Positive Semi-Definite Sparse Low-Rank Matrix Decomposition

Abstract

In recent years, various research fields have been incorporating machine learning techniques to interpret high-dimensional data. Often, most of the information in such datasets is contained in a small subspace of the data. General algorithms that find these subspaces are found but do not make use of prior knowledge of the problem. An example of a situation where we have prior knowledge about both the initial dataset and its relevant subset is found in Structural Equation Modeling, where both sets are represented by positive semi-definite matrices. This study aimed to improve the performance of these general algorithms on positive semi-definite matrices.

We first showed that the mathematical properties on which the general algorithms are based hold in our constrained problem. We then derived both the optimality conditions of our problem and a closed-form algorithm that can find the positive semi-definite decomposition. When comparing the efficiency of our algorithm to its general counterpart, the results showed that our constrained algorithm converges roughly 29% faster. This suggests that our algorithm performs significantly better on the space of positive semi-definite matrices than the general algorithm does.