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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 newmetric, having the following properties. For any pair of points from the pointset that are not too close, the distance between their images is a strictlyconcave increasing function of their original distance, up to multiplicativeerror. The target dimension $k$ need only be quadratic in the logarithm of thesize of the point set to ensure the result holds with high probability. Thelinear embeddings are random matrices composed of standard Cauchy randomvariables, and the proofs rely on Chernoff bounds for sums of iid randomvariables. The new metric is translation invariant, but is not induced by anorm.

Linear Dimension Reduction Approximately Preserving a Function of the 1-Norm