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For two nonzero vectors with the same features in the same order, cosine similarity is their dot product divided by the product of their Euclidean lengths. Use a short NumPy function for one pair of dense vectors, or scikit-learn’s pairwise function for collections of rows and sparse text data.
What cosine similarity measures
Cosine similarity compares the direction of two vectors rather than their raw magnitude:
similarity(a, b) = dot(a, b) / (||a||₂ × ||b||₂)
Scikit-learn defines it as the L2-normalized dot product. For ordinary real-valued vectors, the result ranges from -1 to 1. With nonnegative features, such as counts or TF-IDF weights, it ranges from 0 to 1. Multiplying a nonzero vector by a positive constant does not change its cosine similarity, so cosine can discard magnitude information that may matter for your task. Scikit-learn’s metrics documentation explains the definition and use.
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Implement cosine similarity for one pair of vectors
This NumPy helper checks that both inputs are one-dimensional, have matching shapes, and are nonzero before calculating the normalized dot product:
import numpy as np
def cosine_similarity(a, b):
a = np.asarray(a, dtype=float)
b = np.asarray(b, dtype=float)
if a.ndim != 1 or b.ndim != 1:
raise ValueError("a and b must be one-dimensional vectors")
if a.shape != b.shape:
raise ValueError("a and b must have the same shape")
norm_a = np.linalg.norm(a)
norm_b = np.linalg.norm(b)
if norm_a == 0 or norm_b == 0:
raise ValueError("cosine similarity is undefined for a zero vector")
return float(np.dot(a, b) / (norm_a * norm_b))For example, [1, 0] and [0, 1] have a dot product of zero, so their cosine similarity is 0. The explicit checks make invalid or incompatible inputs fail clearly instead of producing a misleading score.
Compare rows with scikit-learn
For pairwise comparisons between sets of vectors, use scikit-learn’s cosine_similarity. It returns a matrix whose entry at row i, column j is the similarity between X[i] and Y[j].
from sklearn.metrics.pairwise import cosine_similarity
scores = cosine_similarity(X, Y)The function accepts SciPy sparse matrices, making it suitable for sparse feature representations such as text vectors. See the pairwise metrics API documentation for the accepted inputs and return shape.
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Cosine similarity operates on vectors, not raw strings. First convert documents into vectors in the same feature space—for example, with TF-IDF. Scikit-learn notes that when TF-IDF rows are L2-normalized, their dot product is already cosine similarity. Its preprocessing guide describes the normalized-vector shortcut.
For repeated queries against a fixed collection
If the collection’s rows are L2-normalized, compute their dot products with each query to obtain cosine scores. For a matrix of normalized rows, matrix multiplication can calculate many scores at once. This shortcut is valid only when both sides are normalized in the same way; keep that invariant explicit so normalized and unnormalized vectors are not mixed.
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Zero vectors
A zero vector has a zero norm, making the denominator zero; ordinary cosine similarity is undefined. Reject it, as the helper above does, or define a documented application-specific convention. Do not add an arbitrary epsilon and present the resulting value as the standard formula. Scikit-learn’s normalization implementation handles zero norms internally, but its exact output policy should be checked against the installed release’s documentation if it matters to your application; the main-branch implementation is mutable.
Matching dimensions and feature order
Equal vector lengths are not enough: each coordinate must mean the same feature on both sides. Ensure the vectors share a feature space and ordering. A shape check can catch different lengths, but it cannot detect a semantic mismatch between coordinates.
Negative coordinates and magnitude
Negative coordinates can produce negative cosine scores when vectors point in opposing directions. The commonly expected 0-to-1 range applies to nonnegative features, not all real vectors. Also, cosine ignores positive rescaling: if the size of a vector carries useful information, compare with a metric or method that preserves magnitude rather than relying on cosine alone.
Quick Recap
Choose the implementation for your workload
- One pair of small, dense vectors: use the NumPy helper to keep the calculation explicit and easy to inspect.
- Many rows or sparse text features: use scikit-learn’s
cosine_similarity(X, Y)to get pairwise scores. - Rows already L2-normalized: use a dot product or matrix multiplication, provided the normalization is guaranteed for both inputs.
- Embedding vectors: the same formula applies, but whether cosine is appropriate depends on the embedding model and downstream task. A similarity score is not automatically a calibrated probability or universal judgment of meaning.
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