Median-Normalized Weak Pushforward Potential Training

1. Integration point: replace the usual OT map or critic parameterization with an input-convex neural network (ICNN) potential. For source minibatch X={x_i} sampled from μ, compute p_i=∇_x φ_θ(x_i); these are the generated target samples. Use target minibatch Y={y_i} from ν. The training loss is an MMD or Sobolev-IPM between {p_i} and {y_i}, with the potential evaluated only for median normalization and diagnostics. A simple RKHS choice is a mixture of Gaussian kernels k_σ, giving MMD²(P,Q)=mean k(p_i,p_j)+mean k(y_i,y_j)-2 mean k(p_i,y_j). 2. Pseudocode: initialize φ_θ(x)=m||x||²/2+g_θ(x), where g_θ is an ICNN; for each step sample X,Y; compute p=grad_x φ_θ(X); compute median c=median_i φ_θ(x_i); optionally use centered values φ_θ(X)-c; compute MMD²(p,Y); backpropagate through the input gradient and update θ with AdamW. Keep m fixed initially, then optionally learn m through m=softplus(a)+m_min. 3. The paper supplies the median-invariant L1 stability target and the requirement of strong convexity. The weak metric, kernel bandwidths, and minibatch estimator are engineering choices; estimate bandwidth by the median pairwise distance in the target batch and use several bandwidths to avoid scale sensitivity. Monitor the smallest Hessian eigenvalue only on a validation grid or estimate strong monotonicity from gradient pairs: min_{i
e j}(p_i-p_j)^T(x_i-x_j)/||x_i-x_j||². 4. First experiment: use μ=N(0,I_d), ν generated by a known smooth strongly convex potential in d=2 and d=4, and compare against an unconstrained MLP map trained with W2-style pair matching and an ICNN trained with ordinary MMD. At equal parameter count and gradient evaluations, measure centered L1 potential error, pushforward MMD, map W2 error, and loss variance over seeds. Success means lower centered potential error and smoother convergence at equal target-sample budget, with no increase in pushforward discrepancy; test whether the advantage weakens as d grows, consistent with the paper's dimension-dependent rates.