3,854 research outputs found
On-Line Learning with Restricted Training Sets: An Exactly Solvable Case
We solve the dynamics of on-line Hebbian learning in large perceptrons
exactly, for the regime where the size of the training set scales linearly with
the number of inputs. We consider both noiseless and noisy teachers. Our
calculation cannot be extended to non-Hebbian rules, but the solution provides
a convenient and welcome benchmark with which to test more general and advanced
theories for solving the dynamics of learning with restricted training sets.Comment: 19 pages, eps figures included, uses epsfig macr
Dynamics of Learning with Restricted Training Sets I: General Theory
We study the dynamics of supervised learning in layered neural networks, in
the regime where the size of the training set is proportional to the number
of inputs. Here the local fields are no longer described by Gaussian
probability distributions and the learning dynamics is of a spin-glass nature,
with the composition of the training set playing the role of quenched disorder.
We show how dynamical replica theory can be used to predict the evolution of
macroscopic observables, including the two relevant performance measures
(training error and generalization error), incorporating the old formalism
developed for complete training sets in the limit as a
special case. For simplicity we restrict ourselves in this paper to
single-layer networks and realizable tasks.Comment: 39 pages, LaTe
Dynamics of on-line Hebbian learning with structurally unrealizable restricted training sets
We present an exact solution for the dynamics of on-line Hebbian learning in
neural networks, with restricted and unrealizable training sets. In contrast to
other studies on learning with restricted training sets, unrealizability is
here caused by structural mismatch, rather than data noise: the teacher machine
is a perceptron with a reversed wedge-type transfer function, while the student
machine is a perceptron with a sigmoidal transfer function. We calculate the
glassy dynamics of the macroscopic performance measures, training error and
generalization error, and the (non-Gaussian) student field distribution. Our
results, which find excellent confirmation in numerical simulations, provide a
new benchmark test for general formalisms with which to study unrealizable
learning processes with restricted training sets.Comment: 7 pages including 3 figures, using IOP latex2e preprint class fil
Discovery of a very X-ray luminous galaxy cluster at z=0.89 in the WARPS survey
We report the discovery of the galaxy cluster ClJ1226.9+3332 in the Wide
Angle ROSAT Pointed Survey (WARPS). At z=0.888 and L_X=1.1e45 erg/s (0.5-2.0
keV, h_0=0.5) ClJ1226.9+3332 is the most distant X-ray luminous cluster
currently known. The mere existence of this system represents a huge problem
for Omega_0=1 world models.
At the modest (off-axis) resolution of the ROSAT PSPC observation in which
the system was detected, ClJ1226.9+3332 appears relaxed; an off-axis HRI
observation confirms this impression and rules out significant contamination
from point sources. However, in moderately deep optical images (R and I band)
the cluster exhibits signs of substructure in its apparent galaxy distribution.
A first crude estimate of the velocity dispersion of the cluster galaxies based
on six redshifts yields a high value of 1650 km/s, indicative of a very massive
cluster and/or the presence of substructure along the line of sight. While a
more accurate assessment of the dynamical state of this system requires much
better data at both optical and X-ray wavelengths, the high mass of the cluster
has already been unambiguously confirmed by a very strong detection of the
Sunyaev-Zel'dovich effect in its direction (Joy et al. 2001).
Using ClJ1226.9+3332 and ClJ0152.7-1357 (z=0.835), the second-most distant
X-ray luminous cluster currently known and also a WARPS discovery, we obtain a
first estimate of the cluster X-ray luminosity function at 0.8<z<1.4 and
L_X>5e44 erg/s. Using the best currently available data, we find the comoving
space density of very distant, massive clusters to be in excellent agreement
with the value measured locally (z<0.3), and conclude that negative evolution
is not required at these luminosities out to z~1. (truncated)Comment: accepted for publication in ApJ Letters, 6 pages, 2 figures, uses
emulateapj.st
The WARPS survey - IV: The X-ray luminosity-temperature relation of high redshift galaxy clusters
We present a measurement of the cluster X-ray luminosity-temperature relation
out to high redshift (z~0.8). Combined ROSAT PSPC spectra of 91 galaxy clusters
detected in the Wide Angle ROSAT Pointed Survey (WARPS) are simultaneously fit
in redshift and luminosity bins. The resulting temperature and luminosity
measurements of these bins, which occupy a region of the high redshift L-T
relation not previously sampled, are compared to existing measurements at low
redshift in order to constrain the evolution of the L-T relation. We find a
best fit to low redshift (z1 keV, to be L proportional
to T^(3.15\pm0.06). Our data are consistent with no evolution in the
normalisation of the L-T relation up to z~0.8. Combining our results with ASCA
measurements taken from the literature, we find eta=0.19\pm0.38 (for Omega_0=1,
with 1 sigma errors) where L_Bol is proportional to (1 + z)^eta T^3.15, or
eta=0.60\pm0.38 for Omega_0=0.3. This lack of evolution is considered in terms
of the entropy-driven evolution of clusters. Further implications for
cosmological constraints are also discussed.Comment: 11 pages, 7 figures, accepted for publication in MNRA
Order-Parameter Flow in the SK Spin-Glass II: Inclusion of Microscopic Memory Effects
We develop further a recent dynamical replica theory to describe the dynamics
of the Sherrington-Kirkpatrick spin-glass in terms of closed evolution
equations for macroscopic order parameters. We show how microscopic memory
effects can be included in the formalism through the introduction of a dynamic
order parameter function: the joint spin-field distribution. The resulting
formalism describes very accurately the relaxation phenomena observed in
numerical simulations, including the typical overall slowing down of the flow
that was missed by the previous simple two-parameter theory. The advanced
dynamical replica theory is either exact or a very good approximation.Comment: same as original, but this one is TeXabl
Spin-glass model with partially annealed asymmetric bonds
We have considered the two-spin interaction spherical spin-glass model with
asymmetric bonds (coupling constants). Besides the usual interactions between
spins and bonds and between the spins and a thermostat with temperature
there is also an additional factor: the bonds are not assumed
random {\it a priori} but interact with some other thermostat at the
temperature . We show that when the bonds are frozen with respect to the
spins a first order phase transition to a spin-glass phase occurs, and the
temperature of this transition tends to zero if is large. Our analytical
results show that a spin-glass phase can exist in mean-field models with
nonrelaxational dynamics.Comment: 10 pages, late
Application of two-parameter dynamical replica theory to retrieval dynamics of associative memory with non-monotonic neurons
The two-parameter dynamical replica theory (2-DRT) is applied to investigate
retrieval properties of non-monotonic associative memory, a model which lacks
thermodynamic potential functions. 2-DRT reproduces dynamical properties of the
model quite well, including the capacity and basin of attraction.
Superretrieval state is also discussed in the framework of 2-DRT. The local
stability condition of the superretrieval state is given, which provides a
better estimate of the region in which superretrieval is observed
experimentally than the self-consistent signal-to-noise analysis (SCSNA) does.Comment: 16 pages, 19 postscript figure
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