A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation
This provides theoretical insights for tensor approximation methods, which is incremental for researchers in machine learning and signal processing.
The paper tackles the problem of estimating a low-rank signal from a spiked tensor model near the computational threshold, using random matrix theory to analyze spectral behavior and predict reconstruction performance of truncated multilinear SVD, showing that the number of iterations for HOOI convergence tends to 1 in large dimensions.
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.