• Corpus ID: 252531781

Extreme singular values of inhomogeneous sparse random rectangular matrices

@inproceedings{Dumitriu2022ExtremeSV,
  title={Extreme singular values of inhomogeneous sparse random rectangular matrices},
  author={Ioana Dumitriu and Yizhe Zhu},
  year={2022}
}

• Ioana Dumitriu, Yizhe Zhu
• Published 25 September 2022
• Mathematics, Computer Science

. 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 ℓ 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˝os-R´enyi bipartite graphs for a wide range of sparsity. In particular, for Erd˝os-R´enyi bipartite graphs G ( n, m, p ) with p = ω (log n ) /n , and m/n → y ∈ (0 , 1), our sharp bounds imply that there are no outliers outside the support of the Marˇcenko-Pastur law almost surely. This result extends the Bai-Yin theorem to sparse rectangular random matrices.