Smallest singular value of random matrices and geometry of random polytopes
نویسندگان
چکیده
منابع مشابه
Smallest singular value of random matrices and geometry of random polytopes
We study behaviour of the smallest singular value of a rectangular random matrix, i.e., matrix whose entries are independent random variables satisfying some additional conditions. We prove a deviation inequality and show that such a matrix is a “good” isomorphism on its image. Then we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes (absolute...
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Let ξ be a real-valued random variable of mean zero and variance 1. Let Mn(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σn(Mn(ξ)) denote the least singular value of Mn(ξ). The quantity σn(Mn(ξ)) 2 is thus the least eigenvalue of the Wishart matrix MnM ∗ n. We show that (under a finite moment assumption) the probability distribution nσn(Mn(ξ)) 2 is universal in the sen...
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We present a very general chaining method which allows one to control the supremum of the empirical process suph∈H |N−1 ∑N i=1 h (Xi)− Eh2| in rather general situations. We use this method to establish two main results. First, a quantitative (non asymptotic) version of the classical Bai-Yin Theorem on the singular values of a random matrix with i.i.d entries that have heavy tails, and second, a...
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We prove an optimal estimate of the smallest singular value of a random subGaussian matrix, valid for all dimensions. For an N n matrix A with independent and identically distributed sub-Gaussian entries, the smallest singular value of A is at least of the order p N pn 1 with high probability. A sharp estimate on the probability is also obtained. © 2009 Wiley Periodicals, Inc.
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ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 2005
ISSN: 0001-8708
DOI: 10.1016/j.aim.2004.08.004