Optimal pointwise sampling for $L^2$ approximation
Résumé
Given a function $u\in L^2=L^2(D,\mu)$, where $D\subset \mathbb R^d$ and $\mu$ is a measure on $D$,
and a linear subspace $V_n\subset L^2$ of dimension $n$, we show that near-best approximation of $u$ in $V_n$ can be computed from
a near-optimal budget of $Cn$ pointwise evaluations of $u$, with $C>1$ a universal constant. The sampling points are drawn according to some random distribution, the approximation is computed by a weighted least-squares method, and the error is assessed in expected $L^2$ norm. This result improves on the results in [6,8] which require a sampling budget
that is sub-optimal by a logarithmic factor, thanks to a sparsification strategy introduced in [17,18]. As a consequence, we obtain for any compact class $\mathcal K\subset L^2$ that
the sampling number $\rho_{Cn}^{\rm rand}(\mathcal K)_{L^2}$
in the randomized setting is dominated by the Kolmogorov $n$-width $d_n(\mathcal K)_{L^2}$. While our result shows the existence of a randomized sampling
with such near-optimal properties, we discuss remaining issues
concerning its generation by a computationally efficient algorithm.
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