Hierarchical Navigable Small World (HNSW)
Posted: Sun Jan 26, 2025 5:05 am
The construction process of the HNSW algorithm consists of the following steps:
An initial graph is created with a node representing a data point randomly selected from the database.
More nodes are then added to this graph. Each new node is connected to a set of existing nodes, using a proximity function that measures the similarity between the feature vectors of the data points.
The process of node aggregation is repeated until a complete graph is constructed at a given level.
The next level of the hierarchy is then created, using a neighbor selection function to connect nodes in one level to those in the lower level. This function ensures that a hierarchical structure is maintained at higher levels and allows for fast searching through the hierarchy.
The level-building process is repeated until the lowest level is reached, which is usually a complete graph where every node is connected to every other node.
Once the hierarchical structure of the HNSW algorithm has been built, it can be used to perform efficient searches in the vector database. The algorithm allows for approximate searches, where the closest points to a given query are found quickly and with a good level of accuracy.
Similarity measures
Based on the algorithms reviewed, we need to understand the role of similarity measures in vector databases. These measures are the basis of how a vector database compares and identifies the most relevant results for a given query.
Similarity measures are mathematical methods for determining how similar two vectors are in a vector space. Similarity measures are used in vector databases to compare vectors stored in the database and find the ones that are most similar to a given query vector.
Several similarity measures can be used, including:
Cosine similarity: Measures the cosine of the angle between two vectors in a vector space. It ranges from -1 to 1, where 1 represents identical vectors, 0 represents orthogonal vectors, and -1 represents vectors that are diametrically opposite.
Euclidean distance: Measures the straight-line distance between two vectors in a vector space. It ranges from 0 to infinity, where 0 represents identical vectors and larger values represent increasingly different vectors.
Dot product: Measures the product of the magnitudes of two vectors and the cosine of the angle between them. It ranges from -∞ to ∞, where a positive value represents vectors pointing in the same direction, 0 represents orthogonal vectors, and a negative value represents vectors pointing in opposite directions.
The choice of similarity measure will have an effect on the results obtained from a vector database. It is also important to note that each similarity measure has its own advantages and disadvantages, and it is important to choose the right one based on the use case and requirements.
Filters
Each vector stored in the database also includes metadata. In effective poland mobile numbers list addition to the ability to query similar vectors, vector databases can also filter results based on a metadata query. To do this, the vector database typically maintains two indexes: a vector index and a metadata index. It then performs metadata filtering either before or after the vector search, but in either case, there are pitfalls that slow down the query process.
Filtration
The filtering process can be performed before or after the vector search, but each approach has its own challenges that can affect query performance:
Pre-filtering: In this approach, metadata filtering is performed prior to the vector search. While this can help reduce the search space, it can also cause the system to miss relevant results that do not match the metadata filter criteria. Furthermore, extensive metadata filtering can slow down the query process due to additional computational overhead.
Post-filtering: In this approach, metadata filtering is performed after the vector search. This can help ensure that all relevant results are considered, but it can also introduce additional overhead and slow down the query process, as irrelevant results need to be filtered out once the search is complete.
To optimize the filtering process, vector databases use several techniques, such as leveraging advanced indexing methods for metadata or using parallel processing to speed up filtering tasks. Balancing the trade-offs between search performance and filtering accuracy is essential to providing efficient and relevant query results in vector databases.
The HNSW algorithm is a method used to perform efficient searches in high-dimensional vector databases. It was proposed as an alternative to binary search trees and kd trees, which do not perform optimally in high-dimensional spaces.
The HNSW algorithm is based on the idea of building a hierarchical structure that organizes data into different levels. Each level is composed of a set of graphs, where each graph is a set of nodes connected to each other. These nodes represent the data points in the vector database.
An initial graph is created with a node representing a data point randomly selected from the database.
More nodes are then added to this graph. Each new node is connected to a set of existing nodes, using a proximity function that measures the similarity between the feature vectors of the data points.
The process of node aggregation is repeated until a complete graph is constructed at a given level.
The next level of the hierarchy is then created, using a neighbor selection function to connect nodes in one level to those in the lower level. This function ensures that a hierarchical structure is maintained at higher levels and allows for fast searching through the hierarchy.
The level-building process is repeated until the lowest level is reached, which is usually a complete graph where every node is connected to every other node.
Once the hierarchical structure of the HNSW algorithm has been built, it can be used to perform efficient searches in the vector database. The algorithm allows for approximate searches, where the closest points to a given query are found quickly and with a good level of accuracy.
Similarity measures
Based on the algorithms reviewed, we need to understand the role of similarity measures in vector databases. These measures are the basis of how a vector database compares and identifies the most relevant results for a given query.
Similarity measures are mathematical methods for determining how similar two vectors are in a vector space. Similarity measures are used in vector databases to compare vectors stored in the database and find the ones that are most similar to a given query vector.
Several similarity measures can be used, including:
Cosine similarity: Measures the cosine of the angle between two vectors in a vector space. It ranges from -1 to 1, where 1 represents identical vectors, 0 represents orthogonal vectors, and -1 represents vectors that are diametrically opposite.
Euclidean distance: Measures the straight-line distance between two vectors in a vector space. It ranges from 0 to infinity, where 0 represents identical vectors and larger values represent increasingly different vectors.
Dot product: Measures the product of the magnitudes of two vectors and the cosine of the angle between them. It ranges from -∞ to ∞, where a positive value represents vectors pointing in the same direction, 0 represents orthogonal vectors, and a negative value represents vectors pointing in opposite directions.
The choice of similarity measure will have an effect on the results obtained from a vector database. It is also important to note that each similarity measure has its own advantages and disadvantages, and it is important to choose the right one based on the use case and requirements.
Filters
Each vector stored in the database also includes metadata. In effective poland mobile numbers list addition to the ability to query similar vectors, vector databases can also filter results based on a metadata query. To do this, the vector database typically maintains two indexes: a vector index and a metadata index. It then performs metadata filtering either before or after the vector search, but in either case, there are pitfalls that slow down the query process.
Filtration
The filtering process can be performed before or after the vector search, but each approach has its own challenges that can affect query performance:
Pre-filtering: In this approach, metadata filtering is performed prior to the vector search. While this can help reduce the search space, it can also cause the system to miss relevant results that do not match the metadata filter criteria. Furthermore, extensive metadata filtering can slow down the query process due to additional computational overhead.
Post-filtering: In this approach, metadata filtering is performed after the vector search. This can help ensure that all relevant results are considered, but it can also introduce additional overhead and slow down the query process, as irrelevant results need to be filtered out once the search is complete.
To optimize the filtering process, vector databases use several techniques, such as leveraging advanced indexing methods for metadata or using parallel processing to speed up filtering tasks. Balancing the trade-offs between search performance and filtering accuracy is essential to providing efficient and relevant query results in vector databases.
The HNSW algorithm is a method used to perform efficient searches in high-dimensional vector databases. It was proposed as an alternative to binary search trees and kd trees, which do not perform optimally in high-dimensional spaces.
The HNSW algorithm is based on the idea of building a hierarchical structure that organizes data into different levels. Each level is composed of a set of graphs, where each graph is a set of nodes connected to each other. These nodes represent the data points in the vector database.