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Empirical Evidence for Simply Connected Decision Regions in Image Classifiers
Arjhun Swaminathan, Mete Akgün
TL;DR
The topology of neural-network decision regions beyond path connectivity remains unclear. This paper develops an adaptive surface-filling procedure for same-label loops and finds finite-resolution label-preserving surfaces for every tested loop, supporting simple connectivity empirically.
Problem
The topology of classifier decision regions beyond path connectivity remains incompletely understood, including whether loops can be filled without leaving their region.
Method
The paper adaptively constructs finite-resolution label-preserving piecewise-bilinear surfaces spanning same-label loops and compares them with natural Coons patches.
Results
100% success across six architectures and 6000 ImageNet loops, with constructed surfaces typically remaining geometrically close to their Coons reference surfaces.
Takeaways & Limitations
The findings provide empirical evidence that deep-network decision regions may be simply connected beyond previously observed path connectivity.
Takeaways & Limitations
The study provides finite-resolution sampled evidence rather than a formal topological proof and cannot exclude smaller-scale holes or untested non-contractible loops.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper examines whether image-classifier decision regions are simply connected, extending prior empirical evidence of path connectivity to the contraction of closed loops. It proposes label-preserving surface filling and compares the resulting surfaces with natural Coons patches.
- Deep image classifiers achieve strong image-classification performance, but the geometry of their learned decision regions remains incompletely understood.
- Prior empirical work found that same-class images could often be connected by continuous, nearly straight paths that stay within the same decision region.
- Path connectivity does not exclude non-contractible loops, so the paper asks whether closed loops inside a decision region can be continuously filled without leaving it.
- The proposed procedure starts with four same-label images connected by label-preserving edges and iteratively builds a parametrised surface spanning the loop.
- The method empirically enforces consistent predicted labels on a sufficiently dense set of sampled surface points and compares the surfaces with natural Coons patches.
2 Related Work
Prior work has examined neural-network decision regions and boundaries through theoretical, empirical, topological-data-analysis, and adversarial-example perspectives. Most closely related studies found empirical path connectedness, while this work extends the question from paths to label-preserving surfaces spanning tested loops.
- 2 Related Work: Theoretical studies show that network architecture and activation assumptions impose nontrivial constraints on decision-region connectivity.Certain pyramidal architectures necessarily produce connected decision regions under suitable activation assumptions.
- 2 Related Work: Persistent homology summarizes connected components, holes, and higher-dimensional topological features in sampled data for studying decision boundaries and representations.Prior work has proposed persistent-topology methods for analyzing decision boundaries from labelled samples.
- 2 Related Work: Adversarial-example research treats neural-network decision boundaries as geometric objects that can be approximated, searched, or traversed.Small perturbations can change predictions, and methods such as DeepFool iteratively linearise classifiers to estimate boundary-crossing perturbations.
- 2 Related Work: Fawzi et al. empirically found path connected decision regions by constructing same-label polygonal paths that remained within the corresponding region.Their procedure repaired path segments that left the decision region.
- 2 Related Work: This work extends prior path-based analysis by testing whether same-label loops admit label-preserving surfaces, providing a finite-resolution probe of loop contractibility.Loop contractibility is the additional condition underlying simple connectedness.
3 Method
The method fills a closed same-label loop with a finite-resolution, label-preserving piecewise-bilinear surface by adaptively refining a quadrilateral mesh and repairing off-label vertices. It also compares this classifier-constrained surface with a canonical Coons patch to assess geometric closeness.
- Boundary construction: The procedure starts from four same-label corner images, repairs off-label boundary vertices, and represents the loop with four polygonal curves.Straight-line segments between same-label images may leave the decision region, so boundary vertices are repaired using DeepFool.
- Finite-resolution guarantee: At termination, accepted quads define a continuous piecewise-bilinear surface whose sampled interior remains in the target decision region at the chosen finite resolution.Quads are accepted after passing the label check or becoming smaller than the grey-RMS resolution threshold.
- Coons reference: A classical Coons patch supplies a canonical boundary-induced reference, but it is not guaranteed to remain inside the target decision region.The Coons patch blends the four boundary curves and subtracts bilinear corner interpolation.
- Adaptive mesh refinement: The classifier-constrained surface is built on a dyadic quadrilateral mesh using bilinear patches, shared vertices, adaptive sampling, and four-way subdivision.Shared vertices ensure continuity across quad boundaries, while failing quads are subdivided and checked on grids whose resolution is chosen from geometric size.
- Vertex verification and repair: Off-label vertices are repaired by a targeted DeepFool-style update followed by bisection, acting as a local projection back into the target decision region.Vertices already classified as y enter the verified set; others receive a nearby replacement while keeping the ℓ2 displacement small.
- Geometric comparison: Values ρ ≈1 indicate that the label-preserving surface has nearly the same area as the Coons patch, suggesting geometric closeness to a natural loop interpolation.The area ratio ρ is used to compare the constructed surface with the boundary-induced Coons reference.
4 Experiments
Experiments evaluate the surface-filling procedure on 6000 model-specific four-point loops from six modern image classifiers. The procedure successfully filled all tested loops after a stronger repair pass, while diagnostics characterized root-level acceptance, refinement, acceptance mechanisms, geometric deviation, and mesh complexity.
- Experimental setup: The evaluation spans six classifiers and 6000 model-specific four-point loops built from same-label ImageNet validation images.Each model contributes 1000 loops and 4000 endpoint images, with quad sets generated separately per model.
- Cross-model success: All 6000 tested loops were successfully filled after failures under the computationally efficient default repair setting received a stronger repair pass.The stronger setting uses 200 maximum iterations, overshoot 0.05, and 20 bisection steps.
- Root-level diagnostic: Root-level acceptance was limited for most models, ranging from 6.9% for DenseNet-121 to 42.2% for Swin-T before subdivision or vertex repair.The diagnostic tests whether the initial bilinear quadrilateral passes the finite grid check entirely within the target decision region.
- Coverage by refinement level: Refinement-level coverage tracks how quickly the algorithm accepts parameter-domain area and whether difficult regions are concentrated at deeper levels.At level d, one accepted quad represents a fraction 4^-d of the original parameter-domain area; results report medians and interquartile ranges plus median levels for fixed coverage thresholds.
- Geometric diagnostics: Acceptance and geometric diagnostics distinguish direct grid-check certification from size-threshold acceptance and compare final mesh area with a boundary-matched Coons reference using ρ = Aours/ACoons.Ratios near 1 indicate nearly equal surface areas, while the acceptance decomposition identifies reliance on finite-resolution stopping.
- Final mesh complexity: Final mesh complexity measures the refinement required by the average numbers of accepted quads and label-verified vertices, with Swin-T and ConvNeXt-Tiny requiring the smallest meshes on average.These trends are reported as consistent with the coverage-depth results.
5 Conclusion
Across six diverse architectures and 6000 ImageNet loops, the procedure constructed finite-resolution label-preserving surfaces for every tested loop, providing empirical evidence beyond path connectivity. The findings remain finite-resolution evidence rather than a formal topological proof and cannot exclude smaller-scale holes or untested non-contractible loops.
- Conclusion: 100% success rate: finite-resolution label-preserving surfaces were constructed for all 6000 tested ImageNet loops across six diverse architectures.This provides empirical evidence for a surface-level analogue of previously observed path connectivity.
- Limitations and future work: The procedure verifies sampled label preservation at the chosen grey-RMS resolution, so it cannot exclude smaller-scale holes or non-contractible loops outside the tested loop family.The authors characterize the results as finite-resolution empirical evidence rather than a formal topological proof.
A Technical appendices and supplementary material
The appendices quantify the method’s computational cost and vertex-repair behavior, and test sensitivity to the grey-RMS stopping threshold. Repairs are frequent but typically easy, while all tested ResNet-50 loops succeed across the evaluated thresholds.
- Computational cost: Runtime is summarized as median elapsed time per loop over 1000 successful loops per model, including grid computation, subdivision, and DeepFool repair.Each loop is processed independently, with total cost depending on grid evaluations, subdivision, and repaired vertices.
- Vertex repair difficulty: Vertex repairs are frequent but typically easy, averaging about two DeepFool iterations per repair attempt.Proposed edge and centre vertices outside the target decision region are repaired using targeted DeepFool followed by bisection.
- Grey-RMS threshold ablation: All 100 randomly sampled ResNet-50 loops succeed at every tested grey-RMS threshold τ under the stronger repair setting.Smaller τ values require geometrically smaller quads before acceptance and therefore impose a stricter criterion.