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

• URL: http://arxiv.org/abs/2402.03169v2
• Date: Thu, 6 Jun 2024 12:22:18 GMT
• Title: A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation
• Authors: Hugo Lebeau, Florent Chatelain, Romain Couillet,
• Abstract summary: 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. Results allow to accurately predict the reconstruction performance of truncated multilinear SVD (MLSVD) in the non-trivial regime.
• Score: 24.558241146742205
• License: http://creativecommons.org/licenses/by/4.0/
• 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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