Universal Approximation Constraints of Narrow ResNets: The Tunnel Effect

2026-03-30

AI Summary

This work investigates the approximation limitations of narrow residual networks (ResNets) in the absence of input augmentation, where an imbalance between skip connections and residual signals drives key features toward infinity—a phenomenon termed the “tunneling effect”—hindering accurate target function approximation. By integrating functional analysis, approximation theory, and numerical experiments, the study provides the first quantitative characterization of approximation error bounds for ResNets under both residual-dominant and skip-dominant regimes. It reveals that the channel ratio and uniform weight bounds critically govern expressive power, and demonstrates that architectural mismatch with the target function substantially amplifies approximation error. Clear performance disparities between the two regimes are illustrated through low-dimensional examples.

Abstract

We analyze the universal approximation constraints of narrow Residual Neural Networks (ResNets) both theoretically and numerically. For deep neural networks without input space augmentation, a central constraint is the inability to represent critical points of the input-output map. We prove that this has global consequences for target function approximations and show that the manifestation of this defect is typically a shift of the critical point to infinity, which we call the ``tunnel effect'' in the context of classification tasks. While ResNets offer greater expressivity than standard multilayer perceptrons (MLPs), their capability strongly depends on the signal ratio between the skip and residual channels. We establish quantitative approximation bounds for both the residual-dominant (close to MLP) and skip-dominant (close to neural ODE) regimes. These estimates depend explicitly on the channel ratio and uniform network weight bounds. Low-dimensional examples further provide a detailed analysis of the different ResNet regimes and how architecture-target incompatibility influences the approximation error.

Problem

• Universal Approximation
• Residual Neural Networks
• Critical Points
• Tunnel Effect
• Approximation Constraints

Innovation

• Residual Neural Networks
• universal approximation
• tunnel effect
• critical points
• skip-residual ratio