Optimal pointwise sampling for $L^2$ approximation

Abstract

Given a function u ∈ L 2 = L 2 (D, µ), where D ⊂ R d and µ is a measure on D, and a linear subspace V n ⊂ 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 K ⊂ L 2 that the sampling number ρ rand Cn (K) L 2 in the randomized setting is dominated by the Kolmogorov n-width d n (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.

Sampling discretization and related problems

“…Note, that it is pointed out in [88] (see Theorem 3.4 of v2) that in the case of real spaces the constant b > 1 can be taken arbitrarily close to 1 and in the case of complex spaces -the constant b > 2 arbitrarily close to 2 with the constant B allowed to depend on b. We refer the reader to the very recent paper [27] for a discussion of the L 2 recovery when sampling points are drawn according to some random distribution. R.2.…”

“…The following conditional result is proved in [110] with the aid of a lemma from [13]: Suppose Ω is such that 27. We now formulate one recent result for the case 1 ≤ q ≤ ∞.…”

Lower bounds for integration and recovery in $L_2$

“…We want to stress that results in the randomized setting are quite different. In particular, the results do not depend strongly on the assumption whether the singular values are in ℓ 2 or not; see [1,2,11,19,25].…”

Towards optimal sampling for learning sparse approximation in high dimensions

“…Substituting this into (31) and recalling the definition of τ, we see that the discrete measure µ is given by…”

“…We remark in passing recent work that strives to go even further, by removing the log factor in the sample complexity bound. See [31] and references therein. We note, however, that such procedures may not be feasible in practice, or may not guarantee quasi-optimal error bounds.…”