2,458 research outputs found
Phenomenological Models of Socio-Economic Network Dynamics
We study a general set of models of social network evolution and dynamics.
The models consist of both a dynamics on the network and evolution of the
network. Links are formed preferentially between 'similar' nodes, where the
similarity is defined by the particular process taking place on the network.
The interplay between the two processes produces phase transitions and
hysteresis, as seen using numerical simulations for three specific processes.
We obtain analytic results using mean field approximations, and for a
particular case we derive an exact solution for the network. In common with
real-world social networks, we find coexistence of high and low connectivity
phases and history dependence.Comment: 11 pages, 8 figure
Half the entanglement in critical systems is distillable from a single specimen
We establish that the leading critical scaling of the single-copy
entanglement is exactly one half of the entropy of entanglement of a block in
critical infinite spin chains in a general setting, using methods of conformal
field theory. Conformal symmetry imposes that the single-copy entanglement for
critical many-body systems scales as E_1(\rho_L)=(c/6) \log L- (c/6)
(\pi^2/\log L) + O(1/L), where L is the number of constituents in a block of an
infinite chain and c corresponds to the central charge. This proves that from a
single specimen of a critical chain, already half the entanglement can be
distilled compared to the rate that is asymptotically available. The result is
substantiated by a quantitative analysis for all translationally invariant
quantum spin chains corresponding to general isotropic quasi-free fermionic
models. An analytic example of the XY model shows that away from criticality
the above simple relation is only maintained near the quantum phase transition
point.Comment: 4 pages RevTeX, 1 figure, final versio
Choose your path wisely: gradient descent in a Bregman distance framework
We propose an extension of a special form of gradient descent --- in the
literature known as linearised Bregman iteration -- to a larger class of
non-convex functions. We replace the classical (squared) two norm metric in the
gradient descent setting with a generalised Bregman distance, based on a
proper, convex and lower semi-continuous function. The algorithm's global
convergence is proven for functions that satisfy the Kurdyka-\L ojasiewicz
property. Examples illustrate that features of different scale are being
introduced throughout the iteration, transitioning from coarse to fine. This
coarse-to-fine approach with respect to scale allows to recover solutions of
non-convex optimisation problems that are superior to those obtained with
conventional gradient descent, or even projected and proximal gradient descent.
The effectiveness of the linearised Bregman iteration in combination with early
stopping is illustrated for the applications of parallel magnetic resonance
imaging, blind deconvolution as well as image classification with neural
networks
Gradient descent in a generalised Bregman distance framework
We discuss a special form of gradient descent that in the literature has become known as the so-called linearised Bregman iteration. The idea is to replace the classical (squared) two norm metric in the gradient descent setting with a generalised Bregman distance, based on a more general proper, convex and lower semi-continuous functional. Gradient descent as well as the entropic mirror descent by Nemirovsky and Yudin are special cases, as is a specific form of non-linear Landweber iteration introduced by Bachmayr and Burger. We are going to analyse the linearised Bregman iteration in a setting where the functional we want to minimise is neither necessarily Lipschitz-continuous (in the classical sense) nor necessarily convex, and establish a global convergence result under the additional assumption that the functional we wish to minimise satisfies the so-called Kurdyka-Łojasiewicz property
Transparent boundary conditions for the nonlocal nonlinear Schroedinger equation: A model for reflectionless propagation of PT-symmetric solitons
We consider the problem of reflectionless propagation of PT-symmetric
solitons described by the nonlocal nonlinear Schroedinger equation on a line in
the framework of the concept of transparent boundary conditions for evolution
equations. Transparent boundary conditions for the nonlocal nonlinear
Schroedinger equation are derived. The absence of backscattering at the
artificial boundaries is confirmed by the numerical implementation of the
transparent boundary conditions
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