Profile Consistency Regularizer

Attach a profile-prediction head to the encoder before the task head. For each training example, generate two views using independent point dropout, small Gaussian coordinate noise, random rigid rotations, and color-preserving subsampling. Compute the exact or voxelized ECP for both views on the same R radius tuples. The direct absolute discrepancy is useful for evaluation, but gradients cannot pass through the integer Euler computation. Therefore let g_theta(X) be an R-dimensional prediction head and map each integer profile value p to phi(p)=tanh(p/s), with s=4 initially. Optimize L_soft, using stop-gradient on the topological target; optionally include the symmetric loss for both views. Keep the task loss unchanged and ramp lambda from zero to 0.1 over the first 10 percent of training. The paper-derived quantities are the unions of balls, their intersections, and the Euler-characteristic profile; voxel resolution, radius grid, squashing scale, and lambda are empirical choices. Begin with a DGCNN on ModelNet40 or a synthetic colored dataset containing separated, overlapping, and encircling configurations. Compare ordinary augmentation consistency, ECP regularization, and their combination. Measure clean accuracy, accuracy under point dropout, representation variance between equivalent views, profile prediction error, and training overhead. A successful result is improved corruption robustness and lower view variance without more than 10 percent additional training time.