The Future Fast Fourier Transform?

Abstract

It seems likely that improvements in arithmetic speed will continue to outpace advances in communications bandwidth. Furthermore, as more and more problems are working on huge datasets, it is becoming increasingly likely that data will be distributed across many processors because one processor does not have sufficient storage capacity. For these reasons, we propose that an inexact DFT such as an approximate matrixvector approach based on singular values or a variation of the Dutt-Rokhlin fastmultipole-based algorithm [9] may outperform any exact parallel FFT. The speedup may be as large as a factor of three in situations where FFT run time is dominated by communication. For the multipole idea we further propose that a method of "virtual charges" may improve accuracy, and we provide an analysis of the singular values that are needed for the approximate matrix-vector approaches.

A Framework for Low-Communication 1-D FFT

“…This shows that W (exact) is a permuted form of the matrix M in the factorization used in [14]. In this sense, that factorization is included in our framework.…”

“…Theoretically, our DFT factorizations can be made exact with these window functions. We can also rederive the factorization in [14] by one particular compactly supported window. Considerŵ that is 1 on [0, M − 1] and zero outside (−1, M ).…”

The Eigenvalue Distribution of Discrete Periodic Time-Frequency Limiting Operators

“…Edelman et al [18] proposed an approximate algorithm for DFT computations with lower communication cost based on the compressibility (low rank) of the blocks of F M , i.e., …”

“…The special distribution of the PDPSS eigenvalues (See Figure 1) has been exploited for fast computing Fourier extensions of arbitrary extension length in [12]. In this paper, we provide a finer non-asymptotic result that improves upon the expression in [18]. We also characterize the spectrum of submatrices of the DFT matrix (see Corollary 1), which is of independent interest in signal processing.…”

“…In [17], it was shown that unlike the PSWF and DPSS eigenvalues, the PDPSS eigenvalues can exactly achieve 0 and 1 and are degenerate in some cases. A non-asymptotic result on the distribution of the PDPSS eigenvalues was given in [18]. The special distribution of the PDPSS eigenvalues (See Figure 1) has been exploited for fast computing Fourier extensions of arbitrary extension length in [12].…”