Hyperbolic curvature-cone neural field

Integrate the method into the scalar-output head of a hyperbolic neural field or hyperbolic autoencoder decoder. Represent input points \(x\) in the Poincare ball or hyperboloid model of \(\mathbb H^n\), and let an MLP output \(v_\theta(x)\). If negative values are required, use \(v_\theta(x)=-\operatorname{softplus}(a_\theta(x))\). For each minibatch, sample interior points and boundary points near a large Poincare radius. Use automatic differentiation to compute the first and second derivatives of \(v_\theta\). Convert the coordinate Hessian to the hyperbolic covariant Hessian using the metric and Christoffel symbols, then construct \(\Lambda=\nabla^2v-vg\). Raise one index with \(g^{-1}\), symmetrize numerically, and compute its eigenvalues. Evaluate elementary symmetric polynomials with the stable recurrence \(s_0=1\), followed for each eigenvalue \(\lambda_i\) by descending updates \(s_j\leftarrow s_j+\lambda_i s_{j-1}\). Use \(s_n\) and \(s_{n-k}\) in the residual formula, add the softplus penalty for eigenvalues below \(\epsilon>0\), and impose \(\mathcal L_{\partial}=\mathbb E_{x_b}\left|v_\theta(x_b)-\varphi(x_b)\right|^2\). Pseudocode: sample \(x,x_b\); compute \(v,\nabla v,\nabla^2v\); build \(g,\nabla^2v,\Lambda\); eigendecompose \(g^{-1}\Lambda\); compute \(s_n,s_{n-k}\); form \(r_\theta\); backpropagate the combined loss. The geometric derivatives and symmetric-polynomial values are computed directly; the boundary trace, sampling measure \(\mu\), and penalty weights are empirical design choices. Begin with \(n=2,k=1\), a small tanh or SIREN MLP, and sinusoidal boundary data on a circle of radius \(0.95\) in the Poincare disk. Compare against an unconstrained MLP and an MLP using ordinary Euclidean Hessian regularization. Record PDE residual, boundary error, the fraction of sampled points with any \(\lambda_i<0\), Hessian numerical condition number, and interpolation or extrapolation error. The first success criterion is lower boundary-conditioned extrapolation error and fewer curvature-cone violations at equal parameter count; failure is no improvement over Euclidean regularization or unstable second-derivative optimization.