In computer science and data mining, MinHash (or the min-wise independent permutations locality sensitive hashing scheme) is a technique for quickly estimating how similar two sets are. The scheme was invented by Andrei Broder (1997),^[1] and initially used in the AltaVista search engine to detect duplicate web pages and eliminate them from search results.^[2] It has also been applied in large-scale clustering problems, such as clustering documents by the similarity of their sets of words.^[1]

Jaccard similarity and minimum hash values

The Jaccard similarity coefficient is a commonly used indicator of the similarity between two sets. Let U be a set and A and B be subsets of U, then the Jaccard index is defined to be the ratio of the number of elements of their intersection and the number of elements of their union:

${\displaystyle J(A,B)={{|A\cap B|} \over {|A\cup B|}}.}$

This value is 0 when the two sets are disjoint, 1 when they are equal, and strictly between 0 and 1 otherwise. Two sets are more similar (i.e. have relatively more members in common) when their Jaccard index is closer to 1. The goal of MinHash is to estimate J(A,B) quickly, without explicitly computing the intersection and union.

Let h be a hash function that maps the members of U to distinct integers, let perm be a random permutation of the elements of the set U, and for any subset S of U define h_min(S) to be the minimal member of S with respect to h ∘ perm—that is, the member x of S with the minimum value of h(perm(x)). (In cases where the hash function used is assumed to have pseudo-random properties, the random permutation would not be used.)

Now, applying h_min to both A and B, and assuming no hash collisions, we see that the values are equal (h_min(A) = h_min(B)) if and only if among all elements of ${\displaystyle A\cup B}$, the element with the minimum hash value lies in the intersection ${\displaystyle A\cap B}$. The probability of this being true is exactly the Jaccard index, therefore:

Pr[ h_min(A) = h_min(B) ] = J(A,B),

That is, the probability that h_min(A) = h_min(B) is true is equal to the similarity J(A,B), assuming drawing perm from a uniform distribution. In other words, if r is the random variable that is one when h_min(A) = h_min(B) and zero otherwise, then r is an unbiased estimator of J(A,B). r has too high a variance to be a useful estimator for the Jaccard similarity on its own, because ${\displaystyle r}$ is always zero or one. The idea of the MinHash scheme is to reduce this variance by averaging together several variables constructed in the same way.

Algorithm

Incorporating weights

A variety of techniques to introduce weights into the computation of MinHashes have been developed. The simplest extends it to integer weights.^[4] Extend our hash function h to accept both a set member and an integer, then generate multiple hashes for each item, according to its weight. If item i occurs n times, generate hashes ${\displaystyle h(i,1),h(i,2),\ldots ,h(i,n)}$. Run the original algorithm on this expanded set of hashes. Doing so yields the weighted Jaccard Index as the collision probability.

${\displaystyle J_{\mathcal {W}}(x,y)={\frac {\sum _{i}\min(x_{i},y_{i})}{\sum _{i}\max(x_{i},y_{i})}}}$

Further extensions that achieve this collision probability on real weights with better runtime have been developed, one for dense data,^[5] and another for sparse data.^[6]

Another family of extensions use exponentially distributed hashes. A uniformly random hash between 0 and 1 can be converted to follow an exponential distribution by CDF inversion. This method exploits the many beautiful properties of the minimum of a set of exponential variables.

${\displaystyle H(x)={\underset {i}{\operatorname {arg\,min} }}{\frac {-\log(h(i))}{x_{i}}}}$

This yields as its collision probability the probability Jaccard index^[7]

${\displaystyle J_{\mathcal {P}}(x,y)=\sum _{x_{i}\neq 0 \atop y_{i}\neq 0}{\frac {1}{\sum _{j}\max \left({\frac {x_{j}}{x_{i}}},{\frac {y_{j}}{y_{i}}}\right)}}}$

Min-wise independent permutations

In order to implement the MinHash scheme as described above, one needs the hash function h to define a random permutation on n elements, where n is the total number of distinct elements in the union of all of the sets to be compared. But because there are n! different permutations, it would require Ω(n log n) bits just to specify a truly random permutation, an infeasibly large number for even moderate values of n. Because of this fact, by analogy to the theory of universal hashing, there has been significant work on finding a family of permutations that is "min-wise independent", meaning that for any subset of the domain, any element is equally likely to be the minimum. It has been established that a min-wise independent family of permutations must include at least

${\displaystyle \operatorname {lcm} (1,2,\cdots ,n)\geq e^{n-o(n)}}$

different permutations, and therefore that it needs Ω(n) bits to specify a single permutation, still infeasibly large.^[2]

Applications

The original applications for MinHash involved clustering and eliminating near-duplicates among web documents, represented as sets of the words occurring in those documents.^[1][2][11] Similar techniques have also been used for clustering and near-duplicate elimination for other types of data, such as images: in the case of image data, an image can be represented as a set of smaller subimages cropped from it, or as sets of more complex image feature descriptions.^[12]

In data mining, Cohen et al. (2001) use MinHash as a tool for association rule learning. Given a database in which each entry has multiple attributes (viewed as a 0–1 matrix with a row per database entry and a column per attribute) they use MinHash-based approximations to the Jaccard index to identify candidate pairs of attributes that frequently co-occur, and then compute the exact value of the index for only those pairs to determine the ones whose frequencies of co-occurrence are below a given strict threshold.^[13]

The MinHash algorithm has been adapted for bioinformatics, where the problem of comparing genome sequences has a similar theoretical underpinning to that of comparing documents on the web. MinHash-based tools^[14][15] allow rapid comparison of whole genome sequencing data with reference genomes (around 3 minutes to compare one genome with the 90000 reference genomes in RefSeq), and are suitable for speciation and maybe a limited degree of microbial sub-typing. There are also applications for metagenomics ^[14] and the use of MinHash derived algorithms for genome alignment and genome assembly.^[16] Accurate average nucleotide identity (ANI) values can be generated very efficiently with MinHash-based algorithms.^[17]

Other uses

The MinHash scheme may be seen as an instance of locality-sensitive hashing, a collection of techniques for using hash functions to map large sets of objects down to smaller hash values in such a way that, when two objects have a small distance from each other, their hash values are likely to be the same. In this instance, the signature of a set may be seen as its hash value. Other locality sensitive hashing techniques exist for Hamming distance between sets and cosine distance between vectors; locality sensitive hashing has important applications in nearest neighbor search algorithms.^[18] For large distributed systems, and in particular MapReduce, there exist modified versions of MinHash to help compute similarities with no dependence on the point dimension.^[19]

Evaluation and benchmarks

A large scale evaluation was conducted by Google in 2006 ^[20] to compare the performance of Minhash and SimHash^[21] algorithms. In 2007 Google reported using Simhash for duplicate detection for web crawling^[22] and using Minhash and LSH for Google News personalization.^[23]

See also

• Bloom filter – Data structure for approximate set membership
• Count–min sketch
• w-shingling

References