Non-Gaussian Perron–Frobenius Latent Filter

Integrate this as the uncertainty state of a small neural state-space model. Use an encoder $z_k=E_\theta(o_{0:k})$ for the observed latent state and represent uncertainty over $z_k$ with coefficients $c_k$ in a basis of radial functions, tensor-product splines, or Gaussian radial basis functions. Use nonnegative normalized basis functions so that the reconstructed density is nonnegative and has unit mass. First collect short trajectories from a known simulator or from a trained deterministic latent model. At each time, draw particles from the empirical latent distribution, fit coefficients $c_k$ by nonnegative least squares, and construct $C$ and $Y$. Compute $K_P$ with the ridge formula above, then project or optimize it onto the affine mass-preserving set $q^\top K_P=q^\top$; optionally clip unstable eigenvalues to modulus $1-\epsilon$. During inference, update the neural mean state with the learned latent transition and update uncertainty by $c_{k+1}=K_Pc_k$, reconstructing density values only when needed. Pseudocode: encode observations; initialize $c_0$; for each horizon step, set $c\leftarrow K_Pc$; renormalize coefficients; decode density moments or samples; and score the observation likelihood. The paper mechanism is the Perron–Frobenius transport and flexible eDMD basis. Empirical quantities are basis coefficients, $K_P$, and the spectral radius. The first cheap experiment should use a 2-D nonlinear double-well or Henon-map dataset, comparing an EKF-GRU, particle filter, and this model at equal compute budgets. Measure held-out negative log likelihood, Wasserstein distance, and mass error over 1, 5, 10, and 25 steps. The quantitative prediction is that density coefficient norms remain bounded when the fitted $\rho(K_P)<1$, while models with $\rho(K_P)>1$ show exponential coefficient growth; the asymptotic growth rate should approach $\log\rho(K_P)$, and mass error should remain near numerical tolerance after enforcing $q^\top K_P=q^\top$.