Singular value decomposition of large random matrices (for two-way classification of microarrays)

TL;DR

This paper studies the asymptotic behavior of singular values in large random matrices with noise, providing theoretical insights and an algorithm for two-way microarray classification.

Contribution

It introduces new asymptotic results for singular values of large noisy matrices and presents an algorithm for microarray two-way classification.

Findings

Large singular values are of order √mn, others are of order √(m+n).

Almost sure concentration results for singular values and subspaces.

An effective algorithm for microarray two-way classification.

Abstract

Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to Wigner-noise is investigated.It is proved that such an m\times n matrix almost surely has a constant number of large singular values (of order \sqrt{mn}), while the rest of the singular values are of order \sqrt{m+n} as m,n\to\infty. Concentration results of Alon et al. for the eigenvalues of large symmetric random matrices are adapted to the rectangular case, and on this basis, almost sure results for the singular values as well as for the corresponding isotropic subspaces are proved. An algorithm, applicable to two-way classification of microarrays, is also given that finds the underlying block structure.