Nonlinear Mapping and Distance Geometry

Alain Franc 1, 2 Pierre Blanchard 2, 3 Olivier Coulaud 3
2 PLEIADE - from patterns to models in computational biodiversity and biotechnology
LaBRI - Laboratoire Bordelais de Recherche en Informatique, Inria Bordeaux - Sud-Ouest, BioGeCo - Biodiversité, Gènes & Communautés
3 HiePACS - High-End Parallel Algorithms for Challenging Numerical Simulations
LaBRI - Laboratoire Bordelais de Recherche en Informatique, Inria Bordeaux - Sud-Ouest
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.
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https://hal.inria.fr/hal-01897104
Contributor : Olivier Coulaud <>
Submitted on : Friday, May 3, 2019 - 2:40:42 PM
Last modification on : Monday, May 27, 2019 - 11:58:08 AM

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  • HAL Id : hal-01897104, version 2
  • ARXIV : 1810.08661

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Alain Franc, Pierre Blanchard, Olivier Coulaud. Nonlinear Mapping and Distance Geometry. [Research Report] RR-9210, Inria Bordeaux Sud-Ouest. 2018, pp.14. ⟨hal-01897104v2⟩

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