Faster Walsh-Hadamard and Discrete Fourier Transforms From Matrix Non-Rigidity

Abstract

We give algorithms with lower arithmetic operation counts for both the Walsh-Hadamard Transform (WHT) and the Discrete Fourier Transform (DFT) on inputs of power-of-2 size $N$. For the WHT, our new algorithm has an operation count of $\frac{23}{24}N \log N + O(N)$. To our knowledge, this gives the first improvement on the $N \log N$ operation count of the simple, folklore Fast Walsh-Hadamard Transform algorithm. For the DFT, our new FFT algorithm uses $\frac{15}{4}N \log N + O(N)$ real arithmetic operations. Our leading constant $\frac{15}{4} = 3.75$ improves on the leading constant of $5$ from the Cooley-Tukey algorithm from 1965, leading constant $4$ from the split-radix algorithm of Yavne from 1968, leading constant $\frac{34}{9}=3.777\ldots$ from a modification of the split-radix algorithm by Van Buskirk from 2004, and leading constant $3.76875$ from a theoretically optimized version of Van Buskirk's algorithm by Sergeev from 2017. Our new WHT algorithm takes advantage of a recent line of work on the non-rigidity of the WHT: we decompose the WHT matrix as the sum of a low-rank matrix and a sparse matrix, and then analyze the structures of these matrices to achieve a lower operation count. Our new DFT algorithm comes from a novel reduction, showing that parts of the previous best FFT algorithms can be replaced by calls to an algorithm for the WHT. Replacing the folklore WHT algorithm with our new improved algorithm leads to our improved FFT.