Even-Norm Group Diffusion Augmentation

Integrate the method at the data-augmentation and representation-regularization boundary, not inside the attention matrix. Choose a small finite transformation group with cheap exact actions, initially the dihedral group D4 of rotations and reflections of a square image patch. Store one nonnegative rate for each transformation and tie inverse elements by setting lambda[s]=lambda[inverse(s)]. For each training example x, sample a short continuous-time walk: draw N from Poisson(tau times sum_s lambda[s]), then draw each jump independently with probability lambda[s]/sum_r lambda[r], and apply the transformations in sequence to obtain the final transform g*x. Optionally retain all visited states as augmented views. Encode the transformed views h_theta(g*x), subtract their mean over g, and add L_inv to the task loss. Also compute the empirical endpoint histogram q_hat(g) over sampled group states and the diagnostic M_2m=sum_g abs(q_hat(g)-1/|G|)^(2m). Parameterize rates as lambda[s]=softplus(a[s])+epsilon and enforce inverse tying after every optimizer step. Use a schedule in which tau, or all symmetric rate pairs, only increases. The mathematical quantities taken directly from the paper are the Poisson generator, inverse-symmetric rate constraint, and even-norm monotonicity; the feature penalty and Monte Carlo estimator are neural-network adaptations. First test on CIFAR-10 with a small ResNet-18, comparing ordinary random D4 augmentation, fixed uniform D4 augmentation, and learned-rate group diffusion at equal augmentation calls. Measure clean accuracy, transformed-test accuracy, calibration, M_2m versus tau, and training variance. A successful result is improved D4 robustness or accuracy at equal compute, no optimization instability as tau increases, and a nonincreasing empirical M_2m curve.