A supervised learning problem is to find a function in a hypothesis function space given values on isolated data points. Inspired by the frequency principle in neural networks, we propose a Fourier-domain variational formulation for supervised learning problem. This formulation circumvents the difficulty of imposing the constraints of given values on isolated data points in continuum modelling. Under a necessary and sufficient condition within our unified framework, we establish the well-posedness of the Fourier-domain variational problem, by showing a critical exponent depending on the data dimension. In practice, a neural network can be a convenient way to implement our formulation, which automatically satisfies the well-posedness condition.
@article{arxiv.2012.03238,
title = {Fourier-domain Variational Formulation and Its Well-posedness for Supervised Learning},
author = {Tao Luo and Zheng Ma and Zhiwei Wang and Zhi-Qin John Xu and Yaoyu Zhang},
journal= {arXiv preprint arXiv:2012.03238},
year = {2020}
}