Numerical Algorithms on the Affine Grassmannian

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

doi:10.1137/18m1169321

Cited by 10 publications

(6 citation statements)

References 23 publications

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“…46 It is worth noting that  is identified as an equivalence class of n × p matrices under orthogonal transformation of the Stiefel manifold. [46][47][48] Therefore, a point on the Grassmann manifold is represented by an orthonormal matrix 𝚿 ∈ R n×p (the Stiefel representation).…”

Section: Grassmann Manifoldmentioning

confidence: 99%

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Santos

Giovanis

Kontolati

et al. 2022

Numerical Meth Engineering

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A novel surrogate model based on the Grassmannian diffusion maps (GDMaps) and utilizing geometric harmonics (GH) is developed for predicting the response of complex physical phenomena. The method utilizes GDMaps to obtain a low-dimensional representation of the underlying behavior of physical/ mathematical systems with respect to uncertain input parameters. Using this representation, GH, an out-of-sample extension technique, is employed to create a global map from the input parameter space to a Grassmannian diffusion manifold. GH is further employed to locally map points on the diffusion manifold onto the tangent space of a Grassmann manifold. The exponential map is then used to project the points in the tangent space onto the Grassmann manifold, where reconstruction of the full solution is performed. The performance of the proposed surrogate model is verified with three examples. The first problem is a toy example used to illustrate the technique. In the second example, errors associated with the various mappings are assessed by studying response predictions of the electric potential of a dielectric cylinder in a homogeneous electric field. The last example applies the method for uncertainty prediction in the strain field evolution in a model amorphous material using the shear transformation zone theory of plasticity.

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Section: Grassmann Manifoldmentioning

confidence: 99%

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Santos

Giovanis

Kontolati

et al. 2022

Numerical Meth Engineering

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“…The affine Grassmann manifold is a smooth manifold that consists of all d -dimensional affine subspaces in R d , thus it is also called the Grassmannian of the affine subspaces [43]. For any matrix, its representation on affine Grassmann manifold is the affine combination of orthonormal d -frames U with displacement µ, where µ is the mean of the matrix.…”

Section: Affine Grassmann Manifoldmentioning

confidence: 99%

Unsupervised Domain Adaptation via Discriminative Manifold Propagation

Luo,

Ren,

Dai

et al. 2020

Preprint

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Unsupervised domain adaptation is effective in leveraging rich information from a labeled source domain to an unlabeled target domain. Though deep learning and adversarial strategy made a significant breakthrough in the adaptability of features, there are two issues to be further studied. First, hard-assigned pseudo labels on the target domain are arbitrary and error-prone, and direct application of them may destroy the intrinsic data structure. Second, batch-wise training of deep learning limits the characterization of the global structure. In this paper, a Riemannian manifold learning framework is proposed to achieve transferability and discriminability simultaneously. For the first issue, this framework establishes a probabilistic discriminant criterion on the target domain via soft labels. Based on pre-built prototypes, this criterion is extended to a global approximation scheme for the second issue. Manifold metric alignment is adopted to be compatible with the embedding space. The theoretical error bounds of different alignment metrics are derived for constructive guidance. The proposed method can be used to tackle a series of variants of domain adaptation problems, including both vanilla and partial settings. Extensive experiments have been conducted to investigate the method and a comparative study shows the superiority of the discriminative manifold learning framework.

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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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Numerical Algorithms on the Affine Grassmannian

Cited by 10 publications

References 23 publications

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Unsupervised Domain Adaptation via Discriminative Manifold Propagation

On the Sensitivity of Singular and Ill-Conditioned Linear Systems

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