When random tensors meet random matrices

Abstract

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 (In Proc. IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (2005) 129–132), we show that the analysis of the considered model boils down to the analysis of an equivalent spiked symmetric blockwise random matrix, that is constructed from 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 $\frac{{\mathit{n}}_{\mathit{i}}}{{\sum }_{\mathit{j}=1}^{\mathit{d}}{\mathit{n}}_{\mathit{j}}}\to {\mathit{c}}_{\mathit{i}}\in (0,1)$ with ${\mathit{n}}_{\mathit{i}}$’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.