Learning low-complexity autoregressive models with limited time sequence data

We consider the estimation of the state transition matrix in vector autoregressive models when the time sequence data is limited but nonsequence steady-state data is abundant. To leverage both sources of data, we formulate the problem as the least-squares minimization regularized by a Lyapunov penalty. Explicit cardinality or rank constraints are imposed to reduce the complexity of the model. The resulting nonconvex, nonsmooth problem is solved by using the proximal alternating linearization method (PALM). An advantage of PALM is that for the problem under investigation, it is globally convergent to a critical point and the objective value is monotonically decreasing. Furthermore, the proximal operators therein can be efficiently computed by using explicit formulas. Our numerical experiments demonstrate the effectiveness of the developed approach.