Gaussian Barycentric Constraint Layer

Insert the layer after a neural network head whose output must satisfy a convex constraint, such as portfolio weights in a simplex, resource allocations in a box or polytope, bounded graph signals, or safe-control actions. For each input vector x, draw K reparameterized samples y_k=x+sqrt(delta) epsilon_k. Evaluate the exact feasibility predicate q_k=1{y_k in C} and return p_hat=sum_k q_k y_k/(sum_k q_k+epsilon). For boxes and simplices, implement predicates and reductions in vectorized GPU code. If feasible samples are rare, define a convex violation score v_C(y)>=0 and use soft weights w_k=exp(-v_C(y_k)/tau), returning sum_k w_k y_k/sum_k w_k; this is the paper's exponential barycenter with f=v_C and temperature tau. Use common random numbers for neighboring inputs in Jacobian diagnostics. Track constraint violations, feasible-sample rate, output variance, gradient norms, and the empirical Jacobian spectral norm. The paper supplies the conditional-mean construction, covariance Jacobian identity, and firm-nonexpansiveness target; acceptance probability and temperature are empirical choices. First test simplex-constrained regression on synthetic and tabular data with an MLP. Compare post-hoc clipping and renormalization, exact Euclidean simplex projection, and this stochastic layer using K in {8,16,32}. Match wall-clock budgets and evaluate prediction error, constraint satisfaction, robustness to noisy inputs, and training stability. Success means near-zero violations with smoother optimization than clipping and competitive error relative to exact projection.