In this paper we extend the deterministic sublinear FFT algorithm in Plonka et al. (2018) for fast reconstruction of $M$-sparse vectors ${\mathbf x}$ of length $N= 2^J$, where we assume that all components of the discrete Fourier transform $\hat{\mathbf x}= {\mathbf F}_{N} {\mathbf x}$ are available. The sparsity of ${\mathbf x}$ needs not to be known a priori, but is determined by the algorithm. If the sparsity $M$ is larger than $2^{J/2}$, then the algorithm turns into a usual FFT algorithm with runtime ${\mathcal O}(N \log N)$. For $M^{2} < N$, the runtime of the algorithm is ${\mathcal O}(M^2 \, \log N)$. 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 $M>20$ because of numerical instabilities, the modified algorithm is still numerically stable for $M=200$.
翻译:在本文中,我们将Plonka等人(2018年)的确定性亚线底线FFT算法用于快速重建长度为N=2 ⁇ J$的美元(2018年),用于快速重建耗氧矢量($$$ mathbfx $#mathbfff F ⁇ FQN}),我们假设离散的Fourier变换的所有部件都存在 $@hat_mathbfxxxx$(#mathbfxxxal) 。 美元($ mathbfx}) 的宽度并不需要事先知道,而是由算法决定。如果夸度($$ $)大于2 ⁇ J/2美元,那么算法将转换成通常的FFT算法算法算法,运行时间为$@mathcal O}(N\log N) 。 对于美元算法的运行时间,运行时间为$@maxblexal O}(20美元),提议对Plonka等人(2018美元)的方法进行修改后, 将大大改进。