Beyond Affine Layers: Bounded Ball Coverings to Preclude Gradient Vanishing

Traditional deep learning architectures relying on standard affine layers partition the representation space into unbounded polyhedral regions, which inherently introduce systemic structural vulnerabilities when learning complex, non-convex data manifolds. To resolve these limitations, we introduce an alternative "bounded-by-design" geometric framework that completely moves beyond traditional affine transformations. By replacing hyperplanes entirely with variable-radius ball coverings, we reformulate layer mechanics around the fundamental metric of a feature point's proximity to localized multi-spherical anchor centers. Crucially, by mapping this distance metric directly onto localized, parallel inner products, the formulation fully preserves standard batch-level GPU parallelization performance without higher-order computational overhead. Mathematical and visual validation demonstrates that our bounded formulation effectively encapsulates complex topological boundaries, eliminates representational redundancy in empty background spaces, and structurally stabilizes error propagation at the mathematical root.