Rigid-Motion-Quotient Covariance Loss

1. Integration point: attach the loss to the penultimate embedding tensor of a classifier, contrastive encoder, graph neural network, or autoencoder. For each minibatch, generate two augmentations v=1 and v=2, encode them as z_{v,j} in R^d, and compute one covariance per view. Add lambda_rigid times L_rigid to the ordinary task or contrastive loss during training; inference is unchanged. 2. Pseudocode: z1 = encoder(view1(x)); z2 = encoder(view2(x)); mu1 = z1.mean(0); mu2 = z2.mean(0); C1 = (z1-mu1).T @ (z1-mu1)/(n-1); C2 = (z2-mu2).T @ (z2-mu2)/(n-1); C1 = C1 + eps*I; C2 = C2 + eps*I; lam1 = sort(eigvalsh(C1), descending=True); lam2 = sort(eigvalsh(C2), descending=True); loss_rigid = ((sqrt(lam1)-sqrt(lam2))**2).sum(); loss = loss_task + lambda_rigid*loss_rigid; loss.backward(). Use a symmetric eigensolver, clamp eigenvalues below zero, and set eps near 1e-5. 3. The paper supplies the quotient construction, spectral invariance, and square-root metric. The minibatch means and covariances are empirical estimates. For large d, use a randomized eigensolver or retain only the largest k eigenvalues. Monitor eigenvalue crossings and covariance collapse; optionally add a variance floor. 4. First experiment: train a small ResNet-18 on CIFAR-10 or a two-layer GNN on ModelNet40 with two standard augmentations. Compare no distribution loss, Frobenius covariance matching, standard Bures covariance matching, and this quotient loss at matched compute. At evaluation, multiply one view’s embeddings by an independently sampled orthogonal matrix and add a random translation. Success is unchanged prediction or contrastive loss under these transformations, improved augmentation or domain robustness, and lower validation error at equal training steps without covariance collapse. Report eigendecomposition time and covariance condition numbers.