A renormalization-group inspired lattice-based framework for piecewise generalized linear models

Chang JC (2026). ArXiv [Preprint] 2605.05493. https://arxiv.org/abs/2605.05493

Overview

This paper provides the theoretical foundation for the piecewise (quilt) models that underlie much of our applied work. Drawing on ideas from the renormalization group in physics, it introduces a lattice-based framework for hierarchically decomposing generalized linear models into piecewise components.

The framework formalizes how regression coefficients can vary across a lattice of covariate partitions, enabling models that capture complex nonlinear relationships while remaining fully interpretable. A key contribution is a generalization analysis based on the Widely Applicable Information Criterion (WAIC) derived via replica theory, which characterizes how model complexity trades off against fit. The paper also establishes scaling laws for prior regularization across interaction order, providing principled guidance on how to set priors as model complexity grows. Together, these results give a rigorous mathematical basis for building interpretable piecewise models that scale to high-dimensional problems.