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Translation-Invariant Kernels for Multivariable Approximation
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DOI：10.1109/TNNLS.2020.3026720.png)
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Suitability of shallow (one-hidden-layer) networks with translation-invariant kernel units for function approximation and classification tasks is investigated. It is shown that a critical property influencing the capabilities of kernel networks is how the Fourier transforms of kernels converge to zero. The Fourier transforms of kernels suitable for multivariable approximation can have negative values but must be almost everywhere nonzero. In contrast, the Fourier transforms of kernels suitable for maximal margin classification must be everywhere nonnegative but can have large sets where they are equal to zero (e.g., they can be compactly supported). The behavior of the Fourier transforms of multivariable kernels is analyzed using the Hankel transform. The general results are illustrated by examples of both univariable and multivariable kernels (such as Gaussian, Laplace, rectangle, sinc, and cut power kernels).
Keywords:
Kernel
Fourier transforms
Support vector machines
Function approximation
Dictionaries
Task analysis
Classification
Fourier and Hankel transforms
function approximation
radial kernels
translation-invariant kernels
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