On finding optimal collective variables for complex systems by minimizing the deviation between effective and full dynamics

This paper is concerned with collective variables, or reaction coordinates, that map a discrete-in-time Markov process Xn in ℝd to a (much) smaller dimension k ll d. We define the effective dynamics under a given collective variable map xi as the best Markovian representation of Xn under xi. The novelty of the paper is that it gives strict criteria for selecting optimal collective variables via the properties of the effective dynamics. In particular, we show that the transition density of the effective dynamics of the optimal collective variable solves a relative entropy minimization problem from certain family of densities to the transition density of Xn. We also show that many transfer operator-based data-driven numerical approaches essentially learn quantities of the effective dynamics. Furthermore, we obtain various error estimates for the effective dynamics in approximating dominant timescales / eigenvalues and transition rates of the original process Xn and how optimal collective variables minimize these errors. Our results contribute to the development of theoretical tools for the understanding of complex dynamical systems, e.g. molecular kinetics, on large timescales. These results shed light on the relations among existing data-driven numerical approaches for identifying good collective variables, and they also motivate the development of new methods.