For the past 30 years or so, machine learning has stimulated a great deal of research in the study of approximation capabilities (expressive power) of a multitude of processes, such as approximation by shallow or deep neural networks, radial basis function networks, and a variety of kernel based methods. Motivated by applications such as invariant learning, transfer learning, and synthetic aperture radar imaging, we initiate in this paper a general approach to study the approximation capabilities of kernel based networks using non-symmetric kernels. While singular value decomposition is a natural instinct to study such kernels, we consider a more general approach to include the use of a family of kernels, such as generalized translation networks (which include neural networks and translation invariant kernels as special cases) and rotated zonal function kernels. Naturally, unlike traditional kernel based approximation, we cannot require the kernels to be positive definite. Our results apply to the approximation of functions with small smoothness compared to the dimension of the input space.

翻译：过去约三十年间，机器学习极大地推动了各类过程逼近能力（表达力）的研究，包括浅层/深层神经网络逼近、径向基函数网络逼近以及多种基于核的方法。受不变学习、迁移学习和合成孔径雷达成像等应用的启发，本文开创性地提出了一种基于非对称核的核网络逼近能力通用研究框架。虽然奇异值分解是研究此类核的天然工具，但我们采用了更通用的方法，涵盖广义平移网络（包含神经网络和平移不变核作为特例）与旋转带状球函数核等核族。与传统基于核的逼近方法不同，我们自然无法要求核具有正定性。研究结果适用于输入空间维度下具有小光滑度函数的逼近问题。