On Koopman Mode Decomposition and Tensor Component Analysis

TL;DR

This paper establishes a theoretical connection between Koopman mode decomposition and tensor component analysis, showing they are equivalent under certain conditions and offering new computational insights for high-dimensional data analysis.

Contribution

It demonstrates the equivalence of Koopman mode decomposition and tensor component analysis under specific data conditions, and proposes error bounds when conditions are not met.

Findings

Theoretical equivalence between the two methods under certain conditions.

Koopman mode decomposition can serve as an alternative when tensor methods are non-convex.

Provides a new perspective on analyzing dynamical systems with tensor and Koopman techniques.

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.