Recursive simplex stars
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This paper proposes a new method which approximates a classification function separating a d dimensional compact set into two parts. The approach starts by estimating the intersection between the classification boundary and the edges of a regular grid covering the compact set. Then it builds a classification surface made of recursive simplex stars (resistars) defined in the grid cubes containing such boundary points. A first variant, the simple resistar (s-resistar) defines a single star of simplices which share the barycentre of the cube boundary points and include stars of simplices defined similarly in cube facets, and so on recursively until a face boundary points define a single simplex. This definition is simple and easy to apply when the dimensionality increases. However, s-resistars sometimes "glue" together surfaces that should be separated and this deteriorates the local classification performance. The second variant, the multi-boundary resistar (or m-resistar) addresses this problem by defining several simplex stars in a cube or in its faces when necessary, which is shown to increase the local classification performance. With both s-resistars and m-resistars, classifying a point requires only a small number of simple tests without explicitly computing the simplices. It is thus possible to use resistar classification in spaces of relatively high dimensionality (up to 9 in our tests) and for resistar surfaces including a large number of simplices (up to several trillions in our tests). The paper provides a theoretical argument and empirical evidence suggesting that, when the surface to approximate is smooth enough, the error of resistar classification decreases as O(n_G^-2) for a grid of size n_G^d in d dimensions, whereas this error decreases as O(n_G^-1) when classifying with the sign of the nearest vertex of the grid.
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