Angles between subspaces and their tangents

Zhu, Peizhen; Knyazev, Andrew

doi:10.1515/jnum-2013-0013

Cited by 56 publications

(35 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We review the definition of angles between subspaces, and the connections between angles and projectors. Definition 1.4 (Section 6.4.3 in [11] and Section 2 in [28]). Let Z ∈ R m×k and Z ∈ R m×ℓ with ℓ ≥ k have orthonormal columns so that Z T Z = I k and Z T Z = I ℓ .…”

Section: Singular Value Decompositionmentioning

confidence: 99%

Low-Rank Matrix Approximations Do Not Need a Singular Value Gap

Drineas¹,

Ipsen²

2019

SIAM J. Matrix Anal. Appl.

View full text Add to dashboard Cite

This is a systematic investigation into the sensitivity of low-rank approximations of real matrices. We show that the low-rank approximation errors, in the two-norm, Frobenius norm and more generally, any Schatten p-norm, are insensitive to additive rank-preserving perturbations in the projector basis; and to matrix perturbations that are additive or change the number of columns (including multiplicative perturbations). Thus, low-rank matrix approximations are always wellposed and do not require a singular value gap. In the presence of a singular value gap, connections are established between low-rank approximations and subspace angles.

show abstract

Section: Singular Value Decompositionmentioning

confidence: 99%

Low-Rank Matrix Approximations Do Not Need a Singular Value Gap

Drineas¹,

Ipsen²

2019

SIAM J. Matrix Anal. Appl.

View full text Add to dashboard Cite

show abstract

“…and we (again) see that D 2 = I m − D 2 = 1 + TanΨ 2 2 . For further insights on the tangents between subspaces, see e.g., [67].…”

Section: Together This Givesmentioning

confidence: 99%

The Discrete Empirical Interpolation Method: Canonical Structure and Formulation in Weighted Inner Product Spaces

Drmač¹,

Saibaba²

2018

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

New contributions are offered to the theory and numerical implementation of the Discrete Empirical Interpolation Method (DEIM). A substantial tightening of the error bound for the DEIM oblique projection is achieved by index selection via a strong rank revealing QR factorization. This removes the exponential factor in the dimension of the search space from the DEIM projection error, and allows sharper a priori error bounds. Well-known canonical structure of pairs of projections is used to reveal canonical structure of DEIM. Further, the DEIM approximation is formulated in weighted inner product defined by a real symmetric positive-definite matrix W . The weighted DEIM (W -DEIM) can be interpreted as a numerical implementation of the Generalized Empirical Interpolation Method (GEIM) and the more general Parametrized-Background Data-Weak (PBDW) approach. Also, it can be naturally deployed in the framework when the POD Galerkin projection is formulated in a discretization of a suitable energy (weighted) inner product such that the projection preserves important physical properties such as e.g. stability. While the theoretical foundations of weighted POD and the GEIM are available in the more general setting of function spaces, this paper focuses to the gap between sound functional analysis and the core numerical linear algebra. The new proposed algorithms allow different forms of W -DEIM for point-wise and generalized interpolation. For the generalized interpolation, our bounds show that the condition number of W does not affect the accuracy, and for point-wise interpolation the condition number of the weight matrix W enters the bound essentially as min D=diag κ 2 (DW D), where κ 2 (W ) = W 2 W −1 2 is the spectral condition number.

show abstract

“…The concept of principal angles between subspaces was first presented by Jordan in 1875 [15]. The principal angle or canonical principal angle gives information about the relative position of two subspaces in a Euclidean space.…”

Section: Principal Anglesmentioning

confidence: 99%

EEG subspace analysis and classification using principal angles for Brain-Computer Interfaces

Ashari

Anderson

2014

2014 IEEE Symposium on Computational Intelligence in Brain Computer Interfaces (CIBCI)

View full text Add to dashboard Cite

This dissertation studies and examines different ideas using principal angles and subspaces concepts. It introduces a novel mathematical approach for comparing sets of EEG signals for use in new BCI technology. The success of the presented results show that principal angles are also a useful approach to the classification of EEG signals that are recorded during a BCI typing application. In this application, the appearance of a subject's desired letter is detected by identifying a P300-wave within a one-second window of EEG following the flash of a letter. Smoothing the signals before using them is the only preprocessing step that was implemented in this study. The smoothing process based on minimizing the second derivative in time is implemented to increase the classification accuracy instead of using the bandpass filter that relies on assumptions on the frequency content of EEG. This study examines four different ways of removing outliers that are based on the principal angles and shows that the outlier removal methods did not help in the presented situations.ii One of the concepts that this dissertation focused on is the effect of the number of trials on the classification accuracies. The achievement of the good classification results by using a small number of trials starting from two trials only, should make this approach more appropriate for online BCI applications.In order to understand and test how EEG signals are different from one subject toanother, different users are tested in this dissertation, some with motor impairments. Furthermore, the concept of transferring information between subjects is examined by training the approach on one subject and testing it on the other subject using the training subject's EEG subspaces to classify the testing subject's trials.iii

show abstract

Angles between subspaces and their tangents

Cited by 56 publications

References 22 publications

Low-Rank Matrix Approximations Do Not Need a Singular Value Gap

Low-Rank Matrix Approximations Do Not Need a Singular Value Gap

The Discrete Empirical Interpolation Method: Canonical Structure and Formulation in Weighted Inner Product Spaces

EEG subspace analysis and classification using principal angles for Brain-Computer Interfaces

Contact Info

Product

Resources

About