Effective Dimension of Some Weighted Pre-Sobolev Spaces with Dominating Mixed Partial Derivatives

Abstract

This paper considers two notions of effective dimension for quadrature in weighted pre-Sobolev spaces with dominating mixed partial derivatives. We begin by finding a ball in those spaces just barely large enough to contain a function with unit variance. If no function in that ball has more than ε of its variance from ANOVA components involving interactions of order s or more, then the space has effective dimension at most s in the superposition sense. A similar truncation sense notion replaces the cardinality of the ANOVA component by the largest index it contains. Some Poincaré type inequalities are used to bound variance components by multiples of these space's squared norm and those in turn provide bounds on effective dimension. Very low effective dimension in the superposition sense holds for some spaces defined by product weights in which quadrature is strongly tractable. The superposition dimension is O(log(1/ε)/ log(log(1/ε))) just like the superposition dimension used in the multidimensional decomposition method. Surprisingly, even spaces where all subset weights are equal, regardless of their cardinality or included indices, have low superposition dimension in this sense. This paper does not require periodicity of the integrands.

Interpretable Approximation of High-Dimensional Data

“…characterizes smoothness trough the decay of the basis coefficients c k (f ), cf. [33,30]. Now, we still have the curse of dimensionality and need to find a way around it for efficient approximation.…”

“…This means that the significant part of the function is explained by letting only up to a number of variables interact simultaneously, see e.g. [7,25,11,18], relating to the concept of the superposition dimension, see [7,30]. These assumptions are in general not very restricting and allows for a broad range of functions.…”

On the effective dimension and multilevel Monte Carlo

“…Methods that reduce the effective dimension and improve the performance of QMC are described in (Wang and Sloan 2011, Wang and Tan 2013, Xiao and Wang 2019. Owen (2019) gives a recent survey on the effective dimension. Kahalé (2020b) studies the relationship between the truncation dimension and the randomized dimension reduction method, a recent variance reduction technique applicable to high-dimensional problems.…”

Comment: Unreasonable Effectiveness of Monte Carlo

“…Effective dimension is not usually defined as a sum of γ u . That sum might not be smaller than d. For a brief history of effective dimension in QMC, going back to the 1950s, see Owen (2018). The error in higher order digital nets can be reduced by a factor of about n −1/2 by scrambling the digits.…”