Lectures on Discrete Geometry
Fast Nearest-Neighbor Search in Dissimilarity Spaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
On the impossibility of dimension reduction in l1
Journal of the ACM (JACM)
Advances in metric embedding theory
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Embedding metric spaces in their intrinsic dimension
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
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We investigate the possibility of embedding an n-point metric space into a constant dimensional vector space with the maximum norm, such that the embedding is almost isometric, that is, the distortion of distances is kept arbitrarily close to 1. When the source metric is generated by any fixed norm on a finite dimensional vector space, we prove that this embedding is always possible, such that the dimension of the target space remains constant, independent of n. While this possibility has been known in the folklore, we present the first fully detailed proof, which, in addition, is significantly simpler and more transparent, then what was available before. Furthermore, our embedding can be computed in deterministic linear time in n, given oracle access to the norm.