Learning Stable Linear Dynamical Systems Learning Stable Linear Dynamical Systems

Stability is a desirable characteristic for linear dynamical systems, but it is often ignored by algorithms that learn these systems from data. We propose a novel method for learning stable linear dynamical systems: we formulate an approximation of the problem as a convex program, start with a solution to a relaxed version of the program, and incrementally add constraints to improve stability. Rather than continuing to generate constraints until we reach a feasible solution, we test stability at each step; because the convex program is only an approximation of the desired problem, this early stopping rule can yield a higher-quality solution. We employ both maximum likelihood and subspace ID methods to the problem of learning dynamical systems with exogenous inputs directly from data. Our algorithm is applied to a variety of problems including the tasks of learning dynamic textures from image sequences, learning a model of laser and vision sensor data from a mobile robot, learning stable baseline models for drug-sales data in the biosurveillance domain, and learning a model to predict sunspot data over time. We compare the constraint generation approach to learning stable dynamical systems to the best previous stable algorithms (Lacy and Bernstein, 2002, 2003), with positive results in terms of prediction accuracy, quality of simulated sequences, and computational efficiency. Source code and video results of stable dynamic textures are available online at http://www.select.cs.cmu.edu/projects/stableLDS.