The curse of dimensionality for numerical integration of smooth functions II

Abstract

We prove the curse of dimensionality in the worst case setting for numerical integration for a number of classes of smooth d-variate functions. Roughly speaking, we consider different bounds for the directional or partial derivatives of f ∈ C k (D d ) and ask whether the curse of dimensionality holds for the respective classes of functions. We always assume that D d ⊂ R d has volume one and we often assume additionally that D d is either convex or that its radius is proportional to √ d. In particular, D d can be the unit cube. We consider various values of k including the case k = ∞ which corresponds to infinitely differentiable functions. We obtain necessary and sufficient conditions, and in some cases a full characterization for the curse of dimensionality. For infinitely differentiable functions we prove the curse if the bounds on the successive derivatives are appropriately large. The proof technique is based on a volume estimate of a neighborhood of the convex hull of n points which decays exponentially fast in d. For k = ∞, we also study conditions for quasi-polynomial, weak and uniform weak tractability. In particular, weak tractability holds if all directional derivatives are bounded by one. It is still an open problem if weak tractability holds if all partial derivatives are bounded by one.

The curse of dimensionality for numerical integration on general domains

“…In [13], improving and extending previous results obtained in [12], Hinrichs, Novak, Ullrich and Woźniakowski proved the curse of dimensionality in the worst case setting for numerical integration for a number of classes of smooth d-variate functions. They considered different bounds on the Lipschitz constants for the directional or partial derivatives of f ∈ C k (K d ) (k ∈ N ∪ {+∞}), where C k (K d ) denotes the space of k-times continuously differentiable functions on a domain K d ⊆ R d with vol d (K d ) = 1.…”

“…The results from their paper [13] are sharp and characterize the curse of dimensionality if the domain of integration is either the cube, i.e., K d = [0, 1] d , or a convex body with the property lim sup…”

“…Historically, the central question was the rate of error convergence for fixed dimension d. Only recently, motivated by modern applications, the emphasis was shifted to the study of the dependence of the error on the dimension. Specifically, in [12,13] the authors discussed necessary and sufficient conditions on the parameters A d and B d as well as on the domain K d such that the curse of dimensionality holds for the class C 1…”

A Novel Distance Definition Based on Multi Partial Distance Information

“…Conventional distance definitions tend to distort the true closeness, and affect the subsequent learning results. That phenomenon is known in the statistical literatures as the curse of dimensionality [4].…”

A heuristic supervised Euclidean data difference dimension reduction for KNN classifier and its application to visual place classification

“…The ''curse of dimensionality'' term is usually used to describe the problems associated with having high mathematical dimensional space [6]. Dimension reduction algorithms are used to eliminate data redundancy and deal with the curse of dimensionality in order to gain higher speed and less memory usage in applications.…”