1,823 research outputs found
Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem
We study the approximation of the smallest eigenvalue of a Sturm-Liouville
problem in the classical and quantum settings. We consider a univariate
Sturm-Liouville eigenvalue problem with a nonnegative function from the
class and study the minimal number n(\e) of function evaluations
or queries that are necessary to compute an \e-approximation of the smallest
eigenvalue. We prove that n(\e)=\Theta(\e^{-1/2}) in the (deterministic)
worst case setting, and n(\e)=\Theta(\e^{-2/5}) in the randomized setting.
The quantum setting offers a polynomial speedup with {\it bit} queries and an
exponential speedup with {\it power} queries. Bit queries are similar to the
oracle calls used in Grover's algorithm appropriately extended to real valued
functions. Power queries are used for a number of problems including phase
estimation. They are obtained by considering the propagator of the discretized
system at a number of different time moments. They allow us to use powers of
the unitary matrix , where is an
matrix obtained from the standard discretization of the Sturm-Liouville
differential operator. The quantum implementation of power queries by a number
of elementary quantum gates that is polylog in is an open issue.Comment: 33 page
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
Discriminative Features via Generalized Eigenvectors
Representing examples in a way that is compatible with the underlying
classifier can greatly enhance the performance of a learning system. In this
paper we investigate scalable techniques for inducing discriminative features
by taking advantage of simple second order structure in the data. We focus on
multiclass classification and show that features extracted from the generalized
eigenvectors of the class conditional second moments lead to classifiers with
excellent empirical performance. Moreover, these features have attractive
theoretical properties, such as inducing representations that are invariant to
linear transformations of the input. We evaluate classifiers built from these
features on three different tasks, obtaining state of the art results
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Estimating a largest eigenvector by polynomial algorithms with a random start
In 7 and 8
the power and Lanczos algorithms with random start for estimating the largest eigenvalue of an n x n large symmetric positive definite matrix were analyzed.
In this paper we continue this study by estimating an eigenvector corresponding to the largest eigenvalue.
We analyze polynomial algorithms using Krylov information for two error criteria the randomized error and the randomized residual error
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