1,293 research outputs found
Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules
This paper is concerned with the numerical approximation of Fredholm integral equa-
tions of the second kind. A Nyström method based on the anti-Gauss quadrature
formula is developed and investigated in terms of stability and convergence in appro-
priate weighted spaces. The Nyström interpolants corresponding to the Gauss and
the anti-Gauss quadrature rules are proved to furnish upper and lower bounds for the
solution of the equation, under suitable assumptions which are easily verified for a
particular weight function. Hence, an error estimate is available, and the accuracy of
the solution can be improved by approximating it by an averaged Nyström interpolant.
The effectiveness of the proposed approach is illustrated through different numerical
tests
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
Distributed multi-agent Gaussian regression via finite-dimensional approximations
We consider the problem of distributedly estimating Gaussian processes in
multi-agent frameworks. Each agent collects few measurements and aims to
collaboratively reconstruct a common estimate based on all data. Agents are
assumed with limited computational and communication capabilities and to gather
noisy measurements in total on input locations independently drawn from a
known common probability density. The optimal solution would require agents to
exchange all the input locations and measurements and then invert an matrix, a non-scalable task. Differently, we propose two suboptimal
approaches using the first orthonormal eigenfunctions obtained from the
\ac{KL} expansion of the chosen kernel, where typically . The benefits
are that the computation and communication complexities scale with and not
with , and computing the required statistics can be performed via standard
average consensus algorithms. We obtain probabilistic non-asymptotic bounds
that determine a priori the desired level of estimation accuracy, and new
distributed strategies relying on Stein's unbiased risk estimate (SURE)
paradigms for tuning the regularization parameters and applicable to generic
basis functions (thus not necessarily kernel eigenfunctions) and that can again
be implemented via average consensus. The proposed estimators and bounds are
finally tested on both synthetic and real field data
Learning with SGD and Random Features
Sketching and stochastic gradient methods are arguably the most common
techniques to derive efficient large scale learning algorithms. In this paper,
we investigate their application in the context of nonparametric statistical
learning. More precisely, we study the estimator defined by stochastic gradient
with mini batches and random features. The latter can be seen as form of
nonlinear sketching and used to define approximate kernel methods. The
considered estimator is not explicitly penalized/constrained and regularization
is implicit. Indeed, our study highlights how different parameters, such as
number of features, iterations, step-size and mini-batch size control the
learning properties of the solutions. We do this by deriving optimal finite
sample bounds, under standard assumptions. The obtained results are
corroborated and illustrated by numerical experiments
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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