On the Expectation of Operator Norms of Random Matrices

Guédon, Olivier; Hinrichs, Aicke; Litvak, Alexander E.; Prochno, Joscha

doi:10.1007/978-3-319-45282-1_10

Cited by 10 publications

(23 citation statements)

References 27 publications

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“…In [23], Guédon, Hinrichs, Litvak, and Prochno studied our main and motivating question on the order of the expected operator norm of structured random matrices considered as operators between ℓ n p and ℓ m q in the special case where p ≤ 2 ≤ q and the random entries are Gaussian. In this situation, where we are not dealing with the spectral norm, the moment method cannot be employed.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In this situation, where we are not dealing with the spectral norm, the moment method cannot be employed. The approach in [23] was therefore different and based on a majorizing measure construction combining the works [24] and [25]. In [23,Theorem 1.1], the authors proved that if…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The approach in [23] was therefore different and based on a majorizing measure construction combining the works [24] and [25]. In [23,Theorem 1.1], the authors proved that if…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…where γ r := (E|g| r ) 1/r for a standard Gaussian random variable g and p * denotes the Hölder conjugate of p defined by the relation 1/p + 1/p * = 1. Moreover, for p = 1 and q ≥ 2, it was noted in [23,Remark 1.4] (see also [45,Twierdzenie 2]) that…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Norms of structured random matrices

Adamczak¹,

Prochno²,

Strzelecka³

et al. 2021

Preprint

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For m, n ∈ N let X = (X ij ) i≤m,j≤n be a random matrix, A = (a ij ) i≤m,j≤n a real deterministic matrix, and X A = (a ij X ij ) i≤m,j≤n the corresponding structured random matrix. We study the expected operator norm of X A considered as a random operator between ℓ n p and ℓ m q for 1 ≤ p, q ≤ ∞. We prove optimal bounds up to logarithmic terms when the underlying random matrix X has i.i.d. Gaussian entries, independent mean-zero bounded entries, or independent mean-zero ψr (r ∈ (0, 2]) entries. In certain cases, we determine the precise order of the expected norm up to constants. Our results are expressed through a sum of operator norms of Hadamard products A • A and (A • A) T .

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The approach in [23] was therefore different and based on a majorizing measure construction combining the works [24] and [25]. In [23,Theorem 1.1], the authors proved that if…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Norms of structured random matrices

Adamczak¹,

Prochno²,

Strzelecka³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…nu. In [6] Guédon, Hinrichs, Litvak, and Prochno proved that for a structured Gaussian matrix X " pa ij X ij q iďm,jďn and p, q ě 2,…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Estimates of norms of log-concave random matrices with dependent entries

Strzelecka¹

2019

Electron. J. Probab.

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We prove estimates for E}X : ℓ n p 1 Ñ ℓ m q } for p, q ě 2 and any random matrix X having the entries of the form aijYij, where Y " pYij q1ďiďm,1ďjďn has i.i.d. isotropic log-concave rows. This generalises the result of Guédon, Hinrichs, Litvak, and Prochno for Gaussian matrices with independent entries. Our estimate is optimal up to logarithmic factors. As a byproduct we provide the analogue bound for mˆn random matrices, which entries form an unconditional vector in R mn . We also prove bounds for norms of matrices which entries are certain Gaussian mixtures.

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Norms of structured random matrices

et al. 2023

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For $$m,n\in \mathbb {N}$$ m , n ∈ N , let $$X=(X_{ij})_{i\le m,j\le n}$$ X = ( X ij ) i ≤ m , j ≤ n be a random matrix, $$A=(a_{ij})_{i\le m,j\le n}$$ A = ( a ij ) i ≤ m , j ≤ n a real deterministic matrix, and $$X_A=(a_{ij}X_{ij})_{i\le m,j\le n}$$ X A = ( a ij X ij ) i ≤ m , j ≤ n the corresponding structured random matrix. We study the expected operator norm of $$X_A$$ X A considered as a random operator between $$\ell _p^n$$ ℓ p n and $$\ell _q^m$$ ℓ q m for $$1\le p,q \le \infty $$ 1 ≤ p , q ≤ ∞ . We prove optimal bounds up to logarithmic terms when the underlying random matrix X has i.i.d. Gaussian entries, independent mean-zero bounded entries, or independent mean-zero $$\psi _r$$ ψ r ($$r\in (0,2]$$ r ∈ ( 0 , 2 ] ) entries. In certain cases, we determine the precise order of the expected norm up to constants. Our results are expressed through a sum of operator norms of Hadamard products $$A\circ A$$ A ∘ A and $$(A\circ A)^T$$ ( A ∘ A ) T .

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On the Expectation of Operator Norms of Random Matrices

Abstract: Abstract. We prove estimates for the expected value of operator norms of Gaussian random matrices with independent and mean-zero entries, acting as operators from ℓ n p * to ℓ n q , 1 ≤ p * ≤ 2 ≤ q ≤ ∞.

Cited by 10 publications

References 27 publications

Norms of structured random matrices

Norms of structured random matrices

Estimates of norms of log-concave random matrices with dependent entries

Norms of structured random matrices

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