On Koopman Mode Decomposition and Tensor Component Analysis

Abstract

Koopman mode decomposition and tensor component analysis (also known as CANDECOMP/PARAFAC or canonical polyadic decomposition) are two popular approaches of decomposing high dimensional data sets into low dimensional 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 formulated in distinct mathematical languages. We examine the two together and show that, under a certain (reasonable) condition on the data, the theoretical decomposition given by tensor component analysis is the \textit{same} as that given by Koopman mode decomposition. This provides a "bridge" with which the two communities should be able to more effectively communicate. When this condition is not met, Koopman mode decomposition still provides a tensor decomposition with an \textit{a priori} computable error, providing an alternative to the non-convex optimization that tensor component analysis requires. Our work provides new possibilities for algorithmic approaches to Koopman mode decomposition and tensor component analysis, provides a new perspective on the success of tensor component analysis, and builds upon a growing body of work showing that dynamical systems, and Koopman operator theory in particular, can be useful for problems that have historically made use of optimization theory.