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Invertibility via distance for noncentered random matrices with continuous distributions
Random Structures and Algorithms ( IF 1 ) Pub Date : 2020-05-01 , DOI: 10.1002/rsa.20920
Konstantin Tikhomirov 1
Affiliation  

Let A be an n ×n random matrix with independent rows R 1(A ),…,R n (A ), and assume that for any i  ≤ n and any three‐dimensional linear subspace urn:x-wiley:rsa:media:rsa20920:rsa20920-math-0001 the orthogonal projection of R i (A ) onto F has distribution density urn:x-wiley:rsa:media:rsa20920:rsa20920-math-0002 satisfying urn:x-wiley:rsa:media:rsa20920:rsa20920-math-0003 (x F ) for some constant C 1>0. We show that for any fixed n ×n real matrix M we have
urn:x-wiley:rsa:media:rsa20920:rsa20920-math-0004(1)
where C >0 is a universal constant. In particular, the above result holds if the rows of A are independent centered log‐concave random vectors with identity covariance matrices. Our method is free from any use of covering arguments, and is principally different from a standard approach involving a decomposition of the unit sphere and coverings, as well as an approach of Sankar‐Spielman‐Teng for noncentered Gaussian matrices.


中文翻译:

具有连续分布的非中心随机矩阵的距离可逆性

Ñ × Ñ随机矩阵具有独立的行- [R 1),...,[R Ñ),并且假定对于任何 ≤  Ñ和任何三维线性子空间骨灰盒:x-wiley:rsa:media:rsa20920:rsa20920-math-0001的正投影ř)到˚F有分布密度骨灰盒:x-wiley:rsa:media:rsa20920:rsa20920-math-0002满足骨灰盒:x-wiley:rsa:media:rsa20920:rsa20920-math-0003X˚F)对于某一常数c ^ 1 > 0。我们证明对于任何固定的n × n实矩阵中号,我们有
骨灰盒:x-wiley:rsa:media:rsa20920:rsa20920-math-0004(1)
其中C ' > 0是一个通用常数。特别是,如果A的行是具有恒等协方差矩阵的独立居中对数凹面随机向量,则上述结果成立。我们的方法完全没有使用覆盖论点,并且与标准方法(涉及分解单位球体和覆盖物)以及Sankar-Spielman-Teng用于非中心高斯矩阵的方法基本不同。
更新日期:2020-05-01
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