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

arXiv: https://arxiv.org/abs/2012.03238 · HTML: https://mlsanity.com/paper/2012.03238/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Dec 2020.

## Built on by (most cited first, 2 of 2)

- [Overview frequency principle/spectral bias in deep learning](https://mlsanity.com/paper/2201.07395/) (2022)
- [Frequency Principle in Deep Learning Beyond Gradient-descent-based Training](https://mlsanity.com/paper/2101.00747/) (2021)

## Abstract (arXiv)

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.