Fast Distance Oracles for Any Symmetric Norm

Yichuan Deng, Zhao Song, Omri Weinstein, Ruizhe Zhang

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations

Abstract

In the Distance Oracle problem, the goal is to preprocess n vectors x1, x2,..., xn in a d-dimensional metric space (Xd, ∥ · ∥l) into a cheap data structure, so that given a query vector q ∈ Xd and a subset S ⊆ [n] of the input data points, all distances ∥q − xil for xi ∈ S can be quickly approximated (faster than the trivial ∼ d|S| query time). This primitive is a basic subroutine in machine learning, data mining and similarity search applications. In the case of ℓp norms, the problem is well understood, and optimal data structures are known for most values of p. Our main contribution is a fast (1 ± ε) distance oracle for any symmetric norm ∥ · ∥l. This class includes ℓp norms and Orlicz norms as special cases, as well as other norms used in practice, e.g. top-k norms, max-mixture and sum-mixture of ℓp norms, small-support norms and the box-norm. We propose a novel data structure with Oe(n(d + mmc(l)2)) preprocessing time and space, and tq = Oe(d + |S| · mmc(l)2) query time, for computing distances to a subset S of data points, where mmc(l) is a complexity-measure (concentration modulus) of the symmetric norm. When l = ℓp, this runtime matches the aforementioned state-of-art oracles.

Original languageAmerican English
Title of host publicationAdvances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022
EditorsS. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, A. Oh
PublisherNeural information processing systems foundation
ISBN (Electronic)9781713871088
StatePublished - 2022
Event36th Conference on Neural Information Processing Systems, NeurIPS 2022 - New Orleans, United States
Duration: 28 Nov 20229 Dec 2022

Publication series

NameAdvances in Neural Information Processing Systems
Volume35
ISSN (Print)1049-5258

Conference

Conference36th Conference on Neural Information Processing Systems, NeurIPS 2022
Country/TerritoryUnited States
CityNew Orleans
Period28/11/229/12/22

Bibliographical note

Publisher Copyright:
© 2022 Neural information processing systems foundation. All rights reserved.

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