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Session II.5 - Random Matrices

Friday, June 16, 18:00 ~ 18:30

Extreme singular values of inhomogeneous, sparse, rectangular random matrices

Ioana Dumitriu

University of California, San Diego, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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/lower bounds for the largest/smallest singular values of a large rectangular random matrix $X$. These bounds are given in terms of the maximal and minimal $\mathcal{l}_2$-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 Erdos-Renyi bipartite graphs for a wide range of sparsity. In particular, for Erdos-Renyi bipartite graphs $G(n,m,p)$ with $p=\omega(logn)/n$, and $m/n \rightarrow y \in (0,1)$, our sharp bounds imply that there are no outliers outside the support of the Marcenko-Pastur law almost surely.

Joint work with Yizhe Zhu (University of California, Irvine).

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