Matrix Perturbation Theory

By V. N. Bogaevski, A. Povzner. 1991

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

Beyond CCA: Moment Matching for Multi-View Models (2016, neutral): “This kind of analysis can potentially be extended with the techniques outlined in (Chapter 4, Stewart & Sun, 1990).”
A tutorial on spectral clustering (2007, background): “In particular for k = 2, mincut is a relatively easy problem and can be solved efficiently, see Stoer and Wagner (1997) and the discussion therein.”
Community detection in graphs (2009, methodology): “The scenario we have described is expected from perturbation theory (Bhatia, 1997; Stewart and Sun, 1990).”
Two-sided and alternating Jacobi-Davidson (2001, background): “(Cf. [12, p. 145]) We de(cid:12)ne the e(cid:11)ective condition number of a nonzero matrix C as where C + is the pseudo-inverse of C , and the (cid:27) s are the singular values of C .”
Sparse Principal Component Analysis and Iterative Thresholding (2011, methodology): “Geometrically, it equals the squared sine of the largest canonical angle between S and S [30, Theorem 5.5]. b Throughout the paper, we use the loss function (2.1) for principal subspace estimation.”
More on pseudospectra for polynomial eigenvalue problems and applications in control theory (2002, methodology): “In the cased = 1 this representation reduces to the well-known βAx = αBx form of the generalized eigenproblem Ax = λBx, which is used to good effect in [18], for example.”
Optimization and pole assignment in control system design (2001, background): “(For more general results on condition numbers and perturbation analysis of ordinary and generalized eigenvalue problems, see (Chu, 1986b; 1987; Stewart and Sun, 1990).”
Random Design Analysis of Ridge Regression (2011, unknown): “where the second inequality follows from Lemma 3, the third inequality follows from Cauchy–Schwarz, and the fourth inequality follows from Mirsky’s theorem [21].”
Handbook of Cluster Analysis (2015, background): “It is known from matrix perturbation theory [42] that eigenvectors with smaller λk are more affected by numerical errors and noise in the similarities.”
On the Nyström Method for Approximating a Gram Matrix for Improved Kernel-Based Learning (2005, background): “For more details about general linear algebra, see [23, 27, 7]; for more details about matrix perturbation theory, see [35]; and for more details about generalized inverses, see [29, 5].”
Balanced truncation model reduction for semidiscretized Stokes equation (2006, methodology): “In this case λE − A can be reduced to the Weierstrass canonical form [34].”
Proto-value Functions: A Laplacian Framework for Learning Representation and Control in Markov Decision Processes (2007, methodology): “We are also currently exploring the stability of the subspaces defined by proto-value functions using the tools of matrix perturbation theory (Stewart and Sun, 1990; Sato, 1995), which quantifies the degree to which small perturbations of (positive definite) matrices lead to bounded changes in the s”
A useful variant of the Davis--Kahan theorem for statisticians (2014, methodology): “ 1V T 1 Vˆk F, (7) where the ﬁrst inequality follows because VTV 1 = 0, and the second from another application of Lemma 5. For real matrices A and B, we write A ⊗ B for their Kronecker product (e.g. Stewart and Sun, 1990, p. 30) and vec(A) for the vectorisation of A, i.e. the vector formed by stack”
Aggregation methods in dynamical systems and applications in population and community dynamics (2008, background): “This last result implies, see [38], that matrix SF has a strictly dominant eigenvalue of the form λ+O(αk), an associated left eigenvector U w̄l +O(αk) and an associated right eigenvector SV w̄r +O(αk).”
Gradient Primal-Dual Algorithm Converges to Second-Order Stationary Solutions for Nonconvex Distributed Optimization (2018, methodology): “2 in [43]] md(F, F̃ ) := min Π max j∈[N ] |σ̃Π(j) − σj | (50)”
Principal component analysis for second-order stationary vector time series (2014, background): “It can be shown that D(M(H 1),M(H 2)) ∈ [0,1]. Furthermore it is equal to 0 if and only if M(H 1) = M(H 2), and to 1 if and only if M(H 1) and M(H 2) are orthogonal with each other. See, for example, Stewart and Sun (1990) and Pan and Yao (2008). We always assume that the weakly stationary process y”