Spiked Singular Values and Vectors under Extreme Aspect Ratios

The behavior of the leading singular values and vectors of noisy low-rank matrices is fundamental to many statistical and scientific problems. Theoretical understanding currently derives from asymptotic analysis under one of two regimes: {\it classical}, with a fixed number of rows, large number of columns or vice versa; and {\it proportional}, with large numbers of rows and columns, proportional to one another. This paper is concerned with the {\it disproportional} regime, where the matrix is either `tall and narrow' or `short and wide': we study sequences of matrices of size $n \times m_n$ with aspect ratio $n/m_n \rightarrow 0$ or $n/m_n \rightarrow \infty$ as $n \rightarrow \infty$. This regime has important `big data' applications. Theory derived here shows that the displacement of the empirical singular values and vectors from their noise-free counterparts and the associated phase transitions -- well-known under proportional growth asymptotics -- still occur in the disproportionate setting. They must be quantified, however, on a novel scale of measurement that adjusts with the changing aspect ratio as the matrix size increases. In this setting, the top singular vectors corresponding to the longer of the two matrix dimensions are asymptotically uncorrelated with the noise-free signal.