When comparing cosine similarity and Euclidean distance on normalized embeddings in a search system, the primary differences stem from how each metric interprets vector relationships. Normalized embeddings have unit length (magnitude of 1), which simplifies the mathematical relationship between the two metrics but does not eliminate practical distinctions in how they rank results.
1. Sensitivity to Vector Direction vs. Position
Cosine similarity measures the angle between vectors, focusing on their directional alignment. This makes it inherently invariant to magnitude, which is ideal for comparing embeddings where semantic similarity correlates with direction (e.g., text embeddings). For example, in a document search system, two articles about "climate change" might have high cosine similarity even if their embeddings are positioned differently on the unit sphere. In contrast, Euclidean distance measures the straight-line distance between vectors in the embedding space. On normalized embeddings, this distance is mathematically tied to cosine similarity via the formula:
Euclidean Distance² = 2 * (1 - Cosine Similarity).
While this relationship ensures that similar rankings emerge in simple cases, Euclidean distance can behave differently in edge cases. For instance, vectors with small angular differences but large coordinate-wise deviations (e.g., sparse embeddings) might be ranked slightly differently due to the squared error emphasis in Euclidean calculations.
2. Impact on Ranking and Thresholding While both metrics produce similar rankings for normalized embeddings, their score ranges differ. Cosine similarity outputs values between -1 and 1, with 1 indicating perfect alignment. Euclidean distance, on the other hand, ranges from 0 (identical vectors) to 2 (opposite directions). This affects how similarity thresholds are set. For example, a cosine similarity threshold of 0.8 corresponds to a Euclidean distance of ~0.63. Systems using fixed thresholds for filtering results (e.g., "show matches above 0.7 cosine similarity") would need to adjust these values if switching to Euclidean distance, even though the relative rankings remain consistent. Additionally, interfaces displaying raw scores to users might need normalization or explanation to avoid confusion.
3. Computational and Implementation Considerations
Cosine similarity is often computationally cheaper to calculate for normalized embeddings since it reduces to a dot product (u · v). Euclidean distance requires computing squared differences and a square root, though optimizations like skipping the root (since rankings depend on squared distances) can mitigate this. However, infrastructure choices might influence practical outcomes. For example, approximate nearest neighbor (ANN) libraries like FAISS or HNSW often optimize for one metric. While cosine similarity can be simulated with Euclidean distance on normalized data (by preprocessing embeddings), using the "wrong" algorithm without adjustments could degrade performance or accuracy in large-scale systems.
In summary, while cosine similarity and Euclidean distance yield equivalent rankings on normalized embeddings, practical differences arise in threshold interpretation, score presentation, and infrastructure optimization. The choice between them often depends on system requirements, such as computational efficiency or compatibility with existing tools, rather than algorithmic performance.