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arXiv · 2012.03238

Fourier-domain Variational Formulation and Its Well-posedness for Supervised Learning

Tao Luo·Zheng Ma·Zhiwei Wang·Zhi-Qin John Xu·Yaoyu Zhang
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Abstract

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.

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Tao Luo, Zheng Ma, Zhiwei Wang, Zhi-Qin John Xu, Yaoyu Zhang. 2020-12-06. Fourier-domain Variational Formulation and Its Well-posedness for Supervised Learning. https://arxiv.org/abs/2012.03238

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