Description
Invert the linear branch as (output-intercept)/slope and return its two input breakpoints.
Hard sigmoid clamps slope*x+intercept into [0,1]. Only outputs strictly between the saturation limits belong to the invertible linear branch; saturated endpoint outputs correspond to rays rather than one finite inverse. Implementations can choose different slopes and intercepts.
Invert the linear branch as (output-intercept)/slope and return its two input breakpoints.
When to use this tool
Use HardSigmoid Linear Branch Preimage when checking the stated hard-sigmoid linear inverse convention.
How to use the tool and how it works
- Enter output, slope, intercept. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read input, branchLower, branchUpper. Use the output convention below.
Require slope>0 and output strictly between zero and one for clamp(slope*x+intercept,0,1). Invert the linear branch as (output-intercept)/slope and return its two input breakpoints.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| output | Strictly between zero and one; saturated output rays are excluded. | Required. Finite number; strictly above 0; strictly below 1. |
| slope | Slope | Required. Finite number; strictly above 0. |
| intercept | Intercept | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| input | Input | Required. Finite number. |
| branchLower | Branch Lower | Required. Finite number. |
| branchUpper | Branch Upper | Required. Finite number. |
{
"input": {
"output": 0.5,
"slope": 0.25,
"intercept": 0.5
},
"output": {
"input": 0,
"branchLower": -2,
"branchUpper": 2
}
}Limitations and assumptions
- Saturated output rays at zero/one are excluded; the tool does not invert a particular library's hard-sigmoid convention automatically.
- 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: HardTanh Inverse Preimage
Outputs outside [lower,upper] have no preimage; lower gives a lower-ray, upper an upper-ray and strict interior a singleton.
- Activation: Softsign Scalar Inverse
Invert softsign x/(1+abs(x)) as x=y/(1-abs(y)).