We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis on the torus. With a variable transformation we are able to transform the approximation rates and fast algorithms from the torus to other domains. We perform and analyze scattered-data approximation for smooth but arbitrary density functions by using a least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. For non-periodic functions we propose a new extension method. A proper choice of the extension parameter together with the piece-wise polynomial Chui-Wang wavelets extends the functions appropriately. In every case we are able to bound the approximation error with high probability. Additionally, if the function has low effective dimension (i.e. only interactions of few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to decrease the approximation error. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.

翻译：本文采用双曲小波回归方法，对仅具有低维变量交互作用的高维函数进行快速重构。我们使用紧支撑周期性Chui-Wang小波构造环面上的张量化双曲小波基。通过变量变换，我们能够将逼近速率和快速算法从环面域推广到其他区域。针对光滑但具有任意密度函数的散乱数据逼近问题，我们采用最小二乘法进行分析与计算。由于小波的紧支撑特性，相应的系统矩阵具有稀疏性，从而显著加速了矩阵向量乘法运算。对于非周期函数，我们提出了一种新的延拓方法：通过合理选择延拓参数，结合分段多项式Chui-Wang小波，可实现函数的有效延拓。在所有情况下，我们都能以高概率界定逼近误差。此外，若函数具有较低的有效维度（即仅存在少数变量间的交互作用），我们将在第二步中定性识别变量交互关系，并剔除低方差的ANOVA项以降低逼近误差，从而构建自适应的逼近模型。数值实验结果验证了所提方法的有效性。