Euclidean Distance

The L2 norm of the difference between two vectors. For vectors a and b with n dimensions:

d(a,b) = √(Σ(aᵢ - bᵢ)²)

• Computed as the square root of the sum of squared differences across all dimensions.
• Equivalent to the Pythagorean theorem generalized to n-dimensional space.
• In FAISS, this is MetricType.METRIC_L2 and is the default distance metric.

Unlike cosine similarity, Euclidean distance is highly sensitive to the absolute magnitude of vectors, not just their direction.

• Two vectors pointing in the exact same direction but with different lengths will have a non-zero Euclidean distance.
• This makes it suitable for applications where intensity matters, such as comparing raw image pixel intensities or audio spectrograms.
• For semantic search using embeddings, this property is often undesirable, which is why cosine similarity is preferred—it normalizes out magnitude differences.

Euclidean distance satisfies all four conditions of a proper metric space:

• Non-negativity: d(a,b) ≥ 0 (distance is never negative).
• Identity of indiscernibles: d(a,b) = 0 if and only if a = b.
• Symmetry: d(a,b) = d(b,a).
• Triangle Inequality: d(a,c) ≤ d(a,b) + d(b,c).

These properties guarantee consistent behavior in spatial indexing structures like VP-trees and M-trees, which rely on the triangle inequality for pruning.

Euclidean distance degrades rapidly in high-dimensional spaces due to the curse of dimensionality:

• As dimensionality increases, the ratio between the distance to the nearest neighbor and the farthest neighbor approaches 1, making distances lose discriminative power.
• In 100+ dimensions, all points appear nearly equidistant under L2, rendering exact Euclidean nearest-neighbor search meaningless without dimensionality reduction.
• This phenomenon is why ANN algorithms with cosine or inner product similarity dominate modern semantic search over dense embeddings.

Euclidean distance has direct mathematical relationships to other common vector metrics:

• Cosine Similarity: For L2-normalized vectors (unit length), Euclidean distance and cosine similarity have a monotonic relationship: d²(a,b) = 2(1 - cos(a,b)).
• Dot Product: ||a - b||² = ||a||² + ||b||² - 2(a·b). For normalized vectors, maximizing dot product is equivalent to minimizing Euclidean distance.
• Manhattan Distance (L1): Always greater than or equal to Euclidean (L2) distance: L1 ≥ L2 ≥ L∞.

The computational complexity of exact Euclidean distance calculation is O(d) for d dimensions:

• Requires d subtractions, d multiplications, and (d-1) additions, plus one square root.
• Modern libraries like FAISS accelerate this using SIMD instructions (AVX-512, NEON) to process multiple dimensions in parallel.
• For brute-force search over N vectors, total complexity is O(Nd). ANN indices reduce this to O(log N) by avoiding distance computations against the entire dataset.

Measures the cosine of the angle between two vectors, focusing purely on orientation and ignoring magnitude. Unlike Euclidean distance, it is invariant to vector length, making it ideal for comparing document embeddings where the frequency of terms matters less than their relative distribution. Values range from -1 (diametrically opposed) to 1 (identical direction). In normalized vector spaces, Euclidean distance and cosine similarity have a direct mathematical relationship.

A phenomenon where data becomes increasingly sparse as the number of dimensions grows, causing the contrast between the nearest and farthest neighbor to converge toward zero. In high-dimensional spaces, Euclidean distance loses its discriminative power, as all points appear nearly equidistant. This fundamentally necessitates the use of approximate nearest neighbor algorithms and dimensionality reduction techniques before applying distance metrics.

The exact method that computes the Euclidean distance between a query vector and every single vector in the database, guaranteeing perfect recall. The time complexity is O(N * D), where N is the number of vectors and D is the dimensionality. While computationally prohibitive for large-scale production systems, it serves as the ground-truth benchmark against which approximate methods are measured for recall accuracy.

The optimization problem of finding the database vector that maximizes the dot product with a query vector. Unlike Euclidean distance, MIPS is sensitive to both magnitude and direction, making it critical for matrix factorization models and attention mechanisms. Euclidean distance can be converted to an inner product problem via a simple algebraic transformation, allowing MIPS-optimized indices to be used for Euclidean nearest neighbor search.

A preprocessing step that projects high-dimensional vectors into a lower-dimensional space to mitigate the curse of dimensionality. Techniques include Principal Component Analysis (PCA) for linear projection and t-SNE or UMAP for non-linear visualization. Reducing dimensions before computing Euclidean distance can improve both the speed and meaningfulness of similarity comparisons by discarding noise and redundant features.

The process of scaling a vector to have a unit norm (magnitude of 1) by dividing each component by the vector's L2 norm. When all vectors are L2-normalized, Euclidean distance becomes monotonically related to cosine similarity, effectively making the two metrics interchangeable. This normalization is a standard preprocessing step in dense passage retrieval to ensure that distance comparisons are not biased by embedding magnitude.