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 Hermitian matrices with a bipartite block structure. Our main results are 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 Erd\H{o}s-R\'{e}nyi bipartite graphs for a wide range of sparsity. In particular, for Erd\H{o}s-R\'{e}nyi bipartite graphs $\mathcal G(n,m,p)$ with $p=\omega(\log n)/n$, and $m/n\to y \in (0,1)$, our sharp bounds imply that there are no outliers outside the support of the Mar\v{c}enko-Pastur law almost surely. This result extends the Bai-Yin theorem to sparse rectangular random matrices.

翻译：我们根据非回溯跟踪操作员和Ihara-Bass公式,为具有双面块结构的通用 Hermitian 矩阵制定统一的方法,将最大(分别为最小的)大矩形随机矩阵的最大(分别为最小的)单数值捆绑起来。这些界限是以美元差异剖面的最大值和最小值 $\ ell_ 2美元为单位。 证据涉及在相关非回溯矩阵矩阵的光谱半径上找到几乎具有稳定性的上界值 $B$。 双面界限可以应用到极不相近的大型矩形(分别为最小的、最小的、最小的)上界值。 这些边框以最大值和最小值的双面图表为单位, 用于范围很广的 。 特别是, Erd\ H 美元- R\ cal_ 美元, rentral\ mal\ mural\ 外的双面框值结果。