Smooth analysis of the condition number and the least singular value

Tao, Terence; Vu, Van

doi:10.1090/S0025-5718-2010-02396-8

Smooth analysis of the condition number and the least singular value
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by Terence Tao and Van Vu;

Math. Comp. 79 (2010), 2333-2352

DOI: https://doi.org/10.1090/S0025-5718-2010-02396-8

Published electronically: June 4, 2010

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Abstract:

Let $x$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $x$ and let $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when $x$ is gaussian.

Our investigation reveals an interesting fact that the “core” matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n}$. This does not occur in Spielman-Teng studies when ${x}$ is gaussian. Consequently, our general estimate involves the norm $\|M\|$. In the special case when $\|M\|$ is relatively small, this estimate is nearly optimal and extends or refines existing results.

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