A key problem in approximation theory is the recovery of high-dimensional functions from samples. In many cases, the functions of interest exhibit anisotropic smoothness, and, in many practical settings, the nature of this anisotropy may be unknown a priori. Therefore, an important question involves the development of universal algorithms, namely, algorithms that simultaneously achieve optimal or near-optimal rates of convergence across a range of different anisotropic smoothness classes. In this work, we consider universal approximation of periodic functions that belong to anisotropic Sobolev spaces and anisotropic dominating mixed smoothness Sobolev spaces. Our first result is the construction of a universal algorithm. This recasts function recovery as a sparse recovery problem for Fourier coefficients and then exploits compressed sensing to yield the desired approximation rates. Note that this algorithm is nonadaptive, as it does not seek to learn the anisotropic smoothness of the target function. We then demonstrate optimality of this algorithm up to a dimension-independent polylogarithmic factor. We do this by presenting a lower bound for the adaptive $m$-width for the unit balls of such function classes. Finally, we demonstrate the necessity of nonlinear algorithms. We show that universal linear algorithms can achieve rates that are at best suboptimal by a dimension-dependent polylogarithmic factor. In other words, they suffer from a curse of dimensionality in the rate -- a phenomenon which justifies the necessity of nonlinear algorithms for universal recovery.

翻译：逼近论中的一个关键问题是从样本中恢复高维函数。在许多情况下，待研究函数表现出各向异性光滑性，而在实际应用中这种各向异性的性质可能预先未知。因此，一个重要课题是开发通用算法——即能在不同各向异性光滑类上同时达到最优或接近最优收敛速率的算法。本文考虑属于各向异性索伯列夫空间和各向异性主混合光滑索伯列夫空间的周期函数的通用逼近问题。我们的第一个成果是构建了一种通用算法。该算法将函数恢复转化为傅里叶系数的稀疏恢复问题，进而利用压缩感知技术实现所需的逼近速率。值得注意的是，该算法是非自适应的，因为它无需学习目标函数的各向异性光滑性。随后我们证明该算法的最优性可达与维度无关的多对数因子。为此，我们给出了该类函数单位球自适应$m$宽度的下界。最后，我们证明了非线性算法的必要性：通用线性算法至多能达到被维度相关多对数因子次优的速率——这种速率上的维度灾难现象充分说明了非线性算法在通用恢复中的必要性。