Sparse random tensors: Concentration, regularization and applications

TL;DR

This paper establishes non-asymptotic concentration inequalities for the spectral norm of sparse inhomogeneous random tensors, extending results to various sparsity regimes and providing regularization methods with applications in hypergraph construction and tensor sampling.

Contribution

It introduces new concentration bounds for sparse random tensors, extends these bounds to different sparsity regimes via tensor unfolding, and proposes regularization techniques applicable in hypergraph and tensor sampling contexts.

Findings

Spectral norm concentration bound: O(√(n p_max) log^{k-2}(n))

Extended concentration inequalities for various sparsity regimes

Regularization method maintains concentration at lower sparsity levels

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 $p_{\max}\geq \frac{c\log n}{n }$, we show that $\|T-\mathbb E T\|=O(\sqrt{n p_{\max}}\log^{k-2}(n))$ 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 $p_{\max}\geq \frac{c\log n}{n^{m}}$ with $1\leq m\leq k-1$ 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 $p_{\max}\geq \frac{c}{n^{m}}$ with $k/2\leq m\leq k-1$. 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.