# Bilinear Input-Conditioned Koopman Cell

- ID: 526
- Canonical URL: https://synthcore.org/idea/526/bilinear-input-conditioned-koopman-cell
- API JSON: https://synthcore.org/api/idea/526.json
- API Markdown: https://synthcore.org/api/idea/526.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2607.01819](https://arxiv.org/abs/2607.01819)
- Category: architecture
- Solves: stability, accuracy, sample-efficiency
- ML areas: world-model, ssm, rnn, rl
- Math tags: dynamical-systems, linear-algebra, control-theory, tensor-decomposition
- Ratings: usefulness 6/10; difficulty 5/10; novelty 5/10

## Idea description

Replace an unconstrained input-conditioned recurrent transition with a bilinear latent update, so controls modulate a fixed linear latent dynamics matrix through low-rank state-input interactions. The resulting cell preserves the computational simplicity of linear propagation while representing multiplicative effects of actions that an additive control term cannot capture efficiently.

## Mathematical statement

The paper derives finite-dimensional linear and bilinear Koopman forms for controlled systems. In the displayed bilinear case, the lifted state \(z=[x_1,x_2,x_1^2,1]^{\top}\) evolves as \(z^{+}=Az+Bzu\), where \(A=\begin{bmatrix}a&0&0&0\\0&c&d&h\\0&0&a^2&0\\0&0&0&1\end{bmatrix}\) and \(B=\begin{bmatrix}0&0&0&0\\e&0&0&f\\0&0&0&0\\0&0&0&0\end{bmatrix}\). The symbols a, c, d, h, e, and f are scalar system coefficients and u is the control input. Generalize this structure to a learned latent vector \(z_t\in\mathbb{R}^m\) and action \(u_t\in\mathbb{R}^p\): \(z_{t+1}=Az_t+\sum_{j=1}^{p}u_{t,j}B_jz_t+Du_t+b\). The bilinear term is the transferable asset: it represents action-dependent changes in latent dynamics with parameter cost \(O(pm^2)\), or \(O(prm)\) with rank-r factors \(B_j=U_jV_j^{\top}\).

## Key formulas

- $$\begin{bmatrix}x_{1}\\x_{2}\\x_{1}^{2}\\1\end{bmatrix}^{+}=\begin{bmatrix}a&0&0&0\\0&c&d&h\\0&0&a^{2}&0\\0&0&0&1\end{bmatrix}\begin{bmatrix}x_{1}\\x_{2}\\x_{1}^{2}\\1\end{bmatrix}+\begin{bmatrix}0&0&0&0\\e&0&0&f\\0&0&0&0\\0&0&0&0\end{bmatrix}\begin{bmatrix}x_{1}\\x_{2}\\x_{1}^{2}\\1\end{bmatrix}u.$$
- $$z_{t+1}=Az_t+\sum_{j=1}^{p}u_{t,j}B_jz_t+Du_t+b,$$
- $$B_j=U_jV_j^{\top},\qquad z_{t+1}=Az_t+\sum_{j=1}^{p}u_{t,j}U_j(V_j^{\top}z_t)+Du_t+b.$$

## Implementation notes

Use this as the transition cell in a model-predictive world model, controlled SSM, or action-conditioned recurrent predictor. Encode observation x_t into z_t in R^m. Store a base matrix A, action matrix D, bias b, and either dense B_j matrices or rank-r factors U_j in R^{m x r}, V_j in R^{m x r}. At every step compute q_j=V_j^T z_t, multiply each q_j by the scalar action u_{t,j}, map through U_j, sum across j, and update z_next=A z_t+sum_j u_{t,j} U_j q_j+D u_t+b. Decode z_next to predict the next observation. Train with multi-step teacher-forced and free-running rollout losses; optionally constrain A's spectral radius by parameterizing A=rho*tanh(S)/max(1,sigma_max(S)) with rho<1, while leaving the bilinear term expressive. Pseudocode: z=encoder(x); for t in horizon: interaction=0; for j in range(p): interaction += u[j]*U[j]@(V[j].T@z); z=A@z+interaction+D@u+b; prediction=decoder(z); accumulate loss. The paper supplies the bilinear structure; rank r, spectral constraint, and losses are empirical choices. Begin with m=32, r=4, and 2-4 actions on a simulated pendulum, CartPole, or MuJoCo task. Compare against an equal-parameter MLP transition and an additive linear model. Track rollout MSE, control reward under MPC or model-predictive planning, parameter count, and wall-clock cost. Success means materially lower long-horizon prediction error or higher planned control reward than additive dynamics, with no instability and lower cost than the MLP baseline.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: inconclusive
- Verdict: Built a deterministic bilinear Koopman-style controlled predictor and compared it with an additive linear transition on a system containing an action-state interaction. The paper matrix identity held to 3.6e-15 maximum absolute error; the bilinear model recovered the planted coefficients and achieved essentially zero held-out 24-step rollout error, versus additive RMSE 0.0564. This clearly demonstrates the claimed multiplicative-effect advantage on the toy system, though it does not establish performance on noisy high-dimensional world models.

### Mechanism check

- Verdict: Built a deterministic bilinear Koopman-style controlled predictor and compared it with an additive linear transition on a system containing an action-state interaction. The paper matrix identity held to 3.6e-15 maximum absolute error; the bilinear model recovered the planted coefficients and achieved essentially zero held-out 24-step rollout error, versus additive RMSE 0.0564. This clearly demonstrates the claimed multiplicative-effect advantage on the toy system, though it does not establish performance on noisy high-dimensional world models.
- Confidence: 9/10
- Limitations: Only a scalar, noiseless synthetic plant was tested, with offline least-squares fitting and one action dimension. No neural encoder/decoder, rank-constrained latent implementation, stability constraint, training-cost comparison, MPC reward, noise robustness, or CartPole/pendulum benchmark was evaluated.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0991
- Paired wins: 6/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_experiment.py](https://synthcore.org/code/1371/bench_experiment.py)
- [bench_report.json](https://synthcore.org/code/1371/bench_report.json)
- [experiment.py](https://synthcore.org/code/1371/experiment.py)
- [report.md](https://synthcore.org/code/1371/report.md)
- [report_bench_2026-09-04T173728.md](https://synthcore.org/code/1371/report_bench_2026-09-04T173728.md)
- [results.json](https://synthcore.org/code/1371/results.json)
- [Download all files as ZIP](https://synthcore.org/download/1371)

## Disclaimer

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