On the impossibility of dimension reduction in l1

The Johnson--Lindenstrauss lemma shows that any n points in Euclidean space (i.e., ℝn with distances measured under the ℓ2 norm) may be mapped down to O((log n)/ε2) dimensions such that no pairwise distance is distorted by more than a (1 + ε) factor. Determining whether such dimension reduction is possible in ℓ1 has been an intriguing open question. We show strong lower bounds for general dimension reduction in ℓ1. We give an explicit family of n points in ℓ1 such that any embedding with constant distortion D requires nΩ(1/D2) dimensions. This proves that there is no analog of the Johnson--Lindenstrauss lemma for ℓ1; in fact, embedding with any constant distortion requires nΩ(1) dimensions. Further, embedding the points into ℓ1 with (1+ε) distortion requires n½−O(ε log(1/ε)) dimensions. Our proof establishes this lower bound for shortest path metrics of series-parallel graphs. We make extensive use of linear programming and duality in devising our bounds. We expect that the tools and techniques we develop will be useful for future investigations of embeddings into ℓ1.