Preorderings, monotone functions, and best rank r approximations with applications to classical MDS

Abstract

This paper extends the classical results of Eckart and Young (Psychometrika 1, 1936) and Mirsky (Quarterly Journal of Mathematics in Oxford, Series 2, 11, 1960) concerning the best rank r matrix approximation with respect to certain preorderings defined on the space of complex n x k matrices. A note on some related work of Jensen ( J. Stat. Sim. Com. 39 , No. 3, 1991) is included, his assertions, though, are shown to be flawed. In a second part we turn to the problem of approximating a Hermitian matrix by a positive semidefinite matrix of given rank which is of major relevance in the context of multidimensional scaling (MDS). Further universally optimal properties of the classical MDS solution are provided.