# Locality Sensitive Hashing (lsh)

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The following section offers some explanations of the easy LSH approach and how these systems are constructed. Let's look at an extremely simple math function which would be described as LSH. Figure Two shows one such clustering on the perimeter of the rounded circle. Similar test elements for clusters are shown below: Same 20 as in figure 1 except we use hashed cluster colours the previous time. This helps avoidance of clusters with different points, although that will be more apparent as it shows.

## LSH algorithm for nearest neighbour similarity search

The basic idea behind locality-sensitive hashing is that you want to find similar items in your data set by computing a hash table function on the item's attribute values. This will produce a number that is associated with each item. You can then store this number along with the item in a table or index. When you want to find similar items, you can compute the single hash function on the items you're interested in and see which numbers are closest to the same hash value for the item you're looking for.

## Do you need to group data points

LSH is a valuable tool for data mining that can be used to group data points very efficiently. Despite its critics, LSH is still a valuable tool for data mining and can be used to group data very efficiently. It's important to note that while LSH can be effective, it still isn't guaranteed to work and should only be used when other data mining techniques fail.

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## There are many different algorithms that you can use for LSH, in particular, the following are used in practice

The LSH algorithms that have found widespread use in industry are MinHash and Locality Sensitive Hashing. Both of these hash-based approaches are based on the idea of generating random projections with fixed "k" dimensions which can be stored efficiently to identify similar gene expressions. The large amounts of computation required for nearest neighbor search are performed by projecting small subsets of items onto these fixed "k" dimensions.

This is very similar to what happened with word2vec where it was used to compute vectors for each word using a precomputed matrix consisting of all possible word-word pairs. Then when you wanted to find similar words or documents you would look at how the resulting vector was to another vector. LSH algorithms are similar.

If you have a bunch of points in space, say people, and you want to find all the people who are closest together you first have to define your space. If your space is simply each person's location each person would define their coordinate system concerning themselves. This might consist of letting each person know their distance from some reference point on the planet earth or it could be absolute, like exactly where they are without any regard for how far away they are from some other reference point.

Figure 2 shows one such clustering on the perimeter of the rounded circle. Similar test elements for clusters are shown below: Same 20 as in figure 1 except we use hashed cluster colors the previous time. This helps avoidance of clusters with different points, although that will be more apparent as it shows.

Now let's say we want to find all the people who are closest together in our space. We could simply start at some reference point and walk around until we find the first person, then walk to the next person, and so on. This would be like finding the nearest neighbor in a one-dimensional space.

But what if our space is two-dimensional? In this case, we would need to calculate the distance between each person and the reference point. Then we would find the person who is closest to the reference point and add them to our list. This would be like finding the nearest neighbor in a two-dimensional space.

## Definitions

In mathematics and computer science, locality-sensitive hashing (LSH) is a family of algorithms for indexing data objects in space so that similar objects are stored near each other. LSH takes as input a set of data objects, S, and a hash function, h: S →

A different LSH algorithm is used depending on the type of data being hashed. For example, there are variations of LSH specifically designed for hashing text strings, vectors in Euclidean space, and images.

## LSH has been applied in several practical settings, including

* Text search * Spatial indexing * Social network analysis * Signature verification * Genomic sequence

The basic idea behind LSH is that instead of storing all the points as a set, some points are hashed into a smaller set and then those points are stored instead. That way there is no longer a need to look through every point for an element.

This method is used in many applications such as finding similar images, finding items near each other on a map, and finding documents or words which share some commonalities and having them clustered together. This can all be done without needing to compare every single file or document with another one.

In these cases, you would hash the data object into multiple bins where the number of bins is much less than the size of your dataset. For example, if you had 10 input values they could go into 4 buckets evenly (10/4=2.5). You could then use the hash value collisions as an index into a table that stored the reference points for each same bucket.

When you want to find similar objects, you would simply look up the hash collisions value in the table and get the corresponding set of reference points. From there you can compute the distance between each point and the reference point and find the closest one. This is much faster than looking through every object in your dataset.

There are a few different types of LSH algorithms, but they all work on the same basic principle. The most popular type of LSH algorithm is called a Bloom filter. A Bloom filter is a probabilistic data structure that can be used to test whether an element is a member of a set. It does that by using the properties of some hashing function to create many possible buckets for each element in your dataset. When you add an element to the set, it hashes the element into a bucket (like normal LSH). That way, when you go to look up whether or not an element is a member of a set, it hashes the element and checks which bucket it would fall under. Then depending on how many elements are actually in that bucket at that time determines whether or not you have found a positive match.

Figure 3 shows what this idea looks like with text strings. These strings can be thought of as documents where we want to search for words similar to another word over here so we can if they share any commonalities. In the figure, we have a set of six documents and want to find all the words that are similar to the word "the."

We could do this by using a standard text search where we compare each document with every other document. This would be slow and impractical.

Instead, we can use LSH to create a set of buckets for each word in our dataset. We then add the word "the" to each bucket. When we want to find all the words that are similar to "the," we simply look up the hash value for "the" in our table and get the corresponding set of reference points. From there, we can compute the distance between each point and the reference point and find the closest one. This is much faster.

## Figure 3: Example of LSH using a Bloom filter to find similar words in Google News articles

The reason the hash values are shown as points is that they are vectors instead of just real numbers. That way you could have multiple documents that have close hash values for one word. This method enables looking up more than one document at once which speeds up the search process even further. Another type of popular algorithm used with LSH is called Locality Sensitive Hashing (LSH). LSH works by creating bins for your data-set based on some hashing function. The type of hashing function determines how the data will be split up, but different types work better depending on the type of data you are trying to group. This is done by computing the hash value for each data point and then placing them into bins based on that hash function. Doing this has an interesting effect where similar items are placed in the same bin while dissimilar items are put in different ones. This can be better illustrated with an example, so let's say we have 6 documents or data points represented by the circles in Figure 4 . We want to find objects that are close to these six points so we can group them.

Note how all of the blue circles are clustered together even though they vary widely throughout our space compared to other clusters like yellow and red. What happened here was when we used a hashing function to create for each data point, it kept putting them into the same bin. This is because blue is clustered together in our data set. The hashing function we used was just based on the x-coordinate of each point, so it wasn't very discriminating and didn't take into account the they-coordinate. If we had used a different hashing function that took both the x- and y-coordinates into account, then the clusters would be more evenly distributed.

## Figure 4: Example of LSH using a hash function to group data points

There are a few different types of LSH algorithms, but they all work on the same basic principle. The most popular type of LSH algorithm is called a Bloom filter. A Bloom filter is a probabilistic data structure that is used to determine if an element is a member of a set. This is done by computing the hash value for the element and then adding it to the Bloom filter. If the element is already in the Bloom filter, then the Bloom filter will return a positive result. If the element isn't in the Bloom filter, then the Bloom filter will return a negative result.

The cool thing about Bloom filters is that they are space-efficient and have a very low false-positive rate. This means that you can fit a lot of data into a Bloom filter without having it take up too much space and that the chances of it returning a false positive are very low. This makes them perfect for use with LSH algorithms.

Figure 5 shows an example of how a Bloom filter might work. First, the data set is split up into different buckets. Then each bucket is hashed to acquire a reference point. Finally, each data point is added to the table by checking if its hash value equals any of the reference points. The bins are all shaded light blue because they are empty while the other ones are filled with points representing different documents.

## Figure 5: Example of using a Bloom filter with LSH

The main issue with using Bloom filters is that their size grows exponentially as you add more elements to them. This means that there isn't much room for error in your hashing function which could lead to false positives if the items aren't similar enough. There's also some debate over how useful LSH is since the results aren't guaranteed especially if your randomly chosen hash functions are bad. It's also not very effective against more than two dimensions for any data set that requires you to group things through more than one facet of an object.

If you're interested in trying out LSH yourself, there are some open-source libraries that you can use to run your experiments. We'll be using the code from OpenIntro which includes a python package called pyhash for implementing LSH algorithms. To learn more about pyhash, check out their Github page.

Figure 6 shows how simple it is to create an index with pyhash. First, we define the data set for our experiment. Then, we create a new Bloom filter object. On line 32, we add the documents to the Bloom filter by hashing them and then checking if they already exist in the list of items added. Finally, on line 37, we look up an item in the data set by hashing it and checking the Bloom filter to see if it's in the list.

## Figure 6: Using pyhash to create an index

If you want to try out some different LSH variants, you can easily do that by changing the hashing function on line 32. There are a lot of different hashing functions to choose from and each one will give you different results. You can also play around with the size of the Bloom filter and see how that affects the performance of your application.

## Band and Hash

Despite these criticisms, LSH is still a valuable tool for data mining and can be used to group data points very efficiently. One of the most popular uses of LSH is to create indexes for large data sets. This is done by hashing each document and then placing the hashes into a Bloom filter. When you want to find a document, you just hash tables it and look in the Bloom filter to see if it's there. If it is, then you know that the document is somewhere in the index.

There are other variants of LSH that are worth looking into like locality-sensitive tree hashing (LSTH) and MinHash. These algorithms offer some advantages over the standard Bloom filter algorithm like increased speed and smaller filters while offering the same level of accuracy. Both of these algorithms are based on concepts that extend beyond basic LSH and can be used in other fields like computer science to solve problems related to data mining.

## Conclusion

LSH is a valuable tool for data mining that can be used to group data points very efficiently. Despite its critics, LSH is still a valuable tool for data mining and can be used to group data very efficiently. It's important to note that while LSH can be effective, it still isn't guaranteed to work and should only be used when other data mining techniques fail.

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