Nonlinear Mapping and Distance Geometry

TL;DR

This paper unifies Distance Geometry Problem and Nonlinear Mapping within a common optimization framework, analyzing solution continuity, compactness, and the challenges of numerical solutions with potential local minima.

Contribution

It introduces a unified framework for DGP and NLM as optimization problems, exploring solution properties and numerical difficulties.

Findings

Solutions form a continuous spectrum as weight matrices vary.

The set of solutions is compact after centering.

Numerical optimization can be trapped in local minima.

Abstract

Distance Geometry Problem (DGP) and Nonlinear Mapping (NLM) are two well established questions: Distance Geometry Problem is about finding a Euclidean realization of an incomplete set of distances in a Euclidean space, whereas Nonlinear Mapping is a weighted Least Square Scaling (LSS) method. We show how all these methods (LSS, NLM, DGP) can be assembled in a common framework, being each identified as an instance of an optimization problem with a choice of a weight matrix. We study the continuity between the solutions (which are point clouds) when the weight matrix varies, and the compactness of the set of solutions (after centering). We finally study a numerical example, showing that solving the optimization problem is far from being simple and that the numerical solution for a given procedure may be trapped in a local minimum.