Quantile-space Huber-Wasserstein loss

Use this as the output loss for a distributional predictor, uncertainty head, histogram decoder, or teacher-student distillation model. The network should emit b and m unconstrained increment logits d_1,...,d_m; construct monotonically increasing predicted quantiles with Q_j=b+sum_{k<=j}softplus(d_k), avoiding invalid CDF inversions. Convert every training target distribution to the same m quantile levels once during preprocessing. Compute e_j=Q_pred,j-Q_target,j and apply the piecewise Huber loss independently to every j. Backpropagation then supplies the clipped gradient clamp(e_j,-delta,delta)/m, rather than a single scalar robustification of the whole Wasserstein distance. Start with delta=1.345 times a running robust scale s, where s=1.4826*median(abs(e_j-median_k(e_k))) over the current minibatch; detach s from the gradient and update it using an exponential moving average. Compare fixed delta values {0.25,0.5,1,2} and adaptive scaling. The mathematical operation is exact numerical quadrature of the displacement-wise Huber-Wasserstein objective; target quantile estimation and scale tracking are empirical approximations. First run a synthetic contaminated-mixture regression experiment and a small heteroscedastic regression network. Compare MSE on quantiles, ordinary W2 loss, distance-wise Huber-W2, and this loss under 10-40% target outliers. Record clean Wasserstein error, contaminated-test Wasserstein error, calibration or coverage, gradient-norm distributions, and convergence at equal optimization steps. Success is lower contaminated-test error and better calibration while retaining near-MSE clean performance when contamination is absent.