Spiked Singular Values and Vectors under Extreme Aspect Ratios

Understand

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: (1) the classical regime, with a fixed number of rows and large number of columns, or vice versa, and (2) the proportional regime, with large numbers of rows and columns, proportional to one another.
• This paper is concerned with the 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.