Learning Stable Linear Dynamical Systems with the Weighted Least Square Method

Standard subspace algorithms learn Linear Dynamical Systems (LDSs) from time series with the least-square method, where the stability of the system is not naturally guaranteed. In this paper, we propose a novel approach for learning stable systems by enforcing stability directly on the leastsquare solutions. To this end, we first explore the spectral-radius property of the least-square transition matrix and then determine the key component that incurs the instability of the transition matrix. By multiplying the unstable component with a weight matrix on the right side, we obtain a weighted-least-square transition matrix that is further optimized to minimize the reconstruction error of the state sequence while still maintaining the stable constraint. Comparative experimental evaluations demonstrate that our proposed methods outperform the state-of-the-art methods regarding the reconstruction accuracy and the learning efficiency.