论文标题
不均匀稀疏随机矩形矩阵的极端奇异值
Extreme singular values of inhomogeneous sparse random rectangular matrices
论文作者
论文摘要
我们开发了一种统一的方法，以基于非折线操作员和Ihara-bass公式，以界定不均匀的随机矩形矩阵的最大和最小的奇异值，用于具有双部分块结构的一般随机遗传学矩阵。我们获得了大型矩形随机矩阵$ x $的最大（最小）奇异值的概率上限（分别为下）边界。这些界限是根据$ x $的行和方差配置文件的最大和最小$ \ ell_2 $ norms给出的。证明涉及在相关的非折线矩阵$ b $的光谱半径上找到概率上限。两侧边界可以应用于稀疏不均匀的Erdős-rényi二分子图的中心邻接矩阵，以达到广泛的稀疏性，直到关键。特别是，对于Erdős-rényi二分图$ g（n，m，p）$，带有$ p =ω（\ log n）/n $，以及$ m/n $，$ m/n \ to y in（0,1）$，我们的敏锐界限暗示，在Marmčenko-Pastur Law几乎没有肯定地支持Marmčenko-Pastur Law的支持之外，没有任何异常。该结果将bai-yin定理扩展到稀疏的矩形随机矩阵。
We develop a unified approach to bounding the largest and smallest singular values of an inhomogeneous random rectangular matrix, based on the non-backtracking operator and the Ihara-Bass formula for general random Hermitian matrices with a bipartite block structure. We obtain probabilistic upper (respectively, lower) bounds for the largest (respectively, smallest) singular values of a large rectangular random matrix $X$. These bounds are given in terms of the maximal and minimal $\ell_2$-norms of the rows and columns of the variance profile of $X$. The proofs involve finding probabilistic upper bounds on the spectral radius of an associated non-backtracking matrix $B$. The two-sided bounds can be applied to the centered adjacency matrix of sparse inhomogeneous Erdős-Rényi bipartite graphs for a wide range of sparsity, down to criticality. In particular, for Erdős-Rényi bipartite graphs $G(n,m,p)$ with $p=ω(\log n)/n$, and $m/n\to y \in (0,1)$, our sharp bounds imply that there are no outliers outside the support of the Marčenko-Pastur law almost surely. This result extends the Bai-Yin theorem to sparse rectangular random matrices.
