Hypercontractivity Meets Random Convex Hulls: Analysis of Randomized Multivariate Cubatures
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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 analyze 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.
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