In this paper we approximate high-dimensional functions $f\colon\mathbb T^d\to\mathbb C$ by sparse trigonometric polynomials based on function evaluations. Recently it was shown that a dimension-incremental sparse Fourier transform (SFT) approach does not require the signal to be exactly sparse and is applicable in this setting. We combine this approach with subsampling techniques for rank-1 lattices. This way our approach benefits from the underlying structure in the sampling points making fast Fourier algorithms applicable whilst achieving the good sampling complexity of random points (logarithmic oversampling). In our analysis we show detection guarantees of the frequencies corresponding to the Fourier coefficients of largest magnitude. In numerical experiments we make a comparison to full rank-1 lattices and uniformly random points to confirm our findings.

翻译：本文通过稀疏三角多项式基于函数评估来逼近高维函数 $f\colon\mathbb T^d\to\mathbb C$。近期研究表明，一种维度增量式稀疏傅里叶变换（SFT）方法无需信号严格稀疏，可适用于此类场景。我们将该方法与秩-1格的子采样技术相结合，从而利用采样点中的底层结构，使快速傅里叶算法得以应用，同时实现随机点（对数过采样）的优良采样复杂度。在理论分析中，我们展示了与最大幅值傅里叶系数对应的频率检测保证。数值实验中，我们与完整秩-1格及均匀随机点进行了对比，验证了上述结论。