Deterministic Sparse Sublinear FFT with Improved Numerical Stability

Abstract

In this paper we extend the deterministic sublinear FFT algorithm in Plonka et al. (Numer Algorithms 78:133–159, 2018. 10.1007/s11075-017-0370-5) for fast reconstruction of M-sparse vectors $${\mathbf{x}}$$ x of length $$N= 2^J$$ N = 2 J , where we assume that all components of the discrete Fourier transform $$\hat{\mathbf{x}}= {\mathbf{F}}_{N} {\mathbf{x}}$$ x ^ = F N x are available. The sparsity of $${\mathbf{x}}$$ x needs not to be known a priori, but is determined by the algorithm. If the sparsity M is larger than $$2^{J/2}$$ 2 J / 2 , then the algorithm turns into a usual FFT algorithm with runtime $${\mathcal O}(N \log N)$$ O ( N log N ) . For $$M^{2} < N$$ M 2 < N , the runtime of the algorithm is $${\mathcal O}(M^2 \, \log N)$$ O ( M 2 log N ) . The proposed modifications of the approach in Plonka et al. (2018) lead to a significant improvement of the condition numbers of the Vandermonde matrices which are employed in the iterative reconstruction. Our numerical experiments show that our modification has a huge impact on the stability of the algorithm. While the algorithm in Plonka et al. (2018) starts to be unreliable for $$M>20$$ M > 20 because of numerical instabilities, the modified algorithm is still numerically stable for $$M=200$$ M = 200 .