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.
翻译:我们证明,对于Bernoulli 条目的稀少无色随机发热器的光谱规范来说,我们是一种非非非不显性浓度的不平等。对于一个以纯度为单位的无色随机发热器,只要以纯度为单位,我们证明$T-\max ⁇ QQQQQQqq\\\frac{c\c\c\logn\grogn}美元为单位,我们证明$T-\mathbrbE T ⁇ O(\ sqrt{n\maxlogQQk-2}(n) 可能性很大。这种与多元系数因素结合的最佳性是由一个信息理论性较低约束提供的。对于一个信息,我们用纯度为单位的无色度范围扩大到$maxqQQQQQQQQQQQ\\ mqrqr=crum$rqrum roupulationalationalationality $ration=qleqleqrqr=MQQQQQQQQQrqr=m=maxr=max roupal resental resental resental restime restiax rodudududududududududududul=x=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx=======================================m=========m=m=m=m=mqcal========m======m====mcalcrocal