Distance Metric

Euclidean distance calculates the straight-line distance between two points in vector space. It is the most intuitive geometric distance, computed as the square root of the sum of squared differences between corresponding vector dimensions. It is highly sensitive to the absolute magnitude of vectors.

• Formula: √(Σ(x_i - y_i)²)
• Use Case: Best for data where the absolute magnitude of the feature values is meaningful, such as physical measurements or pixel intensities in images.
• Property: It is a true metric, satisfying the triangle inequality.

Cosine similarity measures the cosine of the angle between two vectors, ignoring their magnitudes. It focuses solely on orientation. A value of 1 means identical direction, 0 means orthogonal, and -1 means diametrically opposite.

• Formula: (A · B) / (||A|| ||B||)
• Use Case: The default for most text embeddings (e.g., from models like OpenAI's text-embedding-ada-002), where the semantic meaning is encoded in the vector's direction, not its length.
• Note: Often converted to a distance via cosine distance = 1 - cosine similarity.

The inner product (or dot product) is the sum of the products of corresponding vector components. When vectors are normalized (unit length), it is equivalent to cosine similarity. For unnormalized vectors, it combines both direction and magnitude.

• Formula: Σ(x_i * y_i)
• Use Case: Used in maximum inner product search (MIPS) for recommendation systems, where the goal is to find items with the highest predicted affinity (score), not just similar direction.
• Performance: Many optimized vector indexes (like FAISS) have dedicated modes for maximum inner product search.

Manhattan distance (or L1 norm, Taxicab distance) sums the absolute differences between vector coordinates. It represents the distance traveled along grid-like paths.

• Formula: Σ|x_i - y_i|
• Use Case: Useful in high-dimensional spaces where the Euclidean distance might become less discriminative, or for data with sparse features. Often used in computer vision and clustering algorithms like K-Medians.
• Interpretation: More robust to outliers than Euclidean distance, as it does not square the differences.

These metrics are designed for binary or set-based data representations.

• Jaccard Distance: Measures dissimilarity between finite sample sets. Calculated as 1 minus the size of the intersection divided by the size of the union of the sets. Ideal for comparing document shingles or user behavior sets.
• Hamming Distance: Counts the number of positions at which the corresponding symbols are different between two strings or binary vectors of equal length. Fundamental for error-correcting codes and comparing encoded categorical data.

These are less common for dense float embeddings but are crucial for specific binary quantization or set-similarity search scenarios.

A true distance metric must satisfy four mathematical properties: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. This last property is crucial for many indexing algorithms to prune the search space efficiently.

Choosing a Metric:

• Text Embeddings: Start with cosine similarity.
• Image/Physical Data: Try Euclidean distance.
• Recommendations: Use inner product.
• The metric must match the property preserved by the embedding model used to generate your vectors. Using the wrong metric is a primary cause of poor search quality.

Cosine similarity measures the cosine of the angle between two vectors, ignoring their magnitudes. It ranges from -1 (perfectly opposite) to 1 (identical orientation). It is the most common metric for text embeddings generated by models like Sentence-BERT, as it focuses on semantic direction rather than vector length.

• Key Use Case: Comparing text embeddings where magnitude is less informative than direction.
• Formula: cos(θ) = (A · B) / (||A|| ||B||)
• Database Implementation: Often optimized by indexing normalized vectors, turning cosine similarity into a dot product or Euclidean distance on normalized vectors for faster computation.

Euclidean distance (L2 norm) measures the straight-line distance between two points in vector space. It is the most intuitive geometric distance. Lower values indicate greater similarity. It is often the default metric for image and other dense embeddings.

• Key Use Case: Computer vision embeddings, general-purpose dense vectors.
• Formula: √(Σ(Aᵢ - Bᵢ)²)
• Database Consideration: Some indexing algorithms like HNSW are highly optimized for L2 distance. For normalized vectors, Euclidean distance is monotonically related to cosine similarity.

Inner product (dot product) is the sum of the products of corresponding vector components. For normalized vectors (unit length), it is equivalent to cosine similarity. For non-normalized vectors, it combines both direction and magnitude, which can be desirable for some ranking tasks.

• Key Use Case: Recommendation systems where both user and item embedding magnitudes signal preference strength.
• Formula: A · B = Σ(Aᵢ * Bᵢ)
• Database Note: Many vector databases internally convert high dot product searches to equivalent cosine or Euclidean searches on normalized data for index compatibility.

Manhattan distance (L1 norm or taxicab distance) is the sum of the absolute differences between vector coordinates. It measures distance along axes at right angles, like navigating city blocks. It can be more robust to outliers than Euclidean distance.

• Key Use Case: Sparse, high-dimensional data, certain computer vision applications, and scenarios where robustness to outliers is critical.
• Formula: Σ|Aᵢ - Bᵢ|
• Performance: Computationally simpler than Euclidean distance but less commonly the primary metric for mainstream vector search indexes.

Jaccard similarity measures the similarity between finite sample sets. It is defined as the size of the intersection divided by the size of the union of the sets. For vectors, it is typically applied to binary or tokenized representations (e.g., word sets).

• Key Use Case: Comparing document token sets, recommendation systems based on binary user-item interactions.
• Formula: J(A,B) = |A ∩ B| / |A ∪ B|
• Vector Database Context: Often used in hybrid search scenarios where sparse, set-based data is combined with dense vector similarity.

Hamming distance counts the number of positions at which the corresponding symbols (or bits) are different between two strings or binary vectors of equal length. It is a fundamental metric for comparing categorical or binary data.

• Key Use Case: Comparing binary hash codes (like locality-sensitive hashing outputs), error detection in information theory, and genomic sequence analysis.
• Example: The Hamming distance between [1,0,1,1] and [1,1,1,0] is 2.
• Database Role: Used in specific indexing techniques for ultra-fast, approximate filtering before a more precise distance calculation.