Experimental Validation of Riemannian Attention Operators: A Comprehensive Study on Circle, Sphere, and Torus Manifolds

This document presents a comprehensive experimental validation of the Riemannian attention operator $\mathcal{O}_{\tau}$, defined as a Hilbert-Schmidt integral operator on compact manifolds. Through five systematic experiments conducted on the circle $S^1$, sphere $S^2$, and torus $T^2$, this study confirms the theoretical predictions regarding geometric thermal diffusion and spectral convergence established in recent geometric deep learning frameworks.

Key Experimental Findings:

- Spectral Accuracy on $S^1$: The study performs a complete spectral analysis on the unit circle. The measured empirical decay rate $\mu_{emp} = 0.24998$ matches the theoretical prediction $\mu_{theo} = \tau/2 = 0.25000$ with a relative error of less than $0.01\%$.

- Eigenvalue Verification: The experiments confirm that the operator acts as a geometric low-pass filter, with eigenvalues following the heat kernel decay law $\Lambda_k(\tau) = \exp(-\frac{\tau k^2}{2})$.

- Exponential Convergence: The evolution dynamics demonstrate a strict exponential convergence to the fixed point $u^*$, satisfying the contraction inequality $||u_n - u^*||_{L^2} \le ||u_0 - u^*||_{L^2} e^{-\lambda n}$.

- Topological Robustness: A comparative analysis with Euclidean attention methods reveals a $94.4\%$ reduction in boundary artifacts when using the geodesic distance $d(x,y)$, proving that respect for the manifold topology is critical for numerical stability.

- Multidimensional Validation: The operator's efficacy is demonstrated across different topologies, including signal smoothing on $S^1$, cluster classification on $S^2$, and trajectory regularization on the torus $T^2$.

This dataset and report definitively establish that Riemannian attention mechanisms naturally extend to non-Euclidean domains while maintaining rigorous mathematical guarantees equivalent to thermal diffusion processes.