7,469 research outputs found
A general class of explicit pseudo two-step RKN methods on parallel computers
AbstractThe aim of this paper is to investigate a general class of explicit pseudo two-step Runge-Kutta-Nyström methods (RKN methods) of arbitrarily high order for nonstiff problems for systems of special second-order differential equations y″(t) = f(y(t)). Order and stability considerations show that we can obtain for any given p, a stable pth-order explicit pseudo two-step RKN method requiring p − 2 right-hand side evaluations per step of which each evaluation can be obtained in parallel. Consequently, on a multiprocessor computer, only one sequential right-hand side evaluation per step is required. By a few widely-used test problems, we show the superiority of the methods considered in this paper over both sequential and parallel methods available in the literature
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
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
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
GraFIX: a semiautomatic approach for parsing low- and high-quality eye-tracking data
Fixation durations (FD) have been used widely as a measurement of information processing and attention. However, issues like data quality can seriously influence the accuracy of the fixation detection methods and, thus, affect the validity of our results (Holmqvist, Nyström, & Mulvey, 2012). This is crucial when studying special populations such as infants, where common issues with testing (e.g., high degree of movement, unreliable eye detection, low spatial precision) result in highly variable data quality and render existing FD detection approaches highly time consuming (hand-coding) or imprecise (automatic detection). To address this problem, we present GraFIX, a novel semiautomatic method consisting of a two-step process in which eye-tracking data is initially parsed by using velocity-based algorithms whose input parameters are adapted by the user and then manipulated using the graphical interface, allowing accurate and rapid adjustments of the algorithms’ outcome. The present algorithms (1) smooth the raw data, (2) interpolate missing data points, and (3) apply a number of criteria to automatically evaluate and remove artifactual fixations. The input parameters (e.g., velocity threshold, interpolation latency) can be easily manually adapted to fit each participant. Furthermore, the present application includes visualization tools that facilitate the manual coding of fixations. We assessed this method by performing an intercoder reliability analysis in two groups of infants presenting low- and high-quality data and compared it with previous methods. Results revealed that our two-step approach with adaptable FD detection criteria gives rise to more reliable and stable measures in low- and high-quality data
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