Some estimates of norms of random matrices

By Rafal Latala. 2005

124 citations recorded by the source corpus.

We show that for any random matrix (X ij ) with independent mean zero entries E∥(X ij )∥ ≤ C(max √ΣEX 2 ij + max √ΣEX 2 ij + 4√ΣEX 4 ij ), where C is some universal constant.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6758cf8a5d6349cde76f46f0fcedcea515de2a37

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistics and Probability

Citations in the corpus

Introduction to the non-asymptotic analysis of random matrices (2010, methodology): “cessarily normal). The largest singular value (the spectral norm) can be estimated by a general Latala’s theorem for matrices with non-identically distributed entries: 24 Theorem 37 (Latala’s theorem [42]). Let Abe a random matrix whose entries aij are independent centered random variables with ﬁnit”
No-gaps delocalization for general random matrices (2015, background): “parts have variances bounded below. We will also assume kAk = O( √ n) with high probability. This natural condition holds, in particular, if the entries of A have mean zero and bounded fourth moments [19]. To make this rigorous, we ﬁx a number M ≥ 1 and introduce the boundedness event BA,M := kAk ≤ ”
The Littlewood-Offord problem and invertibility of random matrices (2007, background): “rices under the fourth moment assumption and the subgaussian moment assumption. The following result is a partial case of a recent result of Latala. Theorem 2.3 (Largest singular value: fourth moment [13]). Let A be an n×n matrix whose entries are independent centered random variables with variances”
Lower bounds for the smallest singular value of structured random matrices (2016, methodology): “ (see (1.1)). We will use standard truncation arguments to deduce Lemma 5.5 from the following bounds on the expected operator norm of random matrices due to Latala and Vershynin. Theorem 5.7 (Latala [18]). Let n,m be suﬃciently large and let Y be an n×m random matrix with independent, centered entr”
The spectral norm of random inner-product kernel matrices (2015, background): “covering net argument: kK(X)kCsup y2Dp 2 yTK(X)y; for a constant C>0 and a nite covering net Dp 2 of the unit ball fy2Rp : kyk1g. We use a particular construction of a covering net due to Latala [37]: Denition 2.1. For m= dlog 2 pe, let D p 2 = n y2R : kyk1;y 2 i 2f0;1;2 1;2 ;:::;2 (m+3)gfor all ”
Spectral norm of products of random and deterministic matrices (2008, methodology): “dean sphere in Rn into classes of vectors according to their “sparsity”, and uses the entropyconcentration tradeoﬀ for each class separately. This general line is already present in Latala’s argument [9], and it was developed extensively in the recent work of M. Rudelson and the author on invertibil”
Invertibility of Sparse non-Hermitian matrices (2015, background): “ variables, and establish the required bound when pn= Ω(n−α). For dense matrices, the ﬁniteness of the fourth moment is suﬃcient (and also necessary) to guarantee the necessary bounds on smax(·) (see [13]). However, for sparse case one needs ﬁniteness of the higher moments depending on the choice of”
Non-asymptotic theory of random matrices: extreme singular values (2010, background): “5 (Largest singular value: fourth moment, non-iid entries [46]).”
No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis (2017, unknown): “By random matrix theory, we know when p ≥ c1 log d min{d1,d2} , with probability at least 1 − d −4, we have ‖Z‖ = ‖A− E[A]‖ ≤ c2 √ pd [Latała, 2005]1.”
How Close is the Sample Covariance Matrix to the Actual Covariance Matrix? (2010, background): “Euclidean norm of any column and any row. For random matrices with independent entries, estimate (1.7) follows (under the fourth moment assumption) from more general bounds by Seginer [26] and Latala [15], and even without the loglogN factor. Without independence of entries, this bound was proved by”
Covariance estimation for distributions with 2+ε moments (2011, background): “s well understood by now. In the latter case, if the entries have zero mean and uniformly bounded fourth moments, the bound EkAk . √ N+ √ nfollows, for example, from a general inequality of R. Latala [10]. 1.7. Smallest eigenvalue. Our proof of Theorem 1.1 consists of two separate arguments for uppe”