Smallest singular value of sparse random matrices

Let x be a complex random variable with mean zero and bounded variance σ2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ1, …, λn be the eigenvalues of [Formula: see text]. Define the empirical spectral distributionμn of Nn by the formula [Formula: see text] The following well-known conjecture has been open since the 1950's: Circular Law Conjecture: μn converges to the uniform distribution μ∞ over the unit disk as n tends to infinity. We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 0. Our method builds and improves upon earlier work of Girko, Bai, Götze–Tikhomirov, and Pan–Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.

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The smallest singular value of inhomogeneous square random matrices

The Annals of Probability ◽

10.1214/20-aop1481 ◽

2021 ◽

Vol 49 (3) ◽

Author(s):

Galyna V. Livshyts ◽

Konstantin Tikhomirov ◽

Roman Vershynin

Keyword(s):

Random Matrices ◽

Singular Value ◽

Smallest Singular Value

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On generic chaining and the smallest singular value of random matrices with heavy tails

Journal of Functional Analysis ◽

10.1016/j.jfa.2012.01.027 ◽

2012 ◽

Vol 262 (9) ◽

pp. 3775-3811 ◽

Cited By ~ 10

Author(s):

Shahar Mendelson ◽

Grigoris Paouris

Keyword(s):

Random Matrices ◽

Heavy Tails ◽

Singular Value ◽

Generic Chaining ◽

Smallest Singular Value

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Random polytopes obtained by matrices with heavy-tailed entries

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500275 ◽

2019 ◽

Vol 22 (04) ◽

pp. 1950027

Author(s):

O. Guédon ◽

A. E. Litvak ◽

K. Tatarko

Keyword(s):

Compressed Sensing ◽

Convex Hull ◽

Random Matrices ◽

Random Matrix ◽

Singular Value ◽

Small Ball ◽

Random Polytopes ◽

Heavy Tailed ◽

Smallest Singular Value ◽

Quotient Property

Let [Formula: see text] be an [Formula: see text] random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope [Formula: see text] in [Formula: see text] (the absolute convex hull of rows of [Formula: see text]). In particular, we show that [Formula: see text] where [Formula: see text] depends only on parameters in small ball inequality. This extends results of [A. E. Litvak, A. Pajor, M. Rudelson and N. Tomczak-Jaegermann, Smallest singular value of random matrices and geometry of random polytopes, Adv. Math. 195 (2005) 491–523] and recent results of [F. Krahmer, C. Kummerle and H. Rauhut, A quotient property for matrices with heavy-tailed entries and its application to noise-blind compressed sensing, preprint (2018); arXiv:1806.04261]. This inclusion is equivalent to so-called [Formula: see text]-quotient property and plays an important role in compressed sensing (see [F. Krahmer, C. Kummerle and H. Rauhut, A quotient property for matrices with heavy-tailed entries and its application to noise-blind compressed sensing, preprint (2018); arXiv:1806.04261] and references therein).

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