1998 ApproximateNearestNeighbors

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Subject Headings: Nearest Neighbor Search Task, Curse of Dimensionality.

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Quotes

Abstract

The nearest neighbor problem is the following: Given a set of [math]\displaystyle{ n }[/math] points P={p_1,...,p_n} in some metric space [math]\displaystyle{ X }[/math], preprocess P/ so as to efficiently answer queries which require finding the point in [math]\displaystyle{ P }[/math] closest to a query point q ∈ X. We focus on the particularly interesting case of the [math]\displaystyle{ d }[/math]-dimensional Euclidean space where X = R^d under some l_p norm. Despite decades of effort, the current solutions are far from satisfactory; in fact, for large [math]\displaystyle{ d }[/math], in theory or in practice, they provide little improvement over the brute-force algorithm which compares the query point to each data point. Of late, there has been some interest in the approximate nearest neighbors problem, which is: Find a point p ∈ P that is an e-approximate nearest neighbor of the query [math]\displaystyle{ q }[/math] in that for all p’ ∈ P, d(p, q) < (1 + e)d(p’, q).

We present two algorithmic results for the approximate version that significantly improve the known bounds: (a) preprocessing cost polynomial in [math]\displaystyle{ n }[/math] and [math]\displaystyle{ d }[/math], and a truly sublinear query time (for [math]\displaystyle{ e }[/math] > 1); and, (b) query time polynomial in log-n and [math]\displaystyle{ d }[/math], and only a mildly exponential preprocessing cost Õ(n) × O(1/e^d). Further, applying a classical geometric lemma on random projections (for which we give a simpler proof), we obtain the first known algorithm with polynomial preprocessing and query time polynomial in [math]\displaystyle{ d }[/math] and log n. Unfortunately, for small [math]\displaystyle{ e }[/math], the latter is a purely theoretical result since the exponent depends on l/e. Experimental results indicate that our first algorithm offers orders of magnitude improvement on running times over real data sets. Its key ingredient is the notion of locality-sensitive hashing which may be of independent interest; here, we give applications to information retrieval, pattern recognition, dynamic closest-pairs, and fast clustering algorithms.


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 AuthorvolumeDate ValuetitletypejournaltitleUrldoinoteyear
1998 ApproximateNearestNeighborsRajeev Motwani
Piotr Indyk		Approximate Nearest Neighbors: Towards removing the curse of dimensionalityProceedings of the Thirtieth Annual ACM Symposium on Theory of Computinghttp://dx.doi.org/10.1145/276698.27687610.1145/276698.2768761998
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