Permutation-Avoiding FFT-Based Convolution

TL;DR

This paper investigates permutation-free FFT-based convolution algorithms that eliminate index-reversal permutations, potentially improving efficiency in multi-dimensional discrete convolutions.

Contribution

It introduces permutation-avoiding convolution procedures within a general radix Cooley-Tukey framework for multi-dimensional FFTs.

Findings

Permutation-avoiding algorithms reduce memory access overhead.

Numerical benchmarks show competitive performance against existing FFT convolution methods.

Results suggest potential for more efficient FFT library implementations.

Abstract

Fast Fourier Transform (FFT) libraries are widely used for evaluating discrete convolutions. Most FFT implementations follow some variant of the Cooley-Tukey framework, in which the transform is decomposed into butterfly operations and index-reversal permutations. While butterfly operations dominate the floating-point operation count, the memory access patterns induced by index-reversal permutations significantly degrade the FFT's arithmetic intensity. When performing discrete convolution, the three sets of index-reversal permutations which occur in FFT-based implementations using Cooley-Tukey frameworks cancel out, thus paving the way to implementations free of any permutation. To the best of our knowledge, such permutation-free variants of FFT-based discrete convolution are not commonly used in practice, making such kernels worth investigating. Here, we look into such permutation-avoiding convolution procedures for multi-dimensional cases within a general radix Cooley-Tukey framework. We perform numerical experiments to benchmark the algorithms presented against state-of-the-art FFT-based convolution implementations. Our results suggest that developers of FFT libraries should consider supporting permutation-avoiding convolution kernels.