Dimension Reduction by Random Hyperplane Tessellations
نویسندگان
چکیده
منابع مشابه
Dimension Reduction by Random Hyperplane Tessellations
Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y ∈ K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m = O(w...
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It is well known that the vertex number of the typical cell of a stationary hyperplane tessellation in R has, under some mild conditions, an expectation equal to 2, independent of the underlying distribution. The variance of this vertex number can vary widely. Under Poisson assumptions, we give sharp bounds for this variance, showing, in particular, that its maximum is attained if and only if t...
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It is proved that the shape of the typical cell of a stationary and isotropic Poisson random hyperplane tessellation is, with high probability, close to the shape of a ball if the kth intrinsic volume (k ≥ 2) of the typical cell is large. The shape of typical cells of large diameter is close to the shape of a segment.
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We derive a central limit theorem for the number of vertices of convex polytopes induced by stationary Poisson hyperplane processes in R d. This result generalizes an earlier one proved by Paroux [Adv. for intersection points of motion-invariant Poisson line processes in R 2. Our proof is based on Hoeffd-ing's decomposition of U-statistics which seems to be more efficient and adequate to tackle...
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We consider tessellations of the Euclidean (d− 1)-sphere by (d− 2)-dimensional great subspheres or, equivalently, tessellations of Euclidean d-space by hyperplanes through the origin; these we call conical tessellations. For random polyhedral cones defined as typical cones in a conical tessellation by random hyperplanes, and for their dual cones, we study expectations of certain geometric funct...
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2013
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-013-9561-6