Average-case complexity without the black swans

TL;DR

This paper proposes a weak average-case analysis framework that better aligns theoretical complexity results with practical experience, especially in scenarios where traditional worst-case assumptions are too pessimistic.

Contribution

It introduces the concept of weak average-case analysis and demonstrates its application to condition numbers, eigenvector computation, and conic optimization.

Findings

Weak average-case analysis provides more realistic complexity estimates.

Application to condition numbers shows a 'numerical null set' interpretation.

Analysis of power iteration and conic optimization illustrates practical relevance.

Abstract

We introduce the concept of weak average-case analysis as an attempt to achieve theoretical complexity results that are closer to practical experience than those resulting from traditional approaches. This concept is accepted in other areas such as non-asymptotic random matrix theory and compressive sensing, and has a particularly convincing interpretation in the most common situation encountered for condition numbers, where it amounts to replacing a null set of ill-posed inputs by a "numerical null set". We illustrate the usefulness of these notions by considering three settings: (1) condition numbers that are inversely proportional to a distance of a homogeneous algebraic set of ill-posed inputs; (2) the running time of power iteration for computing a leading eigenvector of a Hermitian matrix; (3) Renegar's condition number for conic optimisation.