Preorderings, monotone functions, and best rank \(r\) approximations with applications to classical MDS (Q1329694)

Let \(F_{n,k}\) denote the linear space of \(n\times k\)-matrices, \(1\leq k\leq n\), over the complex field. Based upon the singular-value theorem concerning the decomposition of a matrix \(A\) from \(F_{n,k}\), minimum norm rank \(r\), \(1\leq r<k\), approximations \(A_{(r)}\) satisfying \[ \psi(A- A_{(r)})\leq \psi(A- G) \qquad \text{for all } G\in F_{n,k} \] have been obtained in the past, first for \(\psi\) being the Euclidean norm and then for all unitarily invariant norms on the space \(F_{n,k}\). In the present paper, these results are extended to a class of monotone functions \(\psi\) with respect to a certain preordering on \(F_{n,k}\). In a second main part, the authors turn to approximating Hermitian matrices by elements of the cone of positive semidefinite matrices of rank less than or equal to \(r\) and its application in multidimensional scaling (MDS). Finally, universally optimal properties of the MDS solution are provided.