The Angle Between Complementary Subspaces

Ipsen, Ilse C. F.; Meyer, Carl D.

doi:10.2307/2975268

Cited by 45 publications

(14 citation statements)

References 15 publications

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“…The last equality in (28) holds because (I − P i ) E i is a projection and the norm of a nontrivial projection in an inner product space depends only on the angle between its range and its nullspace [10].…”

Section: Multilevel Bddcmentioning

confidence: 99%

Adaptive-Multilevel BDDC and its parallel implementation

2013

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We combine the adaptive and multilevel approaches to the BDDC and formulate a method which allows an adaptive selection of constraints on each decomposition level. We also present a strategy for the solution of local eigenvalue problems in the adaptive algorithm using the LOBPCG method with a preconditioner based on standard components of the BDDC. The effectiveness of the method is illustrated on several engineering problems. It appears that the Adaptive-Multilevel BDDC algorithm is able to effectively detect troublesome parts on each decomposition level and improve convergence of the method. The developed open-source parallel implementation shows a good scalability as well as applicability to very large problems and core counts.Comment: 37 pages, 10 figures, 13 table

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Section: Multilevel Bddcmentioning

confidence: 99%

Adaptive-Multilevel BDDC and its parallel implementation

2013

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“…Returning from the above detour, we recall that the central objects of our analysis are the cosine similarity matrices Q (n) k defined in (19). The dynamics of the cosine similarity matrix Q k is fully analogous to the situation demonstrated by the above toy example.…”

Section: B the Scaling Limits Of Stochastic Processes: Main Ideasmentioning

confidence: 87%

“…This concept can be naturally extended to arbitrary d ≥ 1. In general, the closeness of two d-dimensional subspaces may be quantified by their d principal angles [19], [20]. In particular, the cosines of the principal angles are uniquely specified as the singular values of the d × d matrix…”

Section: A Performance Metric: Principal Anglesmentioning

confidence: 99%

“…Thanks to the streaming nature of all three methods, it is easy to see that the dynamics of their estimates X k can be modeled by homogeneous Markov chains with state space in R n×d . Thus, being a function of X k [see (19)], the dynamics of Q (n) k forms a hidden Markov chain. We then show that, as n → ∞ and with proper time scaling, the family of stochastic processes Q (n) k indexed by n converges weakly to a deterministic function of time that is characterized as the unique solution of some ODEs.…”

Section: B the Scaling Limits Of Stochastic Processes: Main Ideasmentioning

confidence: 99%

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Subspace Estimation From Incomplete Observations: A High-Dimensional Analysis

Wang

Eldar

2018

IEEE J. Sel. Top. Signal Process.

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We present a high-dimensional analysis of three popular algorithms, namely, Oja's method, GROUSE and PE-TRELS, for subspace estimation from streaming and highly incomplete observations. We show that, with proper time scaling, the time-varying principal angles between the true subspace and its estimates given by the algorithms converge weakly to deterministic processes when the ambient dimension n tends to infinity. Moreover, the limiting processes can be exactly characterized as the unique solutions of certain ordinary differential equations (ODEs). A finite sample bound is also given, showing that the rate of convergence towards such limits is O(1/ √ n). In addition to providing asymptotically exact predictions of the dynamic performance of the algorithms, our high-dimensional analysis yields several insights, including an asymptotic equivalence between Oja's method and GROUSE, and a precise scaling relationship linking the amount of missing data to the signalto-noise ratio. By analyzing the solutions of the limiting ODEs, we also establish phase transition phenomena associated with the steady-state performance of these techniques.

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“…Hotelling [10] defines PABS in the form of canonical correlations in statistics in 1936. Traditionally, PABS are introduced and used via their sines and more commonly, because of their connection to canonical correlations, cosines; see, e.g., [4,11,14,21,23]. The properties of sines and cosines of PABS are well investigated; e.g., in [1,13,22].…”

Section: Introductionmentioning

confidence: 99%

Angles between subspaces and their tangents

Zhu¹,

Knyazev²

2013

Journal of Numerical Mathematics

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Abstract. Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomposition. We utilize the same idea for the tangents, i.e., explicitly construct matrices, such that their singular values are equal to the tangents of PABS, using several approaches: orthonormal and non-orthonormal bases for subspaces, as well as projectors. Such a construction has applications, e.g., in analysis of convergence of subspace iterations for eigenvalue problems.

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The Angle Between Complementary Subspaces

Cited by 45 publications

References 15 publications

Adaptive-Multilevel BDDC and its parallel implementation

Adaptive-Multilevel BDDC and its parallel implementation

Subspace Estimation From Incomplete Observations: A High-Dimensional Analysis

Angles between subspaces and their tangents

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