  Factor models are a very efficient way to describe high dimensional vectors
of data in terms of a small number of common relevant factors. This problem,
which is of fundamental importance in many disciplines, is usually reformulated
in mathematical terms as follows. We are given the covariance matrix Sigma of
the available data. Sigma must be additively decomposed as the sum of two
positive semidefinite matrices D and L: D | that accounts for the idiosyncratic
noise affecting the knowledge of each component of the available vector of data
| must be diagonal and L must have the smallest possible rank in order to
describe the available data in terms of the smallest possible number of
independent factors.
  In practice, however, the matrix Sigma is never known and therefore it must
be estimated from the data so that only an approximation of Sigma is actually
available. This paper discusses the issues that arise from this uncertainty and
provides a strategy to deal with the problem of robustly estimating the number
of factors.
