Application of locality-sensitive hashing in approximate nearest neighbor search

Locality Sensitive Hashing (LSH) is a method for approximate nearest neighbor search, especially suitable for data in high-dimensional spaces. In many practical applications, such as text and image data, the dimensionality of data points can be very high. In high-dimensional space, traditional distance measurement methods such as Euclidean distance are no longer effective, and traditional linear search methods are inefficient. Therefore, we need some efficient algorithms to solve this problem. The basic idea of ​​LSH is to map similar data points to similar hash buckets through a hash function. In this way, we only need to search in similar hash buckets instead of traversing the entire data set, thus greatly improving search efficiency. The core of the LSH algorithm is to design a suitable hash function. The hash function should have two characteristics: first, similar data points have a high probability of being mapped to similar hash buckets, that is, they have local sensitivity; second, dissimilar data points

The basic idea of ​​Locality Sensitive Hash (LSH) is to map data points in high-dimensional space to low-dimensional space through a hash function, so as to perform approximate nearest neighbor search in low-dimensional space. By introducing randomization techniques, LSH can increase the probability that similar data points are mapped to the same bucket, thereby reducing the search space. The advantage of LSH is that it greatly reduces the search space while ensuring a certain query accuracy, thereby improving query efficiency.

LSH is widely used, such as similar image search in search engines, similar song recommendations in music recommendation systems, and similar user recommendations in social networks. The following will introduce the principle and implementation process of LSH through a simple example.

Suppose we have a data set, and each data point is a 100-dimensional vector. In order to query this data set for the most similar data points to a given vector, we hope to use LSH (Locality Sensitive Hashing) to improve query efficiency. Since the dimensionality of data points is very high, traditional linear search methods are very time-consuming. LSH can map data points in high-dimensional space to low-dimensional space, keeping similar data points relatively close in low-dimensional space and reducing search time. Therefore, using LSH for query can speed up the search and improve efficiency.

First, we need to define a hash function to map the 100-dimensional vector into a low-dimensional space. There are two commonly used hash functions: Euclidean hash and cosine hash. Euclidean hashing maps vectors onto the real number domain by randomly generating some hyperplanes to map data points into different buckets. Cosine hashing maps vectors to a high-dimensional hypersphere, and also maps data points to different buckets by randomly generating some hyperplanes. In this example, we use Euclidean hashing as an example.

We can express the hash function as h(x)=\lfloor\frac{a^Tx b}{w}\rfloor, where a is a random vector and b is A random constant, w is the width of a bucket, \lfloor\rfloor means rounding down. For any vector x, it will be mapped to a bucket, and the bucket number is h(x).

Now we need to choose some random vector a and random constant b, as well as the width of the bucket w. In order to map similar data points to the same bucket as much as possible, we need to choose some parameters so that the probability of similar data points being mapped to the same bucket is relatively high, and dissimilar data points are mapped to the same bucket. The probability is relatively small. This process can be achieved by adjusting parameters.

Generally speaking, we need to select multiple hash functions and map each hash function once. Through the mapping of these hash functions, we can get multiple buckets. We can regard these buckets as a candidate set, and then perform approximate nearest neighbor search in this candidate set. Specifically, we can calculate the distance between the query vector and each data point in the candidate set, and then select the data point with the smallest distance as the approximate nearest neighbor. Since the size of the candidate set is much smaller than the size of the entire data set, this process is much more efficient than linear search.

It should be noted that LSH is an approximate method and it cannot guarantee the accuracy of query results. There may be some errors in LSH query results, and the size of the error is related to the choice of hash function and parameter settings. Therefore, in practical applications, we need to select appropriate hash functions and parameters according to specific scenarios and requirements to achieve a balance between query accuracy and query efficiency.

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