Learning clusters of partially observed linear dynamical systems

We study the problem of learning clusters of partially observed linear dynamical systems from multiple input-output trajectories. This setting is particularly relevant when there are limited observations (e.g., short trajectories) from individual data sources, making direct estimation challenging. In such cases, incorporating data from multiple related sources can improve learning. We propose an estimation algorithm that leverages different data requirements for the tasks of clustering and system identification. First, short impulse responses are estimated from individual trajectories and clustered. Then, refined models for each cluster are jointly estimated using multiple trajectories. We establish end-to-end finite sample guarantees for estimating Markov parameters and state space realizations and highlight trade-offs among the number of observed systems, the trajectory lengths, and the complexity of the underlying models.