Curved latent coverage regularizer

1. Integration point: insert the module after a backbone projection head, where each example has a latent vector z_i in R^d. Select two coordinates, or learn a 2D projection P in R^(2xd), and define a shared augmentation curve gamma(t)=(t,g(t)) in R^2. Parameterize g(t) with a small monotone-convex spline or MLP; use softplus curvature, for example g''(t)=c_min+softplus(q(t)), and obtain g and g' by automatic differentiation. Generate K points gamma(t_k), and form augmented features z_i+P^T gamma(t_k). 2. Pseudocode: sample a minibatch z_i; sample t_k uniformly from [a,b]; compute y_ik=Pz_i+gamma(t_k); choose random pairs of translations h_ij=y_i-y_j; estimate tube overlap with a differentiable kernel O_hat(h)=mean_{u,v} exp(-||u-v-h||^2/(2 sigma^2)), where u and v are sampled points from the curve or its K-point tube and sigma=delta; add L_curve above with kappa a tunable safety factor and c_min>0; optimize task loss plus L_curve. In the formula, delta is the kernel or tube scale, h_ij is the sampled feature translation, and kappa C is approximated by kappa. 3. Compute from the paper: use the exact distance-dependent target delta^2/(delta+||h||), the near/far regime split, and the requirement of nonzero curvature. Estimate empirically: the constant C, the effective overlap O_hat, and whether the chosen latent projection is genuinely curved. Normalize O_hat and the target by feature dimension or minibatch estimator so their magnitudes are comparable. Stop-gradient through the target denominator to avoid trivial scale games, and optionally constrain ||P||_F=1. 4. First experiment: train a small ResNet-18 or ViT-Tiny on CIFAR-10 and CIFAR-100 with standard feature-space augmentation as the baseline. Compare no regularizer, a straight-line latent augmentation with g''=0, and the curved module with L_curve. Use identical augmentation counts and FLOPs. Measure validation accuracy, corruption accuracy, feature covariance rank, average union occupancy of K translated delta-balls, and the empirical curve-overlap curve versus ||h||. Success means lower occupancy collapse and improved corruption or held-out accuracy at equal training loss, while the learned overlap follows the predicted plateau O(delta) near zero distance and approximate 1/||h|| decay at larger distances. Ablate c_min, delta, and the number of curve samples K.