# Temporal-Window Luenberger Projection

- ID: 1668
- Canonical URL: https://synthcore.org/idea/1668/temporal-window-luenberger-projection
- API JSON: https://synthcore.org/api/idea/1668.json
- API Markdown: https://synthcore.org/api/idea/1668.md
- Verification status: unverified
- Source: [arXiv:2608.01371](https://arxiv.org/abs/2608.01371)
- Category: dynamics
- Solves: stability, accuracy
- ML areas: ssm, rnn, world-model, training-dynamics
- Math tags: control-theory, dynamical-systems, optimization, linear-algebra
- Ratings: usefulness 6/10; difficulty 6/10; novelty 8/10

## Idea description

Insert a constraint-aware observer between a neural state-space transition and its next prediction. The observer propagates latent event times, incorporates partial observations, and projects the result onto the set satisfying both lower-bound causality and upper-bound token-lifetime constraints, preventing impossible latent trajectories from entering the recurrent model.

## Mathematical statement

The paper models dater vectors with $x=x_{0}\oplus Ax\oplus Bu\oplus Rw$, observations $y=Cx$, and P-Time Event Graph constraints $x\geq Ux$. Here $\oplus$ is maximum, multiplication is addition, $\varepsilon=-\infty$, and $e=0$; $x_i(k)$ is the firing time of latent transition $i$ at event index $k$. Matrix $A$ propagates causal lower-bound constraints, while $U$ encodes upper-bound token-lifetime restrictions. For a neural model, let $z_k$ be latent event times, $F_\theta$ the learned transition, and $H$ the observation map. First calculate $\tilde z_{k+1}=F_\theta(z_k,u_k)$, then choose the greatest feasible under-approximation $\hat z_{k+1}\leq\tilde z_{k+1}$ satisfying $\hat z_{k+1}\geq U\hat z_{k+1}$ and $H\hat z_{k+1}=y_{k+1}$. Monotonicity is the useful mechanism: max-plus propagation preserves order, while projection removes estimates that violate temporal windows.

## Key formulas

- $$x=x_{0}\oplus Ax\oplus Bu\oplus Rw,\qquad y=Cx,\qquad x\geq Ux.$$
- $$\tilde z_{k+1}=F_{\theta}(z_k,u_k),\qquad \hat z_{k+1}=\max\left\{z:\ z\leq\tilde z_{k+1},\ z\geq Uz,\ Hz=y_{k+1}\right\},$$
- $$\hat z_{k+1,i}\leq\tilde z_{k+1,i},\qquad \hat z_{k+1,i}\geq\max_j\left(U_{ij}+\hat z_{k+1,j}\right),$$
- $$\Delta_i=\max_j(U_{ij}+\tilde z_{k+1,j})-\tilde z_{k+1,i};\qquad \Delta_i>0\Longrightarrow\text{the unconstrained prediction violates an upper-bound window by at least }\Delta_i.$$

## Implementation notes

1. Integration point: use a recurrent state-space model or world model whose latent vector contains timestamp-like coordinates, such as object arrival times, contact times, queue events, or discretized latent milestones. Keep the learned transition $F_\theta$ unchanged initially. After every transition prediction, apply an observer/projection layer before feeding the state to the next step. Known observation coordinates are clamped to measured event times; unknown coordinates are corrected. 2. Pseudocode: `z_pred = F_theta(z, u)`; compute the constraint residual `r[i] = max_j(U[i,j] + z_pred[j]) - z_pred[i]`; for violated constraints, solve the small linear program `maximize sum_i z[i]` subject to `z[i] <= z_pred[i]`, `z[i] >= U[i,j] + z[j]`, and `z[observed] = y`. For a cheaper approximation, use sparse relaxation sweeps over the difference-constraint graph, lowering violated coordinates and restoring observed coordinates after each sweep. 3. Computed from the paper: monotone propagation, the inequality $z\geq Uz$, and greatest-subapproximation correction. Estimated empirically: the neural transition, observation noise, constraint matrix $U$, and projection convergence. 4. First experiment: train a GRU state-space model on synthetic partially observed timed event graphs with random lower bounds and windows $[l_{ij},u_{ij}]$. Compare vanilla rollout, observation-only correction, and the projection observer. Measure impossible-window violations, hidden-event MAE, and error after each observation. The prediction is that projected rollouts have zero constraint violations up to solver tolerance, while vanilla violations grow with horizon. After an observation touching a connected component, hidden-state error should decrease over subsequent propagation steps. Sweep window tightness and identify the infeasibility threshold; before that threshold, test for at least a 20% reduction in long-horizon MAE.

## Disclaimer

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