Multiscale High-Dimensional Sparse Fourier Algorithms for Noisy Data
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We develop an efficient and robust high-dimensional sparse Fourier algorithm for noisy samples. Earlier in the paper "Multi-dimensional sublinear sparse Fourier algorithm" (2016), an efficient sparse Fourier algorithm with Θ(ds s) average-case runtime and Θ(ds) sampling complexity under certain assumptions was developed for signals that are s-sparse and bandlimited in the d-dimensional Fourier domain, i.e. there are at most s energetic frequencies and they are in [-N/2, N/2)^d∩Z^d. However, in practice the measurements of signals often contain noise, and in some cases may only be nearly sparse in the sense that they are well approximated by the best s Fourier modes. In this paper, we propose a multiscale sparse Fourier algorithm for noisy samples that proves to be both robust against noise and efficient.
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