Empirical Evidence for Simply Connected Decision Regions in Image Classifiers

Technical Summary: Empirical Evidence for Simply Connected Decision Regions in Image Classifiers

Problem Statement
While prior empirical work has established that decision regions in deep neural networks are path connected (i.e., any two images of the same predicted class can be joined by a continuous path within that class), the higher-order topological structure remains unclear. Specifically, it is unknown whether these regions are simply connected. A region is simply connected if every closed loop within it can be continuously contracted to a point without leaving the region. If a decision region contains "holes" (non-contractible loops), it implies a more complex, fragmented topology that could impact interpretability and adversarial robustness. This paper addresses the question: Can closed loops inside a predicted decision region be filled by a label-preserving surface?

Methodology
The authors propose an iterative quad-mesh filling procedure to construct finite-resolution, label-preserving surfaces bounded by a given loop of same-label images.

1. Problem Setup: The input space is normalized image space. Given four images ($x_{00}, x_{10}, x_{01}, x_{11}$) assigned the same label $y$, the goal is to construct a surface $S: [0, 1]^2 \to X$ such that the boundary matches the loop and the interior satisfies $f(S(u, v)) = y$.
2. Boundary Construction: The four corners define a loop. Since straight lines between same-label images may cross decision boundaries, the authors construct a piecewise-linear boundary loop by initializing vertices along the sides and repairing off-label vertices using a targeted DeepFool-style projection back into the target decision region.
3. Surface Filling (Adaptive Subdivision):
   • The method operates on a dyadic grid, starting with a single quadrilateral (quad) defined by the boundary.
   • Resolution Check: A quad is accepted if its vertices are verified to be in the target region and its geometric diameter (measured in grey-RMS units) is below a threshold $\tau$.
   • Label Check: If the size threshold is not met, the quad's bilinear interpolation is sampled on an adaptive grid. If all sampled points retain the target label, the quad is accepted.
   • Subdivision: If the label check fails, the quad is subdivided into four child quads. New edge midpoints and center points are initialized via linear interpolation.
   • Repair: Any newly created vertex that falls outside the target region is repaired using a targeted DeepFool update followed by a bisection step to project it back into the region.
4. Geometric Comparison: To assess the geometric complexity of the constructed surface, the authors compare it to a canonical Coons patch (a standard geometric interpolation of the four boundary curves). They calculate the ratio of the constructed surface area to the Coons patch area ($\rho$).

Experimental Setup

• Datasets & Models: The method was evaluated on the ImageNet validation set across six diverse architectures: ResNet-50, DenseNet-121, EfficientNet-B0, ConvNeXt-Tiny, ViT-B/16, and Swin-T.
• Scope: For each model, 1,000 loops were constructed (one for each of the 1,000 ImageNet classes), totaling 6,000 loops and 24,000 images.
• Parameters: The grey-RMS stopping threshold was set to $\tau = 0.5$. A default repair setting (50 DeepFool iterations) was used, with a stronger setting (200 iterations) applied only to initial failures.

Key Results

1. Universal Success: Across all six models and all 6,000 tested loops, the procedure successfully constructed a finite-resolution label-preserving surface. The final success rate was 100% after applying the stronger repair setting to the few initial failures.
2. Necessity of Adaptive Refinement: A "root-level" diagnostic showed that naive bilinear interpolation (the initial quad without subdivision) only satisfied the label constraint for a minority of loops (ranging from 6.9% for DenseNet-121 to 42.2% for Swin-T). This confirms that the adaptive subdivision and repair mechanism is essential.
3. Geometric Simplicity: The area ratio $\rho$ between the constructed surface and the Coons patch was concentrated near 1. This indicates that the label-preserving surfaces are geometrically close to the natural interpolation of the boundary, rather than being highly distorted or convoluted.
4. Mesh Complexity: The number of quads required to fill the loops varied by architecture. Transformer-based models (Swin-T, ViT-B/16) and ConvNeXt-Tiny required significantly fewer quads (smaller meshes) than ResNet-50 or DenseNet-121, suggesting their decision regions are locally flatter or more coherent in these sampled areas.

Significance and Claims
The paper provides strong empirical evidence (at finite resolution) supporting the hypothesis that decision regions in modern image classifiers are not only path connected but also simply connected.

• Topological Insight: The findings suggest that decision regions are organized into globally coherent structures where loops are contractible, challenging the notion that adversarial vulnerability implies global topological fragmentation.
• Limitations: The authors explicitly state that these results are finite-resolution empirical evidence, not a formal topological proof. The method verifies sampled points at a specific grey-RMS scale and cannot exclude the existence of smaller-scale holes or non-contractible loops outside the tested resolution or loop family.
• Implications: The results offer a new empirical picture of neural network decision regions, suggesting that despite local adversarial fragility, the global topology of these regions may be surprisingly simple and well-behaved.

The work does not claim to prove simple connectedness for all possible inputs or infinite resolution, but rather establishes a robust pattern across diverse architectures and the full ImageNet label space at the tested resolution.