Mode Decomposition for Homogeneous Symmetric Operators
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Finding latent structures in data is drawing increasing attention in diverse fields such as fluid dynamics, signal processing, and machine learning. Dimensionality reduction facilitates the revelation of such structures. For dynamical systems of linear and nonlinear flows, a prominent dimensionality reduction method is DMD, based on the theory of Koopman operators. In this work, we adapt DMD to homogeneous flows and show it can approximate well nonlinear spectral image decomposition techniques. We examine dynamics based on symmetric γ-homogeneous operators, 0 < γ < 1. These systems have a polynomial decay profile and reach steady state in finite time. DMD, on the other hand, can be viewed as an exponential data fitting algorithm. This yields an inherent conflict, causing large approximation errors (and non-existence of solutions in some particular cases). The contribution of this work is threefold. First, we suggest a rescaling of the time variable that solves the conflict between DMD and homogeneous flows. This adaptation of DMD can be performed when the homogeneity and the time step size are known. Second, we suggest the blind homogeneity normalization for time rescaling when neither the homogeneity nor the step size are known. Third, we formulate a new dynamic mode decomposition that constrains the matrix of the dynamics to be symmetric, termed SDMD. With these adaptations, we provide a closed form solution of DMD for dynamics u_t = P(u), u(t=0)=u_0, where P is a nonlinear γ-homogeneous operator, when u_0 admits P(u_0)=λ u_0. Then, we prove the validity of the blind homogeneity normalization. In addition, we show SDMD achieves lower mean square error for the spectrum estimation. Finally, we turn to formulating a discrete nonlinear spectral decomposition, based on SDMD and related to nonlinear eigenfunctions of γ-homogeneous operators.
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