Shape-profile dependence regularizer

Integrate the method at a representation-learning boundary. Let a backbone produce paired minibatch representations $X=f_\phi(a)$ and $Y=g_\phi(b)$, such as two augmented views, two modalities, or query/document embeddings. Add a small auxiliary encoder $W_\theta=q_\theta(X,Y,\epsilon)$; use a categorical $K$-state latent with Gumbel-softmax for the first MVP. Select a grid of $(\alpha_i,\beta_i)$ values, concentrating on the lower-left triangle. Estimate $H(X|W)$ and $H(Y|W)$ with conditional categorical or Gaussian negative log-likelihoods. Estimate $I(X:Y|W)$ using a conditional density-ratio critic: $\widehat I=\mathbb{E}[T_\psi(x,y,w)]-\log\mathbb{E}[\exp T_\psi(x,\widetilde y,w)]$, where $\widetilde y$ is shuffled within the minibatch. Update $\theta$ and critic parameters $\psi$ by ascent on $\widehat{\mathcal S}_\theta$; update backbone parameters $\phi$ by descent on the task loss plus $\lambda$ times a profile penalty. One simple penalty is $\lambda\max_i\widehat{\mathcal S}(\alpha_i,\beta_i)$ over lower-left grid points, while another matches a desired profile between positive and negative pairs. Estimate the supremum by several ascent steps or by maintaining an exponential-moving-average ensemble of auxiliary encoders. The paper supplies the supremum objective and structural convexity/affinity properties; latent cardinality, critic architecture, minibatch shuffling, and estimator stabilization are empirical. Add a diagnostic convexity residual $\max(0,\widehat S(tu+(1-t)v)-t\widehat S(u)-(1-t)\widehat S(v))$ for random grid points $u,v$ and $t\in[0,1]$. First test on CIFAR-10 with a small two-tower encoder and SimCLR augmentations, comparing InfoNCE, a scalar mutual-information penalty, and this shape regularizer at equal FLOPs. Measure linear-probe accuracy, augmentation robustness, estimated profile convexity violation, and run-to-run variance. Success means improved linear-probe accuracy or robustness at fixed contrastive loss, with a stable and reproducible change in lower-left profile values.