Jacobian-Normalized Latent Observer

1. Exact integration point: use a recurrent latent state-space model or learned observer with latent state $\hat{x}_t$, transition network $f_\theta$, and measurement or decoder network $h_\phi$. Keep the usual innovation correction, but multiply it by the adaptive scalar $\alpha_t$ before adding it to the next latent state. Initialize $\tilde K$ as a learned matrix or a fixed low-rank matrix, and constrain its norm if necessary. 2. Pseudocode: compute $\hat y=h_\phi(\hat x)$; compute the innovation $r=y-\hat y$; obtain the Jacobian-vector product or full Jacobian $J=\partial h_\phi(\hat x)/\partial\hat x$ using automatic differentiation; estimate $s=\|\tilde KJ\|_2$ by 3--5 power iterations; set $\alpha=\beta/(s+\varepsilon)$; update $\hat x^+=f_\theta(\hat x)+\alpha\tilde K r$. Stop gradients through $s$ for the first implementation, then test differentiating through the normalization. 3. Computed from the paper: the inverse-Jacobian scaling rule and the contraction bound. Estimated empirically: $L_f$, $L_h$, the operator norm, and the actual error Jacobian norm. Estimate $L_f$ by power-iteration Jacobian probes of $f_\theta$ and log the local ratio $\|A_t\|_2$. 4. First cheap experiment: train a GRU or 2-layer MLP latent observer on noisy pendulum or Lorenz-63 state reconstruction, comparing fixed-gain correction against normalized correction at matched parameter count. Sweep $\beta$ and sequence length, then sweep measurement-network output scale. The predicted signature is that fixed-gain divergence occurs when the measured local correction norm crosses its contraction budget, while normalized models keep $\|\tilde KJ\alpha\|_2\le\beta$ up to numerical error. For a transition with measured $L_f$, test the stability boundary $L_f+\beta<1$; the normalized observer should lose exponential contraction near this boundary, whereas its critical point should shift much less when the measurement representation is rescaled. Verify that the observed one-step error decay follows approximately $\|e_t\|\lesssim(L_f+\beta)^t\|e_0\|$ while errors remain in the local region.