论文标题
在稀疏矩形随机矩阵的最大和最小的奇异值上
On the Largest and the Smallest Singular Value of Sparse Rectangular Random Matrices
论文作者
论文摘要
我们得出稀疏矩形$ n \ times n $随机矩阵的最大和最小的奇异值的估计值，假设$ \ lim_ {n，n \ to \ frac} \ frac nn = y \ in（0,1）$。我们考虑一个具有稀疏参数$ p_n $的模型，以便某些$α> 1 $ $ np_n \ sim \ sim \ log^{α} n $，并假设矩阵元素的矩满足条件$ \ mathbf e | mathbf e | x_ | x_ {jk} | x_ {jk} |^{4+δ} {4+δ} {4+δ} \ le c <\ c <\ c <\ c <\ f infty $。我们还假设，我们考虑的矩阵条目以$（np_n）^{\ frac12- \ varkappa} $截断为$ \ varkappa：= \fracδ{2（4+δ）} $。
We derive estimates for the largest and smallest singular values of sparse rectangular $N\times n$ random matrices, assuming $\lim_{N,n\to\infty}\frac nN=y\in(0,1)$. We consider a model with sparsity parameter $p_N$ such that $Np_N\sim \log^{α}N$ for some $α>1$, and assume that the moments of the matrix elements satisfy the condition $\mathbf E|X_{jk}|^{4+δ}\le C<\infty$. We assume also that the entries of matrices we consider are truncated at the level $(Np_N)^{\frac12-\varkappa}$ with $\varkappa:=\fracδ{2(4+δ)}$.
