The Curse of Instability

TL;DR

This paper introduces the 'curse of instability' as a fundamental concept that explains the classical curse of dimensionality, linking instability to analytical and multiscale complexities in high-dimensional problems.

Contribution

It proposes the 'curse of instability' as a new perspective to understand high-dimensional challenges and offers practical insights and conjectures for mathematical analysis.

Findings

The curse of instability underpins the curse of dimensionality.

Instability indicates analytical difficulties and multiscale complexity.

Practical guidelines for analyzing high-dimensional models are suggested.

Abstract

High-dimensional computational challenges are frequently explained via the curse of dimensionality, i.e., increasing the number of dimensions leads to exponentially growing computational complexity. In this commentary, we argue that thinking on a different level helps to understand, why we face the curse of dimensionality. We introduce as a guiding principle the curse of instability, which triggers the classical curse of dimensionality. Furthermore, we claim that the curse of instability is a strong indicator for analytical difficulties and multiscale complexity. Finally, we suggest some practical conclusions for the analysis of mathematical models and formulate several conjectures.