Harmonic Analysis on Semigroups

By Christian Berg, J. Christensen, P. Ressel. 1984

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In Pure Mathematics

Citations in the corpus

Learning Theory for Distribution Regression (2014, methodology): “If K is pd on X, and h(u) = ∑∞ n=0 anu n with an ≥ 0, then h◦K is pd (Berg et al., 1984, Chapter 3) (Berg et al., 1984, Chapter 3).”
Fastfood: Approximate Kernel Expansions in Loglinear Time (2014, neutral): “For details see e.g. Berg et al. (1984).”
Characteristic Kernels and Infinitely Divisible Distributions (2014, methodology): “…term infinite divisibility on a p.d. kernel is understood in the sense of the usual product: a p.d. kernel ϕ : X ×X → C on a nonempty set X is called infinitely divisible if, for each n ∈ N, there exists a p.d. kernel ϕn : X × X → C such that ϕ = (ϕn) n (Berg et al., 1984, Definition 2.6, p. 76).”
On the optimal estimation of probability measures in weak and strong topologies (2013, background): “Instead of directly showing the radial basis function (RBF) class to be a VC-subgraph, [40, see the proof of Corollary 1] bounded the expected suprema of the Rademacher chaos process of degree 2, i.e., indexed by the RBF class, K , by that of the Gaussian class—see (68) in the proof of Theorem 5(a)—”
Hypothesis testing using pairwise distances and associated kernels (2012, background): “Note that in the terminology of Berg et al. (1984), ρ satisfying (1) is said to be a negative definite function.”
Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, positive): “The result follows from Berg et al. (1984, Corollary 4.3), which says that if P n f → P f, ∀ f ∈ C 0 ( X ), then P n f → P f, ∀ f ∈ C b ( X ), i.e., P n w → P . w ( ⇒ ) Suppose P n → P , i.e., P n f → P f, ∀ f ∈ C b ( X ).”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, methodology): “Then, the associated kernel, k can be obtained from ρ̃ (Berg et al., 1984, Chapter 3, Lemma 2.1) as k(x, y) = 1 2 [ρ̃2(x, x0) + ρ̃ 2(y, x0)− ρ̃2(x, y)], x, y, x0 ∈M, (60) which can then be used to compute γk.”
Graph Kernels (2008, neutral): “kernels are closed under addition and pointwise limits (Berg et al., 1984), k(α, β) is a valid p.”
Sinkhorn Distances: Lightspeed Computation of Optimal Transport (2013, background): “If M is negative deﬁnite, there exists vectors (Berg et al., 1984, §3.3.2).”
Kernel dimension reduction in regression (2009, methodology): “From the Riesz representation theorem for Radon measures (see, for example, [ 3, Chapter 2 ]), on a locally compact space the linear functional f 7! EP [f(X)] on the space of functions of compact support uniquely determines a Radon probability measure P . Thus, if is a compact Polish space, a univer”
Kernel methods in machine learning (2007, neutral): “[18]; indeed, a large part of the material in the present section is based on that work.”
Comparing molecules and solids across structural and alchemical space. (2015, background): “This kernel is also positive-definite, being based on a scalar product [41], and therefore induces a metric D̄(A,B) = √ 2− 2K̄(A,B).”
Optimal Transport Kernels for Sequential and Parallel Neural Architecture Search (2020, methodology): “Therefore, from Proposition 1 and following Theorem 3.2.2 in Berg et al. (1984), we can derive a positive deﬁnite TW kernel upon d NN for neural networks x , z as where the scalar σ l is the length-scale parameter.”
Jensen-Shannon divergence as a measure of distinguishability between mixed quantum states (2005, background): “¿From some results of harmonic analysis it is possible to show that the metric space ( X + N , √ JS ), where X + N denotes the discrete probability distributions space, can be isometri-cally mapped into a subset of a Hilbert space [17], [18].”
Some inequalities for information divergence and related measures of discrimination (2000, background): “[ 4 ] J. C. Mallinson and J. W. Miller, “Optimal codes for digital magnetic”