Matrix Perturbation Theory

By G. W. Stewart, Jiguang Sun. 1990

How researchers see this paper

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

Citations in the corpus

Set2Graph: Learning Graphs From Sets (2020, methodology): “diag(1;2;:::;n). This diagonal addition does not change the 2-edge values assigned by G2, and it guarantees a simple specturm using standard hermitian matrix eigenvalue perturbation theory (see e.g., [36], Section IV:4). Now let G2(X) = U(X) (X)V(X)T be the SVD of G2(X), where U= [u 1;:::;u n] and V”
Learning One-hidden-layer Neural Networks with Landscape Design (2017, neutral): “Theorem A.9 (Stewart and Sun [Ste90]) .”
Proximal Methods for Hierarchical Sparse Coding (2010, background): “The next lemma shows how we can easily add a non-negativity constraint on the proximal operator when the norm W is absolute (Stewart and Sun, 1990, Deﬁnition 1.2), that is, a norm for which the relation W ( u ) ≤ ( w ) holds for any two vectors w and u ∈ R p such that | u j | ≤ | w j | for all j .”
Single View Metrology (2000, unknown): “We now use matrix perturbation theory (Golub and Van Loan, 1989; Stewart and Sun, 1990; Wilkinson, 1965) to compute the covariance Λs of the solution vector s based on this zero approximation.”
Estimating Smooth GLM in Non-interactive Local Differential Privacy Model with Public Unlabeled Data (2019, background): “Lemma 11 (Weyl’s Inequality (Stewart, 1990)) Let X,Y ∈ Rp×p be two symmetric matrices, and E = X − Y .”
The asymptotic variance of subspace estimates (2004, unknown): “According to (Stewart and Sun 1990, Thm 2.3, p. 183), there is an eigenvalue λi of A such that the difference between the i-the eigenvalue, λ̂iN , of (T̂N ) −1ÂN T̂N and λi, satisfies λ̂iN − λi = v>i ˜̃ANui v>i ui +O(‖ ˜̃AN‖2) (4.41) where vi and ui are the normalized left and right eigenvectors…”
Voronoi-Based Curvature and Feature Estimation from Point Clouds (2011, background): “[27]), one then obtains the convergence of the eigenvalues and eigenvectors of VCn ,R ∗ χr(p) to those of VS,R ∗χr(p), provided that the eigenvalues have multiplicity one.”
Making Method of Moments Great Again? -- How can GANs learn the target distribution (2020, background): “bation of Inverses) Given matrices A;E2R k. If Ais invertible, kA 1Ek 2 < 1, then A~ = A+ Eis also invertible, and: kA~ 1 A 1k 2 6 kEk 2kA 1k2 2 1 k A 1Ek 2 The proof of this lemma can be found in [69]. Lemma 8. (Weyl’s Theorem) Let A;E2R nbe two symmetrical matrices. Denote i(X) be the i-th lar”
An Analysis of Random Design Linear Regression (2011, unknown): “Since ∑ j λj(Σw) 2 = ∑ j ( λj(Σ) λj(Σ) + λ )2 = d2,λ and, by Mirsky’s theorem (Stewart and Sun, 1990, Corollary 4.13), ∑ j (λj(Σ̂w)− λj(Σw))2 ≤ ‖Σ̂w −Σw‖2F = ‖∆λ‖2F Hence, tr(M) = 1 n · tr(Σ̂−1λ,wΣ̂wΣ̂−1λ,wΣw) ≤ 1 n · 1 (1− ‖∆λ‖)2 · ( d2,λ + √ d2,λ · ‖∆λ‖2F ) which completes the proof.”
A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization (2008, unknown): “(Stewart and Sun, 1990).”
P4P: Practical Large-Scale Privacy-Preserving Distributed Computation Robust against Malicious Users (2010, background): “Proof The classic Weyl and Mirsky theorems [47] bound the perturbation to A’s singular values in terms of the Frobenius norm ‖ · ‖F of E := A− Ã:”