982 research outputs found
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
Fixed-Rank Approximation of a Positive-Semidefinite Matrix from Streaming Data
Several important applications, such as streaming PCA and semidefinite
programming, involve a large-scale positive-semidefinite (psd) matrix that is
presented as a sequence of linear updates. Because of storage limitations, it
may only be possible to retain a sketch of the psd matrix. This paper develops
a new algorithm for fixed-rank psd approximation from a sketch. The approach
combines the Nystrom approximation with a novel mechanism for rank truncation.
Theoretical analysis establishes that the proposed method can achieve any
prescribed relative error in the Schatten 1-norm and that it exploits the
spectral decay of the input matrix. Computer experiments show that the proposed
method dominates alternative techniques for fixed-rank psd matrix approximation
across a wide range of examples
NFFT meets Krylov methods: Fast matrix-vector products for the graph Laplacian of fully connected networks
The graph Laplacian is a standard tool in data science, machine learning, and
image processing. The corresponding matrix inherits the complex structure of
the underlying network and is in certain applications densely populated. This
makes computations, in particular matrix-vector products, with the graph
Laplacian a hard task. A typical application is the computation of a number of
its eigenvalues and eigenvectors. Standard methods become infeasible as the
number of nodes in the graph is too large. We propose the use of the fast
summation based on the nonequispaced fast Fourier transform (NFFT) to perform
the dense matrix-vector product with the graph Laplacian fast without ever
forming the whole matrix. The enormous flexibility of the NFFT algorithm allows
us to embed the accelerated multiplication into Lanczos-based eigenvalues
routines or iterative linear system solvers and even consider other than the
standard Gaussian kernels. We illustrate the feasibility of our approach on a
number of test problems from image segmentation to semi-supervised learning
based on graph-based PDEs. In particular, we compare our approach with the
Nystr\"om method. Moreover, we present and test an enhanced, hybrid version of
the Nystr\"om method, which internally uses the NFFT.Comment: 28 pages, 9 figure
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