Euclidean distance

As explained in MongoDB 8.0 in Action, Third Edition

Euclidean distance is the straight-line distance between two points or vectors in Euclidean space.

Euclidean distance measures the straight-line distance between two points in vector space. In vector databases, it helps rank stored embeddings by how close they are to a query embedding, supporting semantically similar results. section 12.1.2 section 12.2.3

Use Euclidean distance when spatial separation between vectors is meaningful:

• Apply it to image-recognition and clustering tasks, where the distance between points carries practical significance. section 12.2.3
• Compare it with cosine similarity and dot product when choosing a vector-search similarity metric. section 12.2.3
• Use it as the distance metric guiding nearest-neighbor navigation in HNSW indexes. section 12.2

Embedding search workflow

An embedding model converts stored content and a user query into vectors. The vector database compares the query vector with stored vectors using a similarity measure such as Euclidean distance, then returns the closest results. section 12.1.2

Figure 12.2 Vector database operations. Data is processed by an embedding model, transforming it into vector embeddings such as [1.45, ..., 1.01, 0.89, 0.06]. These embeddings are stored in a vector database. Queries from the application are also transformed into embeddings before the database is searched. The most relevant results based on these query embeddings are then retrieved and returned as query results. (Image © MongoDB 2024 CC BY-NC-SA 3.0)

Trace how source data becomes embeddings, enters a vector database, and produces retrieved results for an embedded query. section 12.1.2

How MongoDB 8.0 in Action, Third Edition covers it

How deep it goes

Defines straight-line distance and connects it to HNSW navigation, but does not present a calculation formula.

Examples

Uses image recognition, clustering, embedding retrieval, and nearest-neighbor graph navigation as applications.

What it assumes

Assumes familiarity with vectors and embeddings while relating them to vector-database similarity search.

What you can do after

Evaluate similarity metrics for vector search and understand how distance guides approximate or exact nearest-neighbor retrieval.

How it connects

Connects Euclidean distance with embeddings, vector databases, cosine similarity, dot product, HNSW, and exact nearest-neighbor search.

liveBookmanning.com

Read MongoDB 8.0 in Action, Third Edition on liveBook Get the book at manning.com

Also explained in

• 

  liveBookmanning.com
  Classic Computer Science Problems in Swift: Essential techniques for practicing programmersEuclidean distance is the straight-line distance between two points calculated from the square root of the sum of squared coordinate differences.
• 

  liveBookmanning.com
  Advanced Algorithms and Data StructuresEuclidean distance is the straight-line distance between points in Euclidean space, computed from the square root of the sum of squared coordinate differences.
• 

  liveBookmanning.com
  Data Without LabelsEuclidean distance is the straight-line distance between points in a coordinate space, computed as the square root of the summed squared coordinate differences.
• 

  liveBookmanning.com
  Natural Language Processing in Action, Second EditionEuclidean distance is the straight-line distance between two points or vectors computed from the square root of the sum of squared coordinate differences.
• 

  liveBookmanning.com
  Machine Learning for Tabular Data: XGBoost, Deep Learning, and AIEuclidean distance is the straight-line distance between two points in Euclidean space.
• 

  liveBookmanning.com
  Statistics Slam DunkEuclidean distance is the straight-line distance between points computed as the square root of the sum of squared coordinate differences.
• 

  liveBookmanning.com
  Optimization Algorithms: AI techniques for design, planning, and control problemsEuclidean distance is the straight-line distance between points calculated from the square root of the sum of squared coordinate differences.

Compare how 8 books explain Euclidean distance →

Related concepts

• K-means clusteringused by · K-means clustering partitions data into a specified number of groups by assigning observations to representative centers and iteratively updating those centers to reduce within-group distances.
• Manhattan distancecontrasts with · Manhattan distance is the sum of the absolute differences between corresponding coordinates of two points or vectors.
• minkowski distanceequivalent to · Minkowski distance is a parameterized metric between points based on a positive-order norm, including Manhattan and Euclidean distance as special cases.
• distancekind of · Distance is a measure of separation between points or entities in a defined space or context.
• l2 normequivalent to · The L2 norm is the nonnegative Euclidean magnitude of a vector, equal to the square root of the sum of its components’ squares, and to absolute value for a scalar.
• cosine distancealternative to · Cosine distance is a measure of dissimilarity between nonzero vectors based on their angular separation, commonly defined as one minus cosine similarity.

This explainer was generated by Manning's knowledge graph from the text of MongoDB 8.0 in Action, Third Edition. The section links point to the parts of the book it draws on.