Locality-sensitive hashing

Template:Short description In computer science, locality-sensitive hashing (LSH) is a fuzzy hashing technique that hashes similar input items into the same "buckets" with high probability. (The number of buckets is much smaller than the universe of possible input items.) Since similar items end up in the same buckets, this technique can be used for data clustering and nearest neighbor search. It differs from conventional hashing techniques in that hash collisions are maximized, not minimized. Alternatively, the technique can be seen as a way to 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).

Locality-preserving hashing was initially devised as a way to facilitate data pipelining in implementations of massively parallel algorithms that use randomized routing and universal hashing to reduce memory contention and network congestion.

Definitions

A finite family $ℱ$ of functions $h:M\to S$ is defined to be an LSH family for

• a metric space $ℳ=(M,d)$,
• a threshold $r>0$,
• an approximation factor $c>1$,
• and probabilities $p_1>p_2$

if it satisfies the following condition. For any two points $a,b\in M$ and a hash function $h$ chosen uniformly at random from $ℱ$:

• If $d(a,b)\le r$, then $h(a)=h(b)$ (i.e., Template:Mvar and Template:Mvar collide) with probability at least $p_1$,
• If $d(a,b)\ge cr$, then $h(a)=h(b)$ with probability at most $p_2$.

Such a family $ℱ$ is called $(r,cr,p_1,p_2)$-sensitive.

LSH with respect to a similarity measure

Alternatively it is possible to define an LSH family on a universe of items Template:Mvar endowed with a similarity function $\varphi :U\times U\to [0,1]$. In this setting, a LSH scheme is a family of hash functions Template:Mvar coupled with a probability distribution Template:Mvar over Template:Mvar such that a function $h\in H$ chosen according to Template:Mvar satisfies $Pr[h(a)=h(b)]=\varphi (a,b)$ for each $a,b\in U$.

Amplification

Given a $(d_1,d_2,p_1,p_2)$-sensitive family $ℱ$, we can construct new families $𝒢$ by either the AND-construction or OR-construction of $ℱ$.

To create an AND-construction, we define a new family $𝒢$ of hash functions Template:Mvar, where each function Template:Mvar is constructed from Template:Mvar random functions $h_1,\dots ,h_k$ from $ℱ$. We then say that for a hash function $g\in 𝒢$, $g(x)=g(y)$ if and only if all $h_i(x)=h_i(y)$ for $i=1,2,\dots ,k$. Since the members of $ℱ$ are independently chosen for any $g\in 𝒢$, $𝒢$ is a $(d_1,d_2,p_1^k,p_2^k)$-sensitive family.

To create an OR-construction, we define a new family $𝒢$ of hash functions Template:Mvar, where each function Template:Mvar is constructed from Template:Mvar random functions $h_1,\dots ,h_k$ from $ℱ$. We then say that for a hash function $g\in 𝒢$, $g(x)=g(y)$ if and only if $h_i(x)=h_i(y)$ for one or more values of Template:Mvar. Since the members of $ℱ$ are independently chosen for any $g\in 𝒢$, $𝒢$ is a $(d_1,d_2,1-(1-p_1{)}^k,1-(1-p_2{)}^k)$-sensitive family.

Applications

LSH has been applied to several problem domains, including:

• Near-duplicate detection
• Hierarchical clustering
• Genome-wide association study
• Image similarity identification
  • VisualRank
• Gene expression similarity identificationTemplate:Citation needed
• Audio similarity identification
• Nearest neighbor search
• Audio fingerprint
• Digital video fingerprinting
• Shared memory organization in parallel computing
• Physical data organization in database management systems
• Training fully connected neural networks
• Computer security
• Machine Learning

Methods

Bit sampling for Hamming distance

One of the easiest ways to construct an LSH family is by bit sampling. This approach works for the Hamming distance over Template:Mvar-dimensional vectors $\{0,1{\}}^d$. Here, the family $ℱ$ of hash functions is simply the family of all the projections of points on one of the $d$ coordinates, i.e., $ℱ=\{h:\{0,1{\}}^d\to \{0,1\}\mid h(x)=x_i\text{ for some }i\in \{1,\dots ,d\}\}$, where $x_i$ is the $i$th coordinate of $x$. A random function $h$ from $ℱ$ simply selects a random bit from the input point. This family has the following parameters: $P_1=1-R/d$, $P_2=1-cR/d$. That is, any two vectors $x,y$ with Hamming distance at most $R$ collide under a random $h$ with probability at least $P_1$. Any $x,y$ with Hamming distance at least $cR$ collide with probability at most $P_2$.

Min-wise independent permutations

Suppose Template:Mvar is composed of subsets of some ground set of enumerable items Template:Mvar and the similarity function of interest is the Jaccard index Template:Mvar. If Template:Mvar is a permutation on the indices of Template:Mvar, for $A\subseteq S$ let $h(A)=\underset{a\in A}{\min }\{\pi (a)\}$. Each possible choice of Template:Mvar defines a single hash function Template:Mvar mapping input sets to elements of Template:Mvar.

Define the function family Template:Mvar to be the set of all such functions and let Template:Mvar be the uniform distribution. Given two sets $A,B\subseteq S$ the event that $h(A)=h(B)$ corresponds exactly to the event that the minimizer of Template:Mvar over $A\cup B$ lies inside $A\cap B$. As Template:Mvar was chosen uniformly at random, $Pr[h(A)=h(B)]=J(A,B)$ and $(H,D)$ define an LSH scheme for the Jaccard index.

Because the symmetric group on Template:Mvar elements has size Template:Mvar!, choosing a truly random permutation from the full symmetric group is infeasible for even moderately sized Template:Mvar. 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 Template:Mvar. It has been established that a min-wise independent family of permutations is at least of size $\mathrm{lcm}\{1,2,\dots ,n\}\ge e^{n-o(n)}$, and that this bound is tight.

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. Restricted min-wise independence is the min-wise independence property restricted to certain sets of cardinality at most Template:Mvar. Approximate min-wise independence differs from the property by at most a fixed Template:Mvar.

Open source methods

Nilsimsa Hash

Nilsimsa is a locality-sensitive hashing algorithm used in anti-spam efforts. 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. The paper suggests that the Nilsimsa satisfies three requirements:

1. The digest identifying each message should not vary significantly for changes that can be produced automatically.
2. The encoding must be robust against intentional attacks.
3. 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.

TLSH

TLSH is locality-sensitive hashing algorithm designed for a range of security and digital forensic applications. 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.

An implementation of TLSH is available as open-source software.

Random projection

$\frac{\theta (u,v)}{\pi }$ is proportional to $1-\cos (\theta (u,v))$ on the interval [0, $\pi$

The random projection method of LSH due to Moses Charikar called SimHash (also sometimes called arccos) uses an approximation of the cosine distance between vectors. The technique was used to approximate the NP-complete max-cut problem.

The basic idea of this technique is to choose a random hyperplane (defined by a normal unit vector Template:Mvar) at the outset and use the hyperplane to hash input vectors.

Given an input vector Template:Mvar and a hyperplane defined by Template:Mvar, we let $h(v)=\mathrm{sgn}(v\cdot r)$. That is, $h(v)=\pm 1$ depending on which side of the hyperplane Template:Mvar lies. This way, each possible choice of a random hyperplane Template:Mvar can be interpreted as a hash function $h(v)$.

For two vectors Template:Mvar with angle $\theta (u,v)$ between them, it can be shown that

$Pr[h(u)=h(v)]=1-\frac{\theta (u,v)}{\pi }.$

Since the ratio between $\frac{\theta (u,v)}{\pi }$ and $1-\cos (\theta (u,v))$ is at least 0.439 when $\theta (u,v)\in [0,\pi ]$, the probability of two vectors being on different sides of the random hyperplane is approximately proportional to the cosine distance between them.

Stable distributions

The hash function $h_{𝐚,b}(\mathit{\upsilon }):{ℛ}^d\to 𝒩$ maps a Template:Mvar-dimensional vector $\mathit{\upsilon }$ onto the set of integers. Each hash function in the family is indexed by a choice of random $𝐚$ and $b$ where $𝐚$ is a Template:Mvar-dimensional vector with entries chosen independently from a stable distribution and $b$ is a real number chosen uniformly from the range [0,r]. For a fixed $𝐚,b$ the hash function $h_{𝐚,b}$ is given by $h_{𝐚,b}(\mathit{\upsilon })=\lfloor \frac{𝐚\cdot \mathit{\upsilon }+b}{r}\rfloor$.

Other construction methods for hash functions have been proposed to better fit the data. 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. The hashcodes are found via training of an artificial neural network or graphical model.Template:Citation needed

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 $ℱ$. The algorithm has two main parameters: the width parameter Template:Mvar and the number of hash tables Template:Mvar.

In the first step, we define a new family $𝒢$ of hash functions Template:Mvar, where each function Template:Mvar is obtained by concatenating Template:Mvar functions $h_1,\dots ,h_k$ from $ℱ$, i.e., $g(p)=[h_1(p),\dots ,h_k(p)]$. In other words, a random hash function Template:Mvar is obtained by concatenating Template:Mvar randomly chosen hash functions from $ℱ$. The algorithm then constructs Template:Mvar hash tables, each corresponding to a different randomly chosen hash function Template:Mvar.

In the preprocessing step we hash all Template:Mvar Template:Mvar-dimensional points from the data set Template:Mvar into each of the Template:Mvar hash tables. Given that the resulting hash tables have only Template:Mvar non-zero entries, one can reduce the amount of memory used per each hash table to $O(n)$ using standard hash functions.

Given a query point Template:Mvar, the algorithm iterates over the Template:Mvar hash functions Template:Mvar. For each Template:Mvar considered, it retrieves the data points that are hashed into the same bucket as Template:Mvar. The process is stopped as soon as a point within distance Template:Mvar from Template:Mvar is found.

Given the parameters Template:Mvar and Template:Mvar, the algorithm has the following performance guarantees:

• preprocessing time: $O(nLkt)$, where Template:Mvar is the time to evaluate a function $h\in ℱ$ on an input point Template:Mvar;
• space: $O(nL)$, plus the space for storing data points;
• query time: $O(L(kt+dn{P}_2^k))$;
• the algorithm succeeds in finding a point within distance Template:Mvar from Template:Mvar (if there exists a point within distance Template:Mvar) with probability at least $1-(1-P_1^k{)}^L$;

For a fixed approximation ratio $c=1+ϵ$ and probabilities $P_1$ and $P_2$, one can set $k=\lceil \frac{\log n}{\log 1/P_2}\rceil$ and $L=\lceil P_1^{-k}\rceil =O(n^{\rho }{P}_1^{-1})$, where $\rho =\frac{\log P_1}{\log P_2}$. Then one obtains the following performance guarantees:

• preprocessing time: $O(n^{1+\rho }{P}_1^{-1}kt)$;
• space: $O(n^{1+\rho }{P}_1^{-1})$, plus the space for storing data points;
• query time: $O(n^{\rho }{P}_1^{-1}(kt+d))$;

Improvements

When Template:Mvar is large, it is possible to reduce the hashing time from $O(n^{\rho })$. This was shown by and which gave

• query time: $O(t{\log }^2(1/P_2)/P_1+n^{\rho }(d+1/P_1))$;
• space: $O(n^{1+\rho }/P_1+{\log }^2(1/P_2)/P_1)$;

It is also sometimes the case that the factor $1/P_1$ 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 it was shown how to reduce the query time to $O(n^{\rho }/P_1^{1-\rho })$ (not including hashing costs) and similarly the space usage.

See also

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• Random indexing
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Further reading

• Samet, H. (2006) Foundations of Multidimensional and Metric Data Structures. Morgan Kaufmann. Template:ISBN
• Template:Cite conference
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External links

• Alex Andoni's LSH homepage
• LSHKIT: A C++ Locality Sensitive Hashing Library
• A Python Locality Sensitive Hashing library that optionally supports persistence via redis
• 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.
• Slash: A C++ LSH library, implementing Spherical LSH by Terasawa, K., Tanaka, Y
• LSHBOX: An Open Source C++ Toolbox of Locality-Sensitive Hashing for Large Scale Image Retrieval, Also Support Python and MATLAB.
• SRS: A C++ Implementation of An In-memory, Space-efficient Approximate Nearest Neighbor Query Processing Algorithm based on p-stable Random Projection
• TLSH open source on Github
• JavaScript port of TLSH (Trend Micro Locality Sensitive Hashing) bundled as node.js module
• Java port of TLSH (Trend Micro Locality Sensitive Hashing) bundled as maven package