Operator $\ell_p\to\ell_q$ norms of random matrices with iid entries
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929697535819776 |
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| author | Latała, Rafał Strzelecka, Marta |
| author_facet | Latała, Rafał Strzelecka, Marta |
| contents | We prove that for every $p,q\in[1,\infty]$ and every random matrix $X=(X_{i,j})_{i\le m, j\le n}$ with iid centered entries satisfying the regularity assumption $\|X_{i,j}\|_{2ρ} \le α\|X_{i,j}\|_ρ$ for every $ρ\ge 1$, the expectation of the operator norm of $X$ from $\ell_p^n$ to $\ell_q^m$ is comparable, up to a constant depending only on $α$, to \[ m^{1/q}\sup_{t\in B_p^n}\Bigl\|\sum_{j=1}^nt_jX_{1,j}\Bigr\|_{ q\wedge \operatorname{Log} m} +n^{1/p^*}\sup_{s\in B_{q^*}^m}\Bigl\|\sum_{i=1}^{m} s_iX_{i,1}\Bigr\|_{ p^*\wedge \operatorname{Log} n}. \] We give more explicit formulas, expressed as exact functions of $p$, $q$, $m$, and $n$, for the asymptotic operator norms in the case when the entries $X_{i,j}$ are: Gaussian, Weibullian, log-concave tailed, and log-convex tailed. In the range $1\le q\le 2\le p$ we provide two-sided bounds under a weaker regularity assumption $(\mathbb{E} X_{1,1}^4)^{1/4}\leq α(\mathbb{E} X_{1,1}^2)^{1/2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09814 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Operator $\ell_p\to\ell_q$ norms of random matrices with iid entries Latała, Rafał Strzelecka, Marta Probability Functional Analysis We prove that for every $p,q\in[1,\infty]$ and every random matrix $X=(X_{i,j})_{i\le m, j\le n}$ with iid centered entries satisfying the regularity assumption $\|X_{i,j}\|_{2ρ} \le α\|X_{i,j}\|_ρ$ for every $ρ\ge 1$, the expectation of the operator norm of $X$ from $\ell_p^n$ to $\ell_q^m$ is comparable, up to a constant depending only on $α$, to \[ m^{1/q}\sup_{t\in B_p^n}\Bigl\|\sum_{j=1}^nt_jX_{1,j}\Bigr\|_{ q\wedge \operatorname{Log} m} +n^{1/p^*}\sup_{s\in B_{q^*}^m}\Bigl\|\sum_{i=1}^{m} s_iX_{i,1}\Bigr\|_{ p^*\wedge \operatorname{Log} n}. \] We give more explicit formulas, expressed as exact functions of $p$, $q$, $m$, and $n$, for the asymptotic operator norms in the case when the entries $X_{i,j}$ are: Gaussian, Weibullian, log-concave tailed, and log-convex tailed. In the range $1\le q\le 2\le p$ we provide two-sided bounds under a weaker regularity assumption $(\mathbb{E} X_{1,1}^4)^{1/4}\leq α(\mathbb{E} X_{1,1}^2)^{1/2}$. |
| title | Operator $\ell_p\to\ell_q$ norms of random matrices with iid entries |
| topic | Probability Functional Analysis |
| url | https://arxiv.org/abs/2401.09814 |