On the impossibility of dimension reduction in l 1

Abstract

The Johnson--Lindenstrauss lemma shows that any n points in Euclidean space (i.e., ℝ n with distances measured under the ℓ 2 norm) may be mapped down to O ((log n )/ϵ 2 ) dimensions such that no pairwise distance is distorted by more than a (1 + ϵ) factor. Determining whether such dimension reduction is possible in ℓ 1 has been an intriguing open question. We show strong lower bounds for general dimension reduction in ℓ 1 . We give an explicit family of n points in ℓ 1 such that any embedding with constant distortion D requires n Ω(1/ D 2 ) dimensions. This proves that there is no analog of the Johnson--Lindenstrauss lemma for ℓ 1 ; in fact, embedding with any constant distortion requires n Ω(1) dimensions. Further, embedding the points into ℓ 1 with (1+ϵ) distortion requires n ½− 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 ℓ 1 .