Matrices, Moments and Quadrature with Applications

By Gene H. Golub, G. Meurant. 2009

380 citations recorded by the source corpus.

Subfields: Applied Mathematics, Natural Language Processing, Numerical Methods & Scientific Computing

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Citations in the corpus

An Investigation into Neural Net Optimization via Hessian Eigenvalue Density (2019, background): “It turns out that if the integrand were a polynomial g of degree d , with d small enough compared to m , it is possible to compute the integral exactly, Theorem 2.1 ([9] Chapter 6) .”
Coupled-cluster response theory for near-edge x-ray-absorption fine structure of atoms and molecules (2012, background): “In the present work, we circumvent this issue by a determination of the linear absorption cross section not from the complete oscillator strength distribution (which corresponds to solving fully a large eigenvalue problem) but from the imaginary part of the electric dipole polarizability [8] σ (ω)…”
The Physics of Communicability in Complex Networks (2011, methodology): “…methods of bounding and estimating arbitrary entries in a matrix function ( )f A have been developed by Golub, Meurant and collaborators (see Golub and Meurant, 2010 and references therein) and were first applied to problems of network analysis by Benzi and Boito (2010) (see also Bonchi et…”
Scalable Log Determinants for Gaussian Process Kernel Learning (2017, methodology): “We can also approximate z log(K̃)z via a Lanczos decomposition; see [32] for discussion of a Lanczos-based computation of z f(K̃)z and [20, 21] for stochastic Lanczos estimation of log determinants.”
A probing method for computing the diagonal of a matrix inverse (2012, methodology): “This approach is developed by Golub and collaborators, see, e.g. [4, 20, 21], and is described in detail in the book by Golub and Meurant [7].”
GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration (2018, background): “Running Lanczos with a starting vector z i ensures that all columns of ˜ Q i are orthogonal to z i except the ﬁrst, so ˜ Q i z i = e 1 [13, 16, 47].”
Quadrature rule-based bounds for functions of adjacency matrices (2010, neutral): “(2010), doi:10.1016/j.laa.2010.03.035”
Randomized numerical linear algebra: Foundations and algorithms (2020, background): “Golub and Meurant (2010) contains a more complete treatment, and Musco et al. (2018) gives a theoretical discussion about stability.”
PyHessian: Neural Networks Through the Lens of the Hessian (2019, methodology): “Instead, it is possible to use stochastic methods from RandNLA to extract this information, without explicitly forming the Hessian matrix.”
Fast Estimation of tr(f(A)) via Stochastic Lanczos Quadrature (2017, methodology): “The vectors w k are orthonormal, and we can show that the Lanczos polynomials are orthogonal with respect to the measure µ ( t ) in (6); see Theorem 4.2 in [20].”