# Bessel-Conical Radial Basis

- ID: 240
- Canonical URL: https://synthcore.org/idea/240/bessel-conical-radial-basis
- API JSON: https://synthcore.org/api/idea/240.json
- API Markdown: https://synthcore.org/api/idea/240.md
- Verification status: unverified
- Source: [arXiv:2608.27398](https://arxiv.org/abs/2608.27398)
- Category: architecture
- Solves: accuracy, stability, sample-efficiency
- ML areas: mlp, initialization, training-dynamics
- Math tags: spectral-theory, pde, special-functions, geometry
- Ratings: usefulness 5/10; difficulty 4/10; novelty 8/10

## Idea description

Replace or augment ordinary radial positional features with Dirichlet Bessel eigenfunctions whose first zero exactly matches the domain boundary. This creates a scale-normalized basis aligned with the lowest Jacobi modes of a cone, potentially improving learning of fields or shapes with radial geometry and hard outer boundaries.

## Mathematical statement

For a stable regular minimal hypercone \(\mathbf C\), let \(\lambda_{\mathbf C}^{D}(R)\) be the first Dirichlet eigenvalue of its Jacobi operator on \(\mathbf C\cap B_R\), let \(\mu_1\) be the first eigenvalue of the link Jacobi operator, and define \(b_1^2=(n-2)^2/4+\mu_1\). If \(J_{b_1}\) is the Bessel function of the first kind and \(j_{b_1,1}\) is its first positive zero, the paper gives the exact law \(\lambda_{\mathbf C}^{D}(R)=j_{b_1,1}^2/R^2\). The transferable properties are inverse-square scaling and a radial mode that vanishes exactly at the boundary. For normalized radius \(\rho=\|x\|/R\), use \(\phi_k(\rho)=J_b(j_{b,k}\rho)\), where \(b\) is selected from the angular spectrum or estimated from a discretized link operator.

## Key formulas

- $$\lambda_{\mathbf C}^{D}(R)=\frac{j_{b_1,1}^{2}}{R^{2}},\qquad b_1^{2}=\frac{(n-2)^{2}}{4}+\mu_{1}.$$
- $$\phi_k(r;R)=J_b\!\left(j_{b,k}\frac{r}{R}\right),\qquad \phi_k(R;R)=J_b(j_{b,k})=0.$$
- $$h_\theta(x)=\operatorname{MLP}_\theta\left([x,\phi_1(\|x\|;R),\ldots,\phi_K(\|x\|;R),\phi_k(\|x\|;R)Y_\ell(\omega)]_{k,\ell}\right),\qquad \omega=x/\|x\|.$$

## Implementation notes

Integrate the basis at the input of a coordinate MLP, neural operator, or polar/spherical CNN patch. Normalize each sample to a known outer radius \(R\), compute \(\rho=\|x\|/(R+\varepsilon)\), and concatenate \(K\) radial features \(\phi_k(\rho)=J_b(j_{b,k}\rho)\) to Cartesian or Fourier features. Start with \(K=8\) and fixed \(b=\max(0,(n-2)/2)\). For a non-spherical cross-section, estimate the link eigenvalue \(\mu_1\) once using a finite-element or graph eigensolver, then set \(b=\sqrt{(n-2)^2/4+\mu_1}\). Compute Bessel zeros with a scientific-library root finder and cache them. Compare against learned sinusoidal radial features and no radial features at equal width. The theorem predicts that rescaling the domain from \(R\) to \(aR\) rescales the lowest spectral variation by \(a^{-2}\); therefore train on several radii and evaluate zero-shot at unseen radii. The cheapest experiment is a coordinate MLP fitting analytic radial fields or Poisson solutions on disks and balls. Record relative L2 error, boundary-condition violation, optimization steps, and zero-shot radius-transfer error. Success is lower error and faster convergence near the Dirichlet boundary without meaningful inference overhead.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.