Affiliation:
1. Mathematical Institute, University of Oxford, Oxford, Oxfordshire, UK
Abstract
Given a probability measure
μ
on a set
X
and a vector-valued function
φ
, a common problem is to construct a discrete probability measure on
X
such that the push-forward of these two probability measures under
φ
is the same. This construction is at the heart of numerical integration methods that run under various names such as quadrature, cubature or recombination. A natural approach is to sample points from
μ
until their convex hull of their image under
φ
includes the mean of
φ
. Here, we analyse the computational complexity of this approach when
φ
exhibits a graded structure by using so-called hypercontractivity. The resulting theorem not only covers the classical cubature case of multivariate polynomials, but also integration on pathspace, as well as kernel quadrature for product measures.
Funder
Engineering and Physical Sciences Research Council
Oxford-Man Institute
CIMDA
Subject
General Physics and Astronomy,General Engineering,General Mathematics
Cited by
1 articles.
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