A new upper bound for sampling numbers

09/30/2020
by   Nicolas Nagel, et al.
0

We provide a new upper bound for sampling numbers (g_n)_n∈ℕ associated to the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants C,c>0 (which are specified in the paper) such that g^2_n ≤Clog(n)/n∑_k≥⌊ cn ⌋σ_k^2 , n≥ 2 , where (σ_k)_k∈ℕ is the sequence of singular numbers (approximation numbers) of the Hilbert-Schmidt embedding Id:H(K) → L_2(D,ϱ_D). The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver's conjecture, which was shown to be equivalent to the Kadison-Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of H^s_mix(𝕋^d) in L_2(𝕋^d) with s>1/2. We obtain the asymptotic bound g_n ≤ C_s,dn^-slog(n)^(d-1)s+1/2 , which improves on very recent results by shortening the gap between upper and lower bound to √(log(n)).

READ FULL TEXT

Authors

page 1

page 2

page 3

page 4

04/26/2022

A sharp upper bound for sampling numbers in L_2

For a class F of complex-valued functions on a set D, we denote by g_n(F...
08/26/2021

Lower bounds for integration and recovery in L_2

Function values are, in some sense, "almost as good" as general linear i...
09/05/2019

Multiple Lattice Rules for Multivariate L_∞ Approximation in the Worst-Case Setting

We develop a general framework for estimating the L_∞(T^d) error for the...
09/10/2020

Non-asymptotic Optimal Prediction Error for RKHS-based Partially Functional Linear Models

Under the framework of reproducing kernel Hilbert space (RKHS), we consi...
03/25/2020

On the worst-case error of least squares algorithms for L_2-approximation with high probability

It was recently shown in [4] that, for L_2-approximation of functions fr...
12/09/2021

Explicit Bounds for Linear Forms in the Exponentials of Algebraic Numbers

In this paper, we study linear forms λ = β_1e^α_1+⋯+β_me^α_m, ...
09/29/2021

Random sections of ℓ_p-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators

We study the circumradius of a random section of an ℓ_p-ellipsoid, 0<p≤∞...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.