Finite-Sample Metric Non-Collapse for Geometrically Supervised Latent World Models in Control

[2608.07265] Finite-Sample Metric Non-Collapse for Geometrically Supervised Latent World Models in Control Skip to main content Search Submit Donate Log in Mathematics > Optimization and Control arXiv:2608.07265 (math) [Submitted on 7 Aug 2026 ( v1 ), last revised 23 Aug 2026 (this version, v2)] Title: Finite-Sample Metric Non-Collapse for Geometrically Supervised Latent World Models in Control Authors: Alain Bensoussan , Minh-Nhat Phung , Minh-Binh Tran View PDF HTML (experimental) 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. Comments: Revised version prepared in response to the editorial assessment. The main manuscript is 32 pages; detailed mathematical derivations have been moved to the accompanying Supplementary Material. The principal results and contributions are strengthened and clarified Subjects: Optimization and Control (math.OC) ; Machine Learning (cs.LG) Cite as: arXiv:2608.07265 [math.OC] (or arXiv:2608.07265v2 [math.OC] for this version) https://doi.org/10.48550/arXiv.2608.07265 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Minh-Nhat Phung [ view email ] [v1] Fri, 7 Aug 2026 14:25:33 UTC (179 KB) [v2] Sun, 23 Aug 2026 02:25:33 UTC (182 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2026-08 Change to browse by: cs cs.LG math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

Related documents

• Finite-Sample Metric Non-Collapse for Geometrically Supervised Latent World Models in Control#670734
• Supervise What Decides Success: Criterion-Aligned Auxiliary Losses for Latent World-Model Planning#1156496
• CAER: Causal Action Effect Reweighting for World Model Training#818935
• CAER: Causal Action Effect Reweighting for World Model Training#808943
• LLM-driven design of physics-constrained constitutive models: two agents are better than one#222588