Asymptotic behavior of the eigenvalues of certain integral equations

By Harold Widom. 1963

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Citations in the corpus

On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions (2015, methodology): “For these kernels, the decay of eigenvalues has been well-studied by Widom (1963), who relates the decay of eigenvalues to the tails of the distribution dρ and the decay of the Fourier transform of t .”
Filtering with State-Observation Examples via Kernel Monte Carlo Filter (2013, methodology): “e rank of low rank matrices: Low rank approximation works well in practice, since eigenvalues of a kernel matrix often decay very rapidly. Indeed this has been theoretically shown for some cases; see Widom (1963, 1964) and discussions in Bach and Jordan (2002). (ii) A data reduction method based on ”
ABJM theory as a Fermi gas (2011, background): “3) [44] and we will verify it in section 5 by a detailed analysis of the WKB expansion.”
On the Equivalence between Quadrature Rules and Random Features (2015, methodology): “For these kernels, the decay of eigenvalues has been well-studied by Widom (1963), who relate the decay of eigenvalues to the tails of the distribution dρ and the decay of the Fourier transform of t .”
Kernel Bayes' rule: Bayesian inference with positive definite kernels (2013, background): “It is known that in some typical cases the eigenspectram of a Gram matrix decays fast (Widom, 1963, 1964; Bach and Jordan, 2002).”
Self-attention with Functional Time Representation Learning (2019, background): “It has been shown that under mild conditions the Fourier coefficients cj decays exponentially to zero [21], and classic approximation theory guarantees a uniform convergence bound for truncated Fourier series [9] (see Appendix for discussions).”
Gradient-Based Kernel Dimension Reduction for Regression (2014, background): “It is known that the eigenspectrum of Gram matrices with Gaussian kernel decays fast for some typical data distributions (Widom, 1963, 1964) so that the low-rank approximation can give good approximation accuracy with signiﬁcant saving of the computational cost.”
Locality defeats the curse of dimensionality in convolutional teacher–student scenarios (2021, background): “Furthermore, for kernels with filter size s (or t) and positive smoothness exponent αs (or αt), the eigenvalues decay with a power −(s+αs) (or −(t+αt)) of the modulus of the wavevector k= √ k · k [38].”