Certified Coarse-to-Fine Coordinate Refinement

Attach the refinement procedure after a neural network that outputs a low-rank feature subspace and needs continuous coordinates. Use a bounded domain \(D\subset\mathbb R^d\), construct a grid with spacing \(c_g/\kappa\), and evaluate \(q(z)=1-\|U_s^*\varphi_z\|_2^2\) in batches. Choose a threshold from synthetic training data, initially setting \(\tau\) halfway between the largest positive score and the smallest negative score; during deployment, calibrate it on a held-out validation set. Accept grid points with \(q(z)\leq\tau\), suppress points within \(c_g/(2\kappa)\) of a lower-score accepted point, and refine each survivor. Pseudocode: for z0 in grid: if q(z0)<=tau: z=z0; for t in range(T): z=project_D(z-eta/kappa**2*grad_q(z)); if norm(grad_q(z))<tol: break; append(z,q(z)); merge points within c_merge/kappa. Start with \(\eta=0.1\) and backtrack if the score increases. For \(\varphi_z[m]=e^{i\kappa\omega_m^Tz}/\sqrt M\), compute \(\nabla_z\varphi_z[m]=i\kappa\omega_m\varphi_z[m]\) exactly and obtain \(\nabla q\) by autodiff or the analytic projector derivative. The paper determines the bandwidth scaling, basin radius, and separation diagnostic; empirical validation determines \(\gamma\), \(\tau\), and merge radii for learned features. First test on 3D synthetic sources with a small MLP or Transformer encoder, comparing exhaustive gradient descent from every grid point and direct coordinate regression. Measure wall-clock score evaluations, gradient steps, missed sources, duplicate count, and localization error. The expected signal is at least a 2x reduction in refinement evaluations with no increase in missed-source rate, plus stable convergence under subspace perturbations.