Numerical Treatment of the Integral Equations

By G. Micula, Sanda Micula. 1999

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Citations in the corpus

Kernel-based Conditional Independence Test and Application in Causal Discovery (2011, neutral): “5 of Baker (1977).”
Nyströmformer: A Nystöm-based Algorithm for Approximating Self-Attention (2021, methodology): “The Nyström method (Baker 1977) was developed as a way of discretizing an integral equation with a simple quadrature rule and remains a widely used approach for approximating the kernel matrix with a given sampled subset of columns (Williams and Seeger 2001).”
Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP) (2015, background): “4 of Baker (1977) proves this eigenvalue approximation is asymptotically consistent (e.”
Kernel Projection Machine: a New Tool for Pattern Recognition (2004, background): “(cid:20) (cid:20) Indeed, it is well known in numerical analysis (see [2]) that the eigenvectors of the kernel matrix approximate the eigenfunctions of the kernel operator.”
Empirical characteristic function approach to goodness-of-fit tests for the Cauchy distribution with parameters estimated by MLE or EISE (2005, result): “These remainder terms satisfy sup t∈S |∆(2) n | P −→ 0, and sup t∈S |∆(3) n | P −→ 0 by conditions (iv) and (vi) of Csörgő (1983).”
Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering (2003, unknown): “To obtain an embedding for a new data point, we propose to use the Nystr¨om formula (Baker, 1977), which has been used successfully to speed-up kernel methods computations by focussing the heavier computations (the eigendecomposition) on a subset of examples.”
On fractional Tikhonov regularization (2015, unknown): “We assume that a bound for the error ‖bδ − b‖Y ≤ δ (1.2) D. Gerth and R. Ramlau were supported by the Austrian Science Fund (FWF):W1214-N15, Project DK-08.”
Learning Eigenfunctions Links Spectral Embedding and Kernel PCA (2004, methodology): “4) is proportional to the so-called Nystr(cid:127)om formula (Baker, 1977; Williams and Seeger, 2000) (eq.”
Trefftz and Collocation Methods (2008, background): “The field flourished in the 1970s with the publication of several monographs—Kagiwada and Kalaba [245] in 1974, Atkinson [8] in 1976, Ivanov [230] in 1976, and Baker [20] in 1977.”
Implementation of Karhunen-Loeve expansion for simulation using a wavelet-Galerkin scheme (2002, methodology): “The proposed method has significant computational advantages over the conventional Galerkin method [7,8].”
Proto-value Functions: A Laplacian Framework for Learning Representation and Control in Markov Decision Processes (2007, methodology): “The Nyström method interpolates the value of eigenvectors computed on sample states to novel states, and is an application of a classical method used in the numerical solution of integral equations (Baker, 1977).”
Finite-Dimensional Projection for Classification and Statistical Learning (2008, unknown): “The empirical eigenfunctions are computed by using formula (18) and a random draw of 500 points.”
Proto-transfer Learning in Markov Decision Processes Using Spectral Methods (2006, methodology): “The Nyström method interpolates the value of eigenvectors computed on known sample states to novel states, and is an application of a classical method used in the numerical solution of integral equations (Baker, 1977).”
Dimensionality Reduction: A Comparative Review (2008, methodology): “The out-of-sample extension for these techniques is performed by means of a so-called Nyström approximation [6,86], which is known to be equivalent to the Kernel PCA projection [97].”
Goodness-of-fit tests for symmetric stable distributions—Empirical characteristic function approach (2006, methodology): “, g(s, t)[2] = g(s, t)+ g(t, s), and Bα = ∫ ∞ −∞ e α |u| log |u|w(u)du, Bσ = α ∫ ∞ −∞ e α |u|w(u)du.”