TY - JOUR

T1 - Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points

AU - Migliorati, Giovanni

AU - Nobile, Fabio

AU - Tempone, Raul

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2015/8/28

Y1 - 2015/8/28

N2 - We study the accuracy of the discrete least-squares approximation on a finite dimensional space of a real-valued target function from noisy pointwise evaluations at independent random points distributed according to a given sampling probability measure. The convergence estimates are given in mean-square sense with respect to the sampling measure. The noise may be correlated with the location of the evaluation and may have nonzero mean (offset). We consider both cases of bounded or square-integrable noise / offset. We prove conditions between the number of sampling points and the dimension of the underlying approximation space that ensure a stable and accurate approximation. Particular focus is on deriving estimates in probability within a given confidence level. We analyze how the best approximation error and the noise terms affect the convergence rate and the overall confidence level achieved by the convergence estimate. The proofs of our convergence estimates in probability use arguments from the theory of large deviations to bound the noise term. Finally we address the particular case of multivariate polynomial approximation spaces with any density in the beta family, including uniform and Chebyshev.

AB - We study the accuracy of the discrete least-squares approximation on a finite dimensional space of a real-valued target function from noisy pointwise evaluations at independent random points distributed according to a given sampling probability measure. The convergence estimates are given in mean-square sense with respect to the sampling measure. The noise may be correlated with the location of the evaluation and may have nonzero mean (offset). We consider both cases of bounded or square-integrable noise / offset. We prove conditions between the number of sampling points and the dimension of the underlying approximation space that ensure a stable and accurate approximation. Particular focus is on deriving estimates in probability within a given confidence level. We analyze how the best approximation error and the noise terms affect the convergence rate and the overall confidence level achieved by the convergence estimate. The proofs of our convergence estimates in probability use arguments from the theory of large deviations to bound the noise term. Finally we address the particular case of multivariate polynomial approximation spaces with any density in the beta family, including uniform and Chebyshev.

UR - http://hdl.handle.net/10754/576075

UR - http://linkinghub.elsevier.com/retrieve/pii/S0047259X15001931

UR - http://www.scopus.com/inward/record.url?scp=84941642510&partnerID=8YFLogxK

U2 - 10.1016/j.jmva.2015.08.009

DO - 10.1016/j.jmva.2015.08.009

M3 - Article

SN - 0047-259X

VL - 142

SP - 167

EP - 182

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

ER -