The smallest singular value of deformed random rectangular matrices

Fan Yang University of Wisconsin-Madison

Probability mathscidoc:1805.28003

We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\le N \le M \le dN$ for some constant $d\ge 1$. Let $X$ be an $M\times n$ random matrix with independent and identically distributed entries, which have zero mean, unit variance and arbitrarily high moments. Let $T$ be an $N\times M$ deterministic matrix with comparable singular values $c\le s_{N}(T) \le s_{1}(T) \le c^{-1}$ for some constant $c>0$. Let $A$ be an $N\times n$ deterministic matrix with $\|A\|=O(\sqrt{N})$. Then we prove that for any $\epsilon>0$, the smallest singular value of $TX-A$ is larger than $N^{-\epsilon}(\sqrt{N}-\sqrt{n-1})$ with high probability. If we assume further the entries of $X$ have subgaussian decay, then the smallest singular value of $TX-A$ is at least of the order $\sqrt{N}-\sqrt{n-1}$ with high probability, which is an essentially optimal estimate.
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@inproceedings{fanthe,
  title={The smallest singular value of deformed random rectangular matrices},
  author={Fan Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180507143800920437085},
}
Fan Yang. The smallest singular value of deformed random rectangular matrices. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180507143800920437085.
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