Random polytopes obtained by matrices with heavy-tailed entries

Guédon, Olivier; Litvak, Alexander E.; Tatarko, Kateryna

doi:10.1142/s0219199719500275

Cited by 8 publications

(5 citation statements)

References 34 publications

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“…where p ′ = p − log(1/λ 1 ). Thanks to the characterization of K p (E) and Theorem 1.6 one immediately recovers Theorem 1.2 as well as the main result from [19].…”

Section: Stochastic Dominationsupporting

confidence: 64%

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On the geometry of polytopes generated by heavy-tailed random vectors

Guédon,

Krahmer,

Kümmerle

et al. 2019

Preprint

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We study the geometry of centrally-symmetric random polytopes, generated by N independent copies of a random vector X taking values in R n . We show that under minimal assumptions on X, for N n and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector-namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on X we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when X is q-stable or when X has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing-noise blind sparse recovery.

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“…where p ′ = p − log(1/λ 1 ). Thanks to the characterization of K p (E) and Theorem 1.6 one immediately recovers Theorem 1.2 as well as the main result from [19].…”

Section: Stochastic Dominationsupporting

confidence: 64%

“…subgaussian centered coordinates ( [26]); when X is an isotropic, log-concave random vector ( [10]); and when X has i.i.d. centered coordinates that satisfy a small-ball condition ( [19]).…”

Section: Introductionmentioning

confidence: 99%

On the geometry of polytopes generated by heavy-tailed random vectors

Guédon,

Krahmer,

Kümmerle

et al. 2019

Preprint

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show abstract

“…mean zero, variance one entries. Very recently, Guedon, Litvak and Tatarko [11] extended the method of Rebrova and Tikhomirov to obtain the bound on the smallest singular values of "tall" matrices, as well as to study the geometry of random polytopes. In all the cases, the crucial point is to bypass the estimate on the operator norm, by discretizing the sphere in a non-standard way.…”

Section: Introductionmentioning

confidence: 99%

The smallest singular value of heavy-tailed not necessarily i.i.d. random matrices via random rounding

Livshyts

2021

JAMA

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We show the existence of a net near the sphere, such that the values of any matrix on the sphere and on the net are compared via a regularized Hilbert-Schmidt norm, which we introduce. This allows to construct an efficient net which controls the length of Ax for any random matrix A with independent columns (no other assumptions are required). As a consequence we show that the smallest singular value σ n (A) of an N × n random matrix A with i.i.d. mean zero, variance one entries enjoys the following small ball estimate, for any > 0:The proof of this result requires working with matrices whose rows are not independent, and, therefore, the fact that the theorem about discretization works for matrices with dependent rows, is crucial. Furthermore, in the case of the square n×n matrix A with independent entries having concentration function separated from 1, i.i.d. rows, and such that E||A|| 2 HS ≤ cn 2 , one hasfor any > 0. In addition, for > c √ n the assumption of i.i.d. rows is not required. Our estimates generalize the previous results of Rudelson and Vershynin [29], [30], which required the sub-gaussian mean zero variance one assumptions, as well as the work of Rebrova and Tikhomirov [25], where mean zero variance 1 and i.i.d. entries were required.

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“…, ±X n } for various classes of X such as Gaussian, Rademacher or vector with i.i.d. subgaussian entries [10,13,14,16,23]. One result about the Rademacher vector is the following: Theorem 2 [13] Let d be a sufficiently large positive integer and X 1 , X 2 , .…”

Section: Inclusion Of Deterministic Convex Bodiesmentioning

confidence: 99%

Estimating the probability that a given vector is in the convex hull of a random sample

Hayakawa

Lyons

Oberhauser

2023

Probab. Theory Relat. Fields

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For a d-dimensional random vector X, let $$p_{n, X}(\theta )$$ p n , X ( θ ) be the probability that the convex hull of n independent copies of X contains a given point $$\theta $$ θ . We provide several sharp inequalities regarding $$p_{n, X}(\theta )$$ p n , X ( θ ) and $$N_X(\theta )$$ N X ( θ ) denoting the smallest n for which $$p_{n, X}(\theta )\ge 1/2$$ p n , X ( θ ) ≥ 1 / 2 . As a main result, we derive the totally general inequality $$1/2 \le \alpha _X(\theta )N_X(\theta )\le 3d + 1$$ 1 / 2 ≤ α X ( θ ) N X ( θ ) ≤ 3 d + 1 , where $$\alpha _X(\theta )$$ α X ( θ ) (a.k.a. the Tukey depth) is the minimum probability that X is in a fixed closed halfspace containing the point $$\theta $$ θ . We also show several applications of our general results: one is a moment-based bound on $$N_X(\mathbb {E}\!\left[ X\right] )$$ N X ( E X ) , which is an important quantity in randomized approaches to cubature construction or measure reduction problem. Another application is the determination of the canonical convex body included in a random convex polytope given by independent copies of X, where our combinatorial approach allows us to generalize existing results in random matrix community significantly.

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Random polytopes obtained by matrices with heavy-tailed entries

Cited by 8 publications

References 34 publications

On the geometry of polytopes generated by heavy-tailed random vectors

On the geometry of polytopes generated by heavy-tailed random vectors

The smallest singular value of heavy-tailed not necessarily i.i.d. random matrices via random rounding

Estimating the probability that a given vector is in the convex hull of a random sample

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