Description
Let z=x/width; return 2*z²*exp(-z²)/width as the width partial derivative, or zero when z is nonfinite or abs(z)>40.
A Gaussian-shaped activation exp(-(x/width)²) depends on the dimensionless ratio of input to positive width. Sensitivity to width is distinct from the input derivative; increasing width changes the spread rather than translating the activation.
Let z=x/width; return 2*z²*exp(-z²)/width as the width partial derivative, or zero when z is nonfinite or abs(z)>40.
When to use this tool
Use Gaussian Activation Width Sensitivity when checking the stated Gaussian width sensitivity convention.
How to use the tool and how it works
- Enter x, width. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read derivative. Use the output convention below.
Require width>0 for exp(-(x/width)²). Let z=x/width; return 2*z²*exp(-z²)/width as the width partial derivative, or zero when z is nonfinite or abs(z)>40.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| x | Finite scalar input coordinate; use the mathematical scale specified above. | Required. Finite number. |
| width | Width | Required. Finite number; strictly above 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| derivative | Width partial of exp(-(x/width)^2); set to zero when x/width is nonfinite or has magnitude>40. | Required. Finite number. |
{
"input": {
"x": 2,
"width": 2
},
"output": {
"derivative": 0.36787944117144233
}
}Limitations and assumptions
- Width is held in the same input-coordinate scale as x; the numerical tail cutoff is explicit and this is not an input derivative.
- 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
- Tanh Temperature Sensitivity
With t=tanh(x/temperature), return -x*(1-t²)/temperature², the partial derivative with respect to temperature; return zero directly when 1-t² is numerically zero.
- Sigmoid Temperature Sensitivity
Evaluate s=sigmoid(x/temperature) using sign-stable exponential branches; return -x*s*(1-s)/temperature², or zero when s rounds to an endpoint.