Deterministic Sparse Fourier Transform with an ell_infty Guarantee
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In this paper we revisit the deterministic version of the Sparse Fourier Transform problem. While the randomized case is well-understood, the main work in the deterministic case is the work of Merhi et al. (J Fourier Anal Appl 2018), which obtains O(k^2 log^5.5 n) samples and similar runtime with the ℓ_2/ℓ_1 guarantee. We focus on the stronger ℓ_∞/ℓ_1 guarantee and the closely related problem of incoherent matrices. We list our contributions as follows. 1. We give a "for-all" scheme with O(k^2 log n) samples. 2. We give a for-all scheme with O(k^2 log^2 n) samples, and O(k^2 log^3 n) time. 3. We derandomize both schemes in polynomial time in n, such that all subsequent Sparse Fourier Transform "queries" can be answered deterministically in O(nk log n) and O(k^2 log^3 n) time, respectively. 4. We give two different deterministic constructions of incoherent matrices, combinatorial objects that are closely related to ℓ_infty/ℓ_1 sparse recovery schemes. The first one keeps rows of the Discrete Fourier Matrix, while the second uses Fourier-friendly measurements with the help of the Weil bound from algebraic geometry. Our constructions match previous constructions by DeVore (J Complexity 2007), Amini and Marvasti (IEEE Trans Info Theory 2011) and Nelson, Nguyen and Woodruff (RANDOM 12), where there was no constraint on the form of the incoherent matrix. Our algorithms are nearly sample optimal, since a lower bound of Ω(k^2 + k log n) is known, even for the case where the sensing matrix can be arbitrarily designed. Similarly, for incoherent matrices, a lower bound of Ω(k^2 log n/ log k) is known, indicating that our constructions are nearly optimal.
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