stdplus-ext 0.0.1

The stdplus activation function

stdplus

stdplus(x) is a smooth activation function for computing Gaussian standard deviations in variational models. It is designed as a numerically stable alternative to softplus / exp, and is introduced in Chapter 6 of Yujia Yan’s PhD dissertation (University of Rochester, 2025).
It resembles a 1-centered softplus (i.e., equals 1 at x = 0) while providing a clean analytic derivative and a stable evaluation recipe.

Definition

stdplus function is defined as a function y = f(x) such that $$y^2- 2 \log(y) - 1 = x^2$$

This is proposed in Chapter 6 of Yujia Yan's dissertation: Slashed Normal Parameterization for Approximate Normal Posterior Distributions

Why stdplus?

In the Slashed Normal / KL-amplitude parameterization, Gaussian parameters are expressed as a complex “KL amplitude” ψ = a + bi, and the standard deviation is computed with:

• μ = sqrt(2) * a
• σ = stdplus(sqrt(2) * b)

This yields an exact closed-form KL term equal to ||ψ||_2^2 (enabling direct, interpretable rate control).

Reference

• Yujia Yan, Structured Analysis and Generation in Music, Audio, and Beyond, PhD Dissertation, University of Rochester, 2025. (See Chapter 6, Sec. 6.6 “More on stdplus Function”.)

Install

pip3 install -e .