A Power Transform

TL;DR

The paper introduces a unifying, self-inverting power transform $f(x,\lambda)$ that generalizes and interrelates a broad set of mathematical tools including robust loss functions, kernel classes for IRLS, parametric probability distributions, bump functions, and neural network activations. By composing $f$ with a base quadratic loss, exponentiating, and normalizing, the author derives a cohesive framework: robust losses $\rho(x,\lambda,c)$, stationary kernels $k(x,\lambda,c)$, distributions $P(x,\lambda,c)$, bump functions $b(x,\lambda)$, and signed-input activations $f_\pm(x,\lambda_+,\lambda_-)$. The work emphasizes numerical stability and provides a fast, stable implementation (e.g., using $\operatorname{expm1}$ and $\operatorname{log1p}$) along with a reference JAX implementation, enabling practical deployment across statistics, machine learning, and neural networks. It also clarifies connections to the Box-Cox transform, showing a bijection between the two and positioning $f$ as a more expressive yet structured alternative. Overall, this framework offers a versatile toolkit for customizing robustness, similarity measures, probabilistic models, and activation shapes within a single, coherent mathematical core.

Abstract

Power transforms, such as the Box-Cox transform and Tukey's ladder of powers, are a fundamental tool in mathematics and statistics. These transforms are primarily used for normalizing and standardizing datasets, effectively by raising values to a power. In this work I present a novel power transform, and I show that it serves as a unifying framework for wide family of loss functions, kernel functions, probability distributions, bump functions, and neural network activation functions.