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

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 .

show abstract

“…Later, an extension of (1.5) to the case of matrices with i.i.d. isotropic log-concave rows was obtained by Strzelecka in [55].…”

Section: History Of the Problem And Known Resultsmentioning

confidence: 99%

Norms of structured random matrices

et al. 2023

Self Cite

View full text Add to dashboard Cite

show abstract

Chevet-type inequalities for subexponential Weibull variables and estimates for norms of random matrices

Latała,

Strzelecka

2024

Electron. J. Probab.

View full text Add to dashboard Cite

Norms of structured random matrices

Adamczak¹,

Prochno²,

Strzelecka³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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 .

show abstract

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

Cited by 3 publications

References 16 publications

Norms of structured random matrices

Norms of structured random matrices

Chevet-type inequalities for subexponential Weibull variables and estimates for norms of random matrices

Norms of structured random matrices

Contact Info

Product

Resources

About