Metric spaces and completely monotone functions

By I. J. Schoenberg. 1938

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In Pure Mathematics

Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Finally, we briefly mention Schoenberg’s characterization (Schoenberg 1938) for a class of radial kernels.”
Minimax Estimation of Kernel Mean Embeddings (2016, background): “From Schönberg’s representation (Schoenberg, 1938; Wendland, 2005, Theorems 7.13 & 7.14) it is known that a kernel k is radial on every Rd if and only if there exists ν ∈M b+([0,∞)) such that the following holds for all x, y ∈ Rd: k(x, y) = φ(‖x− y‖2) = ∫ ∞ 0 e−t‖x−y‖ 2 dν(t).”
On the Schoenberg Transformations in Data Analysis: Theory and Illustrations (2010, background): “Yet, despite its re-spectable age (Schoenberg 1938a), the properties and the very existence of this class of transformations appear to be little known in the Data Analytic community.”
On the mathematical foundations of learning (2001, methodology): “It was originally proved for R by Schoenberg [34] (together with a more difficult converse), but it follows for subsets of R by restricting to such a subset a kernel defined on R.”
Kernel methods in machine learning (2007, neutral): “Bochner’s theorem generalizes earlier work of Mathias, and has itself been generalized in various ways, that is, by Schoenberg [115].”
A Hilbert Space Approach to Effective Resistance Metric (2009, methodology): “The following theorem is inspired by the work of von Neumann and Schoenberg [4,5,45,46,53] on “screw functions”, but is a completely new result.”
Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics (2016, background): “Schoenberg (1938) characterized the class Φd as scale mixtures of the characteristic functions of random vectors uniformly distributed on the spherical shell of Rd, with any positive measure, F :”
Radial kernels and their reproducing kernel Hilbert spaces (2010, methodology): “The equivalence between ii) and iii) is the result of Berstein [8, 35] (see also Schoenberg [25, Thm. 3]).”
Noncommutative Riesz transforms -- Dimension free bounds and Fourier multipliers (2014, methodology): “This is a standard application of the ideas around Schoenberg’s theorem [64], which claims that ψ : G → R+ is a length function if and only if the associated semigroup Sψ = (Sψ,t)t≥0 given by Sψ,t : λ(g) 7→ exp(−tψ(g))λ(g) is Markovian on L(G).”
Nonseparable, Stationary Covariance Functions for Space–Time Data (2002, background): “Speci - cally, the isotropic function C4h5 D 4˜h˜251 h 2 òd1 is a spatial covariance function for all dimensions d if and only if 4t51 t ¶ 0, is completely monotone (Schoenberg 1938; Cressie 1993, p. 86).”
Initialization matters: Orthogonal Predictive State Recurrent Neural Networks (2018, background): “(47) That the above expression is positive is implied by the fact that every positive definite function φ (not parametrized by n) can be rewritten as φ(r) = σ(r2), where σ is completely monotone on [0,+∞] and from the basic properties of completely monotone functions (see: Schoenberg (1938)).”