Norms of structured random matrices
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866929592763154432 |
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| author | Adamczak, Radosław Prochno, Joscha Strzelecka, Marta Strzelecki, Michał |
| author_facet | Adamczak, Radosław Prochno, Joscha Strzelecka, Marta Strzelecki, Michał |
| contents | For $m,n\in\mathbb{N}$ let $X=(X_{ij})_{i\leq m,j\leq n}$ be a random matrix, $A=(a_{ij})_{i\leq m,j\leq n}$ a real deterministic matrix, and $X_A=(a_{ij}X_{ij})_{i\leq m,j\leq n}$ the corresponding structured random matrix. We study the expected operator norm of $X_A$ considered as a random operator between $\ell_p^n$ and $\ell_q^m$ for $1\leq p,q \leq \infty$. 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 $ψ_r$ ($r\in(0,2]$) 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 $A\circ A$ and $(A\circ A)^T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_14413 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Norms of structured random matrices Adamczak, Radosław Prochno, Joscha Strzelecka, Marta Strzelecki, Michał Probability Functional Analysis Primary 60B20, Secondary 46B09, 52A23, 60G15, 60E15 For $m,n\in\mathbb{N}$ let $X=(X_{ij})_{i\leq m,j\leq n}$ be a random matrix, $A=(a_{ij})_{i\leq m,j\leq n}$ a real deterministic matrix, and $X_A=(a_{ij}X_{ij})_{i\leq m,j\leq n}$ the corresponding structured random matrix. We study the expected operator norm of $X_A$ considered as a random operator between $\ell_p^n$ and $\ell_q^m$ for $1\leq p,q \leq \infty$. 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 $ψ_r$ ($r\in(0,2]$) 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 $A\circ A$ and $(A\circ A)^T$. |
| title | Norms of structured random matrices |
| topic | Probability Functional Analysis Primary 60B20, Secondary 46B09, 52A23, 60G15, 60E15 |
| url | https://arxiv.org/abs/2112.14413 |