Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures

• URL: http://arxiv.org/abs/2206.07588v1
• Date: Wed, 15 Jun 2022 15:12:35 GMT
• Title: Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures
• Authors: Johanna Ziegel and David Ginsbourger and Lutz D\"umbgen
• Abstract summary: We discuss radial kernels on separable Hilbert spaces, and introduce broad classes of kernels on Banach spaces and on metric spaces of strong negative type. The general results are used to give explicit classes of kernels on separable $Lp$ spaces and on sets of measures.
• Score: 0.5156484100374059
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present new classes of positive definite kernels on non-standard spaces that are integrally strictly positive definite or characteristic. In particular, we discuss radial kernels on separable Hilbert spaces, and introduce broad classes of kernels on Banach spaces and on metric spaces of strong negative type. The general results are used to give explicit classes of kernels on separable $L^p$ spaces and on sets of measures.

Related papers

• Reproducing kernel Hilbert spaces in the mean field limit [6.844996517347866]
  kernels are function spaces generated by kernels, so called reproducing kernel Hilbert spaces. We show the rigorous mean field limit of kernels and provide a detailed analysis of the limiting reproducing Hilbert space.
  arXiv  Detail & Related papers  (2023-02-28T09:46:44Z)
• Approximation of optimization problems with constraints through kernel Sum-Of-Squares [77.27820145069515]
  We show that pointwise inequalities are turned into equalities within a class of nonnegative kSoS functions. We also show that focusing on pointwise equality constraints enables the use of scattering inequalities to mitigate the curse of dimensionality in sampling the constraints.
  arXiv  Detail & Related papers  (2023-01-16T10:30:04Z)
• Sobolev Spaces, Kernels and Discrepancies over Hyperspheres [4.521119623956821]
  This work provides theoretical foundations for kernel methods in the hyperspherical context. We characterise the native spaces (reproducing kernel Hilbert spaces) and the Sobolev spaces associated with kernels defined over hyperspheres. Our results have direct consequences for kernel cubature, determining the rate of convergence of the worst case error, and expanding the applicability of cubature algorithms.
  arXiv  Detail & Related papers  (2022-11-16T20:31:38Z)
• Continuous percolation in a Hilbert space for a large system of qubits [58.720142291102135]
  The percolation transition is defined through the appearance of the infinite cluster. We show that the exponentially increasing dimensionality of the Hilbert space makes its covering by finite-size hyperspheres inefficient. Our approach to the percolation transition in compact metric spaces may prove useful for its rigorous treatment in other contexts.
  arXiv  Detail & Related papers  (2022-10-15T13:53:21Z)
• Understanding neural networks with reproducing kernel Banach spaces [20.28372804772848]
  Characterizing function spaces corresponding to neural networks can provide a way to understand their properties. We prove a representer theorem for a wide class of reproducing kernel Banach spaces. For a suitable class of ReLU activation functions, the norm in the corresponding kernel Banach space can be characterized in terms of the inverse Radon transform of a bounded real measure.
  arXiv  Detail & Related papers  (2021-09-20T17:32:30Z)
• A Short Note on the Relationship of Information Gain and Eluder Dimension [86.86653394312134]
  We show that eluder dimension and information gain are equivalent in a precise sense for reproducing kernel Hilbert spaces. We show that this is not a coincidence -- eluder dimension and information gain are equivalent in a precise sense for reproducing kernel Hilbert spaces.
  arXiv  Detail & Related papers  (2021-07-06T04:01:22Z)
• A Note on Optimizing Distributions using Kernel Mean Embeddings [94.96262888797257]
  Kernel mean embeddings represent probability measures by their infinite-dimensional mean embeddings in a reproducing kernel Hilbert space. We show that when the kernel is characteristic, distributions with a kernel sum-of-squares density are dense. We provide algorithms to optimize such distributions in the finite-sample setting.
  arXiv  Detail & Related papers  (2021-06-18T08:33:45Z)
• Switch Spaces: Learning Product Spaces with Sparse Gating [48.591045282317424]
  We propose Switch Spaces, a data-driven approach for learning representations in product space. We introduce sparse gating mechanisms that learn to choose, combine and switch spaces. Experiments on knowledge graph completion and item recommendations show that the proposed switch space achieves new state-of-the-art performances.
  arXiv  Detail & Related papers  (2021-02-17T11:06:59Z)
• Metrizing Weak Convergence with Maximum Mean Discrepancies [88.54422104669078]
  This paper characterizes the maximum mean discrepancies (MMD) that metrize the weak convergence of probability measures for a wide class of kernels. We prove that, on a locally compact, non-compact, Hausdorff space, the MMD of a bounded continuous Borel measurable kernel k, metrizes the weak convergence of probability measures if and only if k is continuous.
  arXiv  Detail & Related papers  (2020-06-16T15:49:33Z)
• Reproducing Kernel Hilbert Spaces Cannot Contain all Continuous Functions on a Compact Metric Space [2.7793394375935088]
  Given an uncountable, compact metric space, we show that there exists no reproducing kernel Hilbert space that contains the space of all continuous functions on this compact space.
  arXiv  Detail & Related papers  (2020-02-08T14:24:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.