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.
翻译:在本文中,我们考虑了大型矩形随机矩阵低级别扰动的单值和单向量,在制度内,矩阵是“长”的:我们允许列数(行数)在列数(行数)中多行增长。我们证明存在一个关键的信号对噪音比率(取决于矩阵的维度),极端的单值和单向量显示了BBPP类型过渡阶段。作为一个主要应用,我们调查了不对称的单级单级激增的沙诺模式的演进算法,并获得了一个精确的阈值,这个阈值独立于发光过程。如果信号对噪声比率高于临界值,则发光时检测信号;否则,它无法捕捉信号。