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For Euclidean space ($\ell_2$), there exists the powerful dimension reductiontransform of Johnson and Lindenstrauss, with a host of known applications.Here, we consider the problem of dimension reduction for all $\ell_p$ spaces $1\le p \le 2$. Although strong lower bounds are known for dimension reduction in$\ell_1$, Ostrovsky and Rabani successfully circumvented these by presenting an$\ell_1$ embedding that maintains fidelity in only a bounded distance range,with applications to clustering and nearest neighbor search. However, theirembedding techniques are specific to $\ell_1$ and do not naturally extend toother norms.  In this paper, we apply a range of advanced techniques and produce boundedrange dimension reduction embeddings for all of $1 \le p \le 2$, therebydemonstrating that the approach initiated by Ostrovsky and Rabani for $\ell_1$can be extended to a much more general framework. We also obtain improvedbounds in terms of the intrinsic dimensionality. As a result we achieveimproved bounds for proximity problems including snowflake embeddings andclustering.

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