# Non-Gaussian Perron–Frobenius Latent Filter

- ID: 1457
- Canonical URL: https://synthcore.org/idea/1457/non-gaussian-perron-frobenius-latent-filter
- API JSON: https://synthcore.org/api/idea/1457.json
- API Markdown: https://synthcore.org/api/idea/1457.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2607.26632](https://arxiv.org/abs/2607.26632)
- Category: architecture
- Solves: stability, accuracy, sample-efficiency
- ML areas: world-model, rnn, ssm
- Math tags: dynamical-systems, linear-algebra, probability, spectral-theory, control-theory
- Ratings: usefulness 7/10; difficulty 6/10; novelty 6/10

## Idea description

Replace Gaussian covariance propagation in a neural state-space model with a finite Perron–Frobenius operator acting on coefficients of a learned density basis. A neural encoder maps observations to latent states, while an eDMD-derived matrix transports the full coefficient vector and supports multimodal or skewed uncertainty. This creates a cheap deterministic uncertainty layer that can be rolled forward for long horizons without repeatedly sampling particles.

## Mathematical statement

For a measurable nonlinear map $x_{k+1}=T(x_k)$ and density $p_k$, the Perron–Frobenius operator satisfies $p_{k+1}=\mathscr{P}p_k$ and is defined by $\int_{T^{-1}(A)}f(x)\,d\mu=\int_A(\mathscr{P}f)(x)\,d\mu$ for every measurable set $A$. Choose nonnegative basis functions $\{\psi_i\}_{i=1}^{m}$ and approximate a density by $p(x)\approx\sum_{i=1}^{m}c_i\psi_i(x)$, where $c_k\in\mathbb{R}^{m}$ is the density-coefficient vector. Learn a matrix $K_P$ such that $c_{k+1}\approx K_Pc_k$ from trajectory or particle-density snapshots. If $C=[c_0,\ldots,c_{N-1}]$ and $Y=[c_1,\ldots,c_N]$, a ridge estimate is $K_P=YC^\top(CC^\top+\lambda I)^{-1}$, with regularization $\lambda>0$. Enforce probability-mass preservation using $q_i=\int_X\psi_i(x)d\mu$ and $q^\top K_P=q^\top$. The spectral-radius condition $\rho(K_P)\leq1$ is the discrete-time stability signature; eigenvalues with modulus greater than one predict exponentially growing density coefficients.

## Key formulas

- $$x_{k+1}=T(x_k),\qquad p_{k+1}=\mathscr{P}p_k,$$
- $$\int_{T^{-1}(A)}f(x)\,d\mu=\int_A(\mathscr{P}f)(x)\,d\mu,$$
- $$p(x)\approx\sum_{i=1}^{m}c_i\psi_i(x),\qquad c_{k+1}\approx K_Pc_k,\qquad K_P=YC^\top(CC^\top+\lambda I)^{-1},$$
- $$q^\top K_P=q^\top,\qquad \rho(K_P)=\max_j|\lambda_j(K_P)|\leq1.$$

## Implementation notes

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 $q^\top K_P=q^\top$.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: harms
- Verdict: Built a finite nonnegative RBF Perron–Frobenius latent-density filter with ridge operator learning and affine mass-preserving projection. The spectral sweep confirmed the predicted boundary gamma*=1/0.93=1.0753: gamma=1.075 gave rho=0.99975 and bounded coefficients, while gamma=1.10 gave rho=1.023 and growth; observed log-norm slopes matched log(rho) to numerical precision. Mass error fell from 4.51e-3 before projection to 6.66e-16 after projection. On the nonlinear multimodal toy map, PF W1 was lower than Gaussian closure at all tested horizons and separations, but the learned operator had fitted spectral radius 1.083, so long-horizon stability was not automatically achieved.

### Mechanism check

- Verdict: Built a finite nonnegative RBF Perron–Frobenius latent-density filter with ridge operator learning and affine mass-preserving projection. The spectral sweep confirmed the predicted boundary gamma*=1/0.93=1.0753: gamma=1.075 gave rho=0.99975 and bounded coefficients, while gamma=1.10 gave rho=1.023 and growth; observed log-norm slopes matched log(rho) to numerical precision. Mass error fell from 4.51e-3 before projection to 6.66e-16 after projection. On the nonlinear multimodal toy map, PF W1 was lower than Gaussian closure at all tested horizons and separations, but the learned operator had fitted spectral radius 1.083, so long-horizon stability was not automatically achieved.
- Confidence: 8/10
- Limitations: Only a 1-D nonlinear toy map was tested; no neural encoder, GRU, particle-filter baseline, 2-D Henon/double-well system, held-out observation likelihood, FLOP benchmark, or eigenvalue clipping experiment was included. The spectral sweep uses a controlled diagonal operator, so its exact slope agreement is a verification of the stated linear-algebra claim rather than evidence that noisy eDMD estimates are always stable.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 0/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_report.json](https://synthcore.org/code/571/bench_report.json)
- [math_check.json](https://synthcore.org/code/571/math_check.json)
- [pf_latent_experiment.py](https://synthcore.org/code/571/pf_latent_experiment.py)
- [report.md](https://synthcore.org/code/571/report.md)
- [report_bench_2026-08-31T181415.md](https://synthcore.org/code/571/report_bench_2026-08-31T181415.md)
- [results.json](https://synthcore.org/code/571/results.json)
- [run_output.json](https://synthcore.org/code/571/run_output.json)
- [stage2_pf_bench.py](https://synthcore.org/code/571/stage2_pf_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/571)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.