52,635 research outputs found
Recursive numerical calculus of one-loop tensor integrals
A numerical approach to compute tensor integrals in one-loop calculations is
presented. The algorithm is based on a recursion relation which allows to
express high rank tensor integrals as a function of lower rank ones. At each
level of iteration only inverse square roots of Gram determinants appear. For
the phase-space regions where Gram determinants are so small that numerical
problems are expected, we give general prescriptions on how to construct
reliable approximations to the exact result without performing Taylor
expansions. Working in 4+epsilon dimensions does not require an analytic
separation of ultraviolet and infrared/collinear divergences, and, apart from
trivial integrals that we compute explicitly, no additional ones besides the
standard set of scalar one-loop integrals are needed.Comment: Typo corrected in formula 79. 22 pages, Latex, 1 figure, uses
axodraw.st
An approximate empirical Bayesian method for large-scale linear-Gaussian inverse problems
We study Bayesian inference methods for solving linear inverse problems,
focusing on hierarchical formulations where the prior or the likelihood
function depend on unspecified hyperparameters. In practice, these
hyperparameters are often determined via an empirical Bayesian method that
maximizes the marginal likelihood function, i.e., the probability density of
the data conditional on the hyperparameters. Evaluating the marginal
likelihood, however, is computationally challenging for large-scale problems.
In this work, we present a method to approximately evaluate marginal likelihood
functions, based on a low-rank approximation of the update from the prior
covariance to the posterior covariance. We show that this approximation is
optimal in a minimax sense. Moreover, we provide an efficient algorithm to
implement the proposed method, based on a combination of the randomized SVD and
a spectral approximation method to compute square roots of the prior covariance
matrix. Several numerical examples demonstrate good performance of the proposed
method
New Square-Root Factorization of Inverse Toeplitz Matrices
Abstract-Square-root (in particular, Cholesky) factorization of Toeplitz matrices and of their inverses is a classical area of research. The Schur algorithm yields directly the Cholesky factorization of a symmetric Toeplitz matrix, whereas the Levinson algorithm does the same for the inverse matrix. The objective of this letter is to use results from the theory of rational orthonormal functions to derive square-root factorizations of the inverse of an n × n positive definite Toeplitz matrix. The main result is a new factorization based on the Takenaka-Malmquist functions, that is parameterized by the roots of the corresponding auto-regressive polynomial of order n. We will also discuss briefly the connection between our analysis and some classical results such as Schur polynomials and the Gohberg-Semencul inversion formula
Root finding with threshold circuits
We show that for any constant d, complex roots of degree d univariate
rational (or Gaussian rational) polynomials---given by a list of coefficients
in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a
uniform family of constant-depth polynomial-size threshold circuits). The basic
idea is to compute the inverse function of the polynomial by a power series. We
also discuss an application to the theory VTC^0 of bounded arithmetic.Comment: 19 pages, 1 figur
On Taking Square Roots without Quadratic Nonresidues over Finite Fields
We present a novel idea to compute square roots over finite fields, without
being given any quadratic nonresidue, and without assuming any unproven
hypothesis. The algorithm is deterministic and the proof is elementary. In some
cases, the square root algorithm runs in bit operations
over finite fields with elements. As an application, we construct a
deterministic primality proving algorithm, which runs in
for some integers .Comment: 14 page
New Structured Matrix Methods for Real and Complex Polynomial Root-finding
We combine the known methods for univariate polynomial root-finding and for
computations in the Frobenius matrix algebra with our novel techniques to
advance numerical solution of a univariate polynomial equation, and in
particular numerical approximation of the real roots of a polynomial. Our
analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page
flatIGW - an inverse algorithm to compute the Density of States of lattice Self Avoiding Walks
We show that the Density of States (DoS) for lattice Self Avoiding Walks can
be estimated by using an inverse algorithm, called flatIGW, whose step-growth
rules are dynamically adjusted by requiring the energy histogram to be locally
flat. Here, the (attractive) energy associated with a configuration is taken to
be proportional to the number of non-bonded nearest neighbor pairs (contacts).
The energy histogram is able to explicitly direct the growth of a walk because
the step-growth rule of the Interacting Growth Walk \cite{IGW} samples the
available nearest neighbor sites according to the number of contacts they would
make. We have obtained the complex Fisher zeros corresponding to the DoS,
estimated for square lattice walks of various lengths, and located the
temperature by extrapolating the finite size values of the real zeros to their
asymptotic value, (reasonably close to the known value,
\cite{barkema}).Comment: 18 pages, 7 eps figures; parts of the manuscript are rewritten so as
to improve clarity of presentation; an extra reference adde
On the matrix square root via geometric optimization
This paper is triggered by the preprint "\emph{Computing Matrix Squareroot
via Non Convex Local Search}" by Jain et al.
(\textit{\textcolor{blue}{arXiv:1507.05854}}), which analyzes gradient-descent
for computing the square root of a positive definite matrix. Contrary to claims
of~\citet{jain2015}, our experiments reveal that Newton-like methods compute
matrix square roots rapidly and reliably, even for highly ill-conditioned
matrices and without requiring commutativity. We observe that gradient-descent
converges very slowly primarily due to tiny step-sizes and ill-conditioning. We
derive an alternative first-order method based on geodesic convexity: our
method admits a transparent convergence analysis ( page), attains linear
rate, and displays reliable convergence even for rank deficient problems.
Though superior to gradient-descent, ultimately our method is also outperformed
by a well-known scaled Newton method. Nevertheless, the primary value of our
work is its conceptual value: it shows that for deriving gradient based methods
for the matrix square root, \emph{the manifold geometric view of positive
definite matrices can be much more advantageous than the Euclidean view}.Comment: 8 pages, 12 plots, this version contains several more references and
more words about the rank-deficient cas
- …