L2 norm

As explained in Natural Language Processing in Action, Second Edition

The L2 norm of a vector is the square root of the sum of the squares of its components.

The L2 norm measures a vector’s length: square each component, add the squares, and take the square root. Geometrically, it is the diagonal length of a right triangle extended across every dimension. section 3.3 section 6.4.6

Use the L2 norm to:

• Calculate Euclidean distance by subtracting two vectors first, then measuring the L2 norm of their difference. section 3.3
• Normalize vectors by dividing each vector by its own L2 norm, giving vectors with length 1. section 6.4.6
• Relate vector length to cosine similarity, which divides a dot product by the L2 norm of each vector. section 3.3.1

Compute and normalize a vector

For vector [1, 2, 3, 4], the L2 norm is sqrt(1² + 2² + 3² + 4²), approximately 5.47. NumPy computes it with np.linalg.norm(vector), then normalizes the vector with vector / np.linalg.norm(vector). The same operation scales 300-dimensional sentence embeddings to a common length while preserving their directional differences. section 6.4.6 section 6.4.6

Figure 3.2 2D Euclidean distance

Inspect the two-dimensional geometry of Euclidean distance and the vector difference whose L2 norm supplies that distance. section 3.3

How Natural Language Processing in Action, Second Edition covers it

How deep it goes

Defines the formula, explains geometric consequences, and demonstrates NumPy normalization without deriving deeper linear-algebra theory.

Examples

Uses two-dimensional vector geometry, a four-element NumPy vector, TF vectors, and normalized sentence embeddings.

What it assumes

Assumes vectors, component-wise arithmetic, and the Pythagorean theorem while introducing the norm through geometric intuition.

What you can do after

Apply it via NumPy to calculate vector lengths, normalize embeddings, and support cosine-similarity computations.

How it connects

Connects Euclidean distance, cosine similarity, dot product, vector normalization, sentence embeddings, and L1 norm.

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Also explained in

• 

  liveBookmanning.com
  Grokking Machine LearningThe L2 norm of a vector is the square root of the sum of the squared values of its components.
• 

  liveBookmanning.com
  Math and Architectures of Deep LearningThe Euclidean norm measuring a vector's length by squaring components, summing them, and taking the square root.

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Related concepts

• Euclidean distanceequivalent to · Euclidean distance is the straight-line distance between two points in Euclidean space, calculated as the square root of the sum of squared differences between corresponding coordinates.
• weight decayused by · Weight decay is a model-training regularization technique that discourages large parameter values by penalizing their magnitudes or shrinking them during optimization.
• L1 regularizationused by · L1 regularization is a technique that adds a penalty proportional to the sum of absolute model-parameter values, encouraging sparse parameter estimates.
• ridge regressionconfigures · Ridge regression is linear regression that applies an L2 penalty to coefficient magnitudes, shrinking estimates toward zero to reduce overfitting.

This explainer was generated by Manning's knowledge graph from the text of Natural Language Processing in Action, Second Edition. The section links point to the parts of the book it draws on.