  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.
