An Input-Adaptive Algorithm for High Performance Sparse Fast Fourier Transform

Abstract

Many applications invoke the Fast Fourier Transform (FFT) on sparse inputs, with most of their Fourier coefficients being very small or equal to zero. Compared with the “dense” FFT algorithms, the input sparsity makes it easier to parallelize the sparse counterparts. In general, sparse FFT algorithms filter input into different frequency bins, and then process the bins separately. Clearly, the performance is largely determined by the efficiency and the effectiveness of those filters. However, sparse FFT algorithms are input-oblivious with regard to filter selection, i.e., input characters are not considered in the design and tuning of their sparse filters, which leads to sub-optimal binning and consequently hurts performance. This paper proposes an input-adaptive sparse FFT algorithm that takes advantage of the similarity between input samples to automatically design and customize sparse filters that lead to better parallelism and performance. More specifically, given a sparse signal that has only $k$ non-zero Fourier coefficients similar to another known spectral representation, our algorithm utilizes sparse approximation to estimate the DFT output in the runtime sub-linear to the input size. Moreover, our work automatically adapts to different input characteristics by integrating and tuning several adaptive filters to efficiently package the non-zero Fourier coefficients into a small number of bins which can then be estimated accurately. Therefore, the input-tuned filtering gets rid of recursive coefficient estimation and improves parallelism and performance. We evaluate our input-adaptive sparse FFT implementation in sequential case on Intel i7 CPU and in parallel versions on three NVIDIA GPUs, i.e., NVIDIA GeForce GTX480, Tesla C2070 and Tesla C2075. In particular, our performance is compared to that of the SSE-enabled FFTW and to the results of a highly-influential recently proposed sparse Fourier algorithm. In summary, our algorithm is faster than FFT both in theory and implementation. Furthermore, the range of sparsity $k$ that our approach can outperform dense FFT is larger than that of other sparse Fourier algorithms.