A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation
<|reference_start|>A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation: 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.<|reference_end|>
@article{lebeau2024a,
    title={A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation},
    author={Hugo Lebeau, Florent Chatelain, Romain Couillet},
    journal={arXiv preprint arXiv:2402.03169},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.03169},
    primaryClass={stat.ML cs.LG math.PR}
}