We prove a non-asymptotic concentration inequality for the spectral norm of sparse inhomogeneous random tensors with Bernoulli entries. For an order-k inhomogeneous random tensor T with sparsity , we show that with high probability. The optimality of this bound up to polylog factors is provided by an information theoretic lower bound. By tensor unfolding, we extend the range of sparsity to with and obtain concentration inequalities for different sparsity regimes. We also provide a simple way to regularize T such that concentration still holds down to sparsity with . We present our concentration and regularization results with two applications: (i) a randomized construction of hypergraphs of bounded degrees with good expander mixing properties, (ii) concentration of sparsified tensors under uniform sampling.
Y.Z. is partially supported by NSF DMS-1949617.
We thank anonymous referees for their detailed comments and suggestions, which have improved the quality of this paper. We also thank Arash A. Amini, Nicholas Cook, Ioana Dumitriu, Kameron Decker Harris, and Roman Vershynin for helpful comments.
"Sparse random tensors: Concentration, regularization and applications." Electron. J. Statist. 15 (1) 2483 - 2516, 2021. https://doi.org/10.1214/21-EJS1838