Low-Rank Dynamic Mode Decomposition: Optimal Solution in Polynomial-Time

This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach can be formalised as attempting to solve a low-rank constrained optimisation problem. This problem is non-convex and state-of-the-art algorithms are all sub-optimal. This paper shows that there exists a closed-form solution, which can be computed in polynomial-time, and characterises the $\ell_2$-norm of the optimal approximation error. The theoretical results serve to design low-complexity algorithms building reduced models from the optimal solution, based on singular value decomposition or low-rank DMD. The algorithms are evaluated by numerical simulations using synthetic and physical data benchmarks.