Building a Weighted Complex from Data



In some applications, both the neighborhood structure of the data and the weights of the cells are naturally and directly defined by the problem at hand (e.g., road networks, social networks, communication networks, chemical graph theory or surface simplification). However, in many other applications the appropriate representation of the data to be analyzed is not provided (e.g., machine learning). Therefore, to use the tools of discrete calculus, a practitioner must determine the topology and weights of the graph or complex from the data that is most appropriate for solving the problem. In this chapter, we will discuss different techniques for generating a meaningful weighted complex from an embedding or from the data itself. Our focus will be primarily on generating weighted edges and faces from node and/or edge data, but we additionally demonstrate how these techniques may be applied to weighting higher-order structures.


Edge Weight Rotation System Laplacian Matrix Cycle Basis Locality Preserve Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.Siemens Corporate ResearchPrincetonUSA
  2. 2.Athinoula A. Martinos Center for Biomedical Imaging, Department of RadiologyMassachusetts General Hospital, Harvard Medical SchoolCharlestownUSA

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