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：本文利用基于函数评估的稀疏三角多项式逼近高维函数$f\colon\mathbb T^d\to\mathbb C$。近期研究表明，维度增量式稀疏傅里叶变换方法无需信号严格稀疏即可适用此类场景。我们将该方法与秩1格点的子采样技术相结合，使得所提方法既可借助采样点的潜在结构实现快速傅里叶算法，又能获得随机点采样（对数级过采样）的低采样复杂度。通过理论分析，我们证明了幅度最大的傅里叶系数对应频率的检测保证。数值实验通过与全秩1格点和均匀随机点的对比验证了上述结论。