23,527 research outputs found
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
Revisiting the Nystrom Method for Improved Large-Scale Machine Learning
We reconsider randomized algorithms for the low-rank approximation of
symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel
matrices that arise in data analysis and machine learning applications. Our
main results consist of an empirical evaluation of the performance quality and
running time of sampling and projection methods on a diverse suite of SPSD
matrices. Our results highlight complementary aspects of sampling versus
projection methods; they characterize the effects of common data preprocessing
steps on the performance of these algorithms; and they point to important
differences between uniform sampling and nonuniform sampling methods based on
leverage scores. In addition, our empirical results illustrate that existing
theory is so weak that it does not provide even a qualitative guide to
practice. Thus, we complement our empirical results with a suite of worst-case
theoretical bounds for both random sampling and random projection methods.
These bounds are qualitatively superior to existing bounds---e.g. improved
additive-error bounds for spectral and Frobenius norm error and relative-error
bounds for trace norm error---and they point to future directions to make these
algorithms useful in even larger-scale machine learning applications.Comment: 60 pages, 15 color figures; updated proof of Frobenius norm bounds,
added comparison to projection-based low-rank approximations, and an analysis
of the power method applied to SPSD sketche
Approximate Matrix Multiplication with Application to Linear Embeddings
In this paper, we study the problem of approximately computing the product of
two real matrices. In particular, we analyze a dimensionality-reduction-based
approximation algorithm due to Sarlos [1], introducing the notion of nuclear
rank as the ratio of the nuclear norm over the spectral norm. The presented
bound has improved dependence with respect to the approximation error (as
compared to previous approaches), whereas the subspace -- on which we project
the input matrices -- has dimensions proportional to the maximum of their
nuclear rank and it is independent of the input dimensions. In addition, we
provide an application of this result to linear low-dimensional embeddings.
Namely, we show that any Euclidean point-set with bounded nuclear rank is
amenable to projection onto number of dimensions that is independent of the
input dimensionality, while achieving additive error guarantees.Comment: 8 pages, International Symposium on Information Theor
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