Adaptive algorithms in sampling recovery

TL;DR

This paper investigates optimal adaptive sampling algorithms for recovering smooth functions on a unit cube, establishing their asymptotic error decay rate and linking it to best n-term approximation and nonlinear widths.

Contribution

It introduces a framework for adaptive sampling recovery of smooth functions, deriving the asymptotic order of the minimal recovery error and connecting it to nonlinear approximation measures.

Findings

Asymptotic error decay rate is proportional to n^{-α/d}.

Established the equivalence of recovery error with nonlinear n-widths.

Provided bounds linking adaptive sampling to best n-term approximation.

Abstract

We study optimal algorithms in adaptive sampling recovery of smooth functions defined on the unit $d$-cube ${\II}^d:= [0,1]^d$. The recovery error is measured in the quasi-norm $\|\cdot\|_q$ of $L_q := L_q(\II^d)$. For $B$ a subset in $L_q,$ we define a sampling algorithm of recovery with the free choice of sample points and recovering functions from $B$ as follows. For each $f$ from the quasi-normed Besov space $B^\alpha_{p,\theta}$, we choose $n$ sample points. This choice defines $n$ sampled values. Based on these sample points and sampled values, we choose a function from $B$ for recovering $f$. The choice of $n$ sample points and a recovering function from $B$ for each $f \in B^\alpha_{p,\theta}$ defines a $n$-sampling algorithm $S_n^B$ by functions in $B$. If $\Phi = \{\phi_k\}_{k \in K}$ is a family of elements in $L_q$, let $\Sigma_n(\Phi)$ be the non-linear set of linear combinations of $n$ free terms from $\Phi,$ that is $\Sigma_n(\Phi):= \{\, \phi = \sum_{j=1}^n a_j \phi_{k_j}: \ k_j \in K \, \}$. Denote by ${\mathcal G}$ the set of all families $\Phi$ in $L_q$ such that the intersection of $\Phi$ with any finite dimensional subspace in $L_q$ is a finite set, and by $\Cc(B^\alpha_{p,\theta}, L_q)$ the set of all continuous mappings from $B^\alpha_{p,\theta}$ into $L_q$. We define the quantity $$\nu_n(B^\alpha_{p,\theta},L_q) := \inf_{\Phi \in {\mathcal G}} \inf_{S_n^B \in \Cc(X, L_q): B= \Sigma_n(\Phi)} \sup_{\|f\|_{B^\alpha_{p,\theta}} \le 1} \ \|f - S_n^B(f)\|_q.$$ Let $0 < p,q, \theta \le \infty $ and $\alpha > d/p$. Then we prove the asymptotic order $$ \nu_n(B^\alpha_{p,\theta},L_q) \asymp n^{- \alpha / d}.$$ We also obtained the asymptotic order of quantities of optimal recovery by $S_n^B$ in terms of best $n$-term approximation as well of other non-linear $n$-widths.