2021/10/19 by Gérard Ben Arous, Arous, Gérard Ben, Daniel Zhengyu Huang +3
Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Probability (math.PR) #Random Matrices and Applications #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2110.10210
openalex publication_date 2021/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.