The Curse of Instability

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.