On Koopman Mode Decomposition and Tensor Component Analysis

Abstract

Koopman mode decomposition and tensor component analysis are two tools that decompose high dimensional data sets into low dimensional modes that capture relevant features and/or dynamics. Despite their similar goal, the two methods are largely used by distinct scientific communities. For the first time, 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 same as that given by Koopman mode decomposition. This realization provides possibilities for new algorithmic approaches to both Koopman mode decomposition and tensor component analysis, new insight into what is being captured by the tensor component analysis modes, and a "bridge" with which the two communities can more effectively communicate.Koopman operator theory (KOT) has emerged as a powerful and general framework with which to understand nonlinear dynamical systems in a data-driven manner. Despite its many strengths, there are numerous scientific fields, such as neuroscience and biology, where it remains under employed. As part of this stems from KOT being formulated mathematically in a distinct way from existing, commonly used methods, such as principal component analysis (PCA), there is a need for bridging KOT with such methods. Here, for the first time, we investigate how a major KOT analysis method differs from tensor component analysis (TCA), an extension of PCA that has quickly become adopted by some of those same fields hesitant on KOT. We show that, in certain scenarios, TCA and KOT, in theory, give the same decomposition of data. This not only makes strides in establishing a bridge between TCA and KOT, but also provides new insight into TCA and new possible algorithms for implementing KOT and TCA.