  The dynamic mode decomposition (DMD) is a broadly applicable dimensionality
reduction algorithm that approximates a matrix containing time-series data by
the outer product of a matrix of exponentials, representing Fourier-like time
dynamics, and a matrix of coefficients, representing spatial structures. This
interpretable spatio-temporal decomposition is commonly computed using linear
algebraic techniques in its simplest formulation or a nonlinear optimization
procedure within the variable projection framework. For data with sparse
outliers or data which are not well-represented by exponentials in time, the
standard Frobenius norm fit of the data creates significant biases in the
recovered time dynamics. As a result, practitioners are left to clean such
defects from the data manually or to use a black-box cleaning approach like
robust PCA. As an alternative, we propose a framework and a set of algorithms
for incorporating robust features into the nonlinear optimization used to
compute the DMD itself. The algorithms presented are flexible, allowing for
regu- larizers and constraints on the optimization, and scalable, using a
stochastic approach to decrease the computational cost for data in high
dimensional space. Both synthetic and real data examples are provided.
