Reproducing kernel Banach spaces for machine learning

Zhang, Haizhang; Xu, Yuesheng; Zhang, Jun

doi:10.1109/ijcnn.2009.5179093

Cited by 77 publications

(202 citation statements)

References 27 publications

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“…This reproducing kernel is unique. However, as showed in [49], different RKBSs may have the same reproducing kernel: For 1 < p < ∞, the Paley-Wiener classes…”

Section: Reproducing Kernel Banach Spacesmentioning

confidence: 98%

“…[49], a reproducing kernel Banach space on Ω ⊆ R (or C) is a reflexive Banach space B of functions on Ω for which its dual space B * is isometric to a Banach space B of functions on Ω and the point evaluation is continuous on both B and B for each t ∈ Ω. It has been proved in [49] that there exists a reproducing kernel for an RKBS as defined above. To this end, we introduce the bilinear form on B × B * by setting u, v * B…”

Section: Reproducing Kernel Banach Spacesmentioning

confidence: 99%

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The Kramer Sampling Theorem Revisited

2013

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The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. Besides, it has been the cornerstone for a significant mathematical literature on the topic of sampling theorems associated with differential and difference problems. In this work we provide, in an unified way, new and old generalizations of this result corresponding to various different settings; all these generalizations are illustrated with examples. All the different situations along the paper share a basic approach: the functions to be sampled are obtaining by duality in a separable Hilbert space H through an H-valued kernel K defined on an appropriate domain.

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“…This reproducing kernel is unique. However, as showed in [49], different RKBSs may have the same reproducing kernel: For 1 < p < ∞, the Paley-Wiener classes…”

Section: Reproducing Kernel Banach Spacesmentioning

confidence: 98%

Section: Reproducing Kernel Banach Spacesmentioning

confidence: 99%

The Kramer Sampling Theorem Revisited

2013

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“…They showed that many past sampling formulae can be obtained in this manner. Recently, the approach has been generalized to reproducing kernel Banach spaces [20] by frames for Banach spaces via semi-inner-products, [21].…”

Section: Introductionmentioning

confidence: 99%

Optimal sampling points in reproducing kernel Hilbert spaces

Wang

Zhang

2016

Journal of Complexity

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The recent developments of basis pursuit and compressed sensing seek to extract information from as few samples as possible. In such applications, since the number of samples is restricted, one should deploy the sampling points wisely. We are motivated to study the optimal distribution of finite sampling points. Formulation under the framework of optimal reconstruction yields a minimization problem. In the discrete case, we estimate the distance between the optimal subspace resulting from a general Karhunen-Loève transform and the kernel space to obtain another algorithm that is computationally favorable. Numerical experiments are then presented to illustrate the performance of the algorithms for the searching of optimal sampling points.

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“…In addition, the RKBS given in Definition 4.1 is different from that of [23]. Our RKBS can be one-sided or two-sided and its reproducing kernel K can be defined on nonsymmetric domains, i.e., K : Ω 2 ×Ω 1 → C, where Ω 1 and Ω 2 can be various subsets of R d 1 and R d 2 , respectively (see Definition 4.1).…”

Section: Introductionmentioning

confidence: 99%

Solving support vector machines in reproducing kernel Banach spaces with positive definite functions

Fasshauer

Hickernell

2015

Applied and Computational Harmonic Analysis

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In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using the orthogonality of semi-innerproducts, we can obtain the explicit representations of the dual (normalized-duality-mapping) elements of support vector machine solutions. In addition, we can introduce the reproduction property in a generalized native space by Fourier transform techniques such that it becomes a reproducing kernel Banach space, which can be even embedded into Sobolev spaces, and its reproducing kernel is set up by the related positive definite function. The representations of the optimal solutions of support vector machines (regularized empirical risks) in these reproducing kernel Banach spaces are formulated explicitly in terms of positive definite functions, and their finite numbers of coefficients can be computed by fixed point iteration. We also give some typical examples of reproducing kernel Banach spaces induced by Matérn functions (Sobolev splines) so that their support vector machine solutions are well computable as the classical algorithms. Moreover, each of their reproducing bases includes information from multiple training data points. The concept of reproducing kernel Banach spaces offers us a new numerical tool for solving support vector machines.

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Reproducing kernel Banach spaces for machine learning

Cited by 77 publications

References 27 publications

The Kramer Sampling Theorem Revisited

The Kramer Sampling Theorem Revisited

Optimal sampling points in reproducing kernel Hilbert spaces

Solving support vector machines in reproducing kernel Banach spaces with positive definite functions

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