Long Random Matrices and Tensor Unfolding

Gérard Ben Arous; Daniel Zhengyu Huang; Jiaoyang Huang

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arXiv · 2110.10210

Long Random Matrices and Tensor Unfolding

Gérard Ben Arous·Daniel Zhengyu Huang·Jiaoyang Huang

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Abstract

In this paper, we consider the singular values and singular vectors of low rank perturbations of large rectangular random matrices, in the regime the matrix is "long": we allow the number of rows (columns) to grow polynomially in the number of columns (rows). We prove there exists a critical signal-to-noise ratio (depending on the dimensions of the matrix), and the extreme singular values and singular vectors exhibit a BBP type phase transition. As a main application, we investigate the tensor unfolding algorithm for the asymmetric rank-one spiked tensor model, and obtain an exact threshold, which is independent of the procedure of tensor unfolding. If the signal-to-noise ratio is above the threshold, tensor unfolding detects the signals; otherwise, it fails to capture the signals.

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BibTeX RIS

Gérard Ben Arous, Daniel Zhengyu Huang, Jiaoyang Huang. 2021-10-19. Long Random Matrices and Tensor Unfolding. https://arxiv.org/abs/2110.10210

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