Power-Law Warp

📐 Definition

For $p > 0$ , define the power-law warp

\omega_p(x) = \operatorname{sgn}(x)\,|x|^{p}

with $\operatorname{sgn}(0) = 0$ .

Domain and Codomain

$x \in \mathbb{R}$ (or $\mathbb{R}^d$ applied componentwise). Output is real-valued; monotone increasing for $p > 0$ .

⚙️ Key Properties

\omega_p(-x) = -\omega_p(x), \qquad \omega_p'(x) = p\,|x|^{p-1}\quad (x \ne 0), \qquad \omega_p(\lambda x) = \lambda^{p} \omega_p(x)\ (\lambda > 0)

$p = 1$ yields the identity map; $p = 2$ yields $\omega_2(x)=x|x|$ with slope $\omega_2'(x)=2|x|$ (flattening to $0$ at the origin); $p \to 0^+$ compresses amplitudes toward $\operatorname{sgn}(x)$ .

🎯 Special Cases and Limits

$p=1$ is identity; $p>1$ expands large magnitudes; $0<p<1$ compresses them.
As $p\to 0^+$ , $\omega_p(x)$ approaches $\operatorname{sgn}(x)$ for $x\ne 0$ .

Serves as a simple analytic warp; connects to fractional derivatives through the same power-law exponents.

Usage in Oakfield

Oakfield uses power-law mappings in its built-in warp and measurement operators:

Analytic warp (POWER profile) applies a power-based response with sign/magnitude handling and a computed gradient used in the warp update.
Remainder (POWER nonlinearity) applies a power transform before differencing, with an epsilon guard to avoid singular behavior near zero.
Power laws also appear in spectral operators (e.g. dispersion uses pow(|k-k0|, order)), but those are spectral-phase laws rather than pointwise warps.

Historical Foundations

📜 Power Laws

Power-law mappings are classic nonlinear rescalings that encode multiplicative similarity and are widely used for contrast enhancement and shaping.

🌍 Modern Perspective

They remain a simple, tunable warp with predictable monotonic behavior and easy implementation.

📚 References

Rudin, Real and Complex Analysis