Moment-Corrected Sub-Riemannian Diffusion Layer

Integrate this at the neighbor-aggregation stage of a point-cloud, graph, or geometric transformer block. For every node i, construct a k-nearest-neighbor set N(i), local coordinates \xi_ij=(x_j-x_i)/r_i in a fixed or learned local frame, and normalized positive weights w_ij proportional to exp(-||\xi_ij||^2/ au), with sum_j w_ij=1. Compute the empirical first moment \bar\xi_i=sum_j w_ij\xi_ij and covariance tensor C_i=sum_j w_ij(\xi_ij-\bar\xi_i)(\xi_ij-\bar\xi_i)^T+\epsilon I. Define centered offsets ilde\xi_ij=\xi_ij-\bar\xi_i so the first-order drift term is removed. For node features H, compute d_ij=H_j-H_i and directional derivatives delta_a H_i=sum_j w_ij ilde\xi_ij,a d_ij. Use a second message-passing pass to approximate delta_a(delta_b H_i), then compute D_i H_i=0.5 sum_{a,b}(C_i)_{ab} delta_a(delta_b H_i). Add alpha D_i H_i to the original residual block, initializing alpha at zero and constraining it nonnegative with softplus. The mathematical quantities supplied by the paper are the first-moment drift decomposition and the second-moment principal symbol; moments, local frames, and derivatives are estimated empirically from neighborhoods. First test on an anisotropic synthetic-manifold classification task and ModelNet point-cloud classification, comparing ordinary mean aggregation, isotropic graph diffusion, and this layer at matched parameter count and FLOPs. Evaluate accuracy, sample efficiency, gradient-norm variance, sensitivity to neighborhood radius, and robustness to nonuniform point density. Success means improved accuracy or faster loss descent, especially when neighborhoods are anisotropic or sampling density varies.