Mengnien Wu

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.

This paper is devoted to the study of a partial inverse spectral problem for Sturm-Liouville operators with frozen arguments on a star-shaped graph. The potentials are assumed to be known a priori on all edges except one, and the objective is to reconstruct the unknown potential on the remaining edge using a subset of the spectral data. A constructive algorithm for solving this problem is presented, which relies on the Riesz basis property of a system of vector functions.