A Simple Tool for Bounding the Deviation of Random Matrices on Geometric Sets

Liaw, Christopher; Mehrabian, Abbas; Plan, Yaniv; Vershynin, Roman

doi:10.1007/978-3-319-45282-1_18

Cited by 56 publications

(67 citation statements)

References 25 publications

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“…Let X := Ax ∈ R m with x ∈ S n−1 . The isotropic and sub-Gaussian assumption on A now implies X has independent coordinates satisfying EX 2 i = 1 and X i ψ 2 ≤ K. Lemma 5.3 in [20] states that…”

Section: Concentration Of Random Vectorsmentioning

confidence: 99%

“…That is, Z x − Z y ψ 2 ≤ M x − y 2 for some M and for all x, y ∈ R n . Theorem 1.3 in [20] showed sub-Gaussian increments for B = I m with M = CK 2 . Here we improve and generalize this result to any B with M = CK √ log K B .…”

Section: Sub-gaussian Increments Lemmamentioning

confidence: 99%

“…This is called the "sketched problem". It reduce the dimension from n to m and can be viewed as an approximation to the original problem (20). Moreover, say a solutionx to the sketched problem (21) is δ-optimal to the original optimal solution x * of (20) if…”

Section: Randomized Sketchesmentioning

confidence: 99%

“…There has been a series of recent works [8,20,24] to demonstrate the effectiveness of random mappings outside the Gaussian setup. Unlike the Gaussian case in which we have a rotation invariance property, non-Gaussian setups require more sophisticated arguments to address various new technical challenges.…”

Section: Introductionmentioning

confidence: 99%

“…Normal(0, 1) entries. But when it comes to the dependency on the sub-Gaussian parameter K, whether the K 2 factor can be improved is a question raised but left unanswered in [20]. Other important works regarding this type of bounds are either not explicit [15,24] or at least of the same order K 2 [7,8,23].…”

Section: Introductionmentioning

confidence: 99%

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Non-Gaussian Random Matrices on Sets:Optimal Tail Dependence and Applications

Jeong

LiP

Plan

et al. 2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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Random linear mappings are widely used in modern signal processing, compressed sensing and machine learning. These mappings may be used to embed the data into a significantly lower dimension while at the same time preserving useful information. This is done by approximately preserving the distances between data points, which are assumed to belong to R n . Thus, the performance of these mappings is usually captured by how close they are to an isometry on the data. Random Gaussian linear mappings have been the object of much study, while the sub-Gaussian settings is not yet fully understood. In the latter case, the performance depends on the sub-Gaussian norm of the rows. In many applications, e.g., compressed sensing, this norm may be large, or even growing with dimension, and thus it is important to characterize this dependence.We study when a sub-Gaussian matrix can become a near isometry on a set, show that previous best known dependence on the sub-Gaussian norm was sub-optimal, and present the optimal dependence. Our result not only answers a remaining question posed by Liaw, Mehrabian, Plan and Vershynin in 2017, but also generalizes their work. We also develop a new Bernstein type inequality for sub-exponential random variables, and a new Hanson-Wright inequality for quadratic forms of sub-Gaussian random variables, in both cases improving the bounds in the sub-Gaussian regime under moment constraints. Finally, we illustrate popular applications such as Johnson-Lindenstrauss embeddings, randomized sketches and blind demodulation, whose theoretical guarantees can be improved by our results in the sub-Gaussian case.

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Section: Concentration Of Random Vectorsmentioning

confidence: 99%

Section: Sub-gaussian Increments Lemmamentioning

confidence: 99%

Section: Randomized Sketchesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Non-Gaussian Random Matrices on Sets:Optimal Tail Dependence and Applications

Jeong

LiP

Plan

et al. 2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

Self Cite

View full text Add to dashboard Cite

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Sub‐Gaussian Matrices on Sets: Optimal Tail Dependence and Applications

Jeong

Plan

et al. 2021

Comm Pure Appl Math

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Random linear mappings are widely used in modern signal processing, compressed sensing, and machine learning. These mappings may be used to embed the data into a significantly lower dimension while at the same time preserving useful information. This is done by approximately preserving the distances between data points, which are assumed to belong to R n . Thus, the performance of these mappings is usually captured by how close they are to an isometry on the data. Gaussian linear mappings have been the object of much study, while the sub-Gaussian settings is not yet fully understood. In the latter case, the performance depends on the sub-Gaussian norm of the rows. In many applications, e.g., compressed sensing, this norm may be large, or even growing with dimension, and thus it is important to characterize this dependence.We study when a sub-Gaussian matrix can become a near isometry on a set, show that previous best-known dependence on the sub-Gaussian norm was suboptimal, and present the optimal dependence. Our result not only answers a remaining question posed by Liaw, Mehrabian, Plan, and Vershynin in 2017, but also generalizes their work. We also develop a new Bernstein-type inequality for subexponential random variables, and a new Hanson-Wright inequality for quadratic forms of sub-Gaussian random variables, in both cases improving the bounds in the sub-Gaussian regime under moment constraints. Finally, we illustrate popular applications such as Johnson-Lindenstrauss embeddings, null space property for 0-1 matrices, randomized sketches, and blind demodulation, whose theoretical guarantees can be improved by our results (in the sub-Gaussian case).

show abstract