82 research outputs found
Sufficient conditions for convergence of the Sum-Product Algorithm
We derive novel conditions that guarantee convergence of the Sum-Product
algorithm (also known as Loopy Belief Propagation or simply Belief Propagation)
to a unique fixed point, irrespective of the initial messages. The
computational complexity of the conditions is polynomial in the number of
variables. In contrast with previously existing conditions, our results are
directly applicable to arbitrary factor graphs (with discrete variables) and
are shown to be valid also in the case of factors containing zeros, under some
additional conditions. We compare our bounds with existing ones, numerically
and, if possible, analytically. For binary variables with pairwise
interactions, we derive sufficient conditions that take into account local
evidence (i.e., single variable factors) and the type of pair interactions
(attractive or repulsive). It is shown empirically that this bound outperforms
existing bounds.Comment: 15 pages, 5 figures. Major changes and new results in this revised
version. Submitted to IEEE Transactions on Information Theor
Learning quantum models from quantum or classical data
In this paper, we address the problem how to represent a classical data
distribution in a quantum system. The proposed method is to learn quantum
Hamiltonian that is such that its ground state approximates the given classical
distribution. We review previous work on the quantum Boltzmann machine (QBM)
and how it can be used to infer quantum Hamiltonians from quantum statistics.
We then show how the proposed quantum learning formalism can also be applied to
a purely classical data analysis. Representing the data as a rank one density
matrix introduces quantum statistics for classical data in addition to the
classical statistics. We show that quantum learning yields results that can be
significantly more accurate than the classical maximum likelihood approach,
both for unsupervised learning and for classification. The data density matrix
and the QBM solution show entanglement, quantified by the quantum mutual
information . The classical mutual information in the data ,
with maximal classical correlations obtained by choosing a suitable
orthogonal measurement basis. We suggest that the remaining mutual information
is obtained by non orthogonal measurements that may violate the Bell
inequality. The excess mutual information may potentially be used to
improve the performance of quantum implementations of machine learning or other
statistical methods.Comment: 28 pages, 7 figure
Modeling the structure and evolution of discussion cascades
We analyze the structure and evolution of discussion cascades in four popular
websites: Slashdot, Barrapunto, Meneame and Wikipedia. Despite the big
heterogeneities between these sites, a preferential attachment (PA) model with
bias to the root can capture the temporal evolution of the observed trees and
many of their statistical properties, namely, probability distributions of the
branching factors (degrees), subtree sizes and certain correlations. The
parameters of the model are learned efficiently using a novel maximum
likelihood estimation scheme for PA and provide a figurative interpretation
about the communication habits and the resulting discussion cascades on the
four different websites.Comment: 10 pages, 11 figure
Learning Price-Elasticity of Smart Consumers in Power Distribution Systems
Demand Response is an emerging technology which will transform the power grid
of tomorrow. It is revolutionary, not only because it will enable peak load
shaving and will add resources to manage large distribution systems, but mainly
because it will tap into an almost unexplored and extremely powerful pool of
resources comprised of many small individual consumers on distribution grids.
However, to utilize these resources effectively, the methods used to engage
these resources must yield accurate and reliable control. A diversity of
methods have been proposed to engage these new resources. As opposed to direct
load control, many methods rely on consumers and/or loads responding to
exogenous signals, typically in the form of energy pricing, originating from
the utility or system operator. Here, we propose an open loop
communication-lite method for estimating the price elasticity of many customers
comprising a distribution system. We utilize a sparse linear regression method
that relies on operator-controlled, inhomogeneous minor price variations, which
will be fair to all the consumers. Our numerical experiments show that reliable
estimation of individual and thus aggregated instantaneous elasticities is
possible. We describe the limits of the reliable reconstruction as functions of
the three key parameters of the system: (i) ratio of the number of
communication slots (time units) per number of engaged consumers; (ii) level of
sparsity (in consumer response); and (iii) signal-to-noise ratio.Comment: 6 pages, 5 figures, IEEE SmartGridComm 201
Linear PDEs and eigenvalue problems corresponding to ergodic stochastic optimization problems on compact manifolds
We consider long term average or `ergodic' optimal control poblems with a
special structure: Control is exerted in all directions and the control costs
are proportional to the square of the norm of the control field with respect to
the metric induced by the noise. The long term stochastic dynamics on the
manifold will be completely characterized by the long term density and
the long term current density . As such, control problems may be
reformulated as variational problems over and . We discuss several
optimization problems: the problem in which both and are varied
freely, the problem in which is fixed and the one in which is fixed.
These problems lead to different kinds of operator problems: linear PDEs in the
first two cases and a nonlinear PDE in the latter case. These results are
obtained through through variational principle using infinite dimensional
Lagrange multipliers. In the case where the initial dynamics are reversible we
obtain the result that the optimally controlled diffusion is also
symmetrizable. The particular case of constraining the dynamics to be
reversible of the optimally controlled process leads to a linear eigenvalue
problem for the square root of the density process
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