We study the structure of $$n \times n$$ n × n random matrices with centered i.i.d. entries having only two finite moments. In the recent joint work with R. Vershynin, we have shown that the operator norm of such matrix A can be reduced to the optimal order $$O(\sqrt{n})$$ O ( n ) with high probability by zeroing out a small submatrix of A , but did not describe the structure of this “bad” submatrix nor provide a constructive way to find it. In the current paper, we give a very simple description of a small “bad” subset of entries. We show that it is enough to zero out a small fraction of the rows and columns of A with largest $$L_2$$ L 2 norms to bring the operator norm of A to the almost optimal order $$O(\sqrt{n \log \log n})$$ O ( n log log n ) , under additional assumption that the matrix entries are symmetrically distributed. As a corollary, we also obtain a constructive procedure to find a small submatrix of A that one can zero out to achieve the same norm regularization. The main component of the proof is the development of techniques extending constructive regularization approaches known for the Bernoulli matrices (from the works of Feige and Ofek, and Le, Levina and Vershynin) to the considerably broader class of heavy-tailed random matrices.

中文翻译：

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随机矩阵范数的构造正则化

我们研究了 $$n \times n$$ n × n 个随机矩阵的结构，其中中心 iid 条目只有两个有限矩。在最近与 R. Vershynin 的联合工作中，我们已经证明可以通过清零以高概率将此类矩阵 A 的算子范数降低到最优阶 $$O(\sqrt{n})$$O (n) A 的一个小子矩阵，但没有描述这个“坏”子矩阵的结构，也没有提供找到它的建设性方法。在当前的论文中，我们对一个小的“坏”条目子集进行了非常简单的描述。我们表明，将 A 的具有最大 $$L_2$$L 2 范数的一小部分行和列归零就足以使 A 的算子范数达到几乎最优的阶 $$O(\sqrt{n \ log \log n})$$ O ( n log log n ) ，另外假设矩阵条目是对称分布的。作为推论，我们还获得了一个构造过程来找到 A 的一个小子矩阵，可以将其归零以实现相同的范数正则化。证明的主要组成部分是将伯努利矩阵（来自 Feige 和 Ofek，以及 Le、Levina 和 Vershynin 的作品）已知的建设性正则化方法扩展到更广泛的重尾随机矩阵类别的技术的发展。