# Visible-Time Drift Training

- ID: 1346
- Canonical URL: https://synthcore.org/idea/1346/visible-time-drift-training
- API JSON: https://synthcore.org/api/idea/1346.json
- API Markdown: https://synthcore.org/api/idea/1346.md
- Verification status: unverified
- Source: [arXiv:2607.24088](https://arxiv.org/abs/2607.24088)
- Category: training
- Solves: accuracy, stability, sample-efficiency
- ML areas: diffusion, world-model, loss, training
- Math tags: stochastic-processes, probability, geometry, measure-theory
- Ratings: usefulness 5/10; difficulty 3/10; novelty 7/10

## Idea description

Train a neural drift model for a partially observed diffusion using only increments accumulated at times when the latent process is visible, while feeding the projected observation as the state input. The projection may create boundary finite-variation artifacts, but the paper's visible-time identity implies that these artifacts do not bias stochastic estimating equations restricted by the visibility indicator.

## Mathematical statement

The latent trajectory for experiment i is the d-dimensional diffusion $X^{(i)}(t)=x_0+\int_0^t b(X^{(i)}(s))\,ds+\int_0^t\Sigma(X^{(i)}(s))\,dW^{(i)}(s)$, where $b:\mathbb{R}^d\to\mathbb{R}^d$ is the drift, $\Sigma:\mathbb{R}^d\to\mathbb{R}^{d\times q}$ is the dispersion matrix, and $W^{(i)}$ is q-dimensional Brownian motion. The observed state is $Y^{(i)}(t)=\Pi_{D(\theta^{(i)}(t))}(X^{(i)}(t))$, where $D(\theta)$ is a time-varying closed convex censoring region and $\Pi_D$ is Euclidean projection; the visibility flag is $\delta^{(i)}(t)=\mathbf{1}_{\{X^{(i)}(t)\in\operatorname{Int}(D(\theta^{(i)}(t)))\}}$. The paper establishes that $Y$ is a continuous semimartingale with projection-induced finite-variation corrections, and that these corrections assign no mass to visible times. Consequently, for suitable predictable vector test functions $\varphi$, $\int_0^T\delta^{(i)}(t)\varphi(Y^{(i)}(t))^{\top}dY^{(i)}(t)=\int_0^T\delta^{(i)}(t)\varphi(X^{(i)}(t))^{\top}dX^{(i)}(t)$. Since $\delta(t)=1$ implies $Y(t)=X(t)$, use the neural estimating equation $\sum_i\int\delta^{(i)}(t)\varphi(Y^{(i)}(t))^{\top}[dY^{(i)}(t)-f_\psi(Y^{(i)}(t))dt]$, where $f_\psi:\mathbb{R}^d\to\mathbb{R}^d$ is the learned drift. In discrete time, approximate this with visible increments and a robust quadratic loss.

## Key formulas

- $$X^{(i)}(t)=x_{0}+\int_{0}^{t}b(X^{(i)}(s))\,ds+\int_{0}^{t}\Sigma(X^{(i)}(s))\,dW^{(i)}(s).$$
- $$Y^{(i)}(t)=\Pi_{D(\theta^{(i)}(t))}\!\left(X^{(i)}(t)\right),\qquad\delta^{(i)}(t)=\mathbf{1}_{\{X^{(i)}(t)\in\operatorname{Int}(D(\theta^{(i)}(t)))\}}.$$
- $$\int_0^T\delta^{(i)}(t)\,\varphi(Y^{(i)}(t))^{\top}dY^{(i)}(t)=\int_0^T\delta^{(i)}(t)\,\varphi(X^{(i)}(t))^{\top}dX^{(i)}(t).$$
- $$\mathcal{L}_{\mathrm{vis}}(\psi)=\frac{\sum_{i,k}m_{ik}\,\rho\!\left(\Delta Y^{(i)}_k-f_\psi(\widetilde Y^{(i)}_k,t_k,\theta^{(i)}_k)\Delta t_k\right)}{\sum_{i,k}m_{ik}\Delta t_k},\qquad m_{ik}=\delta^{(i)}(t_k)\delta^{(i)}(t_{k+1}),\quad \widetilde Y^{(i)}_k=\frac{Y^{(i)}(t_k)+Y^{(i)}(t_{k+1})}{2}.$$

## Implementation notes

(1) Integration point: replace the ordinary one-step drift loss in a neural SDE, latent-state world model, or continuous-time sequence model. Let the network be $f_\psi(y,t,\theta)$, predicting drift from the projected observation, time, and optional censoring-region parameters. Do not train on intervals whose endpoints are not both visible, because boundary projections must not be interpreted as physical states. (2) Pseudocode: for each trajectory and adjacent pair $(Y_k,\delta_k),(Y_{k+1},\delta_{k+1})$, set $m_k=\delta_k\delta_{k+1}$, $\Delta Y_k=Y_{k+1}-Y_k$, $\widetilde Y_k=(Y_k+Y_{k+1})/2$, and accumulate $m_k\rho(\Delta Y_k-f_\psi(\widetilde Y_k,t_k,\theta_k)\Delta t_k)$. Divide by $\sum_k m_k\Delta t_k$, backpropagate, and optionally add a diffusion covariance head trained on the same mask. Use Huber loss $\rho$ initially for robustness. (3) Computed from the paper: the projection observation rule, visibility mask, and restriction of stochastic estimating equations to visible time. Estimated empirically: $f_\psi$, diffusion covariance, and any correction for unequal visibility rates. If visibility is available only at sample points, use consecutive-visible intervals conservatively; if censoring switches rapidly, subdivide intervals or use midpoint visibility. (4) First experiment: simulate 2D Ornstein–Uhlenbeck and nonlinear double-well diffusions, censor them with moving disks, rectangles, and nonsmooth polytopes, and compare an MLP drift model trained with all projected increments against this visible-time loss. Evaluate drift MSE on uncensored states, rollout error, and stability near boundaries. The expected signal is lower drift bias and more stable rollouts at the same number of observed samples, especially for moving or nonsmooth boundaries.

## Disclaimer

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