Directional Brenier Curvature Penalty

(1) Integration point: add the penalty to the loss used to train a neural OT potential, an OT decoder, or a transport-based diffusion bridge. The tensor being modified is the scalar potential output before taking its input gradient; do not penalize arbitrary decoder Hessians unless the decoder is explicitly represented as a gradient map. (2) Pseudocode: for each minibatch of source points x and random directions v sampled uniformly from the unit sphere, compute a=grad_x(Phi_theta(x)); compute the Hessian-vector product h=grad_x(dot(a,v)); compute curvature c=dot(h,v); compute q=sqrt(dot(v,Qv)); compute width w=max_{y in K}dot(y,v)-min_{y in K}dot(y,v); set b=0.587*q*w; add relu(c-b)^2 or the smooth R_dir term to the transport loss. For K=[l,u], use w=sum_i |v_i|(u_i-l_i), avoiding optimization. For a polytope, compute the two linear programs over its vertices or constraints. (3) Q and K are supplied from source and preprocessing geometry; Hessian-vector products are obtained by automatic differentiation, while the expectation over directions is Monte Carlo. Use 8-32 directions per batch and normalize v. (4) First experiment: compare no curvature regularization, isotropic Frobenius Hessian penalty, and R_dir on 2D/16D Gaussian-to-box OT and on a bounded image decoder. Track transport objective, 95th and 99th percentile Jacobian norm, curvature violation rate, gradient variance, and downstream sample quality. Success is a substantial reduction in high-curvature outliers and training spikes at matched transport cost, with better anisotropic fidelity than an isotropic penalty.