Metric spaces and positive definite functions

By I. J. Schoenberg. 1938

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Citations in the corpus

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.”
VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES OF INTEGRABLE FUNCTIONS AND MERCER THEOREM (2006, background): “For the scalar case, it has been obtained by many authors, see [16, 17, 18, 19, 20, 21] and, for a complete list of references, [15, 22, 23, 4].”
Distance covariance in metric spaces (2011, background): “The connection to embeddings is that, as Schoenberg (1937, 1938) showed, negative type is equivalent to a certain property of embeddability into Hilbert space.”
Random Fractals and Markov Processes (2005, background): “Hence, by a theorem of Schoenberg (1938) [see also Theorems 7.8 and 8.4 in Berg and Frost (1975)], the Lévy exponent ψ is a continuous negative definite function.”
Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices (2013, methodology): “Although the origin of this theorem dates back to 1938 [16], it has received little attention in the computer vision community.”
Assouad's theorem with dimension independent of the snowflaking (2010, methodology): “Indeed, as shown by Linial-London-Rabinovich [23] (extending the corresponding classical result of Schoenberg [26] in the isometric category), c2(X) 6 D if and only if for all n ∈ N, x1, . . . , xn ∈ X and every n × n symmetric positive semidefinite matrix Q = (qij), all of whose rows sum to 0, the ”
Hadamard inverses, square roots and products of almost semidefinite matrices (1999, background): “8 to give a simple proof of a well-known result for distance matrices (distance matrices are almost negative semidefinite matrices with positive offdiagonal entries, and zeroes on the diagonal [10],[19]).”
Generalization and similarity in exemplar models of categorization: Insights from machine learning (2008, unknown): “is a so-called positive definite kernel (Schoenberg, 1938).”
Commute Times for a Directed Graph using an Asymmetric Laplacian (2011, background): “[33, 34, 2] Let Z be an n × n symmetric matrix.”
Geometric embeddings of metric spaces (2003, methodology): “Definite functions on metric spaces were introduced long ago by Schoenberg and von Neumann (3)[58], [78], who studied isometric embeddability of metric spaces in Hilbert spaces via Fourier analysis.”