Terminal Embeddings in Sublinear Time

Recently (Elkin, Filtser, Neiman 2017) introduced the concept of a terminal embedding from one metric space to another with a set of designated terminals. In the case where both metric spaces are Euclidean, recently (Narayanan, Nelson 2019), following work of (Mahabadi, Makarychev, Makarychev, Razenshteyn 2018), constructed a terminal embedding with optimal embedding dimension. This generalizes the Johnson-Lindenstrauss lemma, which only preserves distances within the terminal set and not to the set from the rest of space. The downside is that contructing the embedding for a new point required solving a semidefinite program incurring large runtime. Our main contribution in this work is to give a new data structure for computing terminal embeddings. We show how to pre-process the terminal set to obtain an almost linear-space data structure that supports computing the terminal embedding image of any input point in sublinear time. To accomplish this, we leverage tools developed in the context of approximate nearest neighbor search.