# Minhash lsh

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In computer science and data miningMinHash or the min-wise independent permutations locality sensitive hashing scheme is a technique for quickly estimating how similar two sets are. 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 Uthen 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 :.

This value is 0 when the two sets are disjoint1 when they are equal, and strictly between 0 and 1 otherwise. Two sets are more similar i. The goal of MinHash is to estimate J AB quickly, without explicitly computing the intersection and union. In cases where the hash function used is assumed to have pseudo-random properties, the random permutation would not be used.

The probability of this being true is exactly the Jaccard index, therefore:. The idea of the MinHash scheme is to reduce this variance by averaging together several variables constructed in the same way. The simplest version of the minhash scheme uses k different hash functions, where k is a fixed integer parameter, and represents each set S by the k values of h min S for these k functions.

For example, hashes would be required to estimate J AB with an expected error less than or equal to. It may be computationally expensive to compute multiple hash functions, but a related version of MinHash scheme avoids this penalty by using only a single hash function and uses it to select multiple values from each set rather than selecting only a single minimum value per hash function. Let h be a hash function, and let k be a fixed integer.

If S is any set of k or more values in the domain of hdefine h k S to be the subset of the k members of S that have the smallest values of h. This subset h k S is used as a signature for the set Sand the similarity of any two sets is estimated by comparing their signatures. Specifically, let A and B be any two sets. The difference between this estimator and the estimator produced by multiple hash functions is that X always has exactly k members, whereas the multiple hash functions may lead to a smaller number of sampled elements due to the possibility that two different hash functions may have the same minima.

However, when k is small relative to the sizes of the sets, this difference is negligible. The signature of each set can be computed in linear time on the size of the set, so when many pairwise similarities need to be estimated this method can lead to a substantial savings in running time compared to doing a full comparison of the members of each set. Specifically, for set size n the many hash variant takes O n k time. A variety of techniques to introduce weights into the computation of MinHashes have been developed.

The simplest extends it to integer weights. Run the original algorithm on this expanded set of hashes. Doing so yields the weighted Jaccard Index as the collision probability. Further extensions that achieve this collision probability on real weights with better runtime have been developed, one for dense data,  and another for sparse data.In my previous post I explained how the MinHash algorithm is used to estimate the Jaccard similarity coefficient, because calculating it precisely each time can be expensive.

However, we still need to compare every document to every other document in order to determine which documents are similar or in more fancy words - to cluster the documents by similarity. We can optimize this process by using the concepts of Locality-sensitive hashing LSH. As always, in order to get something we need to give something, this time the trade-off is between accuracy and performance. Saving us time, but taking the risk of missing out on a potential similar document.

### Minhash and locality-sensitive hashing

We will call them from now on the MinHash values of the document. Next we need to create a table that has the n documents as its columns and the k MinHash values as its rows, as can be seen in this example:. For example, if we are using 12 hash functions our k valuewe can split the table into 4 bands our b value with 3 rows each our r valueending up with 4 tables that look something like this:.

And what if a new document shares no identical rows to any other document on any band? In this case we assume that there exists no similar document to it. Note that we did that without actually doing any MinHash comparison. So to recap, we only compare two documents if there exists a band where those two documents share the same values on every row.

### minhash-lsh-algorithm

Therefore, our goal is to find a division into rows and band such that if two documents are not compared share no bandthe probability of them actually being similar is very low. Before I can show you how to choose r rows and b bandswe first need to crunch some numbers:. We can utilize this property to our advantage, by making sure the step occurs as close to the threshold of two documents being similaras we can. We want our step to be as close to 0.

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In this table we can see that if the Jaccard similarity coefficient of two documents is lower than 0. But, if the Jaccard similarity coefficient of the documents is greater than 0. This is of course not a bulletproof method, and you might end up classifying documents that are similar as dissimilar, but this is ok as long as we can tolerate an occasional error.

Java implementation C implementation Python implementation. By adding note 2 we know that the derivative function should have a maximum value in this range Extreme Value Theorem. This maximum value is where the original function f increases the fastest, or in other words, where its slope is the steepest. We can find this maximum point by differentiating again:.

We only found one critical point the maximum pointwhich means that the first derivative should look similar to a parabola, and this is enough to deduce that our original function f will be an s-curve function.

Since A and B are similar, their Jaccard similarity coefficient is at least the step this is how we choose the step :. We get that the probability of a False Negative decreases exponentially:. Background In my previous post I explained how the MinHash algorithm is used to estimate the Jaccard similarity coefficient, because calculating it precisely each time can be expensive.Feature construction is an endless field of study.

For this example fixed length character shingles are used. Shingling is the process of choosing subsets of strings in a document such that the shingles encode word content and word ordering. In practice, depending on document size, k, the shingle size, is tuned to ensure that created shingles are as unique as possible. If chosen well, comparing two sets of shingles to deduce document similarity will have low levels of false positives when sets of shingles from candidate matching documents are compared.

Choosing the size of the shingles for the documents being analyzed is a crucial parameter. This means shingles will be created for every two characters. In English, the average word is 5 letters long. For short documents, choosing 5 or 6 characters as shingle size is a viable choice, while longer documents would benefit from double the word length.

Optimal shingle size will vary based on language and word length. While English word length was used as an example, other alphabets and tokens can just as easily be used with equal success. In the scala session below, we show that the set of shingles generated indicate that the two documents are exactly equal, which is a false positive.

The shingle size parameter should be chosen carefully depending on the application. Given the shingles of two documents, with properly chosen parameters, we could compute their set intersection to determine the similarity measure between the documents. However, as the document size grows, so does the number of shingles required to represent the document. If we can sample the shingles to still accurately reflect the contents of the document, we can compress the size of the signature required to determine document similarity.

In fact, MinHash, as we will discuss later, samples the hash functions for each shingle, always choosing the lowest value, while LSH samples the MinHash signatures to further compress the document signature. We discuss each of these in detail in later sections. In order to convert the shingles to an integer value which can be further hashed by our universal hash functionsa suitable hash function must chosen that is both quick and has a low collision rate.

In our case, we used the djb2a hash function, known for few collisions and super fast computation. Another readily available hash function is MurMurHash. It is capable of producing hashes very quickly and with low collision rates.

A good implementation of MurMurHash can be found in the Algebird source code. However, djb2a is also very simple compared to MurMurHash, so we choose it for ease of understanding and implementation. Below, we show an example of converting a single shingle into a hashed value using djb2a.

This value will be passed to the hash collectionwhich will produce N values from the N hash functions. In this example, the universal hash function is comprised of a and b, which are random numbers, p is a large prime, and N, which serves to divide the space further into buckets. MinHashing requires using collections of hash functions which should produce non-colliding results.

Randomly generating a and b multiple times will create a family of hash functions suitable for sampling when we discuss MinHash.

A universal hash function with input parameters. Generates a function H xwhich given an x computes the hash value. But, given random values for a and b, and a large prime p, we need to determine if this hash function has desirable properties. In order to do this, we can take a look at how integers are hashed into buckets. Given all the hash buckets, a hash function will be considered acceptable if and only if it approximately evenly distributes the input values into hash buckets. In each bucket, there are approximately the same number of hashed values, showing that the universal hash function is a reasonable one.

At 0 we notice that there are approximately two times the number of entries. This is expected because of the way the numbers are partitioned, the 0 value corresponding to those terms both positive and negative hence double the bucket size that are 0 modulo our prime number. Hash collections are used in the MinHash algorithm for generating hash values from shingles and sampling those values.

Below we have a collection of hash functions which accept an initial seed and the number of hashes to be generated.End-to-end earthquake detection pipeline via efficient time series similarity search. A text similarity computation using minhashing and Jaccard distance on reuters dataset. Fast Jaccard similarity search for abstract sets documents, products, users, etc.

An improved method of locality-sensitive hashing for scalable instance matching. In this study, we propose a scalable approach for automatically identifying similar candidate instance pairs in very large datasets utilizing minhash-lsh-algorithm in C.

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Skip to content. Here are 26 public repositories matching this topic Language: All Filter by language. Sort options. Star Code Issues Pull requests. Updated Apr 7, Shell. A Clojure library for querying large data-sets on similarity. Updated Feb 17, Clojure.

## LIST OF ALGORITHMS

There are Python 2. Updated Aug 21, Python. Updated Jun 11, R. Open Improve Minhash Performance. Find a way to improve the performance of Minhash when using multi hash option. Read more. Hacktoberfest enhancement good first issue help wanted. Star 6. Updated Nov 14, Python. Updated Apr 11, Rust. Star 3. Updated Sep 11, Python. Star 2.

Scalable Data Mining - Assignment submissions. Updated Dec 11, Python. Star 1. Updated Dec 24, TeX. Updated Aug 26, Python. Updated Sep 17, Java.The minHash has a property that the probability for two minimum elements are the same is equal to the Jaccard distance between two arrays.

We want to replace large sets by much smaller representations. We call it as signatures. The important property we need for signatures is that we can compare the signatures of two sets and estimate the Jaccard similaritysignatures alone. In other words, if two points are close to each other, the probability that this function hashes them to the same bucket is high. On the contrary, if the points are far apart, the probability that they get hashed to the same bucket is low. There are several distance measures but we opted to use Jaccard distance.

The following code simulates the algorithm for computing the signature matrix. In the code, we'll use very simple hash functions which was suggested in the reference I mentioned earlier. Since the steps are well explained in the reference pdf, I'll skip the details. As we've already guessed, it gives us a reasonable output but not exact outcome. This inaccuracy is due to the fact that we have extremely small size of samples. For the S1,S2the signature matrix estimates the similarity to be 0 which is the correct value. Toggle navigation BogoToBogo. Sponsor Open Source development activities and free contents for everyone.

Thank you.Jaccard similarity is great for measuring resemblance between two sets, however, it can be a biased measure for set intersection. This issue can be illustrated in the following Venn diagrams. This shows that Jaccard similarity is biased when measuring intersection size, as large sets are penalized.

We can use a better measure for intersection, called containment. It is computed as the intersection size divided by the size of one of the set, in this case Q. It soon becomes clear why we use Q in the denominator, when you think about this search problem: suppose you have a large collection of sets, given a query, which is also a set, you want to find sets in your collection that have intersection with the query above a certain threshold.

We can think of the query as set Q, and an arbitrary set in the collection as X. Zhu et al.

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This package implements a slightly simplified version of the index, datasketch. The full implementation is in Go. It can be found at github. The precision of LSH Ensemble increases with the number of partitions, so does the querying time, as the plot below shows.

The experiment code can be found in the benchmark directory of the source code repository. There are other optional parameters that can be used to tune the index to achieve better accuracy or performance. See the documentation of datasketch. How to use LSH Ensemble to find containment-based duplicates? Quick search. Powered by Sphinx 2.The task of finding nearest neighbours is very common.

Approximate algorithms to accomplish this task has been an area of active research.

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These algorithms are faster and scalable. Locality sensitive hashing LSH is one such algorithm. LSH has many applications, including:.

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LSH refers to a family of functions known as LSH families to hash data points into buckets so that data points near each other are located in the same buckets with high probabilitywhile data points far from each other are likely to be in different buckets.

This makes it easier to identify observations with various degrees of similarity. Goal: You have been given a large collections of documents. In the context of this problem, we can break down the LSH algorithm into 3 broad steps:. In this step, we convert each document into a set of characters of length k also known as k-shingles or k-grams.

The key idea is to represent each document in our collection as a set of k-shingles. Now, we need a metric to measure similarity between documents. Jaccard Index is a good choice for this. More number of common shingles will result in bigger Jaccard Index and hence more likely that the documents are similar. Now you may be thinking that we can stop here.

The document matrix is a sparse matrix and storing it as it is will be a big memory overhead. One way to solve this is hashing. The idea of hashing is to convert each document to a small signature using a hashing function H. Suppose a document in our corpus is denoted by d. For Jaccard similarity the appropriate hashing function is min-hashing.

This is the critical and the most magical aspect of this algorithm so pay attention:. Step 2: Hash function is the index of the first in the permuted order row in which column C has value 1. Do this several time use different permutations to create signature of a column. The similarity of the signatures is the fraction of the min-hash functions rows in which they agree. Expected similarity of two signatures is equal to the Jaccard similarity of the columns. The longer the signatures, the lower the error. In the below example you can see this to some extent. There is difference as we have signatures of length 3 only. But if increase the length the 2 similarities will be closer. So using min-hashing we have solved the problem of space complexity by eliminating the sparseness and at the same time preserving the similarity. Min-hash implementation.

The general idea of LSH is to find a algorithm such that if we input signatures of 2 documents, it tells us that those 2 documents form a candidate pair or not i. Remember that we are taking similarity of signatures as a proxy for Jaccard similarity between the original documents. Specifically for min-hash signature matrix:. Now the question is how to create different hash functions. For this we do band partition. There are few considerations here. Ideally for each band we want to take k to be equal to all possible combinations of values that a column can take within a band.