Near G-optimal Tchakaloff designs

Abstract

We show that the notion of polynomial mesh (norming set), used to provide discretizations of a compact set nearly optimal for certain approximation theoretic purposes, can also be used to obtain finitely supported near G-optimal designs for polynomial regression. We approximate such designs by a standard multiplicative algorithm, followed by measure concentration via Caratheodory-Tchakaloff compression.

dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS

“…Following References [18,19], we can however effectively compute a design which has the same G-efficiency of u(k) but a support with a cardinality not exceeding N 2m = dim(P d 2m (X)), where in many applications N 2m card(X), obtaining a remarkable compression of the near optimal design. The theoretical foundation is a generalized version [15] of Tchakaloff Theorem [20] on positive quadratures, which asserts that for every measure on a compact set Ω ⊂ R d there exists an algebraic quadrature formula exact on P d n (Ω)), with positive weights, nodes in Ω and cardinality not exceeding N n = dim(P d n (Ω).…”

“…In a recent paper [19], a connection has been studied between the statistical notion of G-optimal design and the approximation theoretic notion of admissible mesh for multivariate polynomial approximation, deeply studied in the last decade after Reference [13] (see, e.g., References [27,28] with the references therein). In particular, it has been shown that near G-optimal designs on admissible meshes of suitable cardinality have a G-efficiency on the whole d-cube that can be made convergent to 1.…”

InRS: implementing the indicator function of NURBS-shaped planar domains

“…NURBS-shaped domains produced by CAGD algorithms play a central role in digital design and modelling processes. The capability of locating quasi-uniform or random sample points in such domains can be useful in a vast range of applications, for example within several meshfree bivariate approximation algorithms developed in the last twenty years, among which we may quote (without any pretence of completeness) kernel-based and partition-of-unity collocation methods [5,7], construction of algebraic cubature formulas [8,18,19] potentially useful for curved FEM/VEM elements [1,17], compressed MC/QMC integration [2,9], compressed polynomial regression [4,15].…”

Computing optimal experimental designs on finite sets by log-determinant gradient flow

“…, φ N }. Then, using the notation K(x, y; w) := N i=1 φ i (x; w)φ i (y; w), (4) B(x; w) := K(x, x; w), (5) any D-optimal desig w * is also G-optimal, i.e., max x∈X B(x; w * ) = min…”

“…There are several approaches to attack such a problem, treating X as a variable or a fixed parameter of the problem, which is chosen accordingly to certain heuristics as space filling techniques, grid exploration (see e.g., [8]), or minimal spanning tree. Recently, it has been shown that the use of polynomial admissible meshes gives precise quantitative estimates of the approximation intruduced in the discretization of the problem , i.e., when passing from X to X, see [5].…”