Factor analysis with finite data

Factor analysis aims to describe high dimensional random vectors by means of a small number of unknown common factors. In mathematical terms, it is required to decompose the covariance matrix Σ of the random vector as the sum of a diagonal matrix D — accounting for the idiosyncratic noise in the data — and a low rank matrix R — accounting for the variance of the common factors — in such a way that the rank of R is as small as possible so that the number of common factors is minimal. In practice, however, the matrix Σ is unknown and must be replaced by its estimate, i.e. the sample covariance, which comes from a finite amount of data. This paper provides a strategy to account for the uncertainty in the estimation of Σ in the factor analysis problem.