Foundations of Polar Linear Algebra

TL;DR

This paper introduces Polar Linear Algebra, a spectral framework combining radial and angular components, demonstrating its effectiveness and interpretability on a benchmark task like MNIST.

Contribution

It presents a novel spectral operator learning framework based on polar geometry, improving stability, efficiency, and interpretability over traditional methods.

Findings

Polar operators can be trained reliably on MNIST.

Imposing spectral constraints enhances stability and convergence.

The approach reduces parameters and computational complexity.

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.