## 1 Introduction  
Dynamic Mode Decomposition (DMD) is a concept that was first introduced by [PERSON] and [PERSON] to study the spatial dynamic modes of fluid flow [PERSON] and [PERSON] to (1992); [PERSON] and [PERSON] to study the spatial dynamic modes of fluid flow [PERSON] and [PERSON] to (1992). DMD approximates the nonlinear dynamics underlying a given time-varying dataset in terms of a linear auto-regressive model by extracting a set of mode shapes and their corresponding eigenvalues, where the mode shapes represent the spatial spread of dominant features and the eigenvalue associated with each mode shape specifies how that feature evolves over time in terms of the frequency of oscillation and the rate of growth or decay. [PERSON] et al. envisioned DMD as an approximation to the modes of Koopman operator, which is an infinite-dimensional linear representation of nonlinear finite-dimensional dynamics [PERSON] and [PERSON] (1992); [PERSON] and [PERSON] (1992). Even though DMD was initially meant to be used for extracting dynamic information from flow fields [PERSON] and [PERSON] (1992), soon it found new applications in other areas of study as a powerful tool for analyzing the dynamics of nonlinear systems. [PERSON] et al. [PERSON] et al. (2018) expanded the theory of DMD to handle mappingbetween paired datasets. [PERSON] et al. proposed the sparsity-promoting DMD (spDMD) to obtain a sparse representation of the system dynamics by limiting the number of dynamic modes through an \(l_{1}\)-regularization approach [6]. In 2015, the extended DMD (EDMD) was introduced by [PERSON] et al. to approximate the leading eigenvalues, eigenfunctions, and modes of the Koopman operator [7]. The EDMD is a computationally intensive algorithm since it requires the choice of a rich dictionary of basis functions to produce an approximation of the Koopman eigenfunctions. The richer the dictionary is, the more time it takes to compute the inner products which are a key part of EDMD algorithm. In an attempt to overcome this issue, [PERSON] et al. proposed the kernel-based DMD (KDMD) in 2015 [8]. In this approach, rather than choosing the dictionary of the basis functions explicitly, they are defined implicitly by the choice of a kernel function. The kernel function resolves the computational intensity issue of EDMD by finding the inner products of the basis functions without the need to having them defined explicitly. An initial attempt for incorporating compressed sensing in DMD was made by [PERSON] et al. [9], where a subset of an originally-large dataset was taken by non-uniform sampling and was used for finding the temporal coefficients (eigenvalues) through solving an optimization problem. Further, the corresponding modes were found by solving a set of linear equations which involved the fully-sampled dataset. This makes the proposed algorithm (known as NU-DMD) impractical in the case the fully-sampled dataset is not available. Another approach for incorporating compressed sensing in DMD (known as csDMD) was developed by [PERSON] et al. [10]. In csDMD, the DMD eigenvalues are obtained from a sub-sampled dataset (similar to NU-DMD), which has the advantage of reducing computation time, and then the full DMD mode shapes are reconstructed through using an \(l_{1}\)-minimization scheme based on a chosen set of basis vectors. In contrast to NU-DMD, csDMD does not need the fully-sampled dataset in order to recover the mode shapes. One of the initial attempts to deal with the issues involved in recovering a dataset from gappy data is presented in [11]. The proposed method relies on the presence of a set of empirical eigenfunctions, which represent an ensemble of similar datasets, and hence, the fully-sampled dataset is reconstructed based on these empirical eigenfunctions. In the case there is no such set available, they described a technique to build one from an ensemble of marred samples. In the case of marred samples, it is assumed there are several marred samples taken from each face, each one taken with a different mask.