Description
Return scaledInput=beta*x and useLinearTail when scaledInput is strictly greater than switchThreshold.
Beta-scaled softplus is log(1+exp(beta*x))/beta for positive beta. Its input slope is a sigmoid, while curvature and parameter sensitivities are different derivatives. Numerical libraries may switch to a linear tail; such a switch is distinct from the smooth mathematical function.
Return scaledInput=beta*x and useLinearTail when scaledInput is strictly greater than switchThreshold.
When to use this tool
Use Activation: Softplus Linear Tail Switch when checking the stated softplus curvature, parameter sensitivity or branch convention.
How to use the tool and how it works
- Enter x, beta, switchThreshold. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read result. Expand result to inspect the returned fields.
Require beta>0 and switchThreshold>=0. Return scaledInput=beta*x and useLinearTail when scaledInput is strictly greater than switchThreshold.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| x | Finite scalar input coordinate; use the mathematical scale specified above. | Required. Finite number. |
| beta | Supplied scalar shape parameter under the explicitly stated function convention. | Required. Finite number. |
| switchThreshold | Switch Threshold | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Object containing the operation-specific returned members listed below. | Required. Object. |
| result.scaledInput | Scaled Input | Calculated result member: number. |
| result.useLinearTail | Use Linear Tail | Calculated result member: boolean. |
{
"input": {
"x": 10,
"beta": 2,
"switchThreshold": 20
},
"output": {
"result": {
"scaledInput": 20,
"useLinearTail": false
}
}
}Limitations and assumptions
- Equality remains on the nonlinear side; this only selects a caller-supplied branch and does not evaluate softplus or prescribe a library threshold.
- Outputs are nested inside result. Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use the complete explicit function and the intended parameter, branch and numerical conventions for symbolic, tensor or model-level differentiation; assess training behavior independently.
References
Similar or alternative tools
- Activation: Softplus Scalar Second Derivative
With s=sigmoid(beta*x), return beta*s*(1-s), the second derivative with respect to input x.
- Softplus Curvature Inflection Location
Return peakCurvatureLocation=0, peakCurvature=beta/4 and atPeak only when supplied input equals zero exactly.