Random matrix approximation of spectra of integral operators

By Vladimir Koltchinskii, Evarist Gin'e. 2000

221 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9e67f806ac70b732e87688c34a038091576ec64e

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Citations in the corpus

Statistical properties of kernel principal component analysis (2004, methodology): “Compared to the work of [ 12 ] and [11], we are interested in non-asymptotic (i.e.”
Nonasymptotic upper bounds for the reconstruction error of PCA (2016, background): “ it [32, 30, 10]. Asymptotically more precise results can be obtained by using linear or higher order expansions of empirical spectral projectors, see [25, 18, 19, 20] and also Koltchinskii and Gine [16] and Koltchinskii [21]. These methods can also be applied to the excess risk, leading to several ”
On Learning with Integral Operators (2010, background): “Theorem 4.2 and the following corollaries of Koltchinskii and Giné (2000) provide the same convergence rate (in expectation) under a different setting (the kernel K is only symmetric, but with some assumption on the decay of the eigenvalues of L K ).”
Nonlinear sufficient dimension reduction for functional data (2017, unknown): “By a well-known result in perturbation theory [see Koltchinskii and Giné (2000), Lemma 5.2 and Zwald and Blanchard (2006), Theorem 2] the eigenspaces of M̂ converge to those of M at the same rate.”
Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering (2003, unknown): “The use of this formula can be justified by considering the convergence of eigenvectors and eigenvalues, as the number of examples increases (Baker, 1977; Williams and Seeger, 2000; Koltchinskii and Giné, 2000; Shawe-Taylor and Williams, 2003).”
On the Effectiveness of Richardson Extrapolation in Machine Learning (2020, unknown): “This is equal to, for σ ( α ) = 1 α> 0 (see Cho and Saul, 2009; Bach, 2017): When the number of observations n tends to inﬁnity, the eigenvalues of 1 n K are known to converge to the eigenvalues of a certain inﬁnite-dimensional operator (Koltchinskii and Gin´e, 2000).”
The spectral norm of random inner-product kernel matrices (2015, background): “Koltchinskii and Giné [35] studied the spectra of kernel matrices in a regime where each row of X is sampled from a probability distribution over a fixed space (for example Rn for fixed n), showing that under suitable conditions, as p → ∞, the spectrum converges to that of a limiting infinite-dimen”
THE SPECTRUM OF RANDOM INNER-PRODUCT KERNEL MATRICES (2012, background): “j are friends, and j and k are also friends, then chances are greater that i and k are also friends. When the Xi’s are i.i.d uniformly distributed on the unit sphere in Rp and p is ﬁxed, according to [11], as n grows to inﬁnity the top p eigenvectors of the kernel matrix A converge, up to a multiply”
Learning Eigenfunctions Links Spectral Embedding and Kernel PCA (2004, methodology): “Let us start from the Nyström formula and insert fk,∞, taking advantage of Koltchinskii and Giné (2000), theorem 3.”
Geometry on Probability Spaces (2009, background): “Some analysis related to Proposition 1 can be found in [19, 14, 5, 12].”
The spectrum of kernel random matrices (2010, background): “By contrast, in lowdimension, results such as Koltchinskii and Giné (2000) show that the intuition behind kernel-PCA is correct and that the algorithm then performs a genuinely non-linear PCA. Being aware of the difference between the two setting should be helpful to practitioners in that it will in”
Convergence of Laplacian Eigenmaps (2006, result): “We also note results on approximating empirical eigenfunctions in [21] and work in [6].”
High-dimensional geometry of population responses in visual cortex (2018, unknown): “This convergence is guaranteed to occur if population variance is finite, Ec,s [ φ(c, s)(2) ] <∞, and can also happen under less restrictive conditions [34].”
On singular values of matrices with independent rows (2006, background): “This question was explored by Koltchinskii (1998) and Koltchinskii and Giné (2000) and some partial results were obtained on the expected distance (with respect to the distance d(x, y) ¼ inf ( P1 i¼1(xi y (i))2)1=2, with the infimum taken with respect to all permutations) between the set of…”
On Relevant Dimensions in Kernel Feature Spaces (2008, background): “As a consequence, the eigenvalues and eigenvectors of T̃k, which are equal to those of the kernel matrix, converge to those of Tk (see Koltchinskii and Giné, 2000; Koltchinskii, 1998).”