A Laplacian is the trace of a spatial Hessian. Building every Hessian entry is one way to compute that trace, but a supported model can propagate the contraction directly.
The one-layer identity
Δᵏf(x) = Σ_h c_h σ⁽²ᵏ⁾(w_h·x + β_h) ‖w_h‖²ᵏActivation derivative recurrences and weight norms give this specialized expression. The expression removes a nested differentiation graph; its arithmetic and input processing still have costs.
Deep networks have their own contract
The deep Laplacian path retains parameter gradients and has no fixed input-dimension ceiling. For fixed layer widths and depth, work and memory grow with input dimension. Deep repeated Laplacians may use exact support enumeration or a reported directional estimator.
Check a small case first
Our tutorial compares a direct Laplacian with an independent dense-Hessian trace, then verifies finite gradients into the weights.