The Grassmannian of affine subspaces

Lim, Lek-Heng; Wong, Ken Sze-Wai; Yao, Ke

doi:10.48550/arxiv.1807.10883

Cited by 2 publications

(2 citation statements)

References 35 publications

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“…The set of k-dimensional affine subspaces of n is called the affine Grassmannian [16, §7.1] [21,22] of index k of n denoted by…”

Section: Preliminariesmentioning

confidence: 99%

On the Sensitivity of Singular and Ill-Conditioned Linear Systems

Zeng¹

2019

SIAM J. Matrix Anal. Appl.

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Solving a singular linear system for an individual vector solution is an ill-posed problem with a condition number infinity. From an alternative perspective, however, the general solution of a singular system is of a bounded sensitivity as a unique element in an affine Grassmannian. If a singular linear system is given through empirical data that are sufficiently accurate with a tight error bound, a properly formulated general numerical solution uniquely exists in the same affine Grassmannian, enjoys Lipschitz continuity and approximates the underlying exact solution with an accuracy in the same order as the data. Furthermore, any backward accurate numerical solution vector is an accurate approximation to one of the solutions of the underlying singular system.

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“…The set of k-dimensional affine subspaces of n is called the affine Grassmannian [16, §7.1] [21,22] of index k of n denoted by…”

Section: Preliminariesmentioning

confidence: 99%

On the Sensitivity of Singular and Ill-Conditioned Linear Systems

Zeng¹

2019

SIAM J. Matrix Anal. Appl.

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“…Classical examples of metric spacevalued data are directional data such as sphere-valued data, orthonormal frame data and subspace data [31,12]. Detailed expositions of metric space structures on the Stiefel manifold V k pR n q of orthonormal k-frames in R n and the real Grassmannian manifold Gr k pR n q of k-dimensional subspaces in R n can be found in [15,29]. Other notable examples of metric space-valued data include positive definite covariance matrices [30], shape space modelling on the complex Grassmannian [22,25], and hierarchical structures that can be represented in hyperbolic spaces [36].…”

Section: Introductionmentioning

confidence: 99%

Equivariant Estimation of Fréchet Means

McCormack¹,

Hoff²

2021

Preprint

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The Fréchet mean generalizes the concept of a mean to a metric space setting. In this work we consider equivariant estimation of Fréchet means for parametric models on metric spaces that are Riemannian manifolds. The geometry and symmetry of such a space is encoded by its isometry group. Estimators that are equivariant under the isometry group take into account the symmetry of the metric space. For some models there exists an optimal equivariant estimator, which necessarily will perform as well or better than other common equivariant estimators, such as the maximum likelihood estimator or the sample Fréchet mean. We derive the general form of this minimum risk equivariant estimator and in a few cases provide explicit expressions for it. In other models the isometry group is not large enough relative to the parametric family of distributions for there to exist a minimum risk equivariant estimator. In such cases, we introduce an adaptive equivariant estimator that uses the data to select a submodel for which there is an MRE. Simulations results show that the adaptive equivariant estimator performs favorably relative to alternative estimators.

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The Grassmannian of affine subspaces

Cited by 2 publications

References 35 publications

On the Sensitivity of Singular and Ill-Conditioned Linear Systems

On the Sensitivity of Singular and Ill-Conditioned Linear Systems

Equivariant Estimation of Fréchet Means

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