Phase-Delay Spectral Margin for Attractor RNNs

1. Integration point: use a phase-valued recurrent cell or neural ODE with hidden state \(\theta\in\mathbb R^n\). Parameterize nonnegative edge magnitudes by \(A_{ij}=\operatorname{softplus}(a_{ij})\), and learn delays \(\alpha_{ij}\) modulo \(2\pi\). Add an input-dependent frequency term \(\omega_i=W_i x_t+b_i\). 2. Pseudocode: estimate a candidate locked state \(\psi\) by averaging the latent phase trajectory after teacher forcing; compute \(C_{ij}=A_{ij}\cos(\psi_j-\psi_i-\alpha_{ij})\); form \(L_C=\operatorname{diag}(C\mathbf 1)-C\); estimate eigenvalues using a dense eigensolver for small models or Arnoldi/power iterations for sparse models; add \(\mathcal L_{\mathrm{spec}}\) to the task loss; backpropagate through the eigensolver or use a detached spectral estimate for a slower control update. At inference, choose \(\eta\) below the largest value satisfying \(\max_k|1-\eta K\lambda_k|<1\). 3. Computed from the paper's mechanism: the cosine phase-delay composite matrix and its spectral stability test. Estimated empirically: the candidate locked state and basin volume. 4. First cheap experiment: train a 16-unit phase RNN on adding-machine sequences and compare a standard recurrent cell with the spectral-regularized cell at matched parameter count. Sweep \(\eta K\), record hidden-state norms, gradient norms, and recovery after perturbing a locked trajectory. The quantitative prediction is a sharp instability boundary at \(|1-\eta K\lambda_k|=1\); for an approximately real spectrum, measured divergence should begin within 20% of \(\eta_c=2/(K\lambda_{\max}(L_C))\). Increasing the minimum real eigenvalue by \(\gamma\) should increase the perturbation-decay rate approximately as \(K\min_{k\ne\mathrm{gauge}}\operatorname{Re}\lambda_k(L_C)\).