Normalization

📐 Definition

For vector or field $x$ with norm $\|x\|$ and small $\varepsilon \ge 0$ to avoid division by zero,

\operatorname{normalize}(x) = \frac{x}{\|x\| + \varepsilon}

Domain and Codomain

Applies to real or complex vectors/fields with finite norm. Output has unit (or near-unit) norm when $\varepsilon=0$ and $\|x\| \ne 0$ .

⚙️ Key Properties

Scale-invariant: $\operatorname{normalize}(c x) = \operatorname{normalize}(x)$ for $c>0$ when $\varepsilon=0$ . Nonlinear and undefined at $x=0$ unless regularized by $\varepsilon$ .

With $\varepsilon>0$ , $\|x\| \ll \varepsilon$ yields strong attenuation. For $\varepsilon=0$ and $x \ne 0$ , the operation projects onto the unit sphere.

🎯 Special Cases and Limits

$\varepsilon=0$ normalizes exactly except at $x=0$ .
$\varepsilon>0$ provides a safe “soft” normalization near zero norm.

Energy norms define $\|x\|$ ; gain control applies additional scaling after normalization.

Usage in Oakfield

Oakfield relies more on pointwise (phasor-style) normalization than on global “normalize the whole field” operators:

Unit phasors: phase-aware paths form $z/|z|$ in phase_feature, analytic_warp (polar mode), and remainder (polar mode).
Diagnostics compute norms and norm-like summaries (mean_abs, rms, max_abs) via SimFieldStats; these are often used to choose thresholds and scaling.
Energy targeting: the thermostat operator enforces a prescribed mean $|u|^2$ level, which acts like a global normalization constraint.

Historical Foundations

📜 Unit-Norm Projections

Normalization is projection onto a unit-norm constraint (with regularization near the origin), a common step in constrained optimization and numerical stabilization.

🌍 Modern Perspective

It is widely used to keep amplitudes bounded and to standardize feature scales in numerical pipelines.

📚 References

Boyd & Vandenberghe, Convex Optimization