis for Riemannian cubics is very limited in references due to its highly nonlinearity. This paper focuses on presenting analytical and geometrical structures for the so-called Lie quadratics associated with Riemannian cubics. By analysing asymptotics of Lie quadratics, we find asymptotics of Riemannian cubics in Gr2(R4). Finally, we illustrate our results by numerical simulations.

Original languageEnglish
Pages (from-to)363-383
Number of pages21
JournalMathematics of Control, Signals, and Systems
Volume31
Issue number3
DOIs
Publication statusPublished - 1 Sep 2019

Fingerprint

Grassmann Manifold
Interpolants
Interpolation
Interpolate
Pontryagin Maximum Principle
Maximum principle
Symmetric Spaces
Image Analysis
Image analysis
n-dimensional
Theoretical Analysis
Differential equations
Subspace
Statistics
Nonlinearity
Differential equation
Numerical Simulation
Curve
Optimization
Computer simulation

Cite this

@article{0c1e5cbf74664f608c1f79e593606ae6,
title = "Optimal interpolants on Grassmann manifolds",
abstract = "The Grassmann manifoldGrm(Rn) of all m-dimensional subspaces of the n-dimensional space Rn(m< n) is widely used in image analysis, statistics and optimization. Motivated by interpolation in the manifold Gr2(R4) , we first formulate the differential equation for desired interpolation curves called Riemannian cubics in symmetric spaces by the Pontryagin maximum principle (PMP) and then narrow down to it in Gr2(R4). Although computation on this low-dimensional manifold may not occur heavy burden for modern machines, theoretical analysis for Riemannian cubics is very limited in references due to its highly nonlinearity. This paper focuses on presenting analytical and geometrical structures for the so-called Lie quadratics associated with Riemannian cubics. By analysing asymptotics of Lie quadratics, we find asymptotics of Riemannian cubics in Gr2(R4). Finally, we illustrate our results by numerical simulations.",
keywords = "Asymptotics, Grassmann manifold, Optimal control, Riemannian cubic, Symmetric space",
author = "Erchuan Zhang and Lyle Noakes",
year = "2019",
month = "9",
day = "1",
doi = "10.1007/s00498-019-0241-9",
language = "English",
volume = "31",
pages = "363--383",
journal = "Mathematics of Control Signals and Systems",
issn = "0932-4194",
publisher = "Springer",
number = "3",

}

Optimal interpolants on Grassmann manifolds. / Zhang, Erchuan; Noakes, Lyle.

In: Mathematics of Control, Signals, and Systems, Vol. 31, No. 3, 01.09.2019, p. 363-383.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Optimal interpolants on Grassmann manifolds

AU - Zhang, Erchuan

AU - Noakes, Lyle

PY - 2019/9/1

Y1 - 2019/9/1

N2 - The Grassmann manifoldGrm(Rn) of all m-dimensional subspaces of the n-dimensional space Rn(m< n) is widely used in image analysis, statistics and optimization. Motivated by interpolation in the manifold Gr2(R4) , we first formulate the differential equation for desired interpolation curves called Riemannian cubics in symmetric spaces by the Pontryagin maximum principle (PMP) and then narrow down to it in Gr2(R4). Although computation on this low-dimensional manifold may not occur heavy burden for modern machines, theoretical analysis for Riemannian cubics is very limited in references due to its highly nonlinearity. This paper focuses on presenting analytical and geometrical structures for the so-called Lie quadratics associated with Riemannian cubics. By analysing asymptotics of Lie quadratics, we find asymptotics of Riemannian cubics in Gr2(R4). Finally, we illustrate our results by numerical simulations.

AB - The Grassmann manifoldGrm(Rn) of all m-dimensional subspaces of the n-dimensional space Rn(m< n) is widely used in image analysis, statistics and optimization. Motivated by interpolation in the manifold Gr2(R4) , we first formulate the differential equation for desired interpolation curves called Riemannian cubics in symmetric spaces by the Pontryagin maximum principle (PMP) and then narrow down to it in Gr2(R4). Although computation on this low-dimensional manifold may not occur heavy burden for modern machines, theoretical analysis for Riemannian cubics is very limited in references due to its highly nonlinearity. This paper focuses on presenting analytical and geometrical structures for the so-called Lie quadratics associated with Riemannian cubics. By analysing asymptotics of Lie quadratics, we find asymptotics of Riemannian cubics in Gr2(R4). Finally, we illustrate our results by numerical simulations.

KW - Asymptotics

KW - Grassmann manifold

KW - Optimal control

KW - Riemannian cubic

KW - Symmetric space

UR - http://www.scopus.com/inward/record.url?scp=85069690065&partnerID=8YFLogxK

U2 - 10.1007/s00498-019-0241-9

DO - 10.1007/s00498-019-0241-9

M3 - Article

VL - 31

SP - 363

EP - 383

JO - Mathematics of Control Signals and Systems

JF - Mathematics of Control Signals and Systems

SN - 0932-4194

IS - 3

ER -