Conserved Poisson Feature Noise

Integrate this at the residual-stream input of a vision transformer, patch transformer, or CNN feature grid during training only. Build a fixed sparse graph over spatial tokens: use a four-neighbor grid for images, or a local-window/ring graph for sequences. Store its oriented edge-incidence matrix B and Laplacian L=B Bᵀ. Maintain one scalar density vector ρ of length N per batch example, initialized to ones with total mass N. At each training step, sample standard-normal edge noise ξ and channel noise g. Compute edge noise as √ρ_edge times ξ, scatter it through B, and update ρ using the displayed Euler-Maruyama rule. Use ΔtD₀λ_max(L)≤1 for an initial stable step-size choice. Clamp negative entries to zero and rescale the vector so its sum is exactly N. Detach ρ from autograd. Add α(ρ−1)g to patch embeddings or residual activations, optionally after normalizing g so the added variance matches the iid Gaussian baseline. Pseudocode: initialize ρ=ones; sample ξ,g; ρ_tmp=ρ−dtD₀(Lρ)+sqrt(2D₀dt)B@(sqrt(ρ_edge)*ξ); ρ=positive_mass_projection(ρ_tmp,N); h=h+α(ρ−1)[:,None]*g; run the network and backpropagate. The mathematical ingredients are the conservative diffusion and square-root noise; estimate activation variance, token covariance, and validation metrics empirically. First test ViT-Tiny on CIFAR-100 or ImageNet-100 against no noise, dropout, and iid Gaussian activation noise at matched variance and FLOPs. Success means lower validation error, better corruption robustness, or equally good accuracy with fewer training steps, together with stable loss curves and longer-range token covariance than iid noise.