Task-Targeted Spectral Excitation for Dynamics Learning

1. Integration point: use this during data collection for a neural state-space model, recurrent world model, or nonlinear SSM, rather than as an ordinary minibatch loss. Begin with a controllable simulator such as a damped oscillator, CartPole, or MuJoCo pendulum. Generate trajectories using a nominal policy plus an additive colored excitation generator. 2. Pseudocode: collect a trajectory; construct H_L(u) and compute its smallest singular value; compute FFTs U and Y; estimate q(ω) by adding a small sinusoid at each of K frequency bins to validation rollouts and measuring the increase in H-step prediction loss or closed-loop tracking loss; update excitation amplitudes using E_k ← |U_k|²+λ_y|Y_k|² and a_k ← a_k+αq_k/(E_k+δ); inverse FFT the updated spectrum, clip actions to valid bounds, and accept the dataset only if σ_min(H_L(u))≥ε_rank and Σ_kq_kE_k≥ε_task. 3. Computed from the paper: the Hankel rank test, delayed Fourier representation, and the distinction between rank excitation and frequency-relevant excitation. Estimated empirically: q(ω), output spectra, thresholds, and the neural predictor. 4. First cheap experiment: identify a two-dimensional damped oscillator with a GRU or S4-style SSM from 50 trajectories, comparing white noise, rank-only colored noise, and proposed rank-plus-task excitation. Evaluate 50-step prediction and reference tracking. The quantitative prediction is that rank-only data can have σ_min(H_L(u)) above ε_rank but low Σ_kq_kE_k and poor performance; adding energy to high-q bins should reduce task loss. Plot performance against S_task and test whether the failure transition occurs near S_task=ε_task, with a stronger correlation between S_task and test error than between Hankel rank and test error.