Relying on random matrix theory (RMT), this paper studies asymmetric order-$d$ spiked tensor models with Gaussian noise. Using the variational definition of the singular vectors and values of [Lim, 2005], we show that the analysis of the considered model boils down to the analysis of an equivalent spiked symmetric \textit{block-wise} random matrix, that is constructed from \textit{contractions} of the studied tensor with the singular vectors associated to its best rank-1 approximation. Our approach allows the exact characterization of the almost sure asymptotic singular value and alignments of the corresponding singular vectors with the true spike components, when $\frac{n_i}{\sum_{j=1}^d n_j}\to c_i\in [0, 1]$ with $n_i$'s the tensor dimensions. In contrast to other works that rely mostly on tools from statistical physics to study random tensors, our results rely solely on classical RMT tools such as Stein's lemma. Finally, classical RMT results concerning spiked random matrices are recovered as a particular case.
翻译:根据随机矩阵理论(RMT),本文研究使用高森噪声的不对称定值-美元峰值高尔夫模型。使用[Lim, 2005] 单向量和值的变异定义,我们显示,对考虑的模型的分析归结为对等的峰值对齐性对称矩阵分析,该模型的构造来自所研究的与与其最高一级近似值相关的单一矢量的单向量阵列(RMT),我们的方法允许对几乎肯定的单向量单向值和对应的单向量与真正峰值组成部分的匹配进行精确定性,而当 $\frac{n_in_isum_j ⁇ j=1 ⁇ d n_j ⁇ toc_i\in [0,1]$n_i_i]$n_is destrongor维度时,与主要依赖统计物理工具研究随机发量的其他工作相比,我们的结果只依赖于Stein's Lemma等典型RMT工具。最后,关于钉压随机矩阵的典型RMT结果作为特定案例被恢复。