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| '''Locality-sensitive hashing''' ('''LSH''') is a method of performing probabilistic [[dimension reduction]] of high-dimensional data. The basic idea is to [[Hash Function|hash]] the input items so that similar items are mapped to the same buckets with high probability (the number of buckets being much smaller than the universe of possible input items). This is different from the conventional hash functions, such as those used in [[cryptography]], as in the LSH case the goal is to maximize probability of "collision" of similar items rather than avoid collisions.
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| <ref name=MOMD>{{cite web
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| | author = A. Rajaraman and J. Ullman
| |
| | url = http://infolab.stanford.edu/~ullman/mmds.html
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| | title=Mining of Massive Datasets, Ch. 3.
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| | year = 2010
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| }}</ref>
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| Note how locality-sensitive hashing, in many ways, mirrors [[Cluster analysis|data clustering]] and [[Nearest neighbor search]].
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|
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| ==Definition==
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|
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|
| An ''LSH family''
| | Pas celine sac cher Outlet Store me demande quel pourrait résister à la tentation de la créatrice de bijoux glamour. C'est sûrement une idée brillante pour cette maison de mode haut de gamme pour contenir ces boucles d'oreille place dans le Pas de celine sac cher de collecte 2011. Inspiré par des colliers traditionnels masaï, la collection trique ornement est glamour avec des cercles harmonieuses et de pierres précieuses aux couleurs vives, y compris les saphirs, sperratite et de diamants.<br><br>Les boucles d'oreilles clips font la paire pour vos oreilles. Les boucles d'oreilles tribales ornement disposent d'un design spectaculaire de cercles qui se chevauchent, et dont l'un contient une fleur classique de monogramme. La boucle d'oreille en or jaune est décorée avec des saphirs bleus, blancs et roses. Je peux à peine attendre de l'avoir vitrage sur mes oreilles. Et puis je me hésité quand ont le Pas celine sac traverses tribales cher ornement dans les mains. Ces pendants d'oreilles raffinées ont tact combinée jaune chaud avec grenats spessartite de feu, et ils mettent en évidence une fleur de monogramme dans un cercle. Les traverses sont disponibles au prix de 7,450.00 $, tandis que les boucles d'oreilles demandent 16,200.00 $.<br><br><br>Je jure que je ne m'attends pas à un dilemme plus compliqué d'avoir le tiers-celine sac burberry pas cher ornement cercles tribaux boucles d'oreilles. En or jaune avec saphirs jaunes et grenats spessartite, ces boucles d'oreilles élégantes simples sont mis en valeur par une bande contrastante de diamants sur or blanc. Dieu, combien j'aime conception simple. Les arceaux sont au prix de $ 14,500.00. Ce ne sera pas un choix facile, je suppose.<br><br>Bien que la plupart du temps je porte simplement minuscules laiteux boucles d'oreilles perle blanche, je reçois toujours excitant tout en voyant les couleurs Pas celine sac de boucles d'oreilles tribales cher ornemnt. Je me demande quel pourrait résister à la tentation de la créatrice de bijoux glamour. C'est sûrement une idée brillante pour cette maison de mode haut de gamme pour contenir ces boucles d'oreille place dans le Pas de celine sac cher de collecte 2011. Inspiré par des colliers traditionnels Masai, la collection trique ornement est glamour avec des cercles harmonieuses et de pierres précieuses aux couleurs vives, y compris les saphirs, sperratite et de diamants.<br><br>Les boucles d'oreilles clips font la paire pour vos oreilles. Les boucles d'oreilles tribales ornement disposent d'un design spectaculaire de cercles qui se chevauchent, et dont l'un contient une fleur classique de monogramme. La boucle d'oreille en or jaune est décorée avec des saphirs bleus, blancs et roses. Je peux à peine attendre de l'avoir vitrage sur mes oreilles. Et puis je me hésité quand ont le Pas celine sac traverses tribales cher ornement dans les mains. Ces pendants d'oreilles raffinées ont tact combinée or jaune chaud avec grenats spessartite de feu, et ils mettent en évidence une fleur de monogramme dans un cercle. Les traverses sont disponibles au prix de 7,450.00 $, tandis que les boucles d'oreilles demandent 16,200.00 $.<br><br>Je jure que je ne m'attends pas à un dilemme plus compliqué d'avoir le tiers-celine sac burberry pas cher ornement cercles tribaux boucles d'oreilles. En or jaune avec saphirs jaunes et grenats spessartite, ces boucles d'oreilles élégantes simples sont mis en valeur par une bande contrastante de diamants sur or blanc. Dieu, combien j'aime conception simple. Les arceaux sont au prix de $ 14,500.00. Ce ne sera pas un choix facile, je suppose.<br><br>Should you loved this information and you would like to receive details relating to [http://tinyurl.com/mkbuxg7 http://tinyurl.com/mkbuxg7] please visit our website. |
| <ref name=MOMD />
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| <ref name=GIM1999>{{cite journal
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| | author = Gionis, A.
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| | coauthors = [[Piotr Indyk|Indyk, P.]], [[Rajeev Motwani|Motwani, R.]]
| |
| | year = 1999
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| | title = Similarity Search in High Dimensions via Hashing
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| | url = http://people.csail.mit.edu/indyk/vldb99.ps ,
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| | journal = Proceedings of the 25th Very Large Database (VLDB) Conference
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| }}</ref>
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| <ref name=IndykMotwani98>{{cite journal | |
| | author = [[Piotr Indyk|Indyk, Piotr]].
| |
| | coauthors = [[Rajeev Motwani|Motwani, Rajeev]].
| |
| | year = 1998
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| | title = Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality.
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| | url = http://people.csail.mit.edu/indyk/nndraft.ps ,
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| | journal = Proceedings of 30th Symposium on Theory of Computing
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| }}</ref>
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| <math>\mathcal F</math> is defined for a [[metric space]] <math>\mathcal M =(M, d)</math>, a threshold <math>R>0</math> and an approximation factor <math>c>1</math>. This family <math>\mathcal F</math> is a family of functions <math>h:{\mathcal M}\to S</math> which map elements from the [[metric space]] to a bucket <math>s \in S</math>. The LSH family satisfies the following conditions for any two points <math>p, q \in {\mathcal M}</math>, using a function <math>h \in \mathcal F</math> which is chosen uniformly at random:
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| * if <math>d(p,q) \le R</math>, then <math>h(p)=h(q)</math> (i.e.,<math>p</math> and <math>q</math> collide) with probability at least <math>P_1</math>,
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| * if <math>d(p,q) \ge cR</math>, then <math>h(p)=h(q)</math> with probability at most <math>P_2</math>.
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| | |
| A family is interesting when <math>P_1>P_2</math>. Such a family <math>\mathcal F</math> is called ''<math>(R,cR,P_1,P_2)</math>-sensitive''.
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| | |
| Alternatively<ref name=Charikar2002>{{cite journal
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| | author = Charikar, Moses S..
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| | coauthors =
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| | year = 2002
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| | title = Similarity Estimation Techniques from Rounding Algorithms
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| | journal = Proceedings of the 34th Annual ACM Symposium on Theory of Computing 2002
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| | pages = (ACM 1–58113–495–9/02/0005)…
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| | url = http://portal.acm.org/citation.cfm?id=509965
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| | accessdate = 2007-12-21
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| | doi = 10.1145/509907.509965
| |
| }}</ref> it is defined with respect to a universe of items <math>U</math> that have a [[String metric|similarity]] function <math>\phi : U \times U \to [0,1]</math>. An LSH scheme is a family of [[hash function]]s <math>H</math> coupled with a probability distribution <math>D</math> over the functions such that a function <math>h \in H</math> chosen according to <math>D</math> satisfies the property that <math>Pr_{h \in H} [h(a) = h(b)] = \phi(a,b)</math> for any <math>a,b \in U</math>.
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| ===Amplification===
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| Given a <math>(d_1, d_2, p_1, p_2)</math>-sensitive family <math>\mathcal F</math>, we can construct new families <math>\mathcal G</math> by either the AND-construction or OR-construction of <math>\mathcal F</math>.<ref name=MOMD />
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| To create an AND-construction, we define a new family <math>\mathcal G</math> of hash functions <math>g</math>, where each function <math>g</math> is constructed from <math>k</math> random functions <math>h_1, ..., h_k</math> from <math>\mathcal F</math>. We then say that for a hash function <math>g \in \mathcal G</math>, <math>g(x) = g(y)</math> if and only if all <math>h_i(x) = h_i(y)</math> for <math>i = 1, 2, ..., k</math>. Since the members of <math>\mathcal F</math> are independently chosen for any <math>g \in \mathcal G</math>, <math>\mathcal G</math> is a <math>(d_1, d_2, p_{1}^r, p_{2}^r)</math>-sensitive family.
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| To create an OR-construction, we define a new family <math>\mathcal G</math> of hash functions <math>g</math>, where each function <math>g</math> is constructed from <math>k</math> random functions <math>h_1, ..., h_k</math> from <math>\mathcal F</math>. We then say that for a hash function <math>g \in \mathcal G</math>, <math>g(x) = g(y)</math> if and only if <math>h_i(x) = h_i(y)</math> for one or more values of <math>i</math>. Since the members of <math>\mathcal F</math> are independently chosen for any <math>g \in \mathcal G</math>, <math>\mathcal G</math> is a <math>(d_1, d_2, 1- (1 - p_1)^r, 1 - (1 - p_2)^r)</math>-sensitive family.
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| ==Applications==
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| LSH has been applied to several problem domains including{{citation needed|date=August 2011}}
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| *[[Near-duplicate detection]]<ref>
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| {{citation
| |
| | last1 = Gurmeet Singh | first1 = Manku
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| | last2 = Jain | first2 = Arvind
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| | last2 = Das Sarma | first2 = Anish
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| | title = Detecting near-duplicates for web crawling
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| | journal = Proceedings of the 16th international conference on World Wide Web. ACM,
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| | year = 2007}}.</ref><ref>
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| {{citation
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| | author = Das, Abhinandan S., et al.
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| | title = Google news personalization: scalable online collaborative filtering
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| | journal = Proceedings of the 16th international conference on World Wide Web. ACM,
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| | year = 2007|doi=10.1145/1242572.1242610}}.</ref>
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| *[[Hierarchical clustering]]<ref>
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| {{citation
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| | author = Koga, Hisashi, Tetsuo Ishibashi, and Toshinori Watanabe
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| | title = Fast agglomerative hierarchical clustering algorithm using Locality-Sensitive Hashing
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| | journal = Knowledge and Information Systems 12.1: 25-53,
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| | year = 2007}}.</ref>
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| *[[Genome-wide association study]]<ref>
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| {{citation
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| | author = Brinza, Dumitru, et al.
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| | title = RAPID detection of gene–gene interactions in genome-wide association studies
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| | journal = Bioinformatics 26.22 (2010): 2856-2862.}}
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| </ref>
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| *[[Image similarity identification]]
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| **[[VisualRank]]
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| *[[Gene expression similarity identification]]{{citation needed|date=October 2013}}
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| *[[Audio similarity identification]]
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| *[[Nearest neighbor search]]
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| ==Methods==
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| ===Bit sampling for Hamming distance===
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| One of the easiest ways to construct an LSH family is by bit sampling.<ref name=IndykMotwani98 /> This approach works for the [[Hamming distance]] over d-dimensional vectors <math>\{0,1\}^d</math>. Here, the family <math>\mathcal F</math> of hash functions is simply the family of all the projections of points on one of the <math>d</math> coordinates, i.e., <math>{\mathcal F}=\{h:\{0,1\}^d\to \{0,1\}\mid h(x)=x_i,i =1 ... d\}</math>, where <math>x_i</math> is the <math>i</math>th coordinate of <math>x</math>. A random function <math>h</math> from <math>{\mathcal F}</math> simply selects a random bit from the input point. This family has the following parameters: <math>P_1=1-R/d</math>, <math>P_2=1-cR/d</math>.
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| ===Min-wise independent permutations===
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| {{main|MinHash}}
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| Suppose <math>U</math> is composed of subsets of some ground set of enumerable items <math>S</math> and the similarity function of interest is the [[Jaccard index]] <math>J</math>. If <math>\pi</math> is a permutation on the indices of <math>S</math>, for <math>A \subseteq S</math> let <math>h(A) = \min_{a \in A} \{ \pi(a) \}</math>. Each possible choice of <math>\pi</math> defines a single hash function <math>h</math> mapping input sets to integers.
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| Define the function family <math>H</math> to be the set of all such functions and let <math>D</math> be the uniform distribution. Given two sets <math>A,B \subseteq S</math> the event that <math>h(A) = h(B)</math> corresponds exactly to the event that the minimizer of <math>\pi</math> lies inside <math>A \bigcap B</math>. As <math>h</math> was chosen uniformly at random, <math>Pr[h(A) = h(B)] = J(A,B)\,</math> and <math>(H,D)\,</math> define an LSH scheme for the Jaccard index.
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| Because the symmetric group on n elements has size n!, choosing a truly random permutation from the full symmetric group is infeasible for even moderately sized n. Because of this fact, there has been significant work on finding a family of permutations that is "min-wise independent" - a permutation family for which each element of the domain has equal probability of being the minimum under a randomly chosen <math>\pi</math>. It has been established that a min-wise independent family of permutations is at least of size <math>lcm(1, 2, ..., n) \ge e^{n-o(n)}</math>.<ref name=Broder1998>{{cite journal
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| | author = Broder, A.Z.
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| | coauthors = Charikar, M.; Frieze, A.M.; Mitzenmacher, M.
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| | year = 1998
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| | title = Min-wise independent permutations
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| | journal = Proceedings of the thirtieth annual ACM symposium on Theory of computing
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| | pages = 327–336
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| | url = http://www.cs.princeton.edu/~moses/papers/minwise.ps
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| | accessdate = 2007-11-14
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| | doi = 10.1145/276698.276781
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| }}</ref> and that this boundary is tight<ref>
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| {{cite journal
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| | title=An optimal construction of exactly min-wise independent permutations
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| | coauthors=Takei, Y. and Itoh, T. and Shinozaki, T.
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| | journal=Technical Report COMP98-62, IEICE, 1998
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| }}
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| </ref> | |
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| Because min-wise independent families are too big for practical applications, two variant notions of min-wise independence are introduced: restricted min-wise independent permutations families, and approximate min-wise independent families.
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| Restricted min-wise independence is the min-wise independence property restricted to certain sets of cardinality at most k.<ref name=Matousek2002>{{cite journal
| |
| | author = [[Jiří Matoušek (mathematician)|Matoušek]], J.
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| | coauthors = Stojakovic, M.
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| | year = 2002
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| | title = On Restricted Min-Wise Independence of Permutations
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| | journal = Preprint
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| | url = http://citeseer.ist.psu.edu/689217.html
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| | accessdate = 2007-11-14
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| }}</ref>
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| Approximate min-wise independence differs from the property by at most a fixed <math>\epsilon</math>.<ref name=Saks2000>{{cite journal
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| | author = Saks, M.
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| | coauthors = Srinivasan, A.; Zhou, S.; Zuckerman, D.
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| | year = 2000
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| | title = Low discrepancy sets yield approximate min-wise independent permutation families
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| | journal = Information Processing Letters
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| | volume = 73
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| | issue = 1-2
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| | pages = 29–32
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| | url = http://citeseer.ist.psu.edu/saks99low.html
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| | accessdate = 2007-11-14
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| | doi = 10.1016/S0020-0190(99)00163-5
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| }}</ref>
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| | |
| ===Nilsimsa Hash===
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| {{main|Nilsimsa Hash}}
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| '''Nilsimsa''' is an [[Anti-spam techniques|anti-spam]] focused locality-sensitive hashing algorithm.<ref>{{cite web|authors=Damiani et. al|title=An Open Digest-based Technique for Spam Detection|year=2004|url=http://spdp.di.unimi.it/papers/pdcs04.pdf|accessdate=2013-09-01}}</ref> The goal of Nilsimsa is to generate a hash digest of an email message such that the digests of two similar messages are similar to each other. Nilsimsa satisfies three requirements outlined by the paper's authors: | |
| | |
| # The digest identifying each message should not vary signicantly (sic) for changes that can be produced automatically.
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| # The encoding must be robust against intentional attacks.
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| # The encoding should support an extremely low risk of false positives.
| |
| | |
| ===Random projection===
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| The random projection method of LSH<ref name=Charikar2002 /> (termed arccos by Andoni and Indyk <ref name=Andoni2008>{{cite journal
| |
| | author = Alexandr Andoni
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| | coauthors = [[Piotr Indyk|Indyk, P.]]
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| | year = 2008
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| | title = Near-Optimal Hashing Algorithms for Approximate Nearest Neighbor in High Dimensions
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| | journal = Communications of the ACM
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| | volume = 51
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| | number = 1
| |
| | pages = 117–122.
| |
| }}</ref>) is designed to approximate the [[cosine distance]] between vectors. The basic idea of this technique is to choose a random [[hyperplane]] (defined by a normal unit vector <math>r</math>) at the outset and use the hyperplane to hash input vectors.
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| Given an input vector <math>v</math> and a hyperplane defined by <math>r</math>, we let <math>h(v) = sgn(v \cdot r)</math>. That is, <math>h(v) = \pm 1</math> depending on which side of the hyperplane <math>v</math> lies.
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| Each possible choice of <math>r</math> defines a single function. Let <math>H</math> be the set of all such functions and let <math>D</math> be the uniform distribution once again. It is not difficult to prove that, for two vectors <math>u,v</math>, <math>Pr[h(u) = h(v)] = 1 - \frac{\theta(u,v)}{\pi}</math>, where <math>\theta(u,v)</math> is the angle between <math>u</math> and <math>v</math>. <math>1 - \frac{\theta(u,v)}{\pi}</math> is closely related to <math>\cos(\theta(u,v))</math>.
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| In this instance hashing produces only a single bit. Two vectors' bits match with probability proportional to the cosine of the angle between them.
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| ===Stable distributions===
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| The hash function
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| <ref name=DIIM04>{{cite journal
| |
| | author = Datar, M.
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| | coauthors = Immorlica, N., [[Piotr Indyk|Indyk, P.]], Mirrokni, V.S.
| |
| | year=2004
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| | title = Locality-Sensitive Hashing Scheme Based on p-Stable Distributions
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| | url = http://theory.csail.mit.edu/~mirrokni/pstable.ps
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| | journal = Proceedings of the Symposium on Computational Geometry
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| }}</ref> <math>h_{\mathbf{a},b} (\boldsymbol{\upsilon}) :
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| \mathcal{R}^d
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| \to \mathcal{N} </math> maps a ''d'' dimensional vector
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| <math>\boldsymbol{\upsilon}</math> onto a set of integers. Each hash function
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| in the family is indexed by a choice of random <math>\mathbf{a}</math> and
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| <math>b</math> where <math>\mathbf{a}</math> is a ''d'' dimensional
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| vector with
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| entries chosen independently from a [[stable distribution]] and
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| <math>b</math> is
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| a real number chosen uniformly from the range [0,r]. For a fixed
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| <math>\mathbf{a},b</math> the hash function <math>h_{\mathbf{a},b}</math> is
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| given by <math>h_{\mathbf{a},b} (\boldsymbol{\upsilon}) = \left \lfloor
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| \frac{\mathbf{a}\cdot \boldsymbol{\upsilon}+b}{r} \right \rfloor </math>.
| |
| | |
| Other construction methods for hash functions have been proposed to better fit the data.
| |
| <ref name=PJA10>{{cite journal
| |
| | author = Pauleve, L.
| |
| | coauthors = Jegou, H., Amsaleg, L.
| |
| | year=2010
| |
| | title = Locality sensitive hashing: A comparison of hash function types and querying mechanisms
| |
| | url = http://hal.inria.fr/inria-00567191/en/
| |
| | journal = Pattern recognition Letters
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| }}</ref>
| |
| In particular k-means hash functions are better in practice than projection-based hash functions, but without any theoretical guarantee.
| |
| | |
| ==LSH algorithm for nearest neighbor search==
| |
| | |
| One of the main applications of LSH is to provide a method for efficient approximate [[nearest neighbor search]] algorithms. Consider an LSH family <math>\mathcal F</math>. The algorithm has two main parameters: the width parameter <math>k</math> and the number of hash tables <math>L</math>.
| |
| | |
| In the first step, we define a new family <math>\mathcal G</math> of hash functions <math>g</math>, where each function <math>g</math> is obtained by concatenating <math>k</math> functions <math>h_1, ..., h_k</math> from <math>\mathcal F</math>, i.e., <math>g(p) = [h_1(p), ..., h_k(p)]</math>. In other words, a random hash function <math>g</math> is obtained by concatenating <math>k</math> randomly chosen hash functions from <math>\mathcal F</math>. The algorithm then constructs <math>L</math> hash tables, each corresponding to a different randomly chosen hash function <math>g</math>.
| |
| | |
| In the preprocessing step we hash all <math>n</math> points from the data set <math>S</math> into each of the <math>L</math> hash tables. Given that the resulting hash tables have only <math>n</math> non-zero entries, one can reduce the amount of memory used per each hash table to <math>O(n)</math> using standard [[hash functions]].
| |
| | |
| Given a query point <math>q</math>, the algorithm iterates over the <math>L</math> hash functions <math>g</math>. For each <math>g</math> considered, it retrieves the data points that are hashed into the same bucket as <math>q</math>. The process is stopped as soon as a point within distance <math>cR</math> from <math>q</math> is found.
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| | |
| Given the parameters <math>k</math> and <math>L</math>, the algorithm has the following performance guarantees:
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| * preprocessing time: <math>O(nLkt)</math>, where <math>t</math> is the time to evaluate a function <math>h \in \mathcal F</math> on an input point <math>p</math>;
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| * space: <math>O(nL)</math>, plus the space for storing data points;
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| * query time: <math>O(L(kt+dnP_2^k))</math>;
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| * the algorithm succeeds in finding a point within distance <math>cR</math> from <math>q</math> (if it exists) with probability at least <math>1 - ( 1 - P_1^k ) ^ L</math>;
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| | |
| For a fixed approximation ratio <math>c=1+\epsilon</math> and probabilities <math>P_1</math> and <math>P_2</math>, one can set <math>k={\log n \over \log 1/P_2}</math> and <math>L = n^{\rho}</math>, where <math>\rho={\log P_1\over \log P_2}</math>. Then one obtains the following performance guarantees:
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| * preprocessing time: <math>O(n^{1+\rho}kt)</math>;
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| * space: <math>O(n^{1+\rho})</math>, plus the space for storing data points;
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| * query time: <math>O(n^{\rho}(kt+d))</math>;
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| | |
| ==See also==
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| *[[Curse of dimensionality]]
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| *[[Feature hashing]]
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| *[[Fourier-related transforms]]
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| *[[Multilinear subspace learning]]
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| *[[Principal component analysis]]
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| *[[Singular value decomposition]]
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| *[[Wavelet compression]]
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| *[[Rolling hash]]
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| *[[Bloom Filter]]
| |
| | |
| ==References==
| |
| | |
| {{reflist}}
| |
| | |
| ==Further reading==
| |
| | |
| *Samet, H. (2006) ''Foundations of Multidimensional and Metric Data Structures''. Morgan Kaufmann. ISBN 0-12-369446-9
| |
| | |
| ==External links==
| |
| * [http://web.mit.edu/andoni/www/LSH/index.html Alex Andoni's LSH homepage]
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| * [http://lshkit.sourceforge.net/ LSHKIT: A C++ Locality Sensitive Hashing Library]
| |
| * [http://www.vision.caltech.edu/malaa/software/research/image-search/ Caltech Large Scale Image Search Toolbox]: a Matlab toolbox implementing several LSH hash functions, in addition to Kd-Trees, Hierarchical K-Means, and Inverted File search algorithms.
| |
| *[http://infolab.stanford.edu/~manku/papers/07www-duplicates.pdf Simhash at Google]
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| * [https://github.com/salviati/slash Slash: A C++ LSH library, implementing Spherical LSH by Terasawa, K., Tanaka, Y]
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| {{DEFAULTSORT:Locality Sensitive Hashing}}
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| [[Category:Search algorithms]]
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| [[Category:Classification algorithms]]
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| [[Category:Dimension reduction]]
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| [[Category:Hashing]]
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| [[Category:Probabilistic data structures]]
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