A survey on sampling recovery
F. Dai, V. Temlyakov
TL;DR
This survey addresses the fundamental problem of reconstructing unknown functions from finite samples by linking optimal recovery errors to nonlinear approximation and Kolmogorov widths. It foregrounds two core techniques—Lebesgue-type inequalities for greedy algorithms and universal sampling discretization—and surveys three algorithmic frameworks: weighted $\ell_p$ minimization, sparse approximation, and greedy methods (WOMP in Hilbert spaces and WCGA in Banach spaces). It demonstrates that nonlinear sampling recovery can outperform linear methods for several multivariate function classes, including ${\mathbf A}^{r,b}_β(\Psi,\mathcal{G})$, ${\mathbf W}^{a,b}_A$, and Sobolev-type spaces with mixed smoothness, and it provides precise bounds for recovery errors under various discretization and incoherence conditions. The work synthesizes discretization theory, compressed sensing, and approximation theory to chart a path toward universal, efficient sampling strategies with rigorous performance guarantees across high-dimensional function classes and norms.
Abstract
The reconstruction of unknown functions from a finite number of samples is a fundamental challenge in pure and applied mathematics. This survey provides a comprehensive overview of recent developments in sampling recovery, focusing on the accuracy of various algorithms and the relationship between optimal recovery errors, nonlinear approximation, and the Kolmogorov widths of function classes. A central theme is the synergy between the theory of universal sampling discretization and Lebesgue-type inequalities for greedy algorithms. We discuss three primary algorithmic frameworks: weighted least squares and $\ell_p$ minimization, sparse approximation methods, and greedy algorithms such as the Weak Orthogonal Matching Pursuit (WOMP) in Hilbert spaces and the Weak Tchebychev Greedy Algorithm (WCGA) in Banach spaces. These methods are applied to function classes defined by structural conditions, like the $A_β^r$ and Wiener-type classes, as well as classical Sobolev-type classes with dominated mixed derivatives. Notably, we highlight recent findings showing that nonlinear sampling recovery can provide superior error guarantees compared to linear methods for certain multivariate function classes.