  The singular value decomposition is widely used to approximate data matrices
with lower rank matrices. Feng and He [Ann. Appl. Stat. 3 (2009) 1634-1654]
developed tests on dimensionality of the mean structure of a data matrix based
on the singular value decomposition. However, the first singular values and
vectors can be driven by a small number of outlying measurements. In this
paper, we consider a robust alternative that moderates the effect of outliers
in low-rank approximations. Under the assumption of random row effects, we
provide the asymptotic representations of the robust low-rank approximation.
These representations may be used in testing the adequacy of a low-rank
approximation. We use oligonucleotide gene microarray data to demonstrate how
robust singular value decomposition compares with the its traditional
counterparts. Examples show that the robust methods often lead to a more
meaningful assessment of the dimensionality of gene intensity data matrices.
