Quantum mechanics

Singular integral

Algorithm

Random variable

Unit vector

Singular function

Statistics

Materials science

Applied mathematics

Rank (graph theory)

Distribution (mathematics)

Eigenvalues and eigenvectors

Random matrix

Outlier

Matrix (chemical analysis)

Physics

Singular value decomposition

Singular solution

Mathematics

Mathematical analysis

Composite material

Combinatorics

Subspace topology

Multivariate random variable

Integral equation

Singular value

Singular spectrum analysis

The Annals of Statistics

In this paper, we study the matrix denoising model $Y=S+X$, where $S$ is a low rank deterministic signal matrix and $X$ is a random noise matrix, and both are $M\times n$. In the scenario that $M$ and $n$ are comparably large and the signals are supercritical, we study the fluctuation of the outlier singular vectors of $Y$, under fully general assumptions on the structure of $S$ and the distribution of $X$. More specifically, we derive the limiting distribution of angles between the principal singular vectors of $Y$ and their deterministic counterparts, the singular vectors of $S$. Further, we also derive the distribution of the distance between the subspace spanned by the principal singular vectors of $Y$ and that spanned by the singular vectors of $S$. It turns out that the limiting distributions depend on the structure of the singular vectors of $S$ and the distribution of $X$, and thus they are nonuniversal. Statistical applications of our results to singular vector and singular subspace inferences are also discussed.

Singular vector and singular subspace distribution for the matrix denoising model