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 cubescontaining 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 \mathcal{O n_G^{-2}) for a grid of size n_G^d in d dimensions, whereas this error decreases as \mathcal{O .
This paper proposes a new method which builds a simplex based approximation of a d-1-dimensional manifold M separating a d-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 M 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 and in simplices define a separating surface on the whole grid and classifying a point on one side or the other of this surface requires only a small number (at most d) of simple tests . Under some conditions on the definition of the boundary points and on the reach of M, for both variants the Hausdorff distance between M and its approximation decreases like \mathcal{O n_G^{-2}) , where n_G is the number of points on each axis of the grid. The approximation in cubes requires computing less boundary points than the approximation in simplices but the latter is always a manifold and is more accurate for a given value of n_G. The paper reports tests of the method when varying n_G and the dimensionality of the space (up to 9) .
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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,
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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,
approximation is defined in these simplices from the boundary points computed on their edges, with the same principle. Both the approximation in cubes and in simplices define a separating surface on the whole grid and
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on one side or the other of this surface
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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 \mathcal{O
. Under some conditions on the definition of the boundary points and on the reach of M, for both variants the Hausdorff distance between M and its approximation decreases like \mathcal{O
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for a grid of size n_G^d in d dimensions, whereas this error decreases as \mathcal{O
, where n_G is the number of points on each axis of the grid. The approximation in cubes requires computing less boundary points than the approximation in simplices but the latter is always a manifold and is more accurate for a given value of n_G. The paper reports tests of the method when varying n_G and the dimensionality of the space (up to 9)