This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with structured quadrature points (QMC), which provide stable $L_2$ reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a RKHS of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay. Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the $d$-dimensional torus to subsample rank-1 lattices, where it is known that full rank-1 lattices lose half the optimal order of convergence (expressed in terms of the size of the lattice). In contrast to that, our subsampled version regains the optimal rate since many of the lattice points are not needed. Moreover, we utilize fast and memory efficient Fourier algorithms in order to compute the approximation. Numerical experiments in several dimensions support our findings.

翻译：本文关注从采样点进行函数重构。若干方法中使用的采样点分为两类：（1）与快速算法相关联的结构化点；（2）为提升信息复杂度而通过初始随机抽样等手段生成的非结构化点。我们将两类方法相结合，提出在离线步骤中对结构化点进行子采样。具体而言，我们以具有稳定$L_2$重构性质的结构化求积点（QMC）为起点。该子采样过程包括一个计算成本低廉的随机步骤，随后通过确定性步骤进一步减少点数并保留信息。我们将在这些点（属于有界函数再生核希尔伯特空间）上采样并重构函数，同时实现最优误差衰减。该方法具有维度无关性，且只需已知若干初始求积点即可适用。我们将$d$维环面上的通用结论应用于秩1格点的子采样——已知完整秩1格点会损失半数最优收敛阶（以格点规模衡量）。与之对比，由于许多格点冗余，本文的子采样格式恢复了最优收敛率。此外，我们采用快速且内存高效的傅里叶算法进行数值逼近计算。多维数值实验验证了理论结果。