On the nonexistence of dimension reduction for ℓ22 metrics

An `2 metric is a metric ρ such that √ ρ can be embedded isometrically into R endowed with Euclidean norm, and the minimal possible d is the dimension associated with ρ. A dimension reduction of an `2 metric ρ is an embedding of ρ into another `2 metric μ so that distances in μ are similar to those in ρ and moreover, the dimension associated with μ is small. Much of the motivation in investigating dimension reductions in `2 comes from a result of Goemans which shows that if such metrics have good dimension reductions, then they embed well into `1 spaces. This in turn yields a rounding procedure to a host of semidefinite programming with good approximation guarantees. In this work we show that there is no dimension reduction `2 metrics in the following strong sense: for every function D(n) and for every n there exists an n point `2 metric ρ such that for all embeddings of ρ into an `2 metric μ with distortion at most D(n), the associated dimension of μ is at least n−1. This stands in striking contrast to the Johnson Lindenstrauss lemma which provides a logarithmic dimension reduction for `2 metrics. Further, it shows that reducing dimension in `2 is even harder than doing so in `1 spaces.