🤖 AI Summary
This paper addresses the narrow applicability and lack of unification across existing power transformations (e.g., Box–Cox, Tukey) for modeling diverse multivariate mathematical objects. We propose a generalized power transformation framework based on a differentiable family of generalized power functions, parameterized by a single continuous scalar. For the first time, this framework enables unified parametric representation of loss functions, kernel functions, probability densities, convex/concave functions, and neural network activation functions. Theoretically, we rigorously establish its mathematical equivalence and functional generalization capacity. Empirically, it outperforms conventional methods in data standardization, distribution fitting, and neural activation tasks. Our core contribution is the development of the first unified power transformation paradigm that simultaneously ensures universality, analytical tractability, and computational feasibility—providing a foundational tool for statistical modeling and machine learning.
📝 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.