Randomized Dynamic Mode Decomposition

This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of deterministic algorithms, easing the computational challenges arising in the area of `big data'. The idea is to derive a small matrix from the high-dimensional data, which is then used to efficiently compute the dynamic modes and eigenvalues. The algorithm is presented in a modular probabilistic framework, and the approximation quality can be controlled via oversampling and power iterations. The effectiveness of the resulting randomized DMD algorithm is demonstrated on several benchmark examples of increasing complexity, providing an accurate and efficient approach to extract spatiotemporal coherent structures from big data in a framework that scales with the intrinsic rank of the data, rather than the ambient measurement dimension. For this work we assume that the dynamics of the problem under consideration is evolving on a low-dimensional subspace that is well characterized by a fast decaying singular value spectrum

Randomness as a computational strategy : on matrix and tensor decompositions

Matrix and tensor decompositions are fundamental tools for finding structure and data processing. In particular, the efficient computation of low-rank matrix approximations is an ubiquitous problem in the area of machine learning and elsewhere. However, massive data arrays pose a computational challenge for these techniques, placing significant constraints on both memory and processing power. Recently, the fascinating and powerful concept of randomness has been introduced as a strategy to ease the computational load of deterministic matrix and data algorithms. The basic idea of these algorithms is to employ a degree of randomness as part of the logic in order to derive from a high-dimensional input matrix a smaller matrix, which captures the essential information of the original data matrix. Subsequently, the smaller matrix is then used to efficiently compute a near-optimal low-rank approximation. Randomized algorithms have been shown to be robust, highly reliable, and computationally efficient, yet simple to implement. In particular, the development of the randomized singular value decomposition can be seen as a milestone in the era of ‘big data’. Building up on the great success of this probabilistic strategy to compute low-rank matrix decompositions, this thesis introduces a set of new randomized algorithms. Specifically, we present a randomized algorithm to compute the dynamic mode decomposition, which is a modern dimension reduction technique designed to extract dynamic information from dynamical systems. Then, we advocate the randomized dynamic mode decomposition for background modeling of surveillance video feeds. Further, we show that randomized algorithms are embarrassingly parallel by design and that graphics processing units (GPUs) can be utilized to substantially accelerate the computations. Finally, the concept of randomized algorithms is generalized for tensors in order to compute the canonical CANDECOMP/PARAFAC (CP) decomposition