The alignment distance on Spaces of Linear Dynamical Systems

We introduce a family of group action induced distances on spaces of Linear Dynamical Systems (LDSs) of fixed size and order. The distance between two LDSs is computed by finding the change of basis that best aligns the state-space realizations of the two LDSs, hence the name alignment distance. This distance can be computed efficiently, hence it is particularly suitable for applications in modern dynamic data analysis (e.g., video sequence classification and clustering), where a large number of high-dimensional LDSs may need to be compared. Based on the alignment distance, we also define a notion of average between LDSs of the same size and order with the property that the order and in some cases stability are naturally preserved. Various extensions to the basic notion of alignment distance are also proposed.