Prediction limits for soft geometric state abstractions in dynamical systems
Prediction Limits and Koopman Closure of Geometry-Induced Soft State Abstractions
Artificial IntelligenceMachine Learning
Summary
This work looks at how to predict the future behavior of complex systems using a special way of representing their states with soft labels rather than hard categories. The authors provide a mathematical way to know the best possible accuracy of these predictions, showing when it’s impossible to do better. They also explore connections to Koopman theory, which helps understand system dynamics through linear models. The methods were tested on several simulated control systems.
What this means in practice
- •For control engineers: Assess the fundamental limits of soft-coordinate prediction models when designing controllers for nonlinear dynamical systems.
- •For robotics developers: Use KAHM-based soft state representations combined with prediction error bounds to improve state estimation and planning algorithms.
Tested on simulated data.
Authors
Mohit Kumar, Somayeh Kargaran
Abstract
We study when geometry-induced soft state abstractions admit accurate finite-dimensional linear dynamics. Each state is represented by simplex-valued coordinates obtained from class-specific Kernel Affine Hull Machine (KAHM) reconstruction scores, and a matrix is used to predict the next-state coordinates. Our main result is a computable lower confidence bound on the minimum root-mean-square prediction error over all matrices satisfying a prescribed spectral-norm limit. The bound combines within-class variation of successor coordinates with the deviation of soft coordinates from their one-hot reference labels, and can be evaluated from independent state-successor pairs without fitting a prediction matrix. For fixed coordinates and evaluation distribution, the certificate converges almost surely to a population lower bound as the sample size grows; any tolerance below this limit is eventually certified unattainable. Reconstruction-score margins further control the soft-to-hard assignment error. Under deterministic dynamics and exact coordinate closure, eigenvectors of the closure matrix and its reduced transpose induce Koopman and adjoint Koopman eigenfunctions, respectively. A four-state KAHM construction shows that identical soft coordinates can permit exact closure under one dynamics map yet force positive prediction error under another. Experiments on Duffing, Van der Pol, CartPole, MountainCar, and Acrobot compare direct soft-coordinate prediction with state-space DMD/EDMD baselines and report prediction, representation-variation, and spectral diagnostics. The benchmarks assess fitted models but do not numerically evaluate the exclusion certificate.