The rotation of eigenvectors by a perturbation

By Chandler Davis. 1963

Although the behavior of the eigenvalues of a hermitian matrix under perturbation is fairly well understood, there has been almost nothing done on the behavior of the eigenvectors. It is well known that they vary analytically under analytic perturbations but for some purposes one would prefer sharp bounds on the distance between the eigenvectors of a matrix and those of a matrix approximating it. Two sets of theorems of this sort are given. (auth)

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Citations in the corpus

Sufficient Forecasting Using Factor Models (2015, background): “(b) sin( θ ) Theorem (Davis and Kahan, 1970): 6.1 Proof of Proposition 2.1 Proof.”
Geometric median and robust estimation in Banach spaces (2013, neutral): “It follows from Davis–Kahan perturbation theorem [16] (see also Theorem 3 in [49]) that, whenever ‖̂ − ‖F < (1)4 m, ‖P̂rojm − Projm ‖F ≤ 2‖̂ − ‖F m .”
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).”
Robust Covariance Matrix Estimation via Matrix Depth (2015, unknown): “By Davis-Kahan theorem [11], we have ‖V̂ V̂ T − V V T ‖F ≤ C ′‖Γ̂/β − Σ‖op/λ, and the proof is complete.”
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.”
Operator norm consistent estimation of large-dimensional sparse covariance matrices (2008, methodology): “2 Convergence of eigenvectors Perhaps even more interestingly, controlling the spectral norm allows us to get very good control on the angles between the eigenspaces of the population and sample covariance matrix, through the use of the classical sin(θ) theorems of Davis and Kahan (Davis and Kahan (”
PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming (2011, unknown): “The sin-θ -Theorem [11] implies that |sinθ | ≤ ‖X̂−X‖ |λ̂1| ≤ ε0 1− ε0 ,”
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.”
Critical Points of the Singular Value Decomposition (2005, background): “At this point it is possible to describe the relationship between Theorem 1 and the sin(θ) theorem [4, 12], which bounds the rotation of eigenvectors or singular vectors D ow nl oa de d 01 /0 5/ 13 to 1 50 .”
Sparse PCA: Optimal rates and adaptive estimation (2012, background): “The proofs rely on results in [17, 19, 20, 25, 31, 33] and [51].”
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…”
Non-Stationary Dynamic Factor Models for Large Datasets (2016, unknown): “(A86) Analogous formulas to (A84)-(A86) are in Theorem 1 by Sims et al. (1990) and, by combining them, ∥∥Â1VAR1 −A1 ∥∥ = Op ( 1√ T ) .”
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.”