Faster Walsh-Hadamard and Discrete Fourier Transforms From Matrix Non-Rigidity
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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 23/24N 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 15/4N log N + O(N) real arithmetic operations. Our leading constant 15/4 = 3.75 is the first improvement in over 15 years; it improves on the prior best leading constant 34/9 = 3.777… by Van Buskirk from 2004, which in turn improved on the leading constant of 4 from the split-radix algorithm of Yavne from 1968 and the leading constant of 5 from the Cooley-Tukey algorithm from 1965. 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.
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