Nearly optimal sparse fourier transform

We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an ndimensional signal. We show: • An O(k log n)-time randomized algorithm for the case where the input signal has at most k non-zero Fourier coefficients, and • An O(k log n log(n/k))-time randomized algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k = o(n). They are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = nΩ(1) . We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least Ω(k log(n/k)/ log log n) signal samples, even if it is allowed to perform adaptive sampling.

• Type: Journal
• Year: 2012
• Where: STOC
• Authors: Haitham Hassanieh, Piotr Indyk, Dina Katabi, Eric Price
• Keywords: Adaptive Sampling, Algorithms, Sparse Approximation, Sparse Case, STOC 2012