Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

Rebrova, Elizaveta; Tikhomirov, Konstantin

doi:10.1007/s11856-018-1732-y

Cited by 38 publications

(80 citation statements)

References 32 publications

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“…Remark 3.2. This result extends Theorem A and Corollary A from [25], where the authors considered the case of square matrices, T = S n−1 and k = N, which corresponds to the approximation of Γx in the Euclidean norm.…”

Section: Notationssupporting

confidence: 78%

“…It allows us to construct, with high probability, a diagonal matrix D such that the volume of DB n ∞ remains big enough and such that, according to Lemma 3.4, we have a good control of the operator norm of ΓD from ℓ n ∞ to X k,2 . Comparing to [25], Lemma 3.4 simplifies significantly the proof and allows to extend Theorem 3.1 from [25] to the case of rectangular matrices and to approximations with respect to · k,2 norms.…”

Section: Auxiliary Statementsmentioning

confidence: 99%

“…However it is known that such a bound does not hold in general unless fourth moments are bounded ( [29], see also [20] for quantitative bounds). To avoid using the norm of Γ, we use ideas appearing in [25], where the authors constructed a certain deterministic ε-net (in ℓ 2 -metric) N such that AN is a good net for AB n 2 for most realizations of a square random matrix A. We extend their construction in three directions.…”

Section: Introductionmentioning

confidence: 99%

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Random polytopes obtained by matrices with heavy-tailed entries

Guédon

Litvak

Tatarko

2019

Commun. Contemp. Math.

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Let Γ be an N × n 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 Γ * B N 1 in R n (the absolute convex hull of rows of Γ). In particular, we show thatwhere b depends only on parameters in small ball inequality. This extends results of [18] and recent results of [17]. This inclusion is equivalent to so-called ℓ 1 -quotient property and plays an important role in compressed sensing (see [17] and references therein).

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Section: Notationssupporting

confidence: 78%

Section: Auxiliary Statementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Random polytopes obtained by matrices with heavy-tailed entries

Guédon

Litvak

Tatarko

2019

Commun. Contemp. Math.

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“…elements having sub-gaussian tails ( [23]). Moreover, heavy-tailed models, when the matrix entries have only two finite moments, still have polynomial condition numbers κ(A) with high probability [22]. An alternative (although very similar) estimate can be obtained for all matrices, without a condition number assumption, in trade of the absolute constants: Theorem 2.3.…”

Section: Resultsmentioning

confidence: 98%

On block Gaussian sketching for the Kaczmarz method

Rebrova

Needell

2020

Numer Algor

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The Kaczmarz algorithm is one of the most popular methods for solving large-scale over-determined linear systems due to its simplicity and computational efficiency. This method can be viewed as a special instance of a more general class of sketch and project methods. Recently, a block Gaussian version was proposed that uses a block Gaussian sketch, enjoying the regularization properties of Gaussian sketching, combined with the acceleration of the block variants. Theoretical analysis was only provided for the non-block version of the Gaussian sketch method.Here, we provide theoretical guarantees for the block Gaussian Kaczmarz method, proving a number of convergence results showing convergence to the solution exponentially fast in expectation. On the flip side, with this theory and extensive experimental support, we observe that the numerical complexity of each iteration typically makes this method inferior to other iterative projection methods. We highlight only one setting in which it may be advantageous, namely when the regularizing effect is used to reduce variance in the iterates under certain noise models and convergence for some particular matrix constructions. arXiv:1905.08894v2 [math.PR]

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“…Originally, the core part of the algorithm (Step 2) was presented for Bernoulli random matrices in [13,12]. We will use the following version of [13, Lemma 5.1] (based on the ideas developed in [12, Proposition 2.1]):…”

Section: Proof Of Corollary 13mentioning

confidence: 99%

Constructive Regularization of the Random Matrix Norm

Rebrova¹

2019

J Theor Probab

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We show a simple local norm regularization algorithm that works with high probability. Namely, we prove that if the entries of a n × n matrix A are i.i.d. symmetrically distributed and have finite second moment, it is enough to zero out a small fraction of the rows and columns of A with largest L2 norms in order to bring the operator norm of A to the almost optimal order O( √ log log n · n). As a corollary, we also obtain a constructive procedure to find a small submatrix of A that one can zero out to achieve the same goal.This work is a natural continuation of our recent work with R. Vershynin, where we have shown that the norm of A can be reduced to the optimal order O( √ n) by zeroing out just a small submatrix of A, but did not provide a constructive procedure to find this small submatrix.Our current approach extends the norm regularization techniques developed for the graph adjacency (Bernoulli) matrices in the works of Feige and Ofek, and Le, Levina and Vershynin to the considerably broader class of matrices.√ n with high probability. If we are concerned to get an explicit (non-asymptotic) probability estimate for all large enough n, an application of Bernstein's inequality (see, for example, in [18,19]) gives P{ A ≤ t √ n} ≥ 1 − e −c 0 t 2 n for t ≥ C 0 Affiliation:

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Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

Cited by 38 publications

References 32 publications

Random polytopes obtained by matrices with heavy-tailed entries

Random polytopes obtained by matrices with heavy-tailed entries

On block Gaussian sketching for the Kaczmarz method

Constructive Regularization of the Random Matrix Norm

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