## 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 signalto- 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.

Original language | English (US) |
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Pages (from-to) | 5753-5780 |

Number of pages | 28 |

Journal | Annals of Applied Probability |

Volume | 33 |

Issue number | 6 B |

DOIs | |

State | Published - Dec 2023 |

## Keywords

- Random matrices
- tensor PCA

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty