Cosine Similarity for Face Verification

easyPROFace RecognitionEmbeddingsCV: Recognition

You are given two face embeddings and need to calculate their cosine similarity.

Cosine similarity measures the angle between two vectors, ignoring their magnitudes:

$\cos(\theta) = \frac{a \cdot b}{\|a\| \|b\|} = \frac{\sum_i a_i b_i}{\sqrt{\sum_i a_i^2} \cdot \sqrt{\sum_i b_i^2}}$

Values range from:

1: vectors point in the same direction (very similar)
0: vectors are perpendicular (unrelated)
-1: vectors point in opposite directions

For normalized embeddings (common in face recognition), cosine similarity and Euclidean distance are directly related.

Example

Input

a = [1, 0, 0]
b = [1, 0, 0]

Output

1.0

Reasoning

Computing dot product and magnitudes:

Dot product: 1×1 + 0×0 + 0×0 = 1
||a|| = √(1² + 0² + 0²) = 1
||b|| = √(1² + 0² + 0²) = 1

Cosine similarity = 1 / (1 × 1) = 1.0

Identical vectors have similarity 1.

Constraints

a and b are lists of floats (embeddings) of the same length
Return similarity rounded to 4 decimal places
If either vector has zero magnitude, return 0.0

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Cosine Similarity for Face Verification: Background & Strategy

Background Knowledge

Face Embeddings and Vector Representation

In modern face recognition systems, faces are converted into high-dimensional vectors called embeddings through deep neural networks. These embeddings capture the essential features of a face in a compact numerical form. Rather than comparing raw pixel values, face verification compares these embeddings to determine if two images represent the same person. The key insight is that embeddings from the same person cluster together in vector space, while embeddings from different people are pushed apart.

Why Cosine Similarity for Face Recognition

Cosine similarity is particularly well-suited for face recognition because it measures the angle between vectors rather than their magnitude. This is ideal for normalized embeddings (which are common in face recognition systems) because it focuses on direction—the actual facial features—rather than the scale of the embedding. When embeddings are normalized to unit length, cosine similarity becomes computationally efficient and provides a natural similarity score between -1 and 1, where values closer to 1 indicate very similar faces. This metric has become standard in production systems for face verification tasks.

The Mathematical Foundation

The cosine similarity formula computes the dot product of two vectors divided by the product of their magnitudes. Geometrically, this gives you the cosine of the angle between them. For face verification, you typically set a threshold (often around 0.5-0.7 for normalized embeddings) to decide whether two embeddings represent the same person.

Algorithm/Approach

The general approach to solve cosine similarity problems follows this pattern:

Parse/receive the two embedding vectors
Compute the dot product (sum of element-wise products)
Compute the magnitudes (L2 norms) of both vectors
Apply the formula to get the cosine similarity score
Return or compare the result against a threshold if needed

This is a straightforward mathematical computation with no complex data structures required.

Step-by-Step Strategy

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solution.py

Python 3.13

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