Learning structured linear dynamical systems from missing observations

2026-10-08

AI Summary

This study addresses the challenge of learning structured linear dynamical systems from incomplete observations. It proposes an estimator based on a debiased non-convex objective, optimized efficiently via projected gradient descent with convex set constraints, and establishes non-asymptotic error bounds that depend on local complexity measures. Theoretically, the work proves the algorithm’s convergence under short trajectories and low sampling rates. Empirically, experiments demonstrate that the proposed method achieves significantly higher data efficiency than unconstrained baselines, accurately recovering the system transition matrix even under extremely sparse observation regimes.

Abstract

We consider the problem of learning structured linear dynamical systems over convex sets $\mathcal{K}$, where only a small subset of the observations are available at each time point. An estimator which minimizes a bias-corrected, potentially non-convex objective function is proposed. Non-asymptotic bounds are obtained for the statistical error, which depend on the local complexity of $\mathcal{K}$, the trajectory length $T$, and the sub-sampling probability $p$. Convergence of the projected gradient descent algorithm is also established. The general theory is applied to settings where (i) $\mathcal{K}$ is a subspace, (ii) $\mathcal{K}$ is the set of bi-isotonic matrices, and (iii) $\mathcal{K}$ is the set of matrices whose rows are formed by sampling Lipschitz functions. We show meaningful recovery of the transition matrix is possible for values of $T$ much smaller than what is required in the unconstrained case, and for $p = o(1)$.

Problem

• linear dynamical systems
• missing observations
• structured learning
• transition matrix recovery

Innovation

• linear dynamical systems
• missing observations
• bias-corrected estimator
• non-asymptotic bounds
• projected gradient descent