Signed Theta Geometric Embedding Loss

Attach this regularizer to the penultimate node embeddings of a signed GNN, graph transformer, or relation-aware encoder. Split each embedding z_i into equal-width fixed and anti-fixed parts, z_i=(u_i,v_i), and normalize with r_i=z_i/(||z_i||_2+epsilon). Define J(u_i,v_i)=(u_i,-v_i). For each edge (i,j), set k_ij=(1-sigma_ij)/2, so k_ij=0 for a positive edge and 1 for a negative edge. Compute t_ij=dot(r_i,J^{k_ij}r_j), then add lambda times the mean squared penalty (t_ij-c(q))^2 to the task loss, with c(q)=1/(1-q). Initially fix q, for example q=0 or q=0.25; alternatively learn q through q=q_min+(q_max-q_min)sigmoid(a), using a scalar parameter a and safe bounds that keep the target geometrically feasible. Sample edges in minibatches for large graphs. Generate switched copies using random vertex signs s_i, set sigma'_ij=s_i sigma_ij s_j, and multiply each anti-fixed input feature by s_i; verify that the regularizer and graph output are unchanged. Compare cross-entropy alone and cross-entropy plus this loss on signed stochastic block models and a small signed link-prediction dataset, sweeping lambda in {0.01,0.1,1}. Track task accuracy, link AUC, performance after gauge switching, embedding norms, and gradient variance. A successful result is higher accuracy or AUC at equal parameter count, negligible degradation under switching, and fewer unstable training runs.