Positive Grassmannian subset head

Integrate this as the final subset-distribution head of a model that receives an input embedding h, such as a set encoder, graph neural network, or transformer. The head outputs A=f_theta(h) in R^{d x n}; column i corresponds to item i and each candidate output is a d-subset. For an MVP with n<=32 and d<=4, initialize A from a strictly totally-positive matrix A_{r,i}=exp((r-1)t_i) with t_1<...<t_n, or use this matrix plus small noise. Compute every candidate minor x_I=det(A[:,I]) for I in combinations(range(n),d). Penalize invalid signs with a barrier B(A)=-lambda sum_I log(clamp(x_I,epsilon)), compute Z=sum_I x_I, and set p_I=x_I/Z. Add a gauge penalty lambda_g||AA^T-I||_F^2 to reduce the GL_d parameter redundancy. Pseudocode is: A=f_theta(h); x={I:det(A[:,I]) for I in combinations(items,d)}; loss=-sum_I counts[I]*log(clamp(x[I],eps))/N + log(sum_I x[I]) + barrier + gauge; loss.backward(); optimizer.step(). The paper's probability formula supplies the normalization and the determinant formula supplies the structured coordinates; determinants and Z are computed exactly for small n. For larger n, estimate Z by sampled subsets and compare against exact enumeration on smaller instances. Benchmark against an unconstrained softmax over all subsets and a factorized-logit baseline on synthetic data generated from a positive Grassmannian model, then test a small multilabel graph or set-prediction task. Record validation NLL, calibration, rare-subset accuracy, parameter count, and runtime. Success is lower held-out NLL or better rare-subset prediction with O(dn) output parameters instead of O(binomial(n,d)) independent logits, while maintaining positive normalized probabilities.