Abstract
For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are not too close, the distance between their images is a strictly concave increasing function of their original distance, up to multiplicative error. The target dimension $k$ need only be quadratic in the logarithm of the size of the point set to ensure the result holds with high probability. The linear embeddings are random matrices composed of standard Cauchy random variables, and the proofs rely on Chernoff bounds for sums of iid random variables. The new metric is translation invariant, but is not induced by a norm.
Highlights
The Johnson-Lindenstrauss lemma [8] states that for a finite set of points P ⊂ RD and 0 < < 1, there are random linear maps F : RD → Rk satisfying, for any x, y ∈ P,(1 − ) x − y 2 ≤ F (x) − F (y) 2 ≤ (1 + ) x − y 2 with high probability, provided k = Θ( −2 ln |P |)
It is sufficient to draw the entries of F i.i.d. sub-Gaussian [13]. These random linear projections have provided improved worst case performance bounds for many problems in theoretical computer science, machine learning, and numerical linear algebra
Vempala [19] gives a review of problems that may be reduced to analyzing a set of points P ⊂ RD, so that after the random projection F : RD → Rk is applied, the recovery of approximate solutions is possible with time and space bounds depending on k, the target dimension, instead of D, the ambient dimension
Summary
The Johnson-Lindenstrauss lemma [8] states that for a finite set of points P ⊂ RD and 0 < < 1, there are random linear maps F : RD → Rk satisfying, for any x, y ∈ P ,. 1st and 2nd moment estimates for ξ(λ |W |); because the density for a p-stable random variable W is only implicitly defined, the needed 1st and 2nd moment estimates are not so straightforward, but could be empirically found on the computer using methods such as [3] to draw the p-stable random variables This approach, in which we directly project the points from RD, may be contrasted to embedding. (see [15, chapter 8] and [9, chapter 9]) shows that such embeddings exist with distortion (1 + ), with n proportional to D and depending on p and
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