Foundations of Polar Linear Algebra
For researchers in operator learning and spectral methods, this paper offers a novel conceptual framework that enhances interpretability and parallelization, though its impact is limited by evaluation on a simple benchmark.
This work introduces Polar Linear Algebra, a spectral framework for operator learning that combines linear radial and periodic angular components. On MNIST, it achieves reliable training with reduced parameters and computational complexity, while improving stability and convergence through self-adjoint-inspired spectral constraints.
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.