Several Approaches to Break the Curse of Dimensionality

TL;DR

This paper explores various strategies to overcome the curse of dimensionality in high-dimensional numerical problems by leveraging structural properties, within the framework of information-based complexity.

Contribution

It presents multiple approaches to mitigate the curse of dimensionality and provides a comprehensive introduction to information-based complexity theory.

Findings

Certain structural properties can significantly reduce computational complexity.

Exploiting problem-specific features helps overcome exponential growth in information requirements.

Theoretical insights guide the development of more efficient algorithms for high-dimensional problems.

Abstract

In modern science the efficient numerical treatment of high-dimensional problems becomes more and more important. A fundamental insight of the theory of information-based complexity (IBC for short) is that the computational hardness of a problem can not be described properly only by the rate of convergence. There exist problems for which an exponential number of information operations is needed in order to reduce the initial error, although there are algorithms which provide an arbitrary large rate of convergence. Problems that yield this exponential dependence are said to suffer from the curse of dimensionality. While analyzing numerical problems it turns out that we can often vanquish this curse by exploiting additional structural properties. The aim of this thesis is to present several approaches of this type. Moreover, a detailed introduction to the field of IBC is given.