Combinatorial Sublinear-Time Fourier Algorithms - Iwen, M. A. (Foundations of computational mathematics, 2010)
We study the problem of estimating the best k term Fourier representation for a given frequency sparse signal (i.e., vector) A of length N ≫ k . More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of , and estimate their coefficients, in polynomial( k ,log N ) time. Randomized sublinear-time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem (Gilbert et al. in ACM STOC, pp. 152–161, 2002 ; Proceedings of SPIE Wavelets XI, 2005 ). In this paper we develop the first known deterministic sublinear-time sparse Fourier Transform algorithm which is guaranteed to produce accurate results. As an added bonus, a simple relaxation of our deterministic Fourier result leads to a new Monte Carlo Fourier algorithm with similar runtime/sampling bounds to the current best randomized Fourier method (Gilbert et al. in Proceedings of SPIE Wavelets XI, 2005 ). Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in (Iwen in Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA’08), 2008 ).