Near Neighbor Search via Efficient Average Distortion Embeddings
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A recent series of papers by Andoni, Naor, Nikolov, Razenshteyn, and Waingarten (STOC 2018, FOCS 2018) has given approximate near neighbour search (NNS) data structures for a wide class of distance metrics, including all norms. In particular, these data structures achieve approximation on the order of p for ℓ_p^d norms with space complexity nearly linear in the dataset size n and polynomial in the dimension d, and query time sub-linear in n and polynomial in d. The main shortcoming is the exponential in d pre-processing time required for their construction. In this paper, we describe a more direct framework for constructing NNS data structures for general norms. More specifically, we show via an algorithmic reduction that an efficient NNS data structure for a given metric is implied by an efficient average distortion embedding of it into ℓ_1 or into Euclidean space. In particular, the resulting data structures require only polynomial pre-processing time, as long as the embedding can be computed in polynomial time. As a concrete instantiation of this framework, we give an NNS data structure for ℓ_p with efficient pre-processing that matches the approximation factor, space and query complexity of the aforementioned data structure of Andoni et al. On the way, we resolve a question of Naor (Analysis and Geometry in Metric Spaces, 2014) and provide an explicit, efficiently computable embedding of ℓ_p, for p ≥ 2, into ℓ_2 with (quadratic) average distortion on the order of p. We expect our approach to pave the way for constructing efficient NNS data structures for all norms.
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