Combinatorial Sublinear-Time Fourier Algorithms

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 Â, 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 [23, 24]. In this paper we develop the first known deterministic sublinear time sparse Fourier Transform algorithm. 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 [24]. Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in [29].