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# Locality-sensitive hashing

About: Locality-sensitive hashing is a(n) research topic. Over the lifetime, 1894 publication(s) have been published within this topic receiving 69362 citation(s).

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TL;DR: Experimental results indicate that the novel scheme for approximate similarity search based on hashing scales well even for a relatively large number of dimensions, and provides experimental evidence that the method gives improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition.

Abstract: The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search/index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the \curse of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and/or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50).

3,518 citations

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TL;DR: A novel Locality-Sensitive Hashing scheme for the Approximate Nearest Neighbor Problem under lp norm, based on p-stable distributions that improves the running time of the earlier algorithm and yields the first known provably efficient approximate NN algorithm for the case p<1.

Abstract: We present a novel Locality-Sensitive Hashing scheme for the Approximate Nearest Neighbor Problem under lp norm, based on p-stable distributions.Our scheme improves the running time of the earlier algorithm for the case of the lp norm. It also yields the first known provably efficient approximate NN algorithm for the case p

2,806 citations

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08 Dec 2008

TL;DR: The problem of finding a best code for a given dataset is closely related to the problem of graph partitioning and can be shown to be NP hard and a spectral method is obtained whose solutions are simply a subset of thresholded eigenvectors of the graph Laplacian.

Abstract: Semantic hashing[1] seeks compact binary codes of data-points so that the Hamming distance between codewords correlates with semantic similarity. In this paper, we show that the problem of finding a best code for a given dataset is closely related to the problem of graph partitioning and can be shown to be NP hard. By relaxing the original problem, we obtain a spectral method whose solutions are simply a subset of thresholded eigenvectors of the graph Laplacian. By utilizing recent results on convergence of graph Laplacian eigenvectors to the Laplace-Beltrami eigenfunctions of manifolds, we show how to efficiently calculate the code of a novel data-point. Taken together, both learning the code and applying it to a novel point are extremely simple. Our experiments show that our codes outperform the state-of-the art.

2,506 citations

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TL;DR: It is shown that rounding algorithms for LPs and SDPs used in the context of approximation algorithms can be viewed as locality sensitive hashing schemes for several interesting collections of objects.

Abstract: (MATH) A locality sensitive hashing scheme is a distribution on a family $\F$ of hash functions operating on a collection of objects, such that for two objects x,y, PrheF[h(x) = h(y)] = sim(x,y), where sim(x,y) e [0,1] is some similarity function defined on the collection of objects. Such a scheme leads to a compact representation of objects so that similarity of objects can be estimated from their compact sketches, and also leads to efficient algorithms for approximate nearest neighbor search and clustering. Min-wise independent permutations provide an elegant construction of such a locality sensitive hashing scheme for a collection of subsets with the set similarity measure sim(A,B) = \frac{|A ∩ B|}{|A ∪ B|}.(MATH) We show that rounding algorithms for LPs and SDPs used in the context of approximation algorithms can be viewed as locality sensitive hashing schemes for several interesting collections of objects. Based on this insight, we construct new locality sensitive hashing schemes for:A collection of vectors with the distance between → \over u and → \over v measured by O(→ \over u, → \over v)/π, where O(→ \over u, → \over v) is the angle between → \over u) and → \over v). This yields a sketching scheme for estimating the cosine similarity measure between two vectors, as well as a simple alternative to minwise independent permutations for estimating set similarity.A collection of distributions on n points in a metric space, with distance between distributions measured by the Earth Mover Distance (EMD), (a popular distance measure in graphics and vision). Our hash functions map distributions to points in the metric space such that, for distributions P and Q, EMD(P,Q) ≤ Ehe\F [d(h(P),h(Q))] ≤ O(log n log log n). EMD(P, Q).

2,257 citations

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TL;DR: An algorithm for the c-approximate nearest neighbor problem in a d-dimensional Euclidean space, achieving query time of O(dn 1c2/+o(1)) and space O(DN + n1+1c2 + o(1) + 1/c2), which almost matches the lower bound for hashing-based algorithm recently obtained.

Abstract: In this article, we give an overview of efficient algorithms for the approximate and exact nearest neighbor problem. The goal is to preprocess a dataset of objects (e.g., images) so that later, given a new query object, one can quickly return the dataset object that is most similar to the query. The problem is of significant interest in a wide variety of areas.

1,712 citations