Estimation of Missing Entries in Preference Pairwise Matrices Through Nuclear Norm Minimization

Abstract

This paper aims to evaluate the possibility of applying the nuclear norm minimization technique to the problem of incompleteness of an additive preference pairwise matrix. The proposed technique applies to the case of missing values in any position within a matrix of any order, provided that all the alternatives are connected, or equivalently that there is a preference path between each pair of alternatives. The unknown entries are estimated by finding a matrix that best matches the given entries, assuming that the reconstructed matrix can be well approximated by a lower-rank matrix and that the entries are missing completely at random. A series of applications with simulated data were used to evaluate the validity of the proposed technique, particularly in comparison to another popular technique of missing value estimation based on the minimization of an index measuring matrix inconsistency. The simulation results suggest the effectiveness of nuclear norm minimization as a completion technique in some of the considered setting, in particular if violations of strong transitivity property are considered.