Notes on tractability conditions for linear multivariate problems

TL;DR

This paper investigates conditions under which linear multivariate problems are tractable, providing criteria based on singular value sums that do not depend on ordering, thus broadening applicability.

Contribution

It introduces necessary and sufficient tractability conditions for compact linear operators in Hilbert spaces, avoiding the need to order singular values.

Findings

Derived tractability conditions based on singular value sums

Conditions do not require ordering of singular values

Applicable to a broad class of multivariate problems

Abstract

We study approximations of compact linear multivariate operators defined over Hilbert spaces. We provide necessary and sufficient conditions on various notions of tractability. These conditions are mainly given in terms of sums of certain functions depending on the singular values of the multivariate problem. They do not require the ordering of these singular values which in many cases is difficult to achieve.