Permutation-Avoiding FFT-Based Convolution
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914515711426560 |
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| author | Venkovic, Nicolas Anzt, Hartwig |
| author_facet | Venkovic, Nicolas Anzt, Hartwig |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12718 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Permutation-Avoiding FFT-Based Convolution Venkovic, Nicolas Anzt, Hartwig Numerical Analysis Data Structures and Algorithms Mathematical Software 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. |
| title | Permutation-Avoiding FFT-Based Convolution |
| topic | Numerical Analysis Data Structures and Algorithms Mathematical Software |
| url | https://arxiv.org/abs/2506.12718 |