Triangle-weighted support regularizer

Integrate this as an auxiliary loss in a network that predicts a two-dimensional convex uncertainty set or geometric enclosure. The network receives an input x and outputs nonnegative vertex coordinates z_theta,1,...,z_theta,m in R^2, using either a softplus parameterization or unconstrained coordinates followed by a validity loss. For every minibatch example, define the predicted support function h_theta(phi) as the maximum dot product z_theta,j^T u(phi), where u(phi) = (cos(phi), sin(phi)). During training, replace the hard maximum with tau log sum_j exp(z_j^T u(phi) / tau), using tau = 0.02 or annealing tau from 0.1 to 0.01. Select triangle angles alpha and beta, for example alpha = beta = pi/4, and sample 16 uniformly spaced phases t per example. Compute H1 = h_theta(t + pi + alpha), H2 = h_theta(t + pi - beta), and H3 = h_theta(t). Compute v_t = sin(alpha)sin(beta) - sin(beta)H1 - sin(alpha)H2 - sin(alpha + beta)H3, then add lambda mean_t max(v_t, 0)^2 to the task loss. The theorem supplies the angle weights and exact inequality; phase count, temperature, and lambda are empirical choices. First test on synthetic randomly rotated line segments and trajectories, training an MLP to output an m-vertex convex enclosure. Compare a standard pointwise inclusion loss against the same model plus this regularizer. Evaluate directional violation over 360 test orientations, enclosure area, calibration or coverage, and training throughput. The expected signal is lower worst-case directional violation and improved coverage at equal predicted area, with less computation than dense boundary sampling.