Randomness as a computational strategy : on matrix and tensor decompositions
Date
20/11/2017Author
Supervisor
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Abstract
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.
Type
Thesis, PhD Doctor of Philosophy
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