Approximation by non-symmetric networks for cross-domain learning
This work addresses approximation challenges in cross-domain learning for fields such as radar imaging, though it appears incremental by extending kernel methods to non-symmetric cases.
The paper tackles the problem of approximating functions using non-symmetric kernel networks, motivated by applications like invariant learning and synthetic aperture radar imaging, and obtains accuracy estimates for uniform approximation of functions in Sobolev classes by ReLU^r networks with non-integer r.
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. In particular, we obtain estimates on the accuracy of uniform approximation of functions in a Sobolev class by ReLU$^r$ networks when $r$ is not necessarily an integer. Our general results apply to the approximation of functions with small smoothness compared to the dimension of the input space.