Abstract
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
Funding Statement
I.D. and Y.Z. acknowledge support from NSF DMS-1928930 during their participation in the program Universality and Integrability in Random Matrix Theory and Interacting Particle Systems hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall semester of 2021. I.D. is partially supported by NSF DMS-2154099. Y.Z. is partially supported by NSF-Simons Research Collaborations on the Mathematical and Scientific Foundations of Deep Learning. This work was done in part while Y.Z. was visiting the Simons Institute for the Theory of Computing in the Fall of 2022.
Acknowledgements
We are grateful to Antti Knowles and Ramon van Handel for helpful discussions and to the latter for a detailed explanation of the work (Brailovskaya and van Handel, 2022). Y.Z. thanks Ludovic Stephan for a careful reading of this paper and many useful suggestions. We thank the editor and anonymous referees for helping us improve the presentation.
Citation
Ioana Dumitriu. Yizhe Zhu. "Extreme singular values of inhomogeneous sparse random rectangular matrices." Bernoulli 30 (4) 2904 - 2931, November 2024. https://doi.org/10.3150/23-BEJ1699
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