# Neural Koopman Power-Iteration Latent Space

- ID: 216
- Canonical URL: https://synthcore.org/idea/216/neural-koopman-power-iteration-latent-space
- API JSON: https://synthcore.org/api/idea/216.json
- API Markdown: https://synthcore.org/api/idea/216.md
- Verification status: failed_benchmark
- Source: [arXiv:2608.26943](https://arxiv.org/abs/2608.26943)
- Category: dynamics
- Solves: stability, accuracy, sample-efficiency
- ML areas: world-model, ssm, training, regularization
- Math tags: dynamical-systems, spectral-theory, linear-algebra, optimization
- Ratings: usefulness 7/10; difficulty 5/10; novelty 7/10

## Idea description

Add a latent mode bank whose coordinates are learned by neural power iteration on observed state transitions rather than by jointly fitting an unconstrained latent dynamics model. Each mode is repeatedly regressed toward its one-step pushforward, normalized under the data distribution, and deflated against previously learned modes. The resulting latent coordinates are constrained to have approximately linear, diagonal dynamics, which should improve long-horizon prediction and make the representation less dependent on arbitrary dictionary features.

## Mathematical statement

The paper starts from the nonlinear state transition $x^{+}=F(x)$ and the Koopman operator $(\mathcal{K}f)(x)=f(F(x))$, which is linear in the observable $f$ even when $F$ is nonlinear. For samples $x_i$ and $y_i=F(x_i)$, define the empirical inner product $\langle f,g\rangle_\mu=N^{-1}\sum_{i=1}^{N}f(x_i)g(x_i)$ and norm $\|f\|_\mu=\sqrt{\langle f,f\rangle_\mu}$. The implementable power step is $f_{r+1}\approx \mathcal{K}f_r/\|\mathcal{K}f_r\|_\mu$, learned by minimizing $N^{-1}\sum_i[f_{r+1,\theta}(x_i)-f_r(y_i)]^2$. To prevent convergence to the constant mode and to obtain multiple modes, center every observable and project away previously learned modes: $\widetilde{f}_{r+1}=\mathcal{K}f_r-\sum_{j<r}\langle \mathcal{K}f_r,f_j\rangle_\mu f_j$, followed by $f_{r+1}=\widetilde{f}_{r+1}/(\|\widetilde{f}_{r+1}\|_\mu+\varepsilon)$. The associated empirical Koopman eigenvalue estimate is $\lambda_r=\langle f_r,\mathcal{K}f_r\rangle_\mu/\langle f_r,f_r\rangle_\mu$. The mathematical assumptions being exploited are that the Koopman operator is linear on functions and that normalized power iteration preferentially amplifies dominant spectral components; in practice, approximate regression and finite-sample inner products replace exact operator application.

## Key formulas

- $$x^{+}=F(x),\qquad (\mathcal{K}f)(x)=f(F(x))$$
- $$\widetilde{f}_{r+1}(x_i)=f_r(y_i)-\sum_{j<r}\left\langle f_r\circ F,f_j\right\rangle_\mu f_j(x_i),\qquad y_i=F(x_i)$$
- $$f_{r+1}=\frac{\widetilde{f}_{r+1}}{\|\widetilde{f}_{r+1}\|_\mu+\varepsilon},\qquad \|g\|_\mu=\left(\frac{1}{N}\sum_{i=1}^{N}g(x_i)^2\right)^{1/2}$$
- $$\lambda_r=\frac{\langle f_r,\mathcal{K}f_r\rangle_\mu}{\langle f_r,f_r\rangle_\mu}=\frac{\sum_i f_r(x_i)f_r(y_i)}{\sum_i f_r(x_i)^2}$$

## Implementation notes

(1) Integration point: use an encoder with a mode bank $z_r=f_{r,\theta_r}(x)$ before a small latent dynamics module. For an image world model, $x$ is the encoder feature or observation embedding; for a state-space model, it is the current hidden state. Train the mode bank on batches of paired transitions $(x_i,y_i)$, where $y_i$ is the next state or next embedding. Keep the downstream predictor simple, initially $\widehat z_{t+1,r}=\lambda_r z_{t,r}$, and optionally add a learned small matrix after the diagonal baseline. (2) Pseudocode: initialize an empty list of mode outputs. For mode $r=1,\ldots,R$, compute $u_i=f_{r-1}(y_i)$, subtract its empirical mean, and set $u_i\leftarrow u_i-\sum_{j<r}(N^{-1}\sum_i u_i f_j(x_i))f_j(x_i)$. Normalize $u$ by $s=(N^{-1}\sum_i u_i^2+\varepsilon)^{1/2}$. Update neural parameters $\theta_r$ for several minibatch epochs using $L_r=N^{-1}\sum_i[f_{r,\theta_r}(x_i)-u_i/s]^2$, while adding $\rho(N^{-1}\sum_i f_{r,\theta_r}(x_i))^2$ for centering and $\eta\sum_{j<r}(N^{-1}\sum_i f_{r,\theta_r}(x_i)f_j(x_i))^2$ for orthogonality. Recompute $\lambda_r$ with the Rayleigh quotient above. (3) The formulas determine normalization, deflation, and eigenvalue estimation; empirically estimate all inner products from minibatches and use an exponential moving average to reduce noise. Stop a mode if its residual regression error is high or its post-deflation norm is below a threshold. (4) First experiment: compare a GRU, an unconstrained MLP latent transition, and this $R=16$-mode model on a small Lorenz-63 or MuJoCo trajectory dataset. Train all models for equal parameter count and one-step loss, then evaluate 10-, 50-, and 100-step rollout MSE. Success is lower long-horizon error, slower latent norm growth, and a stable estimated spectrum across random seeds; a useful initial target is at least 20% lower 50-step MSE than the unconstrained latent baseline at equal one-step error.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: harms
- Verdict: Built a reproducible Koopman power-iteration MVP with empirical normalization, sequential deflation, Rayleigh-style diagonal latent dynamics, and a Lorenz-63 rollout comparison. The exact toy sweep confirmed the quantitative predictions: one-mode error ratios matched the dominant non-leading eigenvalue modulus (observed 0.3499/0.5000/0.8000/0.9504 versus predicted 0.35/0.5/0.8/0.95), normalization stayed within 3.3e-16, and deflated residual ratios matched 0.1/0.3/0.6/0.9 within about 0.002. In the small Lorenz comparison, the idea was worse at 10 steps and one-step MSE, but much better at 50 and 100 steps, so the long-horizon stability signal is real though not a uniformly better predictor.

### Mechanism check

- Verdict: Built a reproducible Koopman power-iteration MVP with empirical normalization, sequential deflation, Rayleigh-style diagonal latent dynamics, and a Lorenz-63 rollout comparison. The exact toy sweep confirmed the quantitative predictions: one-mode error ratios matched the dominant non-leading eigenvalue modulus (observed 0.3499/0.5000/0.8000/0.9504 versus predicted 0.35/0.5/0.8/0.95), normalization stayed within 3.3e-16, and deflated residual ratios matched 0.1/0.3/0.6/0.9 within about 0.002. In the small Lorenz comparison, the idea was worse at 10 steps and one-step MSE, but much better at 50 and 100 steps, so the long-horizon stability signal is real though not a uniformly better predictor.
- Confidence: 8/10
- Limitations: The downstream MVP is a linear observed-state analogue rather than a neural encoder with minibatch training, and uses only one fixed Lorenz dataset/seed. It does not test GRUs, image observations, equal parameter counts, multiple random seeds, noisy transitions, or GPU execution.

### 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/106/bench_report.json)
- [koopman_experiment.py](https://synthcore.org/code/106/koopman_experiment.py)
- [report.md](https://synthcore.org/code/106/report.md)
- [report_bench_2026-08-30T142438.md](https://synthcore.org/code/106/report_bench_2026-08-30T142438.md)
- [results.json](https://synthcore.org/code/106/results.json)
- [stage2_koopman_bench.py](https://synthcore.org/code/106/stage2_koopman_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/106)

## Disclaimer

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