Empirical Evidence for Simply Connected Decision Regions in Image Classifiers

TL;DR

This paper provides empirical evidence that decision regions in deep neural networks are not only path connected but also simply connected, enhancing understanding of their topological structure.

Contribution

It introduces a novel iterative quad-mesh filling method to test simple connectivity of decision regions in image classifiers.

Findings

Decision regions are empirically shown to be simply connected.

The quad-mesh filling method effectively tests topological properties.

Results support the hypothesis of simple connectivity in deep neural network decision regions.

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.