Magnitude-Euler Path Signature Regularizer

1. Integration point: apply this during training to a small auxiliary computation graph extracted from each MLP block. Create one node for every neuron in the selected hidden layers, plus source and class-output nodes. For a dense layer from neuron u to v, use an edge cost c_uv=1 for the first experiment; optionally use c_uv=1/(|w_uv|+epsilon) or c_uv=softplus(-|w_uv|) after the unit-cost baseline. Remove edges whose gated activation contribution |w_uv h_u| is below a percentile threshold, producing a sample-dependent active DAG. 2. Pseudocode: build active edges; compute shortest directed distances d[u,v] by dynamic programming in topological order; for each L≤L_max, enumerate or sample ordered tuples (x_0,…,x_k) with adjacent finite distances and sum d[x_{j-1},x_j]=L; accumulate n[k,L]. Set chi[L]=sum_k (-1)^k n[k,L], directly implementing the paper’s Euler decategorification. Add the loss above, where chi(X|y) is the minibatch mean for class y, bar_chi is an exponential moving-average target, beta is small, and rho optionally stabilizes the global signature. 3. Mathematics versus estimates: shortest distances and chain lengths are exact for the chosen graph; chi is estimated by uniform chain sampling when enumeration exceeds a budget. Use K=3, L_max=4, 256 sampled chains per k and class. Since hard thresholding is nondifferentiable, update this statistic every N steps and treat it as a detached monitoring or reinforcement-style regularizer initially; a differentiable surrogate can replace edge inclusion with sigmoid gates and soft chain weights. 4. First experiment: train a 3-layer width-256 MLP on MNIST or CIFAR-10 embeddings, comparing cross-entropy, weight decay, and the Euler-signature penalty at matched FLOPs. Report test accuracy, calibration, classwise signature separation, and robustness to neuron permutation or random pruning. A positive result is higher accuracy or corruption robustness at equal parameter count, with classwise chi vectors separating before accuracy improves and no more than 10% training-time overhead.