Mercer’s Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs

By Ingo Steinwart, Clint Scovel. 2012

271 citations recorded by the source corpus.

Subfields: Applied Mathematics, Control and Systems Engineering, Modeling and Simulation, Statistical Learning Theory

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “Strathmann et al. (2015) introduces a kernel Hamiltonian Monte Carlo (KHMC) to address the case where one cannot run a traditional Hamiltonian Monte Carlo (HMC) since the target is doubly intractable, i.”
Consistent Kernel Mean Estimation for Functions of Random Variables (2016, background): “Now, if θ 1,Theorem 4.6 of Steinwart and Scovel (2012) shows that the range of T θ/2 is ( L2(Q),Hk ) θ,2 .”
Control functionals for Monte Carlo integration (2014, background): “mportant properties that make it well-suited to our purposes: (i) The mixed rst order partial derivatives of kexist, so the associated space H(k 0) contains only dierentiable functions (Cor. 4.36 of Steinwart and Scovel, 2012). (ii) The tail conditions (A1,3) and also (A4) are satised in all our exa”
A Kernel Independence Test for Random Processes (2014, methodology): “Using Mercer’s Theorem we obtain an expansion for s. Statement 1.”
Density Estimation in Infinite Dimensional Exponential Families (2013, neutral): “6 in [36], it is easy to show that for all 0 < γ < 1, R(T γ/2 k ) = [W2(p0), Ik(H)]γ,2, where the r.”
High-dimensional geometry of population responses in visual cortex (2018, unknown): “Natural images have a power-law structure[31, 32] (Fig.”
Distributed Learning with Regularized Least Squares (2016, background): “For example, the domain that does not have a metric (Shen et al., 2014), the kernel that is only bounded and measurable (Steinwart and Scovel, 2012), and so on.”
Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems (2016, unknown): “Since D is open and bounded and k̃ is symmetric positive definite, Mercer’s theorem [Steinwart and Scovel, 2012] guarantees the existence of a countable set of eigenvalues and eigenvectors {λi} and {ei} such that λ1 ≥ λ2 ≥ · · · > 0, ∑ λi ∞, {ei} are an orthonormal basis of L2(D) and k̃(x,x′) = ∑…”
Learning Sets with Separating Kernels (2012, background): “ct sum H= H ˆH ? ˆ, where H ˆ= ran P ˆ= ran T= spanfK xjx2X ˆg H? ˆ = kerP ˆ= kerT= ff2Hjf(x) = 0 8x2X ˆg: Remark 3. The reproducing kernel Hilbert space H ˆ (see Remark 2) has been considered before [70], and in particular in the context of semi-supervised manifold regularization [5], where X ˆ is ”
A glass-box interactive machine learning approach for solving NP-hard problems with the human-in-the-loop (2017, methodology): “Automatically, this is done by the Support Vector Machine (SVM), because under certain conditions a kernel can be represented by a dot product in a high-dimensional space (Mercer’s theorem) (Steinwart and Scovel, 2012).”
Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences (2018, methodology): “This implies that, while GP sample paths are almost surely outside of the corresponding RKHS, they lie in a function space “slightly larger” than the RKHS, which is itself a certain RKHS [Steinwart and Scovel, 2012, Steinwart, 2017].”