Norms of structured random matrices.

IF 1.3 2区数学 Q1 MATHEMATICS

Mathematische Annalen Pub Date : 2024-01-01 Epub Date: 2023-04-09 DOI:10.1007/s00208-023-02599-6

Radosław Adamczak, Joscha Prochno, Marta Strzelecka, Michał Strzelecki

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引用次数: 0

Abstract

For $m, n \in N$ , 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 $ℓ_{p}^{n}$ and $ℓ_{q}^{m}$ 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 ${(A \circ A)}^{T}$ .

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结构化随机矩阵的范数

对于 m , n∈ N，设 X = ( X ij ) i ≤ m , j ≤ n 为随机矩阵，A = ( a ij ) i ≤ m , j ≤ n 为实数确定矩阵，X A = ( a ij X ij ) i ≤ m , j ≤ n 为相应的结构随机矩阵。我们研究在 1 ≤ p , q ≤ ∞ 时，将 X A 视为 ℓ p n 和 ℓ q m 之间的随机算子的期望算子规范。当底层随机矩阵 X 具有 i.i.d. 高斯条目、独立均值为零的有界条目或独立均值为零的ψ r ( r ∈ ( 0 , 2 ] ) 条目时，我们将证明对数项以内的最优边界。在某些情况下，我们可以确定精确到常数的期望规范阶数。我们的结果是通过 Hadamard 乘积 A ∘ A 和 ( A ∘ A ) T 的算子规范之和表达的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematische Annalen 数学-数学

CiteScore

2.90

自引率

7.10%

发文量

181

审稿时长

4-8 weeks

期刊介绍： Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.