Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
ACM Transactions on Graphics (TOG)
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
Three-dimensional alpha shapes
ACM Transactions on Graphics (TOG)
Computational Geometry: Theory and Applications
Topology representing networks
Neural Networks
The quickhull algorithm for convex hulls
ACM Transactions on Mathematical Software (TOMS)
A spatial data mining method by Delaunay triangulation
GIS '97 Proceedings of the 5th ACM international workshop on Advances in geographic information systems
Intrinsic Dimensionality Estimation With Optimally Topology Preserving Maps
IEEE Transactions on Pattern Analysis and Machine Intelligence
Principles of data mining
Neural Networks for Pattern Recognition
Neural Networks for Pattern Recognition
A Tutorial on Support Vector Machines for Pattern Recognition
Data Mining and Knowledge Discovery
Separability Index in Supervised Learning
PKDD '02 Proceedings of the 6th European Conference on Principles of Data Mining and Knowledge Discovery
Rule Extraction from Self-Organizing Networks
ICANN '02 Proceedings of the International Conference on Artificial Neural Networks
Shape Reconstruction with Delaunay Complex
LATIN '98 Proceedings of the Third Latin American Symposium on Theoretical Informatics
Proximity Drawability: a Survey
GD '94 Proceedings of the DIMACS International Workshop on Graph Drawing
Improving Classification by Removing or Relabeling Mislabeled Instances
ISMIS '02 Proceedings of the 13th International Symposium on Foundations of Intelligent Systems
γ-Observable neighbours for vector quantization
Neural Networks - New developments in self-organizing maps
Generalized relevance learning vector quantization
Neural Networks - New developments in self-organizing maps
A Multivariate Two-Sample Test Using the Voronoi Diagram
A Multivariate Two-Sample Test Using the Voronoi Diagram
Pattern Classification (2nd Edition)
Pattern Classification (2nd Edition)
A Nonlinear Mapping for Data Structure Analysis
IEEE Transactions on Computers
Class visualization of high-dimensional data with applications
Computational Statistics & Data Analysis
Curvilinear component analysis: a self-organizing neural network for nonlinear mapping of data sets
IEEE Transactions on Neural Networks
A Toplogy Based Multi-Classifier System
ICDAR '05 Proceedings of the Eighth International Conference on Document Analysis and Recognition
Learning Highly Structured Manifolds: Harnessing the Power of SOMs
Similarity-Based Clustering
Gabriel Graphs in Arbitrary Metric Space and their Cellular Automaton for Many Grids
ACM Transactions on Autonomous and Adaptive Systems (TAAS)
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We propose the use of topology representing graphs for the exploratory analysis of high-dimensional labeled data. The Delaunay graph contains all the topological information needed to analyze the topology of the classes (e.g. the number of separate clusters of a given class, the way these clusters are in contact with each other or the shape of these clusters). The Delaunay graph also allows to sample the decision boundary of the Nearest Neighbor rule, to define a topological criterion of non-linear separability of the classes and to find data which are near the decision boundary so that their label must be considered carefully. This graph then provides a way to analyze the complexity of a classification problem, and tools for decision support. When the Delaunay graph is not tractable in too high-dimensional spaces, we propose to use the Gabriel graph instead and discuss the limits of this approach. This analysis technique is complementary with projection techniques, as it allows to handle the data as they are in the data space, avoiding projection distortions. We apply it to analyze the well-known Iris database and a seismic events database.