Concentration inequalities for polynomials in α-sub-exponential random variables

Götze, Friedrich; Sambale, Holger; Sinulis, Arthur

doi:10.1214/21-ejp606

Cited by 30 publications

(37 citation statements)

References 25 publications

(43 reference statements)

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“…Notable examples are Latala's bound on Gaussian chaoses [17] (see [21] for an alternative proof) and its many extensions and ramifications (see e.g. [18,4,6,14] and the refererences therein), as well as tensorization techniques [2]. A distinct feature of Theorems 1.3 and 1.4 is the optimal dependence on the degree d of tensors.…”

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confidence: 99%

Concentration inequalities for random tensors

Vershynin¹

2019

Preprint

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We show how to extend several basic concentration inequalities for simple random tensors X = x1 ⊗ • • • ⊗ x d where all x k are independent random vectors in R n with independent coefficients. The new results have optimal dependence on the dimension n and the degree d. As an application, we show that random tensors are well conditioned:(1 − o(1))n d independent copies of the simple random tensor X ∈ R n d are far from being linearly dependent with high probability. We prove this fact for any degree d = o( n/ log n) and conjecture that it is true for any d = O(n).

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confidence: 99%

Concentration inequalities for random tensors

Vershynin¹

2019

Preprint

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“…More work on related topics include [19,18] where upper and lower bounds for the case of random variables satisfying the moment condition X 2p ≤ α X p are considered for the case of positive variables of order 2. The recent work [9] provides a similar bound to [2] for functions of the random variables that are not necessarily polynomials.…”

Section: Previous Workmentioning

confidence: 98%

The Hanson-Wright Inequality for Random Tensors

Bamberger¹,

Krahmer²,

Ward³

2021

Preprint

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We provide moment bounds for expressions of the type (Xwhere ⊗ denotes the Kronecker product and X (1) , . . . , X (d) are random vectors with independent, mean 0, variance 1, subgaussian entries. The bounds are tight up to constants depending on d for the case of Gaussian random vectors. Our proof also provides a decoupling inequality for expressions of this type. Using these bounds, we obtain new, improved concentration inequalities for expressions of the form B(X (1) ⊗ • • • ⊗ X (d) ) 2 .

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“…There seem to be no sufficiently powerful concentration inequalities available for such models. Known concentration inequalities for random chaoses [30,32,31,3,4,20,1] exhibit an unspecified (possibly exponential) dependence on the degree d, which is too bad for our purposes. An exception is the recent work [46] on concentration of random tensors with an optimal dependence on d. However, the results of [46] only apply for non-symmetric tensors and positive-semidefinite matrices A.…”

Section: Relaxing Independence?mentioning

confidence: 99%

Marchenko-Pastur law with relaxed independence conditions

Bryson

Vershynin

Zhao

2019

Preprint

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We prove the Marchenko-Pastur law for the eigenvalues of sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario -the block-independent model -the coordinates of the data are partitioned into n blocks each of length d in such a way that the entries in different blocks are independent but the entries from the same block may be dependent. In the second scenario -the random tensor model -the data is the homogeneous random tensor of order d, i.e. the coordinates of the data are all n d different products of d variables chosen from a set of n independent random variables. We show that Marchenko-Pastur law holds for the block-independent model as long as d = o(n) and for the random tensor model as long as d = o(n 1/3 ). Our main technical tools are new concentration inequalities for quadratic forms in random variables with block-independent coordinates, and for random tensors.

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Concentration inequalities for polynomials in α-sub-exponential random variables

Cited by 30 publications

References 25 publications

Concentration inequalities for random tensors

Concentration inequalities for random tensors

The Hanson-Wright Inequality for Random Tensors

Marchenko-Pastur law with relaxed independence conditions

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