Positive definite functions on spheres

By N. H. Bingham. 1973

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Citations in the corpus

A la Carte - Learning Fast Kernels (2014, background): “A similar spectral decomposition holds for inner product kernels k ( x, x (cid:48) ) = κ ( (cid:104) x, x (cid:48) (cid:105) ) [Le et al., 2013, Schoenberg, 1942].”
Universal Kernels (2006, methodology): “To this end, we review the construction of Schoenberg kernels on Sd (Schoenberg, 1942).”
Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations (2017, unknown): “The kernel is positive deﬁnite since it admits a Maclaurin expansion with only non-negative coeﬃcients (Schoenberg, 1942; Sch¨olkopf and Smola, 2001).”
A tutorial on support vector regression (2004, unknown): “Theorem 8 (Schoenberg [1942]) A kernel of dot–product type k(x, x′) = k(〈x, x′〉) defined on an infinite dimensional Hilbert space, with a power series expansion”
Invariance and Stability of Deep Convolutional Representations (2017, unknown): “which is positive definite when κ admits a Maclaurin expansion with only non-negative coefficients [24, 27]—that is, κ(u) = ∑+∞ j=0 bju j with bj ≥ 0 for all j and all u in [−1,+1].”
A Fine-Grained Spectral Perspective on Neural Networks (2019, methodology): “As is long known [51, 22, 65, 54], K is diagonalized by spherical harmonics, and the eigenvalues are given by the coefficients of Φ against a system of orthogonal polynomials called Gegenbuaer polynomials.”
The kissing number in four dimensions (2003, background): “Schoenberg [24] extended this property to Gegenbauer (ultraspherical) polynomials G (n) k of tij .”
Kissing numbers, sphere packings, and some unexpected proofs (2004, methodology): “The first one is the simple computation N∑ i,j=1 f (xij ) = d∑ k=0 ck N∑ i,j=1 G(n)k (xij ) ≥ c0 N∑ i,j=1 G(n)0 (xij ) = c0N2, which rests on the fact that by Schoenberg’s lemma the sum of all entries of the matrix ( G(n)k (xij ) ) is nonnegative.”
AN EXTENSION OF A THEOREM OF SCHOENBERG TO PRODUCTS OF SPHERES (2015, background): “Schoenberg in his influential paper published in 1942 ([18]).”
On linear separability of data sets in feature space (2007, background): “For dot product kernels, the following theorem proposed in [6] is always useful.”