Cross-Channel Scattering Front End

(1) Integration point: place this layer immediately after input normalization and before a temporal CNN, graph neural network, or Transformer. The input is a batch $X\in\mathbb{R}^{B\times C\times T}$, with $C$ sensor channels. Choose 4-8 wavelet bands and a sensor adjacency list containing spatially neighboring pairs; do not form all $C^2$ pairs initially. Use complex Morlet filters implemented as two real 1-D convolutions, or use fixed filters for the first experiment. For each band $\lambda$, compute complex responses $Z[:,m,\lambda,:]=(X_m*\psi_\lambda)$ and pairwise products only for $(m,n)$ in the adjacency list. Concatenate the pooled SNST features with the original input or with channelwise scattering features. (2) Pseudocode: Z = complex_conv1d(X, psi); for lambda in bands: for (m,n) in edges: q = Z[:,m,lambda,:] * conj(Z[:,n,lambda,:]); u = sqrt(real(q)2 + imag(q)2 + eps); s = depthwise_avg_pool(u, kernel_size=2**J); features.append(s); H = concat(features, baseline_features); loss = cross_entropy(classifier(H), y). The displayed $U_1$ formula is the computation of u, and the displayed $S_1$ formula is the averaging operation. (3) Mathematical quantities are the complex wavelet responses, conjugate products, magnitude, and low-pass pooling. Estimate no statistics from the paper; select $J$ so that $2^J$ is between 50 and 250 ms of data, and optionally learn a positive scalar multiplier per band after initialization. Normalize each pair feature using training-set mean and variance, with the normalization statistics frozen for validation and test. To limit redundancy, share a linear projection across all edge features or retain only the top-$K$ spatial edges by validation performance. (4) First experiment: use BCI Competition IV-2a, 4-class motor-imagery EEG, with a small 2-layer temporal CNN or EEGNet baseline. Compare raw-input EEGNet, channelwise scattering plus classifier, and channelwise scattering plus SNST edge features under identical parameter budgets and five subject-wise seeds. The primary success signal is higher held-out accuracy or macro-F1 at equal labels and FLOPs; a useful secondary signal is a larger accuracy gain in the first 10-20% of the training data and lower seed-to-seed variance. Also ablate conjugation, edge locality, and temporal pooling to verify that the gain comes from cross-channel amplitude coupling rather than simply adding filter-bank capacity.