The choice of distance metric directly impacts which vectors are considered "nearest" by prioritizing different geometric or directional relationships. Here’s how each metric behaves:
Euclidean distance measures the straight-line distance between two points in space. It considers both the direction and magnitude of vectors, making it sensitive to differences in scale. For example, in a dataset where vectors represent product ratings scaled by user activity (e.g., frequent vs. infrequent users), Euclidean distance would treat a user who rates everything 5/5 differently from one who rates the same products 3/5, even if their preferences align directionally. This metric is ideal when absolute differences in vector coordinates matter, such as in spatial data or physical measurements. However, it can perform poorly if features aren’t normalized, as large-magnitude features dominate the distance calculation.
Cosine similarity focuses on the angle between vectors, ignoring their magnitudes. It quantifies how similar two vectors are in direction, making it useful for comparing documents in NLP (e.g., TF-IDF vectors) or embeddings where magnitude isn’t meaningful. For instance, two text embeddings about "machine learning" would have high cosine similarity even if one document is longer than the other. However, cosine similarity fails when magnitude carries important information. In a recommendation system, if a user’s interaction frequency (reflected in vector magnitude) indicates engagement strength, cosine similarity might overlook highly active users who have broader but weaker preferences.
Dot product combines both magnitude and direction. It’s equivalent to cosine similarity when vectors are normalized but amplifies the influence of larger magnitudes when they’re not. For example, in a recommendation system, a user vector with high magnitude (indicating strong preferences) and an item vector with high magnitude (indicating popularity) would yield a high dot product even if their angle isn’t perfectly aligned. This makes it useful for ranking tasks where both alignment and intensity matter, like matching users to trending items. However, the lack of normalization can skew results toward high-magnitude vectors, even if they’re less relevant directionally.
In practice, the choice depends on the data’s nature and use case. Euclidean suits absolute differences, cosine works for directional alignment, and dot product balances both when magnitudes are meaningful. Normalization is often critical to ensure fair comparisons across metrics.