In practical terms, what differences might you observe in a search system when using cosine similarity instead of Euclidean distance on the same set of normalized embeddings?

When using cosine similarity instead of Euclidean distance on normalized embeddings in a search system, the primary differences lie in how similarity is quantified, the behavior of the metrics under specific data conditions, and their computational interpretation. Both metrics measure relationships between vectors, but their mathematical foundations lead to distinct practical outcomes, even when embeddings are normalized to unit length.

First, cosine similarity focuses on the angle between vectors, ignoring their magnitudes. For normalized embeddings, this means it directly measures directional alignment. For example, in a text retrieval system, two documents with similar word usage (aligned direction) but slightly different embedding magnitudes due to minor term frequency variations would still score highly with cosine similarity. Euclidean distance, however, measures the straight-line distance between points in space, which indirectly incorporates directional differences. On normalized data, Euclidean distance becomes a function of cosine similarity: smaller distances correspond to higher cosine values (since distance = √(2(1 − cosine_similarity))). This relationship means the ranking of results (e.g., top matches in a search) will be identical for both metrics. However, the interpretation of scores differs: cosine similarity ranges from -1 to 1 (with 1 being identical), while Euclidean distance ranges from 0 to 2 (with 0 being identical).

Second, practical differences emerge in scenarios involving sparse or high-dimensional data. For instance, in recommendation systems using user-item embeddings, cosine similarity might prioritize items with overlapping features (e.g., movies sharing genres), even if their overall popularity (magnitude) differs. Euclidean distance, while producing the same ranked list, could lead to confusion if developers misinterpret the distance values as absolute measures of relevance rather than normalized scores. Additionally, thresholds for filtering results (e.g., “show items with similarity > 0.8”) would need adjustment if switching metrics. For example, a cosine threshold of 0.8 corresponds to a Euclidean distance of √(2(1 − 0.8)) ≈ 0.63, which might not be intuitive during debugging or analysis.

Finally, computational considerations can differ. Cosine similarity for normalized embeddings simplifies to a dot product, which is computationally efficient. Euclidean distance requires squaring and summing differences, adding marginal overhead. While this difference is negligible in small systems, it can compound in large-scale applications processing millions of embeddings. For example, a search engine using approximate nearest neighbor libraries like FAISS might see slight performance variations depending on the metric chosen, even though the underlying ranking remains unchanged. Developers must also ensure consistent metric usage across training and inference stages to avoid mismatches in model behavior or evaluation (e.g., training a model with cosine loss but evaluating with Euclidean distance).

In summary, while the ranking of results remains identical for normalized embeddings, developers will observe differences in score interpretation, threshold tuning, and computational efficiency when choosing between cosine similarity and Euclidean distance. The choice depends on the system’s requirements for intuitiveness, performance, and alignment with downstream processes.