Locality-sensitive hashing: Difference between revisions

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{{Short description|Algorithmic technique using hashing}}

In [[computer science]], '''locality-sensitive hashing''' ('''LSH''') is ana algorithmic[[fuzzy hashing]] technique that hashes similar input items into the same "buckets" with high probability.<ref name="MOMD">{{cite web|url=http://infolab.stanford.edu/~ullman/mmds.html|title=Mining of Massive Datasets, Ch. 3.|last1=Rajaraman|first1=A.|last2=Ullman|first2=J.|author2-link=Jeffrey Ullman|year=2010}}</ref> (The number of buckets is much smaller than the universe of possible input items.)<ref name="MOMD" /> Since similar items end up in the same buckets, this technique can be used for [[Cluster analysis|data clustering]] and [[nearest neighbor search]]. It differs from [[Hash function|conventional hashing techniques]] in that [[hash collision]]s are maximized, not minimized. Alternatively, the technique can be seen as a way to [[dimension reduction|reduce the dimensionality]] of high-dimensional data; high-dimensional input items can be reduced to low-dimensional versions while preserving relative distances between items.

Hashing-based approximate [[nearest -neighbor search]] algorithms generally use one of two main categories of hashing methods: either data-independent methods, such as locality-sensitive hashing (LSH); or data-dependent methods, such as [[locality-preserving hashing]] (LPH).<ref>{{cite conference |last1=Zhao |first1=Kang |last2=Lu |first2=Hongtao |last3=Mei |first3=Jincheng |title=Locality Preserving Hashing |conference=AAAI Conference on Artificial Intelligence | volume=28 | year=2014 |url=https://ojs.aaai.org/index.php/AAAI/article/view/9133/8992 |pages=2874–2880}}</ref><ref>{{cite book |last1=Tsai |first1=Yi-Hsuan |last2=Yang |first2=Ming-Hsuan |title=2014 IEEE International Conference on Image Processing (ICIP) |chapter=Locality preserving hashing |date=October 2014 |pages=2988–2992 |doi=10.1109/ICIP.2014.7025604 |isbn=978-1-4799-5751-4 |s2cid=8024458 |issn=1522-4880}}</ref>

The first family of localityLocality-preserving hash functionshashing was initially devised as a way to facilitate [[Pipeline (computing)|data pipelining]] in implementations of [[Parallel RAM|parallelmassively random-access machine (PRAM)parallel]] algorithms that use [[Routing#Path_selection|randomized routing]] and [[universal hashing]] to reduce memory [[Resource_contentionResource contention|contention]] and [[network congestion]].<ref name=Chin1991>{{cite thesis |last=Chin |first=Andrew |date=1991 |title=Complexity Issues in General Purpose Parallel Computing |pages=87–95 |type=DPhil |publisher=University of Oxford |url=https://perma.cc/E47H-WCVP}}</ref><ref name=Chin1994>{{cite journal |last1=Chin |first1=Andrew |date=1994 |title=Locality-Preserving Hash Functions for General Purpose Parallel Computation |url=http://unclaw.com/chin/scholarship/hashfunctions.pdf |journal=Algorithmica |volume=12 |issue=2–3 |pages=170–181 |doi=10.1007/BF01185209|s2cid=18108051 }}</ref>

==Definitions==

AnA finite family <math> \mathcal F</math> of functions <math>h\colon M \to S</math> is defined to be an ''LSH family''<ref name=MOMD /><ref name=GIM1999>{{cite journal

| author1 = Gionis, A.

| author2-link = Piotr Indyk | author2 = Indyk, P. | author3-link = Rajeev Motwani | author3 = Motwani, R.

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| title-link = Symposium on Theory of Computing

}}</ref>

for

<math>\mathcal F</math> is defined for

* a [[metric space]] <math>\mathcal M =(M, d)</math>,

* a threshold <math>Rr>0</math>,

* an approximation factor <math>c>1</math>,

* and probabilities <math>P_1</mathp_1 > and <math>P_2p_2</math>.

Thisif familyit <math>\mathcal F</math> is a set of functions <math>h\colon M \to S</math> that map elements ofsatisfies the metricfollowing space to buckets <math>s \in S</math>condition. An LSH family must satisfy the following conditions forFor any two points <math>pa, qb \in M</math> and anya hash function <math>h</math> chosen uniformly at random from <math>\mathcal F</math>:

* ifIf <math>d(pa,qb) \le Rr</math>, then <math>h(pa)=h(qb)</math> (i.e., {{mvar|pa}} and {{mvar|qb}} collide) with probability at least <math>P_1p_1</math>,

* ifIf <math>d(pa,qb) \ge cRcr</math>, then <math>h(pa)=h(qb)</math> with probability at most <math>P_2p_2</math>.

A family is interesting when <math>P_1>P_2</math>. Such a family <math>\mathcal F</math> is called ''<math>(Rr,cRcr,P_1p_1,P_2p_2)</math>-sensitive''.

=== LSH with respect to a similarity measure ===

Alternatively<ref name=Charikar2002>{{cite conference

| last = Charikar|first= Moses S.|author-link=Moses Charikar

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| pages = 380–388

| doi = 10.1145/509907.509965

|isbn= 1-58113-495-9| citeseerx = 10.1.1.147.4064}}</ref> it is definedpossible withto respectdefine toan LSH family on a universe of items {{mvar|U}} thatendowed havewith a [[String metric|similarity]] function <math>\phi\colon U \times U \to [0,1]</math>. AnIn this setting, a LSH scheme is a family of [[hash function]]s {{mvar|H}} coupled with a [[probability distribution]] {{mvar|D}} over the functions{{mvar|H}} such that a function <math>h \in H</math> chosen according to {{mvar|D}} satisfies the property that <math>Pr_{h \in H}Pr [h(a) = h(b)] = \phi(a,b)</math> for anyeach <math>a,b \in U</math>.

===Locality-preserving hashing===

A '''locality-preserving hash''' is a [[hash function]] {{mvar|f}} that maps points in a [[metric space]] <math>\mathcal M =(M, d)</math> to a scalar value such that

:<math>d(p,q) < d(q,r) \Rightarrow |f(p) - f(q)| < |f(q) - f(r)|</math>

for any three points <math>p,q,r \in M</math>.{{citation needed|date=November 2022|reason=unusual definition}}

In other words, these are hash functions where the relative distance between the input values is preserved in the relative distance between the output hash values; input values that are closer to each other will produce output hash values that are closer to each other.

This is in contrast to [[cryptography|cryptographic]] hash functions and [[checksum]]s, which are designed to have [[Avalanche effect|random output difference between adjacent inputs]].

The first family of locality-preserving hash functions was devised as a way to facilitate [[Pipeline (computing)|data pipelining]] in implementations of [[Parallel RAM|parallel random-access machine (PRAM)]] algorithms that use [[universal hashing]] to reduce memory [[Resource_contention|contention]] and [[network congestion]].<ref name=Chin1991>{{cite thesis |last=Chin |first=Andrew |date=1991 |title=Complexity Issues in General Purpose Parallel Computing |pages=87–95 |type=DPhil |publisher=University of Oxford |url=https://perma.cc/E47H-WCVP}}</ref><ref name=Chin1994>{{cite journal |last1=Chin |first1=Andrew |date=1994 |title=Locality-Preserving Hash Functions for General Purpose Parallel Computation |url=http://unclaw.com/chin/scholarship/hashfunctions.pdf |journal=Algorithmica |volume=12 |issue=2–3 |pages=170–181 |doi=10.1007/BF01185209|s2cid=18108051 }}</ref>

Locality preserving hashes are related to [[space-filling curve]]s.{{how|date=March 2020}}

===Amplification===

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{{citation

| author = Das, Abhinandan S.

| journaltitle = Proceedings of the 16th Internationalinternational Conferenceconference on World Wide Web | pages = 271

| titlechapter = Google news personalization: scalable online collaborative filtering

| journal = Proceedings of the 16th International Conference on World Wide Web | pages = 271

| pages = 271–280

| year = 2007|doi=10.1145/1242572.1242610|display-authors=etal| isbn = 9781595936547

| s2cid = 207163129

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{{citation

| author = Cochez, Michael |author2=Mou, Hao

| journal title= Proceeding SIGMOD '15 Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data | pages=505–517

| title = Twister Tries: Approximate Hierarchical Agglomerative Clustering for Average Distance in Linear Time

|chapter=Twister Tries

| journal = Proceeding SIGMOD '15 Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data | pages=505–517

| pages=505–517

| year = 2015 | doi=10.1145/2723372.2751521| isbn = 9781450327589

|s2cid=14414777

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**[[VisualRank]]

*[[Gene expression]] similarity identification{{citation needed|date=October 2013}}

*[[Audio similarity]] identification

*[[Nearest neighbor search]]

*[[Audio fingerprint]]<ref>

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| url = https://github.com/worldveil/dejavu| date = 2018-12-19}}

</ref>

*[[Digital video fingerprinting]]<ref>

{{citation

| title = A Simple Introduction to Locality Sensitive Hashing (LSH)

| url =https://www.iunera.com/kraken/fabric/locality-sensitive-hashing-lsh/#7-applications-of-lsh| date = 2025-03-27}}

</ref>

*[[Shared memory]] organization in [[parallel computing]]<ref name=Chin1991 /><ref name=Chin1994 />

*Physical data organization in database management systems<ref>

{{citation

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{{cite conference

| author1 = Oliver, Jonathan| author2 = Cheng, Chun | author3 = Chen, Yanggui

| title = TLSH - a locality sensitive hash

| journaltitle = 4th2013 Fourth Cybercrime and Trustworthy Computing Workshop | chapter = TLSH -- A Locality Sensitive Hash | year = 2013

| year = 2013

| pages = 7–13 | url = https://www.academia.edu/7833902

| doi = 10.1109/CTC.2013.9

| isbn = 978-1-4799-3076-0

}}

</ref>

*[[Machine Learning]]<ref name="NL">

{{citation

| author1 = Fanaee-T, Hadi

| title = Natural Learning

| year = 20132024

| arxiv = 2404.05903

}}

</ref>

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# The encoding must be robust against intentional attacks.

# The encoding should support an extremely low risk of false positives.

Testing performed in the paper on a range of file types identified the Nilsimsa hash as having a significantly higher false positive rate when compared to other similarity digest schemes such as TLSH, Ssdeep and Sdhash.<ref>{{cite journal|author=Oliver|display-authors=etal|title=TLSH - A Locality Sensitive Hash|url=https://www.academia.edu/7833902|journal=4th Cybercrime and Trustworthy Computing Workshop|date=2013|accessdate=2015-06-04}}</ref>

====TLSH====

'''TLSH''' is locality-sensitive hashing algorithm designed for a range of security and digital forensic applications.<ref name="TLSH"/> The goal of TLSH is to generate hash digests for messages such that low distances between digests indicate that their corresponding messages are likely to be similar.

Testing performed in the paper on a range of file types identified the Nilsimsa hash as having a significantly higher false positive rate when compared to other similarity digest schemes such as TLSH, Ssdeep and Sdhash.

An implementation of TLSH is available as [[open-source software]].<ref>{{cite web|url=https://github.com/trendmicro/tlsh |title=TLSH |website=[[GitHub]] |access-date=2014-04-10}}</ref>

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===Random projection===

{{main|Random projection}}

[[File:Cosine-distance.png| thumb | <math>\frac{\theta(u,v)}{\pi}</math> is approximately proportional to <math>1-\cos(\theta(u,v))</math> on the interval [0, <math>\pi</math>]]]

The random projection method of LSH due to [[Moses Charikar]]<ref name=Charikar2002 /> called [[SimHash]] (also sometimes called arccos<ref name=Andoni2008>{{cite journal

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The basic idea of this technique is to choose a random [[hyperplane]] (defined by a normal unit vector {{mvar|r}}) at the outset and use the hyperplane to hash input vectors.

Given an input vector {{mvar|v}} and a hyperplane defined by {{mvar|r}}, 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 {{mvar|v}} lies. This way, each possible choice of a random hyperplane {{mvar|r}} can be interpreted as a hash function <math>h(v)</math>.

For two vectors <math>{{mvar|u,v</math>}} with angle <math>\theta(u,v)</math> between them, it can be shown that

:<math>Pr[h(u) = h(v)] = 1 - \frac{\theta(u,v)}{\pi}.</math>

Since the ratio between <math>\frac{\theta(u,v)}{\pi}</math> is proportional toand <math>1-\cos(\theta(u,v))</math>. Inis factat theleast ratio0.439 iswhen always<math>\theta(u, withinv) a\in factor[0, of .87856\pi]</math>,<ref name=Charikar2002 /><ref name="Goemans Williamson 1995 pp. 1115–1145">{{cite journal | lastlast1=Goemans | firstfirst1=Michel X. | last2=Williamson | first2=David P. | title=Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming | journal=Journal of the ACM | publisher=Association for Computing Machinery (ACM) | volume=42 | issue=6 | year=1995 | issn=0004-5411 | doi=10.1145/227683.227684 | pages=1115–1145| s2cid=15794408 | doi-access=free }}</ref>. Meaning the probability of the two vectors being on the samedifferent sidesides of the random hyperplane is approximately proportional to the [[Cosine_similarityCosine similarity#Cosine_DistanceCosine Distance|cosine distance]] between them.

===Stable distributions===

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| url = http://hal.inria.fr/inria-00567191/en/

| journal = Pattern Recognition Letters

| volume = 31 | issue = 11 | pages = 1348–1358 | doi = 10.1016/j.patrec.2010.04.004 | bibcode = 2010PaReL..31.1348P | s2cid = 2666044 }}</ref>

In particular k-means hash functions are better in practice than projection-based hash functions, but without any theoretical guarantee.

===Semantic hashing===

Semantic hashing is a technique that attempts to map input items to addresses such that closer inputs have higher [[semantic similarity]].<ref>{{Cite journal|last1=Salakhutdinov|first1=Ruslan|last2=Hinton|first2=Geoffrey|date=2008|title=Semantic hashing|journal=International Journal of Approximate Reasoning|language=en|volume=50|issue=7|pages=969–978|doi=10.1016/j.ijar.2008.11.006|doi-access=free}}</ref> The hashcodes are found via training of an [[artificial neural network]] or [[graphical model]].{{cncitation needed|date=September 2021}}

==Algorithm for nearest neighbor search==

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In the preprocessing step we hash all {{mvar|n}} {{mvar|d}}-dimensional points from the data set {{mvar|S}} into each of the {{mvar|L}} hash tables. Given that the resulting hash tables have only {{mvar|n}} 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 {{mvar|q}}, the algorithm iterates over the {{mvar|L}} hash functions {{mvar|g}}. For each {{mvar|g}} considered, it retrieves the data points that are hashed into the same bucket as {{mvar|q}}. The process is stopped as soon as a point within distance <math>{{mvar|cR</math>}} from {{mvar|q}} is found.

Given the parameters {{mvar|k}} and {{mvar|L}}, the algorithm has the following performance guarantees:

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* space: <math>O(nL)</math>, plus the space for storing data points;

* query time: <math>O(L(kt+dnP_2^k))</math>;

* the algorithm succeeds in finding a point within distance <math>{{mvar|cR</math>}} from {{mvar|q}} (if there exists a point within distance {{mvar|R}}) with probability at least <math>1 - ( 1 - P_1^k ) ^ L</math>;

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 = \bigleft\lceil{\tfrac{\log n}{\log 1/P_2}}\bigright\rceil</math> and <math>L = \lceil P_1^{-k}\rceil = O(n^{\rho}P_1^{-1})</math>, where <math>\rho={\tfrac{\log P_1}{\log P_2}}</math>. Then one obtains the following performance guarantees:

* preprocessing time: <math>O(n^{1+\rho}P_1^{-1}kt)</math>;

* space: <math>O(n^{1+\rho}P_1^{-1})</math>, plus the space for storing data points;

* query time: <math>O(n^{\rho}P_1^{-1}(kt+d))</math>;

===Finding nearest neighbor without fixed dimensionality===

To generalize the above algorithm without radius {{mvar|R}} being fixed, we can take the algorithm and do a sort of binary search over {{mvar|R}}. It has been shown<ref>{{cite journal |last1=Har-Peled |first1=Sariel |last2=Indyk |first2=Piotr |last3=Motwani |first3=Rajeev |title=Approximate Nearest Neighbor: Towards Removing the Curse of Dimensionality |journal=Theory of Computing |date=2012 |volume=8 |issue=Special Issue in Honor of Rajeev Motwani |pages=321-350 |doi=10.4086/toc.2012.v008a014 |url=https://theoryofcomputing.org/articles/v008a014/v008a014.pdf |access-date=23 May 2025}}</ref> that there is a data structure for the approximate nearest neighbor with the following performance guarantees:

* space: <math>O(n^{1+\rho}P_1^{-1}d\log^2 n)</math>;

* query time: <math>O(n^{\rho}P_1^{-1}(kt+d)\log n)</math>;

* the algorithm succeeds in finding the nearest neighbor with probability at least <math>1 - (( 1 - P_1^k ) ^ L\log n)</math>;

===Improvements===

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When {{mvar|t}} is large, it is possible to reduce the hashing time from <math>O(n^{\rho})</math>.

This was shown by<ref>Dahlgaard, Søren, Mathias Bæk Tejs Knudsen, and Mikkel Thorup. [https://arxiv.org/pdf/1704.04370 "Fast similarity sketching."] 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2017.</ref> and<ref>Christiani, Tobias. [https://arxiv.org/pdf/1708.07586 "Fast locality-sensitive hashing frameworks for approximate near neighbor search."] International Conference on Similarity Search and Applications. Springer, Cham, 2019.</ref> which gave

* query time: <math>O(t\log^2(1/P_2)/P_1 + n^{\rho}(d + 1/P_1))</math>;

* space: <math>O(n^{1+\rho}/P_1 + \log^2(1/P_2)/P_1)</math>;

It is also sometimes the case that the factor <math>1/P_1</math> can be very large.

This happens for example with [[Jaccard similarity]] data, where even the most similar neighbor often has a quite low Jaccard similarity with the query.

In<ref>Ahle, Thomas Dybdahl. "On the Problem of $$ <math>p_1^{-1} $$</math> in Locality-Sensitive Hashing." International Conference on Similarity Search and Applications. Springer, Cham, 2020.</ref> it was shown how to reduce the query time to <math>O(n^\rho/P_1^{1-\rho})</math> (not including hashing costs) and similarly the space usage.

==See also==

*[[Bloom filter]]

* {{Annotated link |Bloom filter}}

*[[ {{Annotated link |Curse of dimensionality]]}}

*[[ {{Annotated link |Feature hashing]]}}

*[[ {{Annotated link |Fourier-related transforms]]}}

*[[Geohash]]

* {{Annotated link |Geohash}}

*[[ {{Annotated link |Multilinear subspace learning]]}}

*[[ {{Annotated link |Principal component analysis]]}}

*[[Random indexing]]<ref>Gorman, James, and James R. Curran. [https://aclanthology.org/P06-1046.pdf "Scaling distributional similarity to large corpora."] Proceedings of the 21st International Conference on Computational Linguistics and the 44th annual meeting of the Association for Computational Linguistics. Association for Computational Linguistics, 2006.</ref>

*[[ {{Annotated link |Rolling hash]]}}

*[[ {{Annotated link |Singular value decomposition]]}}

*[[ {{Annotated link |Sparse distributed memory]]}}

*[[ {{Annotated link |Wavelet compression]]}}

* {{Annotated link |Locality of reference}}

==References==

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*Samet, H. (2006) ''Foundations of Multidimensional and Metric Data Structures''. Morgan Kaufmann. {{ISBN|0-12-369446-9}}

*{{Cite conference| last1 = Indyk | first1 = Piotr | author1-link= Piotr Indyk| last2 = Motwani | first2 = Rajeev | author2-link = Rajeev Motwani| last3 = Raghavan | first3 = Prabhakar| last4 = Vempala | first4 = Santosh | author4-link = Santosh Vempala| title = Locality-preserving hashing in multidimensional spaces | doi = 10.1145/258533.258656 | book-title = Proceedings of the twenty-ninth annual ACM symposium on Theory of computing| conference = [[Symposium on Theory of Computing|STOC '97]]| pages = 618–625| year = 1997 | isbn = 978-0-89791-888-6| citeseerx = 10.1.1.50.4927| s2cid = 15693787 }}

*{{Cite journal | last1 = Chin | first1 = Andrew| title = Locality-preserving hash functions for general purpose parallel computation | url = http://www.unclaw.com/chin/scholarship/hashfunctions.pdf| doi = 10.1007/BF01185209 | journal = [[Algorithmica]]| volume = 12 | issue = 2–3 | pages = 170–181 | year = 1994 | s2cid = 18108051}}

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==External links==

* [http://web.mit.edu/andoni/www/LSH/index.html Alex Andoni's LSH homepage]

* [httphttps://lshkit.sourceforge.net/ LSHKIT: A C++ Locality Sensitive Hashing Library]

* [https://github.com/simonemainardi/LSHash A Python Locality Sensitive Hashing library that optionally supports persistence via redis]

* [https://web.archive.org/web/20101203074412/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.