Recover full system behavior from partial and sparse observations
Identification and Recovery of Linear Dynamics from Partial Trajectories
Information Theory
Summary
This work looks at how to figure out the rules that govern a system’s behavior when you only see parts of its activity over time. The authors show what conditions must be met to uniquely identify these rules and the system’s starting states from limited snapshots. They also provide a way to solve this problem using math and test their approach using simulations and real data. This helps when you have incomplete measurements but want to understand the full system.
What this means in practice
- •For control engineers: Determine system parameters and initial states when only some sensor data is available from dynamic processes.
- •For signal processing teams: Reconstruct full signal trajectories from incomplete measurements in time-series data for improved analysis and prediction.
Authors
Le Gong
Abstract
We study the joint recovery of an unknown system matrix $A\in\mathbb R^{n\times n}$ and a rank-$r$ initial-state matrix $X_0\in\mathbb R^{n\times m}$ from partial observations of the state matrices $A^tX_0$, for $t=0,\ldots,T$. Each trajectory is observed at a fixed set of state coordinates, which may differ across trajectories. We derive necessary conditions for identifiability, including obstructions arising from incomplete spatial coverage and insufficient dynamical span, as well as lower bounds on the number of observations. We then characterize the Jacobian of the measurement map, account for the intrinsic symmetry of the low-rank factorization, and establish a rank-certificate principle yielding generic local identifiability. When the trajectories whose initial states form a basis of $\operatorname{range}(X_0)$ are fully observed, we obtain necessary and sufficient rank conditions for global recovery, together with explicit reconstruction formulas. We formulate the joint recovery problem as a nonlinear least-squares problem over the system matrix $A$ and the low-rank factors of $X_0$, and solve it using Adam. Numerical experiments on synthetic and real-world data evaluate the recovery performance under different spatial sampling rates and observation horizons.