A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation
Hugo Lebeau Florent Chatelain Romain Couillet
This work presents a comprehensive understanding of the estimation of a
planted low-rank signal from a general spiked tensor model near the
computational threshold. Relying on standard tools from the theory of large
random matrices, we characterize the large-dimensional spectral behavior of the
unfoldings of the data tensor and exhibit relevant signal-to-noise ratios
governing the detectability of the principal directions of the signal. These
results allow to accurately predict the reconstruction performance of truncated
multilinear SVD (MLSVD) in the non-trivial regime. This is particularly
important since it serves as an initialization of the higher-order orthogonal
iteration (HOOI) scheme, whose convergence to the best low-multilinear-rank
approximation depends entirely on its initialization. We give a sufficient
condition for the convergence of HOOI and show that the number of iterations
before convergence tends to $1$ in the large-dimensional limit.