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Near Optimal Dimensionality Reductions That Preserve Volumes
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
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ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
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ACM Transactions on Algorithms (TALG)
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In this paper we show that dimensionality reduction (i.e., Johnson-Lindenstrauss lemma) preserves not only the distances between static points, but also between moving points, and more generally between low-dimensional flats, polynomial curves, curves with low winding degree, and polynomial surfaces. We also show that surfaces with bounded doubling dimension can be embedded into low dimension with small additive error. Finally, we show that for points with polynomial motion, the radius of the smallest enclosing ball can be preserved under dimensionality reduction.