A Riemannian inexact Newton method for solving the orthogonal INDSCAL problem in multidimensional scaling

Abstract

The well-known individual differences scaling (INDSCAL) model is intended for simultaneous metric multidimensional scaling (MDS) of several doubly centered matrices of squared dissimilarities. In this work the problem of fitting the orthogonal INDSCAL model to the data is reformulated and studied as a matrix optimization problem on the product manifold of orthonormal and diagonal matrices. A Riemannian inexact Newton method is proposed to address the underlying problem, with the global and quadratic convergence of the proposed method established under some mild assumptions. Furthermore, the positive definiteness condition of the Riemannian Hessian of the objective function at a solution is derived. Some numerical experiments are provided to illustrate the efficiency of the proposed method.