The 2‑norm (L₂ norm) measures the length of a vector in Euclidean space and is foundational to machine learning, optimization, and vector similarity calculations. It generalizes the Pythagorean theorem to n dimensions and is a building block for search, regularization, and high-dimensional distance computation.
Understanding the 2‑Norm Vector (L₂ Norm)
The 2‑norm (or L₂ norm) is a standard way of measuring the length or magnitude of a vector in Euclidean space. For a real vector x in ℝⁿ, the 2‑norm is defined as:
This is a generalization of the Pythagorean theorem to n dimensions.
Key Properties of the 2‑Norm
A valid norm must satisfy the following properties:
Non-negativity
‖x‖₂ ≥ 0, and equals 0 if and only if x is the zero vector.
Homogeneity
For any scalar α, ‖αx‖₂ = |α| · ‖x‖₂.
Triangle Inequality
‖x + y‖₂ ≤ ‖x‖₂ + ‖y‖₂.
Geometric Intuition
In 2D, the 2‑norm gives the straight-line distance from the origin to a point (x₁, x₂). In higher dimensions, it represents the same concept extended into n-dimensional space.
All vectors with a 2‑norm equal to 1 form a unit circle in 2D or a hypersphere in higher dimensions.
Why the 2‑Norm Matters
The 2‑norm is used widely in:
- Machine learning: Calculating distances between data points.
- Regularization: L₂ regularization (Ridge Regression) penalizes large coefficients.
- Optimization: Gradient-based algorithms often minimize L₂ distances.
- Linear algebra: Relates directly to the dot product: ‖x‖₂ = √(x · x).
L₁ vs. L₂ vs. L∞ Norms
| Norm | Formula | Shape (2D) | Behavior |
|---|---|---|---|
| L₁ | ∑ |xᵢ| | Diamond | Promotes sparsity |
| L₂ | √∑ xᵢ² | Circle | Evenly distributes error |
| L∞ | max |xᵢ| | Square | Dominated by largest component |
Example Calculation
Let x = [1, 2] and y = [9, 8]. The L₂ distance between x and y is:
This represents the straight-line (Euclidean) distance between the two points.
- For complex vectors, the L₂ norm is defined as: ‖x‖₂ = √(xᴴx), where xᴴ is the conjugate transpose of x.
- The matrix 2‑norm (also called spectral norm) is equal to the largest singular value of the matrix.
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