A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation

2024/02/05 by Hugo Lebeau, Lebeau, Hugo, Florent Chatelain +3
Engineering · Medicine · #Sparse and Compressive Sensing Techniques #Advanced Neuroimaging Techniques and Applications #Synthetic Aperture Radar (SAR) Applications and Techniques

paper · pdf · doi:10.48550/arxiv.2402.03169

Abstract

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.

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