
TL;DR
This paper introduces a recursive simplex star method for approximating high-dimensional manifolds and classifying points efficiently, with proven accuracy and practical tests up to 9 dimensions.
Contribution
It presents two novel recursive algorithms for manifold approximation using simplices, improving accuracy and efficiency over existing methods.
Findings
Hausdorff distance decreases as O(d n_G^{-2})
The method is effective up to 9-dimensional spaces
Approximation in simplices is more accurate and always forms a manifold
Abstract
This paper proposes a new method which builds a simplex based approximation of a -dimensional manifold separating a -dimensional compact set into two parts, and an efficient algorithm classifying points according to this approximation. In a first variant, the approximation is made of simplices that are defined in the cubes of a regular grid covering the compact set, from boundary points that approximate the intersection between and the edges of the cubes. All the simplices defined in a cube share the barycentre of the boundary points located in the cube and include simplices similarly defined in cube facets, and so on recursively. In a second variant, the Kuhn triangulation is used to break the cubes into simplices and the approximation is defined in these simplices from the boundary points computed on their edges, with the same principle. Both the approximation in cubes…
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Advanced Theoretical and Applied Studies in Material Sciences and Geometry · Mathematics and Applications
