Smoothed analysis of the condition number under low-rank perturbations

@inproceedings{Shah2021SmoothedAO,
  title={Smoothed analysis of the condition number under low-rank perturbations},
  author={Rikhav Shah and Sandeep Silwal},
  booktitle={APPROX-RANDOM},
  year={2021}
}
Let $M$ be an arbitrary $n$ by $n$ matrix of rank $n-k$. We study the condition number of $M$ plus a \emph{low rank} perturbation $UV^T$ where $U, V$ are $n$ by $k$ random Gaussian matrices. Under some necessary assumptions, it is shown that $M+UV^T$ is unlikely to have a large condition number. The main advantages of this kind of perturbation over the well-studied dense Gaussian perturbation where every entry is independently perturbed is the $O(nk)$ cost to store $U,V$ and the $O(nk… 
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