# Koopman Hankel Dual Autoencoder

- ID: 2467
- Canonical URL: https://synthcore.org/idea/2467/koopman-hankel-dual-autoencoder
- API JSON: https://synthcore.org/api/idea/2467.json
- API Markdown: https://synthcore.org/api/idea/2467.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.17698](https://arxiv.org/abs/2608.17698)
- Category: architecture
- Solves: stability, accuracy, generalization
- ML areas: rnn, ssm, transformer, regularization, training-dynamics
- Math tags: dynamical-systems, control-theory, linear-algebra, statistics, spectral-theory
- Ratings: usefulness 7/10; difficulty 5/10; novelty 6/10

## Idea description

Replace pointwise sequence reconstruction with reconstruction of overlapping past and future Hankel windows in a shared latent manifold. A first encoder compresses the delay-coordinate trajectory, while a second decoder or predictor reconstructs the future block from the latent state; training therefore penalizes representations that fit observations but do not preserve dynamical evolution.

## Mathematical statement

For an observed sequence y_k in R^r, optionally lift each sample with observables epsilon_n(y_k) in R^q, where n indexes the chosen observable set. With delay depth p and window count N, define the past and future Hankel matrices Y^- and Y^+ from consecutive samples; the lifted versions replace y_t by epsilon_n(y_t). A finite Koopman approximation assumes Y^+ is approximately K Y^-, where K is a learned linear evolution operator in lifted coordinates. Train an encoder z=E_phi(Y^-), a past decoder Yhat^-=D^-_psi(z), and a future decoder or predictor Yhat^+=D^+_omega(z), with an optional latent transition z_next=A z. The useful property is temporal consistency: low past reconstruction error alone is insufficient unless the same latent neighborhood also gives low future reconstruction error. The latent spectral radius rho(A) supplies a directly measurable bounded-rollout condition.

## Key formulas

- $$\mathcal{Y}^{-}=\begin{bmatrix}y_0&y_1&\cdots&y_{N-1}\\y_1&y_2&\cdots&y_N\\\vdots&\vdots&\ddots&\vdots\\y_{p-1}&y_p&\cdots&y_{N+p-2}\end{bmatrix},\qquad \mathcal{Y}^{+}=\begin{bmatrix}y_p&y_{p+1}&\cdots&y_{N+p-1}\\y_{p+1}&y_{p+2}&\cdots&y_{N+p}\\\vdots&\vdots&\ddots&\vdots\\y_{2p-1}&y_{2p}&\cdots&y_{N+2p-2}\end{bmatrix}.$$
- $$\mathcal{Y}^{+}\approx K\mathcal{Y}^{-},\qquad K=\arg\min_{K}\left\|\mathcal{Y}^{+}-K\mathcal{Y}^{-}\right\|_{F}^{2}+\lambda_K\|K\|_{F}^{2},$$
- $$\mathcal{L}=\|\mathcal{Y}^{-}-D^-_\psi(E_\phi(\mathcal{Y}^{-}))\|_{F}^{2}+\alpha\|\mathcal{Y}^{+}-D^+_\omega(E_\phi(\mathcal{Y}^{-}))\|_{F}^{2}+\beta\|E_\phi(\mathcal{Y}^{+})-A E_\phi(\mathcal{Y}^{-})\|_{2}^{2}+\gamma\|A\|_{2}^{2},$$
- $$T=(e-\mu)^\top\Sigma^{-1}(e-\mu),\qquad e=\operatorname{vec}\!\left(\mathcal{Y}^{+}-D^+_\omega(E_\phi(\mathcal{Y}^{-}))\right).$$

## Implementation notes

1. Integration point: use this module before an RNN, state-space model, or transformer for windowed sequence representation, or use it as a standalone anomaly detector. Choose delay depth p, construct overlapping windows of length 2p, and arrange each window as a p-by-r past block and a p-by-r future block. Implement E_phi as an MLP, temporal convolution, or transformer over the Hankel block. Implement D^- and D^+ as decoders. Optionally parameterize A as a stable latent transition by computing its eigendecomposition after each update and rescaling it when rho(A) exceeds a target value below one.

2. Pseudocode: for every nominal batch, form Yminus and Yplus; compute z=E(Yminus), past_hat=Dminus(z), and future_hat=Dplus(z); minimize the displayed loss, initially using alpha=1 and beta between 0.1 and 1. After training, collect future residual vectors e on held-out nominal windows and estimate their mean mu and covariance Sigma with shrinkage, Sigma=(1-rho)Sigma_emp+rho I. At test time compute T and flag a window when T exceeds the empirical 0.99 or 0.995 nominal quantile. For forecasting, use A^j z or a learned recurrent rollout and decode each future state.

3. Computed versus estimated: Hankel construction and loss evaluation are deterministic. K or A is estimated by ridge regression or gradient descent, while mu and Sigma are estimated from nominal validation residuals. Report the normalized Koopman residual ||Yplus-K Yminus||_F/||Yplus||_F, future reconstruction error, and latent spectral radius rho(A).

4. First cheap experiment: simulate Lorenz-63 with Gaussian observation noise, train on nominal trajectories, and inject a parameter fault by changing a Lorenz coefficient. Compare a pointwise autoencoder, a delay-window autoencoder, and this dual model at equal parameter count. The predicted signature is an initial decrease in Koopman residual as p increases, followed by a plateau from redundant delays. Long-horizon latent rollouts should remain bounded when rho(A)<1 and diverge or grow rapidly near rho(A)=1. Measure whether the empirical boundedness boundary agrees with rho(A)=1 within 20 percent, and whether fault windows cross the calibrated 0.99 Mahalanobis threshold more often than nominal windows.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: inconclusive
- Verdict: Built a reproducible Koopman Hankel dual autoencoder MVP with Hankel construction, ridge Koopman estimation, past/future reconstruction, and CUDA-safe training. The mechanism manifested: noise residual scaled linearly with sigma (R²=0.99985), rollout growth changed sign exactly at |A|=1, and delay depth p=2 reduced the AR(2) Koopman residual from 0.9992 at p=1 to 8.47e-9. In the small fault experiment, the dual future-error detector achieved AUC 0.6967 versus 0.6562 for the past-reconstruction baseline, a modest positive signal.

### Mechanism check

- Verdict: Built a reproducible Koopman Hankel dual autoencoder MVP with Hankel construction, ridge Koopman estimation, past/future reconstruction, and CUDA-safe training. The mechanism manifested: noise residual scaled linearly with sigma (R²=0.99985), rollout growth changed sign exactly at |A|=1, and delay depth p=2 reduced the AR(2) Koopman residual from 0.9992 at p=1 to 8.47e-9. In the small fault experiment, the dual future-error detector achieved AUC 0.6967 versus 0.6562 for the past-reconstruction baseline, a modest positive signal.
- Confidence: 8/10
- Limitations: The toy system was scalar AR(2), not Lorenz-63. The experiment was small and used 700 optimization steps. The baseline was evaluated with past reconstruction while the dual model was evaluated with future reconstruction, so the AUC comparison is indicative rather than fully matched. Learned long-horizon latent rollouts and Mahalanobis threshold calibration were not tested.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.36915
- Paired wins: 6/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_report.json](https://synthcore.org/code/953/bench_report.json)
- [experiment.py](https://synthcore.org/code/953/experiment.py)
- [report.md](https://synthcore.org/code/953/report.md)
- [report_bench_2026-09-01T163930.md](https://synthcore.org/code/953/report_bench_2026-09-01T163930.md)
- [results.json](https://synthcore.org/code/953/results.json)
- [stage2_bench.py](https://synthcore.org/code/953/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/953)

## Disclaimer

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