CELU Join Curvature Jump Tool

Description

At the zero join, return left curvature=1/alpha, right curvature=0 and right-minus-left jump=-1/alpha.

CELU uses alpha*(exp(x/alpha)-1) for negative input and x otherwise, with positive alpha. Rescaling the exponential argument makes the first input derivative continuous at zero, while second derivatives on its two sides still differ.

At the zero join, return left curvature=1/alpha, right curvature=0 and right-minus-left jump=-1/alpha.

When to use this tool

Use CELU Join Curvature Jump when checking the stated CELU join curvature or alpha sensitivity convention.

How to use the tool and how it works

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

Require alpha>0 for CELU's negative branch alpha*(exp(x/alpha)-1). At the zero join, return left curvature=1/alpha, right curvature=0 and right-minus-left jump=-1/alpha.^[1]

CELU places alpha in the negative exponential’s argument as well as its amplitude. First-derivative continuity at zero does not imply matching second derivatives at the join.^[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.

Output field	Interpretation	Returned shape
leftCurvature	Left Curvature	Required. Finite number.
rightCurvature	Right Curvature	Required. Finite number.
rightMinusLeft	Signed curvature jump -1/alpha at the zero join; not a first-derivative jump.	Required. Finite number.

json

{
  "input": {
    "alpha": 2
  },
  "output": {
    "leftCurvature": 0.5,
    "rightCurvature": 0,
    "rightMinusLeft": -0.5
  }
}

Limitations and assumptions

CELU is continuously differentiable once at the join but lacks a matching two-sided second derivative there.
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/celu-join-curvature-jump — tool.io
Continuously Differentiable Exponential Linear Units — Jonathan T. Barron

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