Dimensionality reduction and volume minimization—generalization of the determinant minimization criterion for reduced rank regression problems

In this article we propose a generalization of the determinant minimization criterion. The problem of minimizing the determinant of a matrix expression has implicit assumptions that the objective matrix is always nonsingular. In case of singular objective matrix the determinant would be zero and the minimization problem would be meaningless. To be able to handle all possible cases we generalize the determinant criterion to rank reduction and volume minimization of the objective matrix. The generalized minimization criterion is used to solve the following ordinary reduced rank regression problem:minrank(X)=kdet(B-XA)(B-XA)T,minrank(X)=kdet(B-XA)(B-XA)T,where A and B are known and X is to be determined. This problem is often encountered in the system identification context.