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Pré-Publication, Document De Travail Année : 2024

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

Résumé

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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hal-04673321 , version 1 (20-08-2024)

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  • HAL Id : hal-04673321 , version 1

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Hugo Lebeau, Florent Chatelain, Romain Couillet, Romain Couillet. A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation. 2024. ⟨hal-04673321⟩
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