The Rotation of Eigenvectors by a Perturbation. III

By Chandler Davis, W. Kahan. 1970

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Citations in the corpus

Sufficient Forecasting Using Factor Models (2015, background): “nvectors. Correspondingly, let fb ig p i=1 be the eigenvalues of bin descending order and fb˘gp i=1 be their associated eigenvectors. Then, (a)Weyl’s Theorem: jb i ijkb k: 26 (b) sin() Theorem (Davis and Kahan, 1970): kb˘ i ˘ ik p 2 kb k min(jb i 1 ij;j i b i+1j) : 6.1 Proof of Proposition 2.1 Pro”
Distributed Robust Learning (2014, methodology): “Then, applying the Davis-Kahan perturbation theorem [6], we have, whenever Note that the above condition gives an upper bound on the outlier fraction: which yields, λ ≤ ∆ r ∆ r + 4 σ 2 x log p .”
Geometric median and robust estimation in Banach spaces (2013, neutral): “assume that ∆m>44 r (EkXk4 −tr(Σ2))log(1.4/δ) n . (4.8) Then Pr kProj\ m−ProjmkF ≥ 22 ∆m r (EkXk4 −tr(Σ2))log(1.4/δ) n ≤δ. 12 S. Minsker Proof. It follows from Davis–Kahan perturbation theorem [16] (see also Theorem 3 in [49]) that, whenever kΣˆ −ΣkF <1 4∆m, kProj\ m−ProjmkF ≤ 2kΣˆ −ΣkF ∆m .”
A gentle guide to the basics of two projections theory (2010, background): “The following is a special case of what is usually called the CS decomposition; see, for example, [27,53,79,97].”
Semiparametric estimation of fractional cointegrating subspaces (2006, unknown): “(6) In keeping with (5), we assume that we can write the spectral density matrix of {zt} as f (ω) = Υ(ω)f†(ω)Υ∗(ω) , (7) where Υ(ω) = diag {( 1− e−iω)−d0 , .”
Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, methodology): “See Le Cam (1973) and Donoho and Liu (1991). In nonparametric estimation problems, Le Cam’s method often provides the minimax lower bound for estimating a real-valued functional.”
Spectral Method and Regularized MLE Are Both Optimal for Top-$K$ Ranking (2017, unknown): “The first important result concerns the eigenvector perturbation for probability transition matrices, which can be treated as the analogue of the celebrated Davis-Kahan sin Θ theorem (Davis and Kahan, 1970) for asymmetric matrices.”
Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview (2018, unknown): “This was first established in the celebrated Davis-Kahan sinΘ Theorem [124].”
Statistical Inference on Random Dot Product Graphs: a Survey (2017, result): “nd the nal term in this product is, trivially, at most 1. To bound the second term, we will follow reasoning similar to that in Lemma 2, combined with the Davis-Kahan theorem. The Davis-Kahan Theorem [9, 27] implies that for a suitable constant C>0, YU AU Š A −U PU Š PY≤ CYA−PY d(P) : By Theorem”
Provable Dynamic Robust PCA or Robust Subspace Tracking (2017, unknown): “To bound SE( P̂rot, Erot), we use the Davis-Kahan sin θ theorem [33] given below.”
Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics (2016, background): “This problem has been widely studied in the literature (Davis and Kahan, 1970; Wedin, 1972; Weyl, 1912; Stewart, 1991, 2006; Yu et al., 2015).”
Large Covariance Estimation by Thresholding Principal Orthogonal Complements (2011, unknown): “(sin θ Theorem, Davis and Kahan, 1970) ‖ξ̂i − ξi‖ ≤ √ 2‖Σ̂−Σ‖ min(|λ̂i−1 − λi|, |λi − λ̂i+1”
Item Response Theory—A Statistical Framework for Educational and Psychological Measurement (2021, unknown): “By making use of (3.4) and by proving extensions of the Davis-Kahan-Wedin sine theorem (Davis and Kahan, 1970; Wedin, 1972) for the perturbation of eigenvectors, it can be shown under suitable conditions that (3.5) Under the typical setting where N is much larger than J , the rates in (3.4) and…”
Finite sample approximation results for principal component analysis: a matrix perturbation approach (2009, background): “Lemma 3 is known as the sin θ theorem of Davis and Kahan [9], see also [31, Theorem 11.7.1].”
Relational Representation Learning for Dynamic (Knowledge) Graphs: A Survey (2019, background): “The Davis-Kahan theorem [55] gives an approximation error for the top d eigen-pairs.”
The multiscale structure of non-differentiable image manifolds (2005, background): “Now from the theory of angles between subspaces [16,17], we have that the angles between the subspaces T (s0, θ ) and T (s1, θ ) are naturally expressed as cos(angle #i) = λs0,s1 , i = 0, 1.”
Spectral radii of sparse random matrices (2017, background): “ssert that also the corresponding eigenspaces of Aand EAare close when kA EAkis small compared to the spectral gap around the eigenvalue of Aunder consideration (for instance, the Davis-Kahan Theorem [12] gives a precise quantitative statement of this kind). We may summarize our discussion with the ”