Recursive Noise-Corrected Latent Dynamics

1. Exact integration point: use a small neural encoder $e_\theta$ to map raw observations to $o_k$, and a decoder or prediction head to reconstruct observations. Every $L$ steps, or once per minibatch in streaming training, run the recursive estimator on the last $L$ encoded features and controls. Use the resulting $U_k$ as a stop-gradient coordinate transform, and train a latent SSM predictor with $x_{k+1}=A_kx_k+B_ku_k$; alternatively initialize an SSM transition from $(A_k,B_k)$ and continue gradient training. 2. Pseudocode: compute $o=e_\theta(y)$ and residual $r=y-\operatorname{decode}(o)$; update $C\leftarrow\gamma C+(1-\gamma)oo^\top$ and $R\leftarrow\gamma R+(1-\gamma)\operatorname{diag}(rr^\top)$; eigendecompose $S=C-R$; retain eigenvectors with $\lambda_i>\tau_i=cR_{ii}$ or a validation-calibrated threshold; project $x=U^\top o$; solve the weighted ridge regression for $A,B$; project $A$ to $\rho(A)\leq1-\epsilon$; train the encoder and decoder using reconstruction plus latent one-step and multistep prediction loss. 3. Computed from the paper's mechanism: recursive covariance, fixed-lag statistics, noise-corrected PCA, and recursive model refitting. Estimated empirically: residual-based $R$, threshold constant $c$, forgetting factor $\gamma$, and ridge coefficient $\lambda$. 4. First cheap experiment: use a noisy Lorenz-63 or switched linear dynamical system, encode observations with a two-layer MLP, and compare a standard neural SSM, online PCA initialization, and this method. Inject a sensor variance jump at time $T$ and a transition-matrix change at $2T$. The quantitative prediction is that corrected PCA will suppress dimensions whose eigenvalues are below the estimated noise floor, while the uncorrected baseline will increase rank after the variance jump. For a scalar parameter jump, fitted $A_k$ should adapt exponentially with characteristic time approximately $1/(1-\gamma)$ samples; measure this against the predicted lag within 20%. Stable rollout divergence should begin only when unconstrained estimated $\rho(A_k)$ exceeds one, and the spectral projection should prevent that boundary.