Similarity Search in Vector Databases

What is Similarity Search?

Similarity search is a fundamental operation in vector databases that finds the most similar vectors to a query vector based on distance metrics. Unlike traditional databases that use exact matching, vector databases excel at finding approximate nearest neighbors (ANN) in high-dimensional spaces.

Distance Metrics

Common distance metrics used in similarity search include:

Euclidean Distance (L2)

• Most intuitive distance metric

  • Represents the straight-line distance between two points in space, similar to using a ruler
  • Matches human intuition about distance and is easy to visualize in 2D or 3D space

• Measures straight-line distance between two points

  • Calculates the shortest possible path between two vectors in n-dimensional space
  • Works by taking the square root of the sum of squared differences between corresponding dimensions

• Formula: sqrt(Σ(x_i - y_i)²)

  • For each dimension i, subtract the coordinates (x_i - y_i) and square the result
  • Sum all squared differences and take the square root to get the final distance

• Best for: General-purpose similarity search

  • Works well when the absolute magnitudes of vectors are meaningful to your comparison
  • Particularly effective for dense vectors where all dimensions contribute similarly to similarity

Cosine Similarity

• Measures angle between vectors

  • Focuses on the orientation/direction of vectors rather than their magnitude
  • Perfect for comparing vectors where scale differences should be ignored

• Range: -1 to 1 (1 being most similar)

  • Value of 1 means vectors point in same direction, -1 means opposite directions, 0 means perpendicular
  • This normalized range makes it easy to interpret and set thresholds for similarity

• Formula: cos(θ) = (A·B)/(||A||·||B||)

  • Calculated by taking dot product of vectors (A·B) divided by product of their magnitudes
  • Normalization by vector lengths makes it scale-invariant, focusing purely on direction

• Best for: Text embeddings, semantic search

  • Excels at comparing text embeddings where relative relationships between dimensions matter more than absolute values
  • Particularly useful when vectors have different magnitudes but similar semantic meaning

Manhattan Distance (L1)

• Named after Manhattan’s grid-like street layout

  • Also known as L1 distance or city block distance, inspired by navigating city blocks
  • Measures distance as if traveling along a rectangular grid path, like a taxi in Manhattan

• Sum of absolute differences between coordinates

  • Calculates total distance by adding absolute differences in each dimension
  • Less sensitive to outliers compared to Euclidean distance due to linear growth

• Formula: Σ|x_i - y_i|

  • For each dimension i, take absolute difference between corresponding coordinates
  • Sum all absolute differences without squaring, making it computationally simpler than L2

• Best for: Sparse vectors and feature comparison

  • Particularly effective when dealing with high-dimensional sparse data where most values are zero
  • Commonly used in computer vision and when differences in individual features should be weighted equally

Indexing Methods

HNSW (Hierarchical Navigable Small World)

• Multi-layer graph structure

  • Creates a hierarchical structure where top layers are sparse and lower layers are dense
  • Each layer is a navigable small world graph, enabling fast traversal and search

• Search algorithm

  • Starts from the top sparse layer and gradually descends to denser layers
  • Uses greedy search at each layer to find the closest neighbors efficiently

• Performance characteristics

  • Provides logarithmic time complexity for search operations
  • Offers excellent balance between search speed and accuracy, making it current state-of-the-art

• Implementation considerations

  • Key parameters include M (max connections per node) and ef_construction (search width during build)
  • Higher values improve accuracy but increase memory usage and construction time

IVF (Inverted File Index)

• Clustering-based approach

  • Divides the vector space into Voronoi cells using k-means clustering
  • Each vector is assigned to its nearest centroid, creating clusters of similar vectors

• Search process

  • First finds the nearest centroids to the query vector
  • Then searches only within the selected clusters, significantly reducing search space

• Trade-offs

  • Faster search times but potentially lower accuracy compared to HNSW
  • Memory efficient as it doesn’t require storing graph connections

• Best practices

  • Number of clusters should scale with dataset size (typically sqrt(n))
  • Multiple probe queries can improve recall at the cost of speed

LSH (Locality-Sensitive Hashing)

• Hashing mechanism

  • Uses special hash functions that map similar vectors to the same buckets
  • Multiple hash tables are used to increase the probability of finding true neighbors

• Probability-based approach

  • Similar items have a higher probability of collision in hash buckets
  • Dissimilar items have a lower probability of collision, enabling efficient filtering

• Performance characteristics

  • Sub-linear search time complexity
  • Memory efficient but typically less accurate than HNSW or IVF

• Use cases

  • Excellent for extremely large-scale datasets where approximate results are acceptable
  • Often used in initial filtering before more accurate methods

Implementation Strategies

Basic Search Flow

Example using a vector database client
from vector_db_client import VectorDB
Create embeddings for query
query_text = "example search"
query_vector = embedding_model.encode(query_text)
Perform similarity search
results = vector_db.search(
vector=query_vector,
top_k=5,
namespace="documents"
)

Hybrid Search

Combines vector similarity with traditional filters:

results = vector_db.search(
vector=query_vector,
filter={
"metadata.category": "technology",
"metadata.date": {"$gt": "2023-01-01"}
},
top_k=5
)

Performance Optimization

Tips for Better Search Results

1. Vector Normalization

   • Normalize vectors before storage
   • Ensures consistent distance calculations

2. Dimension Reduction

   • Use techniques like PCA when appropriate
   • Balance between accuracy and performance

3. Index Parameters

   • Tune HNSW parameters (M, ef_construction)
   • Adjust based on dataset size and requirements

4. Batch Processing

   • Use batch operations for insertions
   • Implement bulk loading for large datasets

Common Challenges

1. Curse of Dimensionality

   • Performance degrades in high dimensions
   • Solution: Use dimension reduction or better indexing

2. Quality-Speed Trade-off

   • Faster search often means less accuracy
   • Solution: Tune index parameters based on needs

3. Scale Issues

   • Large datasets require more resources
   • Solution: Implement sharding and clustering

Best Practices

1. Vector Preparation

   • Normalize vectors consistently
   • Use appropriate embedding models
   • Handle missing values properly

2. Index Selection

   • Choose based on dataset size
   • Consider memory constraints
   • Test with representative data

3. Monitoring

   • Track search latency
   • Monitor recall metrics
   • Implement performance logging

Resources for Further Learning

• Understanding HNSW Algorithm
• Vector Similarity Metrics
• Optimization Techniques