Schrodinger spectral-gap regularizer for learned metrics

(1) Integration point: insert the regularizer after the encoder forward pass during training. For a minibatch x_i, compute embeddings z_i=f_θ(x_i), retain the input or local chart coordinates, and obtain the Jacobian J_f(x_i) with forward- or reverse-mode autodiff. Set g_i=J_f(x_i)^T J_f(x_i)+εI with ε=10^-4. (2) Choose K centers c_k from the current embedding minibatch and define localized Gaussian basis functions φ_k(x_i)=exp(-||z_i-c_k||²/(2σ²)). Compute their derivatives by autodiff. Use minibatch weights w_i=1/B and build M, L, and S. For the first MVP, set S=0; the exact curvature version requires differentiating the metric and estimating scalar curvature, which is substantially more expensive. For each batch, solve the K×K generalized eigenproblem (L+γS)v=λMv, or use 3–5 inverse-iteration steps. Add L_spec=softplus(τ-λ_min) to the task loss, with γ=(m-1)/(4m)+0.05 and target margin τ. (3) The paper-derived quantity is the Rayleigh quotient and its Galerkin discretization. The network Jacobian, RBF centers, bandwidth σ, quadrature weights, and curvature estimate are empirical approximations. Use gradient clipping and stop-gradient through eigenvectors if eigenderivative instability appears. (4) First experiment: train a two-layer MLP encoder/classifier on MNIST or CIFAR-10, comparing baseline, Jacobian Frobenius regularization, and L_spec at matched wall-clock and parameter budgets. Measure accuracy, Gaussian-noise and corruption accuracy, Jacobian norms, estimated minimum Rayleigh quotient, and per-step variance. The signal of success is increased λ_min and improved perturbation robustness at equal clean accuracy, with less than 20% training overhead.