Formalizing the Curvature–Planning Success Relationship: A Geometric Theory of Latent Planning

TL;DR: Can we mathematically model how latent trajectory curvature affects planning success, and use this to guide regularizer design?

Research Question: How does the curvature of latent trajectories quantitatively relate to planning success, and can geometric modeling guide principled design of regularizers?

Hypothesis: There exists a quantifiable relationship between latent space curvature and planning reliability; formal geometric models can predict optimal regularization for a given latent manifold.

Experiment Plan: - Develop a theoretical framework (e.g., leveraging Riemannian geometry) that predicts planning errors as a function of curvature and latent manifold properties.

• Use synthetic datasets with controllable geometry to empirically validate predictions.
• Apply the model to real-world planning tasks, adjusting regularization according to theoretical guidance.
• Assess whether theoretically-informed regularization outperforms heuristic approaches.

References:

• Wang, Y., Bounou, O., Zhou, G., Balestriero, R., Rudner, T. G. J., LeCun, Y., & Ren, M. (2026). Temporal Straightening for Latent Planning.
• Lubold, S., Chandrasekhar, A. G., & McCormick, T. (2020). Identifying the latent space geometry of network models through analysis of curvature. Social Science Research Network.
• Park, Y.-H., Kwon, M., Choi, J., Jo, J., & Uh, Y. (2023). Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry. Neural Information Processing Systems.