Dynamic Mode Decomposition with Reproducing Kernels for Koopman Spectral Analysis
A spectral analysis of the Koopman operator, which is an infinite dimensional linear operator on an observable, gives a (modal) description of the global behavior of a nonlinear dynamical system without any explicit prior knowledge of its governing equations. In this paper, we consider a spectral analysis of the Koopman operator in a reproducing kernel Hilbert space (RKHS). We propose a modal decomposition algorithm to perform the analysis using finite-length data sequences generated from a nonlinear system. The algorithm is in essence reduced to the calculation of a set of orthogonal bases for the Krylov matrix in RKHS and the eigendecomposition of the projection of the Koopman operator onto the subspace spanned by the bases. The algorithm returns a decomposition of the dynamics into a finite number of modes, and thus it can be thought of as a feature extraction procedure for a nonlinear dynamical system. Therefore, we further consider applications in machine learning using extracted features with the presented analysis. We illustrate the method on the applications using synthetic and real-world data.
The manuscript, entitled “Koopman Spectral Analysis of Nonlinear Dynamical Sys- tems with Reproducing Kernels” extended the dynamical mode decomposition (DMD) to reproducing kernel Hilbert space (RKHS). The resulting algorithm (KDMD) is then applied to a few toy problems. It hence allows to use features map as definition of the kernel, in order to link more the DMD to specific features of interest; it is then a powerful generalization of DMD. My main concern is that the paper is really hard to read. Even for a few simple notions, the authors emphasize on the mathematical writings when short sentences would have been more than enough. Some sections, on the other hand, are actually verbose. It might make it difficult to understand even for specialists, and a few points should be clarified. It is too cryptic for publication in the Neural Information Processing Systems conference proceedings, but I strongly encourage the authors to rewrite it in a more pedagogic way; the resulting article should actually be highly cited. 
Referee’s report on: Koopman Spectral Analysis of Nonlinear Dynamical Systems with Reproducing Kernels   The manuscript, entitled “Koopman Spectral Analysis of Nonlinear Dynamical Sys- tems with Reproducing Kernels” extended the dynamical mode decomposition (DMD) to reproducing kernel Hilbert space (RKHS).  The resulting algorithm (KDMD) is then applied to a few toy problems. It hence allows to use features map as definition of the kernel, in order to link more the DMD to specific features of interest; it is then a powerful generalization of DMD. My main concern is that the paper is really hard to read. Even for a few simple notions, they emphasize on the mathematical writings when short sentences would have been more than enough. Some sections, on the other hand, are actually verbose. It might make it difficult to understand even for specialists, and a few points should be clarified. It is too cryptic for publication in the Neural Information Processing Systems conference proceedings, but I strongly encourage the authors to rewrite it in a more pedagogic way; the resulting article should actually be highly cited.  I. MAJOR REMARKS Please find my major remarks in the following: 1. The presentation is really verbose when it could be simplified. For instance: “Con- ceptually, DMD can be considered as producing an approximation of the Koopman eigenfunctions using a set of linear monomials of observables as basis functions, which is analogous to a one-term Taylor expansion at each point.” can be easily summarized in one short and simple sentence. 2. Sec. 4 is actually useless in a short paper and was (partially) treated in the introduc- tion. 3. Sec. 5 should have been illustrated, or be summarized in the conclusion. 4. Sec. 6 should have been developed in the way of Sec. 5! It has been condensed too much. II. MINOR REMARKS My other minor remarks are: 1. there is probably a typo in the description of the toy NL case, as it is actually linear. 2. first two problems are actually polynomial. It is were the KDMD should shine - when the kernel can be guessed, as in 1 . More details should be given, as actually in Sec. 5.  REFERENCES 1 Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor, J. Nathan Kutz, “Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control,” http://arxiv.org/abs/1510.03007v2, (2015)