Covariance Matrix Estimation Under Low-Rank Factor Model With Nonnegative Correlations

Inferring the covariance matrix of multivariate data is of great interest in statistics, finance, and data science. It is often carried out via the maximum likelihood estimation (MLE) principle, which seeks a covariance matrix estimator maximizing the observed data likelihood. However, such estimator is usually poor when number of samples is not sufficiently larger than the number of variables. With the assumption that a covariance matrix can be decomposed into a low-rank matrix and a diagonal matrix, factor analysis (FA) model is a popular dimensionality reduction technique in improving the estimation performance. Recently, more and more evidence shows that the covariance matrix of real-valued data may admit the nonnegative correlation structure, which has attracted a lot of interest in some areas like finance and psychometrics. There does not exist any work estimating the covariance matrix simultaneously satisfying both structures. In this paper, we propose an MLE problem formulation for covariance matrix considering jointly the low-rank FA model and nonnegative correlation structures. Since the proposed problem formulation is an intractable non-convex problem, a block coordinate descent algorithm is further proposed to solve a relaxed version of our proposed formulation. The superior performance of our proposed formulation and the algorithm are verified through numerical simulations on both synthetic data and real market data.