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

Code

• github.com/tancik/fourier-feature-networks
  Officialjax★ 1,363
• github.com/yandex-research/tabular-dl-num-embeddings
  pytorch★ 409
• github.com/Yura52/tabular-dl-num-embeddings
  pytorch★ 409
• github.com/jmclong/random-fourier-features-pytorch
  pytorch★ 113
• github.com/titu1994/tf_fourier_features
  tf★ 44
• github.com/locuslab/impsq
  pytorch★ 30
• github.com/noahtren/Fourier-Feature-Networks-TensorFlow-2
  tf★ 21
• github.com/sanowar-raihan/fourier-feature-superresolution
  pytorch★ 18
• github.com/mmathew23/Fourier-Feature-Networks-Pytorch
  pytorch★ 6
• github.com/mjhwright/error-correction
  pytorch★ 2

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.

Reproductions