🤖 AI Summary
This work addresses the longstanding challenge in operator learning of simultaneously achieving stability, interpretability, and efficient parallelization, which stems from the lack of explicit spectral structure modeling. The authors propose a polar-spectral operator framework that leverages polar-coordinate geometry to map problems into the spectral domain, where they are decomposed into orthogonal eigenmodes processed independently. A self-adjoint-inspired spectral constraint mechanism is introduced, which not only reduces parameter count and computational complexity but also naturally yields a novel mode-based model parallelization strategy. Experiments demonstrate that the method enables stable training on MNIST, significantly improves convergence, and produces more interpretable and computationally efficient model representations.
📝 Abstract
This work revisits operator learning from a spectral perspective by introducing Polar Linear Algebra, a structured framework based on polar geometry that combines a linear radial component with a periodic angular component. Starting from this formulation, we define the associated operators and analyze their spectral properties. As a proof of feasibility, the framework is evaluated on a canonical benchmark (MNIST). Despite the simplicity of the task, the results demonstrate that polar and fully spectral operators can be trained reliably, and that imposing self-adjoint-inspired spectral constraints improves stability and convergence. Beyond accuracy, the proposed formulation leads to a reduction in parameter count and computational complexity, while providing a more interpretable representation in terms of decoupled spectral modes. By moving from a spatial to a spectral domain, the problem decomposes into orthogonal eigenmodes that can be treated as independent computational pipelines. This structure naturally exposes an additional dimension of model parallelization, complementing existing parallel strategies without relying on ad-hoc partitioning. Overall, the work offers a different conceptual lens for operator learning, particularly suited to problems where spectral structure and parallel execution are central.