A useful variant of the Davis--Kahan theorem for statisticians

By Yi Yu, Tengyao Wang, Richard J. Samworth. 2014

702 citations recorded by the source corpus.

The Davis–Kahan theorem is used in the analysis of many statistical procedures to bound the distance between subspaces spanned by population eigenvectors and their sample versions. It relies on an eigenvalue separation condition between certain population and sample eigenvalues. We present a variant of this result that depends only on a population eigenvalue separation condition, making it more natural and convenient for direct application in statistical contexts, and provide an improvement in many cases to the usual bound in the statistical literature. We also give an extension to situations where the matrices under study may be asymmetric or even non-square, and where interest is in the distance between subspaces spanned by corresponding singular vectors.

Subfields: Statistics and Probability, Statistical Learning Theory

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistics and Probability

In Statistical Learning Theory

Citations in the corpus

Error bounds for kernel-based approximations of the Koopman operator (2023, background): “□ The next theorem is a variant of the Davis-Kahan sin(Θ) theorem (cf. [65]).”
Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics (2016, methodology): “This problem has been widely studied in the literature (Davis and Kahan, 1970; Wedin, 1972; Weyl, 1912; Stewart, 1991, 2006; Yu et al., 2015).”
Non-Stationary Dynamic Factor Models for Large Datasets (2016, background): “Then, from (A76) by Theorem 2 in Yu et al. (2015), we can prove that ∥Ŵq − JWqJq ∥∥ = Op(θnT,δ), (A78) where Jq is a diagonal q × q matrix with entries 1 or -1, different from J.”
Nonasymptotic upper bounds for the reconstruction error of PCA (2016, background): “[31] Y. Yu, T. Wang, and R. J. Samworth.”
Statistical Inference on Random Dot Product Graphs: a Survey (2017, methodology): “The impact of matrix perturbations on associated eigenspaces is significantly more complicated, and the Davis-Kahan Theorem [9, 26] provides one approach to the latter.”
Simultaneous multiple change-point and factor analysis for high-dimensional time series (2016, neutral): “For part (ii), evoking Corollary 1 of Yu et al. (2015) and noticing that Γχ has distinct eigenvalues given in (C1), we can further show that S is a diagonal matrix with entries ±1.”
Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains (2015, background): “(A2) is used in combination with (A5) to obtain a convergence rate for the maximal eigenvalue of Z− Ẑ, which, in turn, affects the convergence of the eigenvectors ĉm to cm (Yu et al., 2015).”
Non-asymptotic Identification of LTI Systems from a Single Trajectory (2018, result): “This is essentially similar to results of Wedin and Davis-Kahan [5,32] with the added advantage of simultaneous control which we crucially need for our result.”
Communication-efficient Algorithms for Distributed Stochastic Principal Component Analysis (2017, neutral): “To handle the second part, note that by a variant of the Davis-Kahan sinθ theorem (see Corollary 1 in [25]), if maxi ‖X̂i − X‖ ≤ δ/12, then the leading eigenvectors v̂i 1 of X̂i (after choosing the sign appropriately, i.”
Concentration bounds on response-based vector embeddings of black-box generative models (2025, unknown): “We know, R 3 = U | Λ | where the last inequality follows from Davis-Kahan Theorem (Yu et al., 2015).”