A major open problem in the field of metric embedding is the existence of dimension reduction for @math -point subsets of Euclidean space, such that both distortion and dimension depend only on the doubling constant of the pointset, and not on its cardinality. In this paper, we negate this possibility for @math spaces with @math . In particular, we introduce an @math -point subset of @math with doubling constant O(1), and demonstrate that any embedding of the set into @math with distortion @math must have @math .
There are few positive results for @math , such as @cite_18 who showed that @math admits dimension reduction when the aspect ratio of the point set is bounded, and @cite_54 used @math -stable distributions to obtain similar results for all @math , and the Mazur map to obtain a (relatively high-distortion bound) for @math . A weak form of dimensionality reduction in @math was shown by @cite_32 .
@cite_18 @cite_54 @cite_32
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Given a set of @math points in @math , how many dimensions are needed to represent all pair wise distances within a specific distortion? This dimension-distortion tradeoff question is well understood for the @math norm, where @math dimensions suffice to achieve @math distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in @math . A recent result shows that distortion @math can be achieved with @math dimensions. On the other hand, the only lower bounds known are that distortion @math requires @math dimensions and that distortion @math requires @math dimensions. In this work, we show the first near linear lower bounds for dimension reduction in @math . In particular, we show that @math distortion requires at least @math dimensions. Our proofs are combinatorial, but inspired by linear programming. In fact, our techniques lead to a simple combinatorial argument that is equivalent to the LP based proof of Brinkman-Charikar for lower bounds on dimension reduction in @math . We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We describe an @math -time algorithm for computing the exact probability that two @math -dimensional probabilistic point sets are linearly separable, for any fixed @math . A probabilistic point in @math -space is a normal point, but with an associated probability of existence; the existence probabilities of all points are independent. We also show that the @math -dimensional separability problem is equivalent to a @math -dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of @math probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership problem by a factor of @math . In addition, our algorithms can handle input degeneracies in which more than @math points may lie on a @math -dimensional subspace, thus resolving an open problem in 2013. Finally, we prove lower bounds for the separability problem via a reduction from the @math -SUM problem, which show in particular that our @math algorithms for @math -dimensional separability and @math -dimensional convex hull membership are nearly optimal. Consider an @math matrix @math with the restricted isometry property of order k and level @math ; that is, the norm of any k-sparse vector in @math is preserved to within a multiplicative factor of @math under application of @math . We show that by randomizing the column signs of such a matrix @math , the resulting map with high probability embeds any fixed set of @math points in @math into @math without distorting the norm of any point in the set by more than a factor of @math . Consequently, matrices with the restricted isometry property and with randomized column signs provide optimal Johnson–Lindenstrauss embeddings up to logarithmic factors in N. In particular, our results improve the best known bounds on the necessary embedding dimension m for a wide class of structured random matrices; for partial Fourier and partial Hadamard matrices, we improve the recent bound @math given by Ailon and Liberty to $m...