SELU Negative Asymptote Tool

Description

Return -alpha*scale, the limit of the SELU negative branch as input tends to negative infinity.

SELU multiplies an ELU-shaped function by an outer scale: the positive branch is scale*x and the negative branch is scale*alpha*(exp(x)-1). Input, alpha and scale derivatives differ, and one-sided input slopes at zero match only when alpha=1.

Return -alpha*scale, the limit of the SELU negative branch as input tends to negative infinity.

When to use this tool

Use SELU Negative Asymptote when checking the stated scaled exponential-linear limit or derivative convention.

How to use the tool and how it works

Enter alpha, scale. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read limit. Use the output convention below.

Require alpha>0 and scale>0. Return -alpha*scale, the limit of the SELU negative branch as input tends to negative infinity.^[1]

SELU combines an outer scale with an exponential negative branch. This local parameter or input term does not establish the distributional assumptions needed for network self-normalization.^[2]

Input field	Meaning and units	Accepted input
alpha	Positive shape parameter when required by the function convention above.	Required. Finite number; strictly above 0.
scale	Supplied scale parameter with its operation-specific meaning stated above.	Required. Finite number; strictly above 0.

Output field	Interpretation	Returned shape
limit	Limit	Required. Finite number.

json

{
  "input": {
    "alpha": 1.6732632423543772,
    "scale": 1.0507009873554805
  },
  "output": {
    "limit": -1.7580993408473766
  }
}

Limitations and assumptions

This is an asymptotic output bound, not the value at a finite input or proof of network self-normalization.
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

Calculation and input contract: activation/selu-negative-asymptote — tool.io
Self-Normalizing Neural Networks — Günter Klambauer and coauthors

Similar or alternative tools

Activation: Selu Scalar Derivative
Return scale for x>0 and scale*alpha*exp(x) for x<0; at zero select scale*alpha on the left or scale on the right.
SELU Alpha Partial Derivative
Return scale*expm1(x) for x<0 and zero otherwise, differentiating SELU with respect to alpha at fixed x and scale.