On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions

Lin, Rongrong; Zhang, Haizhang; Zhang, Jun

doi:10.1007/s10114-022-1397-7

Cited by 11 publications

(10 citation statements)

References 37 publications

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“…In an effort to introduce a richer set of geometrical structures into RKHS theory, reproducing kernel Banach spaces (RKBSs) generalise RKHSs by starting with a Banach space of functions [Der and Lee, 2007, Zhang et al, 2009, Song et al, 2013, Xu and Ye, 2014, Lin et al, 2022 etc. Precisely: Definition 1 (Reproducing kernel Banach space (RKBS - [Lin et al, 2022])). A reproducing kernel Banach space B on a set X is a Banach space of functions f : X → Y such that every point evaluation δ…”

Section: Notation In Present Papermentioning

confidence: 99%

“…Reproducing kernel Banach spaces (RKBSs) are a generalisationn of RKHSs which start with a Banach space of functions rather than a Hilbert space [Der and Lee, 2007, Lin et al, 2022, Zhang et al, 2009, Zhang and Zhang, 2012, Song et al, 2013, Sriperumbudur et al, 2011, Xu and Ye, 2014. Precisely: Definition 4 (Reproducing kernel Banach space (RKBS, [Lin et al, 2022])). A reproducing kernel Banach space B on a set X is a Banach space of functions f : X → Y such that every point evaluation δ x : B → Y, x ∈ X, on B is continuous.…”

Section: E Neural Network and Reproducing Kernel Banach Spacesmentioning

confidence: 99%

“…We demonstrate that this leads to a representation as an inner product between two feature maps, from data and weight-step space respectively. This structure underlies the construction of reproducing kernel Banach spaces (RKBS, [Lin et al, 2022]), allowing us to go on to show an equivalence between back-propagation and sequential learning in RKBS, which is similar to NTK but without the constraints of linearity, allowing us to derive new bounds on uniform convergence for networks of arbitrary width.…”

Section: Introductionmentioning

confidence: 99%

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TRF: Learning Kernels with Tuned Random Features

Shilton

Gupta

Rana

et al. 2022

AAAI

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Random Fourier features (RFF) are a popular set of tools for constructing low-dimensional approximations of translation-invariant kernels, allowing kernel methods to be scaled to big data. Apart from their computational advantages, by working in the spectral domain random Fourier features expose the translation invariant kernel as a density function that may, in principle, be manipulated directly to tune the kernel. In this paper we propose selecting the density function from a reproducing kernel Hilbert space to allow us to search the space of all translation-invariant kernels. Our approach, which we call tuned random features (TRF), achieves this by approximating the density function as the RKHS-norm regularised least-squares best fit to an unknown ``true'' optimal density function, resulting in a RFF formulation where kernel selection is reduced to regularised risk minimisation with a novel regulariser. We derive bounds on the Rademacher complexity for our method showing that our random features approximation method converges to optimal kernel selection in the large N,D limit. Finally, we prove experimental results for a variety of real-world learning problems, demonstrating the performance of our approach compared to comparable methods.

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Section: Notation In Present Papermentioning

confidence: 99%

Section: E Neural Network and Reproducing Kernel Banach Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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TRF: Learning Kernels with Tuned Random Features

Shilton

Gupta

Rana

et al. 2022

AAAI

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“…In theory, the function space associated with asymmetric kernels is often built on reproducing kernel Banach spaces (RKBS) (Zhang et al, 2009;Song et al, 2013;Fasshauer et al, 2015;Bartolucci et al, 2021;Lin et al, 2022) instead of the classical reproducing kernel Hilbert spaces (RKHS). For the primal feature analysis, Suykens (2016); Suykens (2017) introduce a variational principle for understanding the singular value decomposition from the perspective of asymmetric kernels associated with different feature mappings.…”

mentioning

confidence: 99%

Random Fourier Features for Asymmetric Kernels

He¹,

He²,

Liu³

et al. 2022

Preprint

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“…Note that geometric quantiles include the geometric median as a special case. Infinite-dimensional normed spaces play a major role in kernel methods [64,96,85] and in functional data analysis [48,65,66], since they are an appropriate setting for the modeling of curves (e.g., radar waveforms, spectrometric data, electricity consumption, stock prices).…”

mentioning

confidence: 99%

Statistical properties of approximate geometric quantiles in infinite-dimensional Banach spaces

Romon¹

2022

Preprint

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Geometric quantiles are location parameters which extend classical univariate quantiles to normed spaces (possibly infinite-dimensional) and which include the geometric median as a special case. The infinite-dimensional setting is highly relevant in the modeling and analysis of functional data, as well as for kernel methods.We begin by providing new results on the existence and uniqueness of geometric quantiles. Estimation is then performed with an approximate Mestimator and we investigate its large-sample properties in infinite dimension.When the population quantile is not uniquely defined, we leverage the theory of variational convergence to obtain asymptotic statements on subsequences in the weak topology. When there is a unique population quantile, we show that the estimator is consistent in the norm topology for a wide range of Banach spaces including every separable uniformly convex space. In separable Hilbert spaces, we establish weak Bahadur-Kiefer representations of the estimator, from which √ n-asymptotic normality follows.

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On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions

Cited by 11 publications

References 37 publications

TRF: Learning Kernels with Tuned Random Features

TRF: Learning Kernels with Tuned Random Features

Random Fourier Features for Asymmetric Kernels

Statistical properties of approximate geometric quantiles in infinite-dimensional Banach spaces

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