Parabolic Riesz Feature Preconditioner

(1) Integration point: add the branch after patch embedding or after each early spatiotemporal encoder block, before the nonlinear MLP. Use it for video, event-camera, optical-flow, traffic, or graph signals where information is concentrated at spatial and temporal discontinuities. Let \(X\in\mathbb{R}^{B\times T\times N\times d}\). Construct a sparse graph Laplacian \(L_A\), a causal temporal difference matrix \(D_t\), and \(H=D_t+L_A+\epsilon I\), with \(\epsilon>0\). Let \(\mathbb{D}\) be the spatial incidence matrix, optionally concatenated with a scaled temporal difference. (2) Pseudocode: for each selected scale \(\lambda\), solve \((I+\lambda^2H)Z=X\) using 5--20 preconditioned conjugate-gradient iterations; compute \(R=\lambda\mathbb{D}Z\); apply RMSNorm to \(R\); return \(X_{out}=X+W_sX+gW_rR\), where \(g\) is initialized to zero or 0.1 and learned. Use multiple scales and concatenate the resulting \(R\)'s if memory permits. (3) The mathematical component is the dimensionless factor \(\lambda\mathbb{D}\) combined with the resolvent inverse, which prevents derivative magnitude from increasing uncontrollably with resolution. Estimate the operator norm of \(\lambda\mathbb{D}(I+\lambda^2H)^{-1}\) using five-step power iteration and reduce \(g\) if the estimate exceeds a target such as 2. Select scales from the empirical feature correlation length; the paper does not provide a neural-network-specific optimal scale. (4) First experiment: compare a six-layer graph transformer or small VideoMamba with and without this branch on Moving-MNIST, traffic forecasting, or UCF101 using 64 nodes and 16 time steps. Measure validation accuracy, sensitivity to 10% feature perturbations, gradient-norm variance, and training loss. Compare against direct \(\mathbb{D}X\) and an ordinary convolutional high-pass branch. Success is improved boundary or motion accuracy and lower corruption sensitivity for no more than 20% additional FLOPs.