Empirical Evidence for Simply Connected Decision Regions in Image Classifiers

2026-05-07

AI Summary

This study investigates whether the decision regions of deep image classifiers are simply connected—meaning any closed loop within such a region can be continuously contracted to a point without leaving the region. To address this, the authors propose an iterative quadrilateral mesh-filling method that constructs a finite-resolution surface bounded by a given loop while preserving label consistency, and they employ Coons surfaces to quantify deviations from standard geometric interpolation. Large-scale experiments across multiple state-of-the-art models provide strong empirical evidence that decision regions are indeed simply connected. This work offers the first empirical support for the simple connectivity of deep neural network decision regions, advancing beyond prior studies limited to path connectivity and deepening the understanding of the geometric and topological properties of decision boundaries.

Abstract

Understanding the topology of decision regions is central to explaining the inner workings of deep neural networks. Prior empirical work has provided evidence that these regions are path connected. We study a stronger topological question: whether closed loops inside a decision region can be contracted without leaving that region. To this end, we propose an iterative quad-mesh filling procedure that constructs a finite-resolution label-preserving surface bounded by a given loop and lying entirely within the same decision region. We further connect this construction to natural Coons patches in order to quantify its deviation from a canonical geometric interpolation of the loop. By evaluating our method across several modern image-classification models, we provide empirical evidence supporting the hypothesis that decision regions in deep neural networks are not only path connected, but also simply connected.

Problem

• decision regions
• simply connected
• topology
• deep neural networks
• image classifiers

Innovation

• simply connected
• decision regions
• quad-mesh filling
• Coons patches
• topological analysis