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

摘要

Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to random noise is investigated. It is proved that such an m × n random matrix almost surely has a constant number of large singular values (of order sqrt(m n)), while the rest of the singular values are of order sqrt(m + n) as m, n → ∞. We prove almost sure properties for the corresponding isotropic subspaces and for noisy correspondence matrices. An algorithm, applicable to two-way classification of microarrays, is also given that finds the underlying block structure.