Gauge-Patched Local Experts

(1) Integration point: insert this into a graph-NN or MoE hidden layer. Partition inputs into K overlapping patches or route tokens to K local experts. Expert n outputs h_n(x) in R^d. Parameterize every transition as g_nm(x)=exp(S_nm(x)), where S_nm is skew-symmetric, so g_nm is orthogonal and preserves feature norm. A transition MLP receives the shared input embedding and predicts the entries of S_nm.

(2) Pseudocode:

for batch x:
    for patch n:
        h[n] = expert_n(x[n])
    for overlap (n,m):
        S = skew(transition_nm(x_overlap))
        g = matrix_exp(S)
        L_patch += mean(||h[n]-g @ h[m]||_2^2)
        L_patch += lambda_id * mean(||g-I||_F^2)
    for triple (n,m,k):
        L_patch += lambda_cocycle * mean(||g_nm @ g_mk-g_nk||_F^2)
    L = task_loss + lambda_patch * L_patch
    update(parameters)

For a global readout, align each h_n to a reference patch using transition products along a spanning tree. (3) The paper contributes the small-overlap condition and cocycle equations. Estimate local Holder size by max_{i,j}||h(x_i)-h(x_j)||/||x_i-x_j||^beta, and monitor whether transition magnitudes and patch residuals remain below a chosen empirical delta_patch. (4) First experiment: train a four-layer GCN on a synthetic planar point-cloud classification task with random independent rotations applied to each spatial patch. Compare a shared GCN, ordinary MoE, and gauge-patched MoE at equal parameter count and FLOPs. Success means higher accuracy under unseen local rotations, lower prediction variance across rotations, lower overlap disagreement, and stable optimization as the number of experts grows.