Metric-affine asymmetric contrast loss

(1) Integrate the method at the embedding-loss stage of a Siamese encoder, supervised-contrastive model, or retrieval model. Let the encoder output $z=f_\theta(s)\in\mathbb R^d$. Attach a small head producing a lower-triangular matrix $L(z)$ with positive diagonal and define $G(z)=L(z)L(z)^{\mathsf T}+\epsilon I$. Initially use globally learned constant connection matrices $A_1,\ldots,A_d$; only later test an MLP that makes them depend on $z$. (2) For every anchor-positive-negative triplet, compute $x=f_\theta(a)$, $y^+=f_\theta(p)$, and $y^-=f_\theta(n)$. For each pair $(x,y)$, set $v=y-x$, initialize $P=I_d$, and use $K$ midpoint steps. At step $k$, evaluate $\gamma_k=x+(k+1/2)v/K$, form $B_k=\sum_i v_iA_i(\gamma_k)$, and update $P\leftarrow(I_d-B_k/K)P$. After the loop solve $Pu=v$ for $u$ rather than explicitly computing an inverse, then evaluate $D(x,y)=u^TG(x)u/2$. Optimize the margin expression in the displayed formula, with $m>0$. (3) The discretized transport ODE and quadratic form come from the mathematical construction; the metric and connection parameters are learned. Estimate numerical stability by monitoring the smallest singular value of every $P$ and clip or spectrally normalize the $A_i$ if it approaches zero. The log-determinant term prevents metric collapse, while the Frobenius penalty controls excessive transport curvature. Use $A_i=0$ and $G=I$ as mandatory ablations. (4) Run a first cheap experiment with a small ResNet or ViT on CIFAR-10 using supervised triplet loss. Compare Euclidean contrastive loss, a learned symmetric Mahalanobis loss, and the metric-affine loss at matched encoder FLOPs and approximately matched parameter counts. Evaluate Recall@1, classification accuracy from a frozen linear probe, and performance when query and gallery examples are subjected to different augmentations or directional domain shifts. Also report the asymmetry gap $|D(x,y)-D(y,x)|$, transport condition numbers, and loss-curve variance. A useful success criterion is a statistically significant retrieval or robustness gain over the Mahalanobis baseline without exploding transport condition numbers or materially slower optimization.