## Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains

### Project Page | Paper

Matthew Tancik*^{1}, Pratul P. Srinivasan*^{1,2}, Ben Mildenhall*^{1}, Sara Fridovich-Keil^{1}, Nithin Raghavan^{1}, Utkarsh Singhal^{1}, Ravi Ramamoorthi^{3}, Jonathan T. Barron^{2}, Ren Ng^{1}

^{1}UC Berkeley, ^{2}Google Research, ^{3}UC San Diego

^{*}denotes equal contribution

## Abstract

We show that passing input points through a simple Fourier feature mapping enables a multilayer perceptron (MLP) to learn high-frequency functions in low-dimensional problem domains. These results shed light on recent advances in computer vision and graphics that achieve state-of-the-art results by using MLPs to represent complex 3D objects and scenes. Using tools from the neural tangent kernel (NTK) literature, we show that a standard MLP fails to learn high frequencies both in theory and in practice. To overcome this spectral bias, we use a Fourier feature mapping to transform the effective NTK into a stationary kernel with a tunable bandwidth. We suggest an approach for selecting problem-specific Fourier features that greatly improves the performance of MLPs for low-dimensional regression tasks relevant to the computer vision and graphics communities.

## Code

We provide a demo IPython notebook as a simple reference for the core idea. The scripts used to generate the paper plots and tables are located in the Experiments directory.