eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-06-02
50:1
50:14
10.4230/LIPIcs.SoCG.2021.50
article
Near Neighbor Search via Efficient Average Distortion Embeddings
Kush, Deepanshu
1
Nikolov, Aleksandar
1
Tang, Haohua
1
University of Toronto, Canada
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 π_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 metric β³ is implied by an efficient average distortion embedding of β³ into πβ or the 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 π_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 π_p, for p β₯ 1, into πβ with average distortion on the order of p. Furthermore, we also give data structures for Schatten-p spaces with improved space and query complexity, albeit still requiring exponential pre-processing when p β₯ 2. We expect our approach to pave the way for constructing efficient NNS data structures for all norms.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol189-socg2021/LIPIcs.SoCG.2021.50/LIPIcs.SoCG.2021.50.pdf
Nearest neighbor search
metric space embeddings
average distortion embeddings
locality-sensitive hashing