Robust and scalable methods for the dynamic mode decomposition

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.