Improved time bounds for near-optimal sparse Fourier representations

We study the problem of finding a Fourier representation R of m terms for a given discrete signal A of length N. The Fast Fourier Transform (FFT) can find the optimal N-term representation in time O(N log N), but our goal is to get sublinear time algorithms when m " N. Suppose ∥A∥ ₂ ≤ M ∥A - R _opt∥ ₂, where R _opt is the optimal output. The previously best known algorithms output R such that ∥ A - R∥ ₂ ² ≤ (1 + ε)∥ A - R _opt ∥ ₂ ² with probability at least 1 - δ in time* polym,(m, log(1/δ), log N, log M, 1/ε). Although this is sublinear in the input size, the dominating expression is the polynomial factor in m which, for published algorithms, is greater than or equal to the bottleneck at m ² that we identify below. Our experience with these algorithms shows that this is serious limitation in theory and in practice. Our algorithm beats this m ² bottleneck. Our main result is a significantly improved algorithm for this problem and the d-dimensional analog. Our algorithm outputs an R with the same approximation guarantees but it runs in time m · poly(log(1/δ), log N, log M, 1/ε). A version of the algorithm holds for all N, though the details differ slightly according to the factorization of N. For the d-dimensional problem of size N ₁ × N ₂ × ⋯× N _d, the linear-in-m algorithm extends efficiently to higher dimensions for certain factorizations of the N _i's; we give a quadratic-in-m algorithm that works for any values of N _i's. This article replaces several earlier, unpublished drafts.