# 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…

## One Citation

### Optimal Smoothed Analysis and Quantitative Universality for the Smallest Singular Value of Random Matrices

- Computer Science, MathematicsArXiv
- 2022

It is proved the first quantitative universality for the smallest singular value and condition number of random matrices, and an optimal smoothed analysis for random matrix analysis is derived in terms of a sharp Gaussian approximation.

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