A Fast Matrix Majorization-Projection Method for Penalized Stress Minimization With Box Constraints

Abstract

Abstract-Kruskal's stress minimization, though nonconvex and nonsmooth, has been a major computational model for dissimilarity data in multidimensional scaling. Semidefinite Programming (SDP) relaxation (by dropping the rank constraint) would lead to a high number of SDP cone constraints. This has rendered the SDP approach computationally challenging even for problems of small size. In this paper, we reformulate the stress optimization as an Euclidean Distance Matrix (EDM) optimization with box constraints. A key element in our approach is the conditional positive semidefinite cone with rank cut. Although nonconvex, this geometric object allows a fast computation of the projection onto it and it naturally leads to a majorization-minimization algorithm with the minimization step having a closed-form solution. Moreover, we prove that our EDM optimization follows a continuously differentiable path, which greatly facilitated the analysis of the convergence to a stationary point. The superior performance of the proposed algorithm is demonstrated against some of the state-of-the-art solvers in the field of sensor network localization and molecular conformation.

Feasibility and a fast algorithm for Euclidean distance matrix optimization with ordinal constraints

“…Now we give the details of majorized penalty approach as shown in (3.2). Similar as in [35,Theorem 3.2], the majorized penalty approach enjoys the following convergence result.…”

“…To deal with the rank constraint, we make use of the majorized technique proposed in [35,36], which is detailed below.…”

“…where ρ > 0 is the penalty parameter. As in [35,16], we design a majorization function of g(D) in the following way. Recall that a majorization function of g(D) at D k ∈ S n , denoted as g m (D, D k ), has to satisfy following conditions…”

“…In our following test, we only consider the situation without anchors, that is, m = 0. ∆ is generated in the same way as [35,1]. Specifically, the generation of the n sensors (x 1 , .…”

“…where R is known as the radio range, ij are independent standard normal random variables and nf is the noise factor (see [35]). This type of perturbation in δ ij is known to be multiplicative and follows the unit-ball rule in defining N x .…”