Robust and scalable methods for the dynamic mode decomposition

TL;DR

This paper introduces robust and scalable algorithms for dynamic mode decomposition (DMD) that effectively handle outliers and noise, improving the accuracy and applicability of DMD in real-world data analysis.

Contribution

The authors develop a flexible, scalable framework for robust DMD that incorporates regularizers and constraints within nonlinear optimization, addressing limitations of standard methods.

Findings

Algorithms effectively handle outliers and noise.

Scalable stochastic approach reduces computational costs.

Validated on synthetic and real datasets.

Abstract

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.