On The Equivalence Of Dynamic Mode Decomposition And Complex Nonnegative Matrix Factorization

Dynamic mode decomposition (DMD) has been applied for analyzing various nonlinear dynamic systems. In this study, we clarify the relationship among the variants of DMD and nonnegative matrix factorization (NMF), which is also widely used for audio source separation and document clustering. We show that the decomposition form of DMD is a special case of complex NMF (CNMF), and both are equivalent when certain constraint conditions are added to CNMF. The equivalence of the objective functions of nonnegative DMD and CNMF under the aforementioned constraints is also proven. This theoretical analysis provides guidelines for an effective initial value setting of CNMF and for a new construction method that is positioned between CNMF and DMD.