A new class of fully discrete sparse Fourier transforms: faster stable implementations with guarantees. (English) Zbl 1431.65253

Summary: In this paper we consider sparse Fourier transform (SFT) algorithms for approximately computing the best \(s\)-term approximation of the discrete Fourier transform (DFT) \({\hat{\mathbf {f}}}\in {{\mathbb {C}}}^N\) of any given input vector \(\mathbf{f}\in {{\mathbb {C}}}^N\) in just \((s \log N) ^{{{\mathcal {O}}}(1)}\)-time using only a similarly small number of entries of \(\mathbf {f}\). In particular, we present a deterministic SFT algorithm which is guaranteed to always recover a near best \(s\)-term approximation of the DFT of any given input vector \(\mathbf {f}\in {{\mathbb {C}}}^N\) in \({{\mathcal {O}}} (s^2 \log ^{\frac{11}{2}} (N)) \)-time. Unlike previous deterministic results of this kind, our deterministic result holds for both arbitrary vectors \(\mathbf {f}\in {{\mathbb {C}}}^N\) and vector lengths \(N\). In addition to these deterministic SFT results, we also develop several new publicly available randomized SFT implementations for approximately computing \({\hat{\mathbf {f}}}\) from \(\mathbf {f}\) using the same general techniques. The best of these new implementations is shown to outperform existing discrete sparse Fourier transform methods with respect to both runtime and noise robustness for large vector lengths \(N\).


65T50 Numerical methods for discrete and fast Fourier transforms
65T40 Numerical methods for trigonometric approximation and interpolation
68W25 Approximation algorithms
Full Text: DOI arXiv


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