Probabilistic Analysis of Conic Condition Numbers: I. The Complex Case

Abstract

The smoothed analysis of condition numbers in the preceding chapters was done on a case-by-case basis. For each considered condition number we proved a result giving bounds on either expectation or probability tails or both. In this chapter and the next we proceed differently—the theme of both chapters is the same, but the focus of this is on problems over $\mathbb {C}$, while the focus on the next is on problems over $\mathbb {R}$. We consider a reasonably large class of condition numbers and obtain smoothed analysis estimates for elements in this class depending only on geometric invariants of the corresponding sets of ill-posed inputs.