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Nonlinear mapping and distance geometry

Abstract

Distance Geometry Problem (DGP) and Nonlinear Mapping (NLM) are two well established questions: DGP 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.

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Fig. 1

Notes

  1. 1.

    In such a case, strictly speaking, the matrix built from the pairwise distances is not a Gram matrix.

  2. 2.

    It is unfortunate that least-square scaling has been proposed with the same name MDS than classical MDS. However, classical texts such as [1, 6, 13] are clear on this matter and agree on setting the vocabulary.

  3. 3.

    Here, it is known as part of the problem that the distances are taken between points living in a Euclidean space, whereas in NLM, such an hypothesis is not required.

  4. 4.

    Rigorously, one should replace \(|X'-X|<\epsilon \) by: there exists an \(X'\) in the set of solutions for \(\varOmega '\) such that \(|X'-X|<\epsilon \).

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Correspondence to Alain Franc.

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Franc, A., Blanchard, P. & Coulaud, O. Nonlinear mapping and distance geometry. Optim Lett 14, 453–467 (2020). https://doi.org/10.1007/s11590-019-01431-y

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Keywords

  • Distance Geometry Problem
  • Nonlinear Mapping
  • Least Square Scaling