Renormalization-Group Principles for Deep Neural Architectures

Deep learning achieves remarkable success across diverse domains, yet a fundamental theoretical framework explaining why depth enables effective multi-scale representation learning remains incomplete. Here we establish a formal correspondence between neural network depth and renormalization group (RG) scale transformations from statistical physics. We derive this correspondence through representation dynamics: each layer implements a learned coarse-graining operator that contracts the Fisher information geometry of the data manifold. This framework yields three falsifiable hypotheses with operational definitions: (H1) layer-k representations correspond to effective theories at correlation scale ξk, measurable through the Jacobian spectral distribution; (H2) required network depth scales logarithmically with the intrinsic correlation length of the data distribution; and (H3) RG-inspired architectures with explicit scale-structure exhibit improved multi-scale generalization. We validate these predictions on both controlled hierarchical datasets and standard computer vision benchmarks, demonstrating that the depth-correlation scaling follows the theoretically derived exponential decay. This work provides a mathematically disciplined framework for understanding deep learning hierarchy, offering both conceptual insights and principled architectural guidance.