Principal coordinates analysis with optimal transformation of the variables: Minimizing the sum of squares of the smallest eigenvalues

Principal coordinates analysis (PCO) is a technique proposed by Gower (1966) to analyse a multivariate data matrix, representing the distances between the rows (the objects) by distances between object points in a low‐dimensional space, through an approximation using scalar products. For Euclidean distances, this paper generalizes the technique to a nonlinear variety, i.e. nonlinear transformations of the variables (the columns of the data matrix) are incorporated. These transformations are optimal according to the scalar product criterion that is minimized. This criterion can be re‐expressed in terms of the eigenvalues of the correlation matrix for the variables: nonlinear principal coordinates analysis then minimizes the sum of squares of a pre‐chosen number of small eigenvalues. The optimal nonlinear transformations are obtained by using iterative majorization.