Sparse random tensors: Concentration, regularization and applications

Abstract

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 $p_{max}\ge \frac{clogn}{n}$, we show that $\Vert T-\mathbb{E}T\Vert =O(\sqrt{n{p}_{max}}{log}^{k-2}(n))$ 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 $p_{max}\ge \frac{clogn}{n^m}$ with $1\le m\le k-1$ and obtain concentration inequalities for different sparsity regimes. We also provide a simple way to regularize T such that $O(\sqrt{n^m{p}_{max}})$ concentration still holds down to sparsity $p_{max}\ge \frac{c}{n^m}$ with $k∕2\le m\le k-1$. 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.