On Koopman mode decomposition and tensor component analysis.

Abstract

Koopman mode decomposition and tensor component analysis [also known as CANDECOMP (canonical decomposition)/PARAFAC (parallel factorization)] are two popular approaches of decomposing high dimensional datasets into modes that capture the most relevant features and/or dynamics. Despite their similar goal, the two methods are largely used by different scientific communities and are formulated in distinct mathematical languages. We examine the two together and show that, under certain conditions on the data, the theoretical decomposition given by the tensor component analysis is the same as that given by Koopman mode decomposition. This provides a "bridge" with which the two communities should be able to more effectively communicate. Our work provides new possibilities for algorithmic approaches to Koopman mode decomposition and tensor component analysis and offers a principled way in which to compare the two methods. Additionally, it builds upon a growing body of work showing that dynamical systems theory and Koopman operator theory, in particular, can be useful for problems that have historically made use of optimization theory.