Foundations of Polar Linear Algebra

Giovanni Guasti

cs.LGMar 30, 2026

Foundations of Polar Linear Algebra

Authors: 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.

Figures & tables

Figure 1.1: A natural example of polar symmetry: a pomegranate cross-section. Radial layers and angular segmentation align with a polar grid.

Figure 1.2: 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 $(3,2,1,0)$ .

Panel of Figure 3 — Figure 3.1: 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.

Panel of Figure 5 — Figure 5.1: End-to-end MNIST pipeline used in the proof-of-concept implementation: polar resampling (cached on disk), training/validation loops, and exported artifacts.

$W$	blocks	$m_{R}$	$m_{T}$	Val Acc (%)	Test Acc (%)
32	3	10	16	94.4	94.5
32	2	12	20	92.4	92.7
32	3	10	20	95.0	94.9

Table 5.1: Effect of channel width, depth, and retained spectral modes on MNIST. Depth has a stronger impact than the number of retained modes, while moderately increasing

m_{T}

improves performance when depth is sufficient.

Figure 5.3: Training and validation curves (loss and accuracy) for the MNIST PolarFNO proof-of-concept. The horizontal axis tracks training progression, and the vertical axes show the corresponding loss and accuracy values.

Figure 5.4: Periodic spectral gate used in the self-adjoint experiment. Left: spatial-domain representation $W(\theta)$ obtained via inverse Fourier transform. Right: magnitude of the corresponding Fourier coefficients $|W_{m}|$ . The gate is fixed by construction and acts as a smooth, band-limited periodic modulation in the angular domain.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 9 — Figure A.1: Fully spectral MNIST pipeline annotated with parallelization axes and communication requirements. DP: data parallelism; TP/CP: tensor/channel parallelism; FP: frequency-domain sharding; PP: pipeline parallelism. Communication depends on the chosen sharding strategy (single-device vs multi-device).

Stage	Main pattern	Parallelization opportunities
Input FFT	batched FFT	batch, channel; intra-FFT parallelism via library
Spectral conv (low)	(complex) batched GEMM per mode	frequency $(k_{r},k_{\theta})$ , channel, batch
Residual + $\alpha$	elementwise	frequency, channel, batch
Spectral norm	reduction + broadcast	parallel partial reductions; sync on reduction dimension
Periodic gate	small-kernel circular conv	batch, channel, $k_{r}$ , and $k_{\theta}$
Block stacking	sequential depth	pipeline parallelism across blocks + intra-block parallelism

Appendix

Table A.1: Operator-wise parallelization summary for the fully spectral MNIST pipeline.

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Operator learning studies models that map functions or fields to functions, enabling fast prediction across scientific and dynamical systems.

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