Amelunxen, Dennis and Lotz, Martin
(2016)
*Average-case complexity without the black swans.*
[MIMS Preprint]

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## 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.

Item Type: | MIMS Preprint |
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Uncontrolled Keywords: | Random matrix theory, computational complexity, average-case analysis, condition numbers, numerical complexity |

Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis MSC 2010, the AMS's Mathematics Subject Classification > 68 Computer science |

Depositing User: | Dr. Martin Lotz |

Date Deposited: | 01 Jan 2016 |

Last Modified: | 20 Oct 2017 14:13 |

URI: | https://eprints.maths.manchester.ac.uk/id/eprint/2428 |

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