2,410 research outputs found
Uncovering Causality from Multivariate Hawkes Integrated Cumulants
We design a new nonparametric method that allows one to estimate the matrix
of integrated kernels of a multivariate Hawkes process. This matrix not only
encodes the mutual influences of each nodes of the process, but also
disentangles the causality relationships between them. Our approach is the
first that leads to an estimation of this matrix without any parametric
modeling and estimation of the kernels themselves. A consequence is that it can
give an estimation of causality relationships between nodes (or users), based
on their activity timestamps (on a social network for instance), without
knowing or estimating the shape of the activities lifetime. For that purpose,
we introduce a moment matching method that fits the third-order integrated
cumulants of the process. We show on numerical experiments that our approach is
indeed very robust to the shape of the kernels, and gives appealing results on
the MemeTracker database
Convolutional Dictionary Learning through Tensor Factorization
Tensor methods have emerged as a powerful paradigm for consistent learning of
many latent variable models such as topic models, independent component
analysis and dictionary learning. Model parameters are estimated via CP
decomposition of the observed higher order input moments. However, in many
domains, additional invariances such as shift invariances exist, enforced via
models such as convolutional dictionary learning. In this paper, we develop
novel tensor decomposition algorithms for parameter estimation of convolutional
models. Our algorithm is based on the popular alternating least squares method,
but with efficient projections onto the space of stacked circulant matrices.
Our method is embarrassingly parallel and consists of simple operations such as
fast Fourier transforms and matrix multiplications. Our algorithm converges to
the dictionary much faster and more accurately compared to the alternating
minimization over filters and activation maps
Moment transport equations for the primordial curvature perturbation
In a recent publication, we proposed that inflationary perturbation theory
can be reformulated in terms of a probability transport equation, whose moments
determine the correlation properties of the primordial curvature perturbation.
In this paper we generalize this formulation to an arbitrary number of fields.
We deduce ordinary differential equations for the evolution of the moments of
zeta on superhorizon scales, which can be used to obtain an evolution equation
for the dimensionless bispectrum, fNL. Our equations are covariant in field
space and allow identification of the source terms responsible for evolution of
fNL. In a model with M scalar fields, the number of numerical integrations
required to obtain solutions of these equations scales like O(M^3). The
performance of the moment transport algorithm means that numerical calculations
with M >> 1 fields are straightforward. We illustrate this performance with a
numerical calculation of fNL in Nflation models containing M ~ 10^2 fields,
finding agreement with existing analytic calculations. We comment briefly on
extensions of the method beyond the slow-roll approximation, or to calculate
higher order parameters such as gNL.Comment: 23 pages, plus appendices and references; 4 figures. v2: incorrect
statements regarding numerical delta N removed from Sec. 4.3. Minor
modifications elsewher
The Dynamics of a Genetic Algorithm for a Simple Learning Problem
A formalism for describing the dynamics of Genetic Algorithms (GAs) using
methods from statistical mechanics is applied to the problem of generalization
in a perceptron with binary weights. The dynamics are solved for the case where
a new batch of training patterns is presented to each population member each
generation, which considerably simplifies the calculation. The theory is shown
to agree closely to simulations of a real GA averaged over many runs,
accurately predicting the mean best solution found. For weak selection and
large problem size the difference equations describing the dynamics can be
expressed analytically and we find that the effects of noise due to the finite
size of each training batch can be removed by increasing the population size
appropriately. If this population resizing is used, one can deduce the most
computationally efficient size of training batch each generation. For
independent patterns this choice also gives the minimum total number of
training patterns used. Although using independent patterns is a very
inefficient use of training patterns in general, this work may also prove
useful for determining the optimum batch size in the case where patterns are
recycled.Comment: 28 pages, 4 Postscript figures. Latex using IOP macros ioplppt and
iopl12 which are included. To appear in Journal of Physics A. Also available
at ftp://ftp.cs.man.ac.uk/pub/ai/jls/GAlearn.ps.gz and
http://www.cs.man.ac.uk/~jl
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