In this paper, we discuss the development of a sublinear sparse Fourier algorithm for high-dimensional data. In 11Adaptive Sublinear Time Fourier Algorithm” by Lawlor et al. (Adv. Adapt. Data Anal.5(01):1350003, 2013), an efficient algorithm with \({\Theta }(k\log k)\) average-case runtime and Θ(k) average-case sampling complexity for the one-dimensional sparse FFT was developed for signals of bandwidth N, where k is the number of significant modes such that k ≪ N. In this work we develop an efficient algorithm for sparse FFT for higher dimensional signals, extending some of the ideas in Lawlor et al. (Adv. Adapt. Data Anal.5(01):1350003, 2013). Note a higher dimensional signal can always be unwrapped into a one-dimensional signal, but when the dimension gets large, unwrapping a higher dimensional signal into a one-dimensional array is far too expensive to be realistic. Our approach here introduces two new concepts: “partial unwrapping” and “tilting.” These two ideas allow us to efficiently compute the sparse FFT of higher dimensional signals.

在本文中，我们讨论了针对高维数据的亚线性稀疏傅里叶算法的开发。在11自适应亚线性时间傅立叶算法”，由劳勒等。（。进阶适应数据分析。5（01）：1350003，2013年），用一个有效的算法\（{\西塔}（K \日志K）\）平均情况运行时和Θ（ķ）平均情况采样复杂对于一维FFT稀疏是为的带宽的信号开发ñ，其中ķ是显著模式使得数量ķ « ñ。在这项工作中，我们为高维信号开发了一种有效的稀疏FFT算法，扩展了Lawlor等人的一些思想。（Adv。Adapt。Data Anal。5（01）：1350003，2013）。请注意，始终可以将高维信号解包为一维信号，但是当维数变大时，将高维信号解包为一维数组实在是太昂贵了，无法实现。在这里，我们的方法引入了两个新概念：“部分展开”和“倾斜”。这两个想法使我们能够有效地计算高维信号的稀疏FFT。