Inlier-aware GPD residual loss

Integrate the method at the regression loss. The prediction head outputs ŷ=fθ(x), four mixture logits, and unconstrained parameters ad, au, aσ, aξ. For each minibatch compute r=abs(y−ŷ)+1e−6; set d=softplus(ad), u=d+softplus(au), σ=softplus(aσ)+1e−5, and ξ=0.4*tanh(aξ). Convert the four logits to π0,...,π3 with softmax. Use a half-normal or narrow Gaussian for the inlier density p0, a lognormal p1 truncated to (0,d), and a lognormal p2 truncated to [d,u); compute each truncated log-density using log CDF differences, specifically log p2=log f2−log(F2(u)−F2(d)). Use the GPD formula for p3 and reject or softly penalize samples violating 1+ξ(r−u)/σ>0. Compute q1, q2, q3 with the displayed sigmoid gates, clip them to nonnegative values, and renormalize q0,...,q3 to sum to one. Minimize the negative soft likelihood −Σk qk(log πk+log pk). Initialize d and u from batch residual quantiles, for example the 20th and 90th percentiles, and anneal τ from 0.1 to 0.01. Add a small penalty on |ξ| and a mixture-entropy floor to avoid early collapse. The paper contributes the regional decomposition, GPD tail, and truncation correction; the neural adaptation supplies smooth gates and conditional parameters. First test a two-layer MLP on synthetic regression with 5% Pareto noise and 10% near-zero inlier noise. Compare MSE, Huber, Gaussian NLL, and a standard mixture-density network. Measure RMSE, 95th and 99th percentile calibration, tail conditional error, and gradient stability. Success means improved extreme-quantile calibration and tail prediction without worse median RMSE or exploding gradients.