  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.
