Identification and Recovery of Linear Dynamics from Partial Trajectories
Abstract
We study the joint recovery of an unknown system matrix A∈ℝn× n and a rank-r initial-state matrix X0∈ℝn× m from partial observations of the state matrices AtX0, for t=0,…,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 range(X0) 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 X0, 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.
Related
- Scalable Identification of Physics-Based and Neural Dynamics from Partial Measurements
- Exact Dynamics and Finite-Sample Trajectory Recovery of Linear Recursive Feature Machines
- Identification through Heteroskedasticity in One Lemma: Partial Identification, an Eigenvalue Construction, and a Caution on Rank Tests
- Quantile-Only Regression: Inference and Partial Identification under Quantile Disclosure
- A partial data coefficient identification inverse problem for a semilinear damped wave operator
BibTeX
@misc{gong2026identification,
title = {{Identification and Recovery of Linear Dynamics from Partial Trajectories}},
author = {Le Gong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04455},
eprint = {2610.04455},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04455},
}
RIS
TY - UNPB
TI - Identification and Recovery of Linear Dynamics from Partial Trajectories
AU - Le Gong
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04455
UR - https://arxiv.org/abs/2610.04455
AB - We study the joint recovery of an unknown system matrix A∈ℝ^n× n and a rank-r initial-state matrix X_0∈ℝ^n× m from partial observations of the state matrices A^tX_0, for t=0,…,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 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.
ER -