Noise-Induced Compact-Support Neural Field

Integrate the module after a CNN, graph neural network, or transformer routing block whose activations are interpreted as a nonnegative field u in R_+^n. Use a 2D grid Laplacian for image feature maps, a fixed chain or token-neighbor graph for sequences, or the task graph for graph data. First create nonnegative initial values with u^0 = softplus(z), where z is the original learned activation or routing logit. Run K stochastic relaxation steps during training. Let L_G be the symmetric graph Laplacian, kappa be the diffusion strength, lambda be optional decay, rho be the noise scale, gamma be in (0,1), epsilon be a small numerical floor, and Delta t be the step size. At each step sample independent standard Gaussian eta_i^k, calculate the displayed Euler-Maruyama update, and project negative values to zero. To obtain actual computational sparsity, form a mask m_i = 1[u_i^K > tau] after relaxation and either use it in the next layer or use a straight-through estimator during training. Add a mild support cost alpha times the number of active sites, but do not use such a large alpha that the stochastic mechanism is replaced by ordinary L0 regularization. Backpropagate through sampled noise using reparameterization. Measure exact-zero fraction, active support size, connected-component count, gradient variance, and sensitivity to gamma, rho, and initial mass sum_i u_i^0. Start with a small CIFAR-10 CNN and compare against Bernoulli dropout, deterministic graph diffusion, hard-concrete gates, and L1 regularization at matched accuracy. Sweep gamma in {0.25, 0.5, 0.75}. A positive result is at least twice the exact-zero fraction of dropout at equal validation accuracy, or equal accuracy with substantially fewer active sites and lower measured inference FLOPs. A negative result is instability, accuracy collapse, or merely producing small nonzero values without a useful reduction in active computation.