The reconstruction of high-dimensional sparse signals is a challenging task in a wide range of applications. In order to deal with high-dimensional problems, efficient sparse fast Fourier transform algorithms are essential tools. The second and third authors have recently proposed a dimension-incremental approach, which only scales almost linear in the number of required sampling values and almost quadratic in the arithmetic complexity with respect to the spatial dimension d. Using reconstructing rank-1 lattices as sampling scheme, the method showed reliable reconstruction results in numerical tests but suffers from relatively large numbers of samples and arithmetic operations. Combining the preferable properties of reconstructing rank-1 lattices with small sample and arithmetic complexities, the first author developed the concept of multiple rank-1 lattices. In this paper, both concepts — dimension-incremental reconstruction and multiple rank-1 lattices — are coupled, which yields a distinctly improved high-dimensional sparse fast Fourier transform. Moreover, the resulting algorithm is analyzed in detail with respect to success probability, number of required samples, and arithmetic complexity. In comparison to single rank-1 lattices, the utilization of multiple rank-1 lattices results in a reduction in the complexities by an almost linear factor with respect to the sparsity. Various numerical tests confirm the theoretical results, the high performance, and the reliability of the proposed method.

在各种应用中，高维稀疏信号的重构是一项艰巨的任务。为了处理高维问题，有效的稀疏快速傅里叶变换算法是必不可少的工具。第二和第三作者最近提出了一种尺寸增量方法，该方法仅在所需采样值的数量上几乎线性缩放，而相对于空间尺寸d在算术复杂度上几乎二次缩放。该方法使用重构的1级格作为采样方案，在数值测试中显示出可靠的重构结果，但存在较大数量的样本和算术运算。结合重建具有少量样本的1级晶格的优选属性和算术复杂性，第一位作者提出了多个1级晶格的概念。在本文中，两个概念-尺寸增量重建和多个1级晶格-耦合在一起，这产生了明显改进的高维稀疏快速傅立叶变换。此外，针对成功概率，所需样本数和算术复杂度，对所得算法进行了详细分析。与1级单晶格相比，相对于稀疏度，利用多个1级晶格会导致复杂度降低几乎线性的因素。各种数值测试证实了该方法的理论结果，高性能和可靠性。