Similarity Metrics

## The two functions Cosine similarity between vectors `a` and `b` is their dot product divided by the product of their lengths: cos(a, b) = (Σ aᵢbᵢ) / (√Σaᵢ² · √Σbᵢ²) Geometrically this is the cosine of the angle between the two arrows. It equals 1 when they point in exactly the same direction, 0 when they are perpendicular, and -1 when they point in exactly opposite directions. Because the length of each vector is divided out, multiplying either vector by any positive constant leaves the value unchanged. Cosine is a *similarity*: higher is better. Euclidean distance, also called L2 distance, is the ordinary straight-line gap between the two points: L2(a, b) = √Σ(aᵢ - bᵢ)² It runs from 0 (identical vectors) upward with no ceiling, and it is a *distance*: lower is better. It responds to both the angle and the lengths. Two vectors pointing in exactly the same direction have cosine 1 but a Euclidean distance of |‖a‖ - ‖b‖|, which can be large. ## Why direction usually carries the meaning in text embeddings An embedding model maps text into a vector space where semantically related texts are placed near one another. What "near" means was fixed by how the model was trained. Most modern text-embedding models are trained with a contrastive objective that scores pairs by cosine similarity: matching pairs are pushed toward a high cosine, mismatched pairs toward a low one. Nothing in that objective constrains how *long* a vector is, so length ends up encoding whatever happens to fall out of the architecture — typically things correlated with token count, token frequency, or how strongly the model reacts to the input. Those are rarely the ranking signal you want. Cosine strips them out and compares meaning-as-direction only. This is why a search over documents of wildly different lengths — a two-line note and a ten-page report — should usually not have the report win simply because its vector came out bigger. ## Where Euclidean is genuinely the right call Three cases come up in interviews. First, when the embedding space was trained so that absolute distance is meaningful. Face-verification embeddings are the classic example: models trained with a margin-based loss are designed so that two images of the same person land within a fixed radius of each other, and the deployed system compares an L2 distance to a fixed acceptance threshold. Here the distance value itself, not just the ordering, is the product. Second, when the vectors are not embeddings at all but genuine coordinates — positions, physical measurements, sensor readings — where length is a real quantity and discarding it would be throwing away data. Third, and most importantly in practice, it does not matter at all when the vectors are unit-normalized: with all lengths equal to 1, Euclidean distance is a strictly decreasing function of cosine, so the two produce exactly the same ordering. Many systems normalize on write precisely so this choice stops mattering. ## The rule that actually decides The interview-grade answer is not "cosine is better for text." It is: use the metric the model was trained with, and if you do not know what that was, check the model's documentation rather than guessing. A model trained with a cosine objective and queried with raw Euclidean distance will still return plausible-looking results — plausible enough that nobody notices — while quietly losing recall on exactly the cases the length variation touches. ## Practical edge cases - Cosine is undefined for a zero vector (division by zero). Empty or whitespace-only inputs can produce degenerate vectors; guard the input rather than the metric. - Cosine of exactly -1 essentially never appears with text encoders; encoders do not place opposites at 180 degrees. Unrelated text tends to land somewhere near perpendicular, so the interesting part of the scale is compressed toward the positive end, and a raw cosine value is not a probability. - Cosine similarity is not a metric in the mathematical sense (it does not satisfy the triangle inequality); some index structures that assume a true metric want the distance form instead. - Whatever you choose, use the same metric everywhere: at index build time, at query time, in your evaluation harness, and in any cached scores. Mixed metrics across those stages is a common and silent source of bad results.