Dimension reduction techniques for $\ell_p$, $1 \le p \le 2$, with applications

For ℓ 2 , there exists the powerful dimension reduction transform of Johnson and Lindenstrauss [JL84], with a host of known applications. Here, we consider the problem of dimension reduction for all ℓ p spaces 1 ≤ p < ∞. Although strong lower bounds are known for dimension reduction in ℓ 1 , Ostrovsky and Rabani [OR02] successfully circumvented these by presenting an ℓ 1 embedding that maintains fidelity in only a bounded distance range, with applications to clustering and nearest neighbor search. However, their embedding techniques are specific to ℓ 1 and do not naturally extend to other norms. In this paper, we apply a range of advanced techniques and produce bounded range dimension reduction embeddings for all of 1 ≤ p < ∞, thereby demonstrating that the approach initiated by Ostrovsky and Rabani for ℓ 1 can be extended to a much more general framework. As a result we achieve improved bounds for a host of proximity problems including nearest neighbor search, snowflake embeddings, and clustering, for various values of p. In particular, our nearest neighbor search structure for p > 2 provides an algorithmic solution to a major open problem.