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Foundations of Polar Linear Algebra

Giovanni Guasti

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.

Foundations of Polar Linear Algebra

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.

Paper Structure

This paper contains 188 sections, 6 theorems, 189 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction to Polar Linear Algebra
  2. Introduction
  3. Notational and Conceptual Conventions.
  4. The Polar Grid and Polar Matrices
  5. Algebra on Polar Matrices
  6. Addition
  7. Scalar multiplication
  8. Hadamard (elementwise) product
  9. The Polar Product (Circular Convolution)
  10. Polar Transpose
  11. Self-Adjoint Polar Matrices
  12. Remark (Real-valued case).
  13. Inner Product and Norm
  14. Fourier View on Polar Operators
  15. Aurora's Theorem
  16. ...and 173 more sections

Key Result

Theorem 1.1

A polar matrix $A$ is invertible under the polar product $\otimes$ if and only if In that case the inverse is uniquely given by (i.e., at each radius $r$ we invert the angular spectrum and apply the inverse DFT).

Figures (8)

  • Figure 1: A natural example of polar symmetry: a pomegranate cross-section. Radial layers and angular segmentation align with a polar grid.
  • Figure 2: A polar matrix represented in its natural radial--angular form. The inner ring contains the values $(2,1,0,1)$ and the outer ring the values $(3,2,1,0)$.
  • Figure 3: Logical structure of three foundational results in polar linear algebra. Elisa's Theorem provides the commutative spectral framework underlying the theory. Within this framework, Aurora's Theorem gives the invertibility criterion for a general polar matrix, while Chiara's Theorem specializes to the self-adjoint case and establishes reality of the spectrum.
  • Figure 4: End-to-end MNIST pipeline used in the proof-of-concept implementation: polar resampling (cached on disk), training/validation loops, and exported artifacts.
  • Figure 5: PolarFNOBlock used in the MNIST experiment: the spectral operator is applied on retained low-frequency modes. The spectral pointwise path is the frequency-space analogue of a 1x1 convolution, performing channel mixing on the intermediate spectral tensor. Normalization and nonlinearity are applied in the spatial domain, thus requiring one FFT/iFFT pair per block.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Theorem 1.1: Aurora's Theorem (Invertibility Criterion)
  • proof
  • Theorem 2.1: Elisa's Theorem (Commutativity of the Polar Product)
  • proof
  • Definition 3.1: Circulant Matrix
  • Proposition 3.1: Fourier diagonalization of circulant matrices
  • proof
  • Remark 3.1: Spectral structure of the rotor algebra
  • Definition 3.2: Rotation Equivariance
  • Proposition 3.2: Rotor expansion of equivariant polar operators
  • ...and 5 more