1,149 research outputs found
Monte Carlo methods in PageRank computation: When one iteration is sufficient
PageRank is one of the principle criteria according to which Google ranks Web pages. PageRank can be interpreted as a frequency of visiting a Web page by a random surfer and thus it reflects the popularity of a Web page. Google computes the PageRank using the power iteration method which requires about one week of intensive computations. In the present work we propose and analyze Monte Carlo type methods for the PageRank computation. There are several advantages of the probabilistic Monte Carlo methods over the deterministic power iteration method: Monte Carlo methods provide good estimation of the PageRank for relatively important pages already after one iteration; Monte Carlo methods have natural parallel implementation; and finally, Monte Carlo methods allow to perform continuous update of the PageRank as the structure of the Web changes
Cortical beta oscillations reflect the contextual gating of visual action feedback
In sensorimotor integration, the brain needs to decide how its predictions should accommodate novel evidence by 'gating' sensory data depending on the current context. Here, we examined the oscillatory correlates of this process by recording magnetoencephalography (MEG) data during a new task requiring action under intersensory conflict. We used virtual reality to decouple visual (virtual) and proprioceptive (real) hand postures during a task in which the phase of grasping movements tracked a target (in either modality). Thus, we rendered visual information either task-relevant or a (to-be-ignored) distractor. Under visuo-proprioceptive incongruence, occipital beta power decreased (relative to congruence) when vision was task-relevant but increased when it had to be ignored. Dynamic causal modelling (DCM) revealed that this interaction was best explained by diametrical, task-dependent changes in visual gain. These novel results suggest a crucial role for beta oscillations in the contextual gating (i.e., gain or precision control) of visual vs proprioceptive action feedback, depending on concurrent behavioral demands
Random polytopes obtained by matrices with heavy tailed entries
Let be an random matrix with independent entries and
such that in each row entries are i.i.d. Assume also that the entries are
symmetric, have unit variances, and satisfy a small ball probabilistic estimate
uniformly. We investigate properties of the corresponding random polytope
in (the absolute convex hull of rows of
). In particular, we show that where depends only on parameters in small ball inequality. This
extends results of \cite{LPRT} and recent results of \cite{KKR}. This inclusion
is equivalent to so-called -quotient property and plays an important
role in compressive sensing (see \cite{KKR} and references therein).Comment: Last version, to appear in Communications in Contemporary Mathematic
Random polytopes obtained by matrices with heavy tailed entries
Let be an random matrix with independent entries and
such that in each row entries are i.i.d. Assume also that the entries are
symmetric, have unit variances, and satisfy a small ball probabilistic estimate
uniformly. We investigate properties of the corresponding random polytope
in (the absolute convex hull of rows of
). In particular, we show that where depends only on parameters in small ball inequality. This
extends results of \cite{LPRT} and recent results of \cite{KKR}. This inclusion
is equivalent to so-called -quotient property and plays an important
role in compressive sensing (see \cite{KKR} and references therein).Comment: Last version, to appear in Communications in Contemporary Mathematic
Stability of two-dimensional spatial solitons in nonlocal nonlinear media
We discuss existence and stability of two-dimensional solitons in media with
spatially nonlocal nonlinear response. We show that such systems, which include
thermal nonlinearity and dipolar Bose Einstein condensates, may support a
variety of stationary localized structures - including rotating spatial
solitons. We also demonstrate that the stability of these structures critically
depends on the spatial profile of the nonlocal response function.Comment: 8 pages, 9 figure
Complex light: Dynamic phase transitions of a light beam in a nonlinear non-local disordered medium
The dynamics of several light filaments (spatial optical solitons)
propagating in an optically nonlinear and non-local random medium is
investigated using the paradigms of the physics of complexity. Cluster
formation is interpreted as a dynamic phase transition. A connection with the
random matrices approach for explaining the vibrational spectra of an ensemble
of solitons is pointed out. General arguments based on a Brownian dynamics
model are validated by the numerical simulation of a stochastic partial
differential equation system. The results are also relevant for Bose condensed
gases and plasma physics.Comment: 11 pages, 20 figures. Small revisions, added a referenc
Algorithmic procedures for Bayesian MEG/EEG source reconstruction in SPM
AbstractThe MEG/EEG inverse problem is ill-posed, giving different source reconstructions depending on the initial assumption sets. Parametric Empirical Bayes allows one to implement most popular MEG/EEG inversion schemes (Minimum Norm, LORETA, etc.) within the same generic Bayesian framework. It also provides a cost-function in terms of the variational Free energyâan approximation to the marginal likelihood or evidence of the solution. In this manuscript, we revisit the algorithm for MEG/EEG source reconstruction with a view to providing a didactic and practical guide. The aim is to promote and help standardise the development and consolidation of other schemes within the same framework. We describe the implementation in the Statistical Parametric Mapping (SPM) software package, carefully explaining each of its stages with the help of a simple simulated data example. We focus on the Multiple Sparse Priors (MSP) model, which we compare with the well-known Minimum Norm and LORETA models, using the negative variational Free energy for model comparison. The manuscript is accompanied by Matlab scripts to allow the reader to test and explore the underlying algorithm
PageRank in scale-free random graphs
We analyze the distribution of PageRank on a directed configuration model and
show that as the size of the graph grows to infinity it can be closely
approximated by the PageRank of the root node of an appropriately constructed
tree. This tree approximation is in turn related to the solution of a linear
stochastic fixed point equation that has been thoroughly studied in the recent
literature
Granger causality revisited
This technical paper offers a critical re-evaluation of (spectral) Granger causality measures in the analysis of biological timeseries. Using realistic (neural mass) models of coupled neuronal dynamics, we evaluate the robustness of parametric and nonparametric Granger causality. Starting from a broad class of generative (state-space) models of neuronal dynamics, we show how their Volterra kernels prescribe the second-order statistics of their response to random fluctuations; characterised in terms of cross-spectral density, cross-covariance, autoregressive coefficients and directed transfer functions. These quantities in turn specify Granger causality - providing a direct (analytic) link between the parameters of a generative model and the expected Granger causality. We use this link to show that Granger causality measures based upon autoregressive models can become unreliable when the underlying dynamics is dominated by slow (unstable) modes - as quantified by the principal Lyapunov exponent. However, nonparametric measures based on causal spectral factors are robust to dynamical instability. We then demonstrate how both parametric and nonparametric spectral causality measures can become unreliable in the presence of measurement noise. Finally, we show that this problem can be finessed by deriving spectral causality measures from Volterra kernels, estimated using dynamic causal modelling
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