Sparse factor analysis via likelihood and ℓ1-regularization

In this note we consider the basic problem to identify linear relations in noise. We follow the viewpoint of factor analysis (FA) where the data is to be explained by a small number of independent factors and independent noise. Thereby an approximation of the sample covariance is sought which can be factored accordingly. An algorithm is proposed which weighs in an ℓ₁-regularization term that induces sparsity of the linear model (factor) against a likelihood term that quantifies distance of the model to the sample covariance. The algorithm compares favorably against standard techniques of factor analysis. Their performance is compared first by simulation, where ground truth is available, and then on stock-market data where the proposed algorithm gives reasonable and sparser models.