70 research outputs found
Learning by a nerual net in a noisy environment - The pseudo-inverse solution revisited
A recurrent neural net is described that learns a set of patterns in the
presence of noise. The learning rule is of Hebbian type, and, if noise would be
absent during the learning process, the resulting final values of the weights
would correspond to the pseudo-inverse solution of the fixed point equation in
question. For a non-vanishing noise parameter, an explicit expression for the
expectation value of the weights is obtained. This result turns out to be
unequal to the pseudo-inverse solution. Furthermore, the stability properties
of the system are discussed.Comment: 16 pages, 3 figure
Combining Hebbian and reinforcement learning in a minibrain model
A toy model of a neural network in which both Hebbian learning and
reinforcement learning occur is studied. The problem of `path interference',
which makes that the neural net quickly forgets previously learned input-output
relations is tackled by adding a Hebbian term (proportional to the learning
rate ) to the reinforcement term (proportional to ) in the learning
rule. It is shown that the number of learning steps is reduced considerably if
, i.e., if the Hebbian term is neither too small nor too
large compared to the reinforcement term
Slowly evolving random graphs II: Adaptive geometry in finite-connectivity Hopfield models
We present an analytically solvable random graph model in which the
connections between the nodes can evolve in time, adiabatically slowly compared
to the dynamics of the nodes. We apply the formalism to finite connectivity
attractor neural network (Hopfield) models and we show that due to the
minimisation of the frustration effects the retrieval region of the phase
diagram can be significantly enlarged. Moreover, the fraction of misaligned
spins is reduced by this effect, and is smaller than in the infinite
connectivity regime. The main cause of this difference is found to be the
non-zero fraction of sites with vanishing local field when the connectivity is
finite.Comment: 17 pages, 8 figure
On Cavity Approximations for Graphical Models
We reformulate the Cavity Approximation (CA), a class of algorithms recently
introduced for improving the Bethe approximation estimates of marginals in
graphical models. In our new formulation, which allows for the treatment of
multivalued variables, a further generalization to factor graphs with arbitrary
order of interaction factors is explicitly carried out, and a message passing
algorithm that implements the first order correction to the Bethe approximation
is described. Furthermore we investigate an implementation of the CA for
pairwise interactions. In all cases considered we could confirm that CA[k] with
increasing provides a sequence of approximations of markedly increasing
precision. Furthermore in some cases we could also confirm the general
expectation that the approximation of order , whose computational complexity
is has an error that scales as with the size of the
system. We discuss the relation between this approach and some recent
developments in the field.Comment: Extension to factor graphs and comments on related work adde
Effective field theory for models defined over small-world networks. First and second order phase transitions
We present an effective field theory method to analyze, in a very general
way, models defined over small-world networks. Even if the exactness of the
method is limited to the paramagnetic regions and to some special limits, it
provides, yielding a clear and immediate (also in terms of calculation)
physical insight, the exact critical behavior and the exact critical surfaces
and percolation thresholds. The underlying structure of the non random part of
the model, i.e., the set of spins filling up a given lattice L_0 of dimension
d_0 and interacting through a fixed coupling J_0, is exactly taken into
account. When J_0\geq 0, the small-world effect gives rise, as is known, to a
second-order phase transition that takes place independently of the dimension
d_0 and of the added random connectivity c. When J_0<0, a different and novel
scenario emerges in which, besides a spin glass transition, multiple first- and
second-order phase transitions may take place. As immediate analytical
applications we analyze the Viana-Bray model (d_0=0), the one dimensional chain
(d_0=1), and the spherical model for arbitrary d_0.Comment: 28 pages, 18 figures; merged version of the manuscripts
arXiv:0801.3454 and arXiv:0801.3563 conform to the published versio
Replica symmetry breaking in the `small world' spin glass
We apply the cavity method to a spin glass model on a `small world' lattice,
a random bond graph super-imposed upon a 1-dimensional ferromagnetic ring. We
show the correspondence with a replicated transfer matrix approach, up to the
level of one step replica symmetry breaking (1RSB). Using the scheme developed
by M\'ezard & Parisi for the Bethe lattice, we evaluate observables for a model
with fixed connectivity and long range bonds. Our results agree with
numerical simulations significantly better than the replica symmetric (RS)
theory.Comment: 21 pages, 3 figure
Spin models on random graphs with controlled topologies beyond degree constraints
We study Ising spin models on finitely connected random interaction graphs
which are drawn from an ensemble in which not only the degree distribution
can be chosen arbitrarily, but which allows for further fine-tuning of
the topology via preferential attachment of edges on the basis of an arbitrary
function Q(k,k') of the degrees of the vertices involved. We solve these models
using finite connectivity equilibrium replica theory, within the replica
symmetric ansatz. In our ensemble of graphs, phase diagrams of the spin system
are found to depend no longer only on the chosen degree distribution, but also
on the choice made for Q(k,k'). The increased ability to control interaction
topology in solvable models beyond prescribing only the degree distribution of
the interaction graph enables a more accurate modeling of real-world
interacting particle systems by spin systems on suitably defined random graphs.Comment: 21 pages, 4 figures, submitted to J Phys
The Little-Hopfield model on a Random Graph
We study the Hopfield model on a random graph in scaling regimes where the
average number of connections per neuron is a finite number and where the spin
dynamics is governed by a synchronous execution of the microscopic update rule
(Little-Hopfield model).We solve this model within replica symmetry and by
using bifurcation analysis we prove that the spin-glass/paramagnetic and the
retrieval/paramagnetictransition lines of our phase diagram are identical to
those of sequential dynamics.The first-order retrieval/spin-glass transition
line follows by direct evaluation of our observables using population dynamics.
Within the accuracy of numerical precision and for sufficiently small values of
the connectivity parameter we find that this line coincides with the
corresponding sequential one. Comparison with simulation experiments shows
excellent agreement.Comment: 14 pages, 4 figure
Replicated Transfer Matrix Analysis of Ising Spin Models on `Small World' Lattices
We calculate equilibrium solutions for Ising spin models on `small world'
lattices, which are constructed by super-imposing random and sparse Poissonian
graphs with finite average connectivity c onto a one-dimensional ring. The
nearest neighbour bonds along the ring are ferromagnetic, whereas those
corresponding to the Poisonnian graph are allowed to be random. Our models thus
generally contain quenched connectivity and bond disorder. Within the replica
formalism, calculating the disorder-averaged free energy requires the
diagonalization of replicated transfer matrices. In addition to developing the
general replica symmetric theory, we derive phase diagrams and calculate
effective field distributions for two specific cases: that of uniform sparse
long-range bonds (i.e. `small world' magnets), and that of (+J/-J) random
sparse long-range bonds (i.e. `small world' spin-glasses).Comment: 22 pages, LaTeX, IOP macros, eps figure
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