Soft Clamping Functions

📐 Definition

For bounds $a < b$ , slope parameter $\beta > 0$ , and center $x_0$ , define

\operatorname{softclamp}(x) = a + (b-a)\,\sigma\!\big(\beta(x - x_0)\big), \qquad \sigma(z) = \frac{1}{1+e^{-z}}

Domain and Codomain

For real inputs, $\operatorname{softclamp}:\mathbb{R}\to(a,b)$ , with $\operatorname{softclamp}(x)\to a$ as $x\to-\infty$ and $\operatorname{softclamp}(x)\to b$ as $x\to+\infty$ .

⚙️ Key Properties

Monotone and $C^\infty$ ; derivative

\operatorname{softclamp}'(x) = (b-a)\,\beta\,\sigma(\beta(x-x_0))\big(1-\sigma(\beta(x-x_0))\big)

peaks at $x = x_0$ . Limits: $\lim_{x\to -\infty}\operatorname{softclamp}(x) = a$ , $\lim_{x\to\infty}\operatorname{softclamp}(x) = b$ .

🎯 Special Cases and Limits

$\beta\to\infty$ approaches a hard clamp centered at $x_0$ .
$a=0$ , $b=1$ , $x_0=0$ reduces to the logistic sigmoid $\sigma(\beta x)$ .

Hyperbolic tangent provides a symmetric clamp to $(-1,1)$ ; smooth saturation functions generalize bounded responses with alternative profiles.

Usage in Oakfield

Oakfield uses soft clamping primarily in the thermostat/regulation path:

Thermostat operator supports smooth min/max limiting of its regulated parameters via a softplus-style clamp (controlled by softplus_k), avoiding hard discontinuities when enforcing bounds.
Other parts of the engine also provide hard clamps (e.g. continuity clamp ranges), but the explicit “soft clamp” behavior is concentrated in the thermostat’s smooth clamp logic.

Historical Foundations

📜 Logistic Transitions

Logistic-style transitions provide smooth approximations to step functions and clamps, enabling bounded mappings with controllable transition sharpness.

🌍 Modern Perspective

Soft clamps are standard tools when boundedness is required but differentiability must be preserved for gradient-based methods.

📚 References

Boyd & Vandenberghe, Convex Optimization
NIST Digital Library of Mathematical Functions (DLMF), §4