Pairwise anisotropic minimality regularizer

Integrate this as an auxiliary loss for an implicit neural representation F_theta(x), such as an SDF, occupancy field, neural surface reconstruction model, or coordinate MLP. Apply it only at sampled points near the target zero level set, where the geometric interpretation is relevant and where ||grad F_theta|| is above a small threshold. Choose n input dimensions and positive exponents g_i with distinct pair sums; search small real-valued exponents numerically and reject choices whose pair-sum matrix is poorly conditioned. Use m=n(n-1)/2 scales t_l log-spaced around 1, for example exp([-1,-0.5,0,0.5,1]) in the three-dimensional case. Precompute V and a damped pseudoinverse P=(V^T V+lambda I)^(-1)V^T. For each minibatch point x and each scale, evaluate z_l=D_{t_l}^{-1}x, F_theta(z_l), its gradient, and Hessian using forward-over-reverse autodiff or Hessian-vector products. Compute y_l = ||grad G_l||^2 trace(H_l) - grad G_l^T H_l grad G_l using the scaled derivatives, then compute C_hat=P y and add alpha sum(C_hat^2) to the reconstruction or surface loss. The formulas and pair coefficients come from the paper; alpha, scale placement, damping, gradient clipping, and near-surface sampling are empirical settings. First test on a small 2D or 3D coordinate MLP fitting planes, spheres, and deliberately curved synthetic surfaces, comparing the ordinary single-scale M[F] penalty against the multi-scale loss at equal wall-clock budget. Measure reconstruction error, zero-level-set curvature, robustness after anisotropic coordinate rescaling, and gradient stability. Success means lower residual curvature and less anisotropic cancellation at comparable reconstruction accuracy.