A finite-sample learning-to-control theory for geometrically supervised latent models of nonlinear deterministic systems is established and an encoder-only local--global metric hinge is introduced that enforces directional resolution and separated-state discrimination.
Abstract
We establish a finite-sample learning-to-control theory for geometrically supervised latent models of nonlinear deterministic systems. Geometric supervision is used only during training: simulator state, proprioception, or state estimates with independently validated metric and directional error bounds supply observable-state distances and tangent directions, while deployment remains observation- and action-conditioned. We introduce an encoder-only local--global metric hinge that enforces directional resolution and separated-state discrimination. Under regular observable-factor, coverage, finite-capacity approximation, and uniform $C^{1,1}$ hypotheses, a computable one-sided regularization regime has a strong selection property: with high probability, every approximate empirical minimizer is simultaneously pointwise co-Lipschitz and uniformly approximately semiconjugate to the controlled dynamics. Approximation, sampling, and optimization errors remain explicit and separate. Norm-constrained tensor-product B-spline classes constructively realize the approximation hypotheses, and the interpolation exponent converting mean residual control into a uniform bound is sharp. A modular deterministic corollary transfers the learned certificates to trajectory, finite-horizon cost, learned-cost-head, and optimizer guarantees, while a validated finite-net result enables sharper model-specific certification. Controlled experiments isolate collapse and folding, quantify the analytic certificate's reserve, and demonstrate the control benefit of restored metric resolution. The principal contribution is a complete finite-sample implication from approximate empirical optimization to metric faithfulness, uniform controlled dynamics, and reliable planning for the same learned model.
A joint identifiability condition for controlled world models with Gaussian latent states with Gaussian latent states is presented, which consists of two coupled components: representation identifiability and transition identifiability, and it is proved that when this condition holds, minimizing the LeJEPA-style predictive objective can recover both latent states and controlled dynamics in the sense of orthogonal transformation.
Xiangteng Zhang, Yang Guan, Bo Zhang et al.· 0 citations
This work proposes PRISM-ZO, a projection-robust framework that samples low-dimensional random tangent subspaces and combines symmetric finite differences with median-of-means or Huber aggregation and establishes the unbiasedness of the correctly rescaled projected direction in expectation over the random subspace.
Yinpu Ma, Cunlin Li, Shiyue Zhang· Journal of King Saud Univers...· 0 citations
Stochastic-process models are, as a rule, far easier to simulate than to condition. Non-linear observations, non-Gaussian likelihoods, black-box information, and global constraints all induce intractable conditional laws, requiring bespoke, model-specific constructions. We introduce LatentFlow, a single framework for conditioning stochastic processes, with no learned neural approximations and no training. Our starting point is to write the stochastic process as the deterministic image of a tractable latent innovation, $f_0 = T_{\vartheta}(\xi_0)$, with $\xi_0$ sampled from a simple reference distribution. This reduces process-level conditioning to latent-space inference: pull the likelihood back through $T_{\vartheta}$, sample the resulting latent law with a tractable guided probability flow, and push the samples forward. This construction is provably exact at the level of the target law; in practice, approximation enters only through finite terminal noising, Monte Carlo guidance, and time discretisation of the continuous-time dynamics, each of which is explicit and systematically reducible. As LatentFlow is training-free, conditioning reduces to solving a single reverse-time SDE. This enables conditional sampling in seconds on a single desktop CPU across model classes that have never shared a scalable method: classical spatial priors, nonlinear stochastic dynamics, mechanistic models from the physical and life sciences, stochastic PDEs, heavy-tails and extremes, point and discrete-state processes, and neural or simulator-defined processes.
Louis Sharrock, L. Astfalck, Henry Moss· 0 citations
This work constructs a proposal that dominates the target by a known constant, generally unavailable for non-Gaussian state space models, yielding independent exact smoothing draws and an unbiased likelihood estimator whose relative variance is at most $1/p-1$ per draw at acceptance probability $p$.
Recovering two-dimensional Ito generators from trajectory data is difficult because drift increments have low signal-to-noise, bivariate weak designs can be ill-conditioned, and unconstrained tensor estimates need not be positive semidefinite. We study WG-SINDy estimator combining covariance-shaped spatial kernels, a ridge-stabilized local-polynomial projection, adaptive-LASSO/STLSQ selection, one in-sample per-component feasible diagonal GLS pass, and a PSD projection--Cholesky read-out with mild isotropic shrinkage. The released estimator uses a data-dependent full-cloud smoother and one in-sample per-component feasible diagonal GLS pass; accordingly, we do not claim exact finite-sample martingale cancellation or a feasible-GLS efficiency theorem for the reported implementation. We evaluate the estimator on 29 synthetic two-dimensional systems: 19 meet their declared per-system recovery contracts, eight are retained as named limits, and two remain scoped reviews. Across the 19 PASS rows, the median central-grid drift metric is 0.204 and the median tensor error is 0.0397. Among the six systems with a finite, non-degenerate off-diagonal target, the median $a_{12}$ cosine is 0.997. Positive-semidefinite validity is imposed by construction. These results are synthetic, in-sample sampled-region diagnostics and do not establish universal or real-data recovery.