On Certain Metric Spaces Arising From Euclidean Spaces by a Change of Metric and Their Imbedding in Hilbert Space

By I. J. Schoenberg. 1937

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Visualizing and Measuring the Geometry of BERT (2019, background): “A result of Schoenberg [22], phrased in our terminology, is that if a metric space X has a power-p embedding into Rn, then it also has a power-q embedding for any q > p.”
Some notes on Property A (2006, methodology): “e ways. It thus gives a clear link between kernels and Hilbert space. All of the results on kernels in this section (that are not speciﬁcally related to coarse embeddings) are from the classic papers [Sch37] and [Sch38]. Theorem 3.2.4. • If k: X× X→ Ris a symmetric normalised kernel of negative type”
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.”
Probability on Trees and Networks (2017, background): “Part (b) is due to Oded Schramm, personal communication (2007), while parts (c)–(f) are due to Mark Braverman, personal communication (2009).”
Spherical and Hyperbolic Embeddings of Data (2014, unknown): “positive [27].”
Kernel techniques: From machine learning to meshless methods (2006, background): “More precisely, any symmetric positive definite kernelK generates a distance metric d : Ω×Ω → [0,∞) via d2(x, y) := K(x, x) − 2K(x, y) +K(y, y) for all x, y ∈ Ω (2.10) on a general set (Schoenberg 1937, Stewart 1976).”