# The Fourier transform is a neural network - follow-up

Two weeks ago, “The Fourier transform is a neural network” [1], reached the front page of Hacker News and gained traction on Twitter. Many thanks to all who took the time to read/comment on the post.

The post ended with the question:

Do we ever benefit from explicitly putting Fourier layers into our models?

And two days ago, Lee-Thorp et al. released FNet: Mixing Tokens with Fourier Transforms [2], which addresses this question. From the abstract:

…we find that replacing the self-attention sublayer in a Transformer encoder with a standard, unparameterized Fourier Transform achieves 92% of the accuracy of BERT on the GLUE benchmark, but pre-trains and runs up to seven times faster on GPUs and twice as fast on TPUs. The resulting model, which we name FNet, scales very efficiently to long inputs, matching the accuracy of the most accurate “efficient” Transformers on the Long Range Arena benchmark, but training and running faster across all sequence lengths on GPUs and relatively shorter sequence lengths on TPUs. Finally, FNet has a light memory footprint and is particularly efficient at smaller model sizes: for a fixed speed and accuracy budget, small FNet models outperform Transformer counterparts.

The paper also uses a technique presented in the blog post, i.e. creating a “Fourier matrix”, to use as a Fourier layer (but they throw away the imaginary part):

On TPUs: for relatively shorter sequences (≤8192 tokens), it is faster to precompute the DFT matrix and then compute the Fourier Transform through matrix multiplications than using the FFT; for longer sequences, the FFT is faster.

Although it seems likely that the FFT will eventually be made faster on TPUs.

Really interesting work, great to see!