4 research outputs found
Storage capacity of a constructive learning algorithm
Upper and lower bounds for the typical storage capacity of a constructive
algorithm, the Tilinglike Learning Algorithm for the Parity Machine [M. Biehl
and M. Opper, Phys. Rev. A {\bf 44} 6888 (1991)], are determined in the
asymptotic limit of large training set sizes. The properties of a perceptron
with threshold, learning a training set of patterns having a biased
distribution of targets, needed as an intermediate step in the capacity
calculation, are determined analytically. The lower bound for the capacity,
determined with a cavity method, is proportional to the number of hidden units.
The upper bound, obtained with the hypothesis of replica symmetry, is close to
the one predicted by Mitchinson and Durbin [Biol. Cyber. {\bf 60} 345 (1989)].Comment: 13 pages, 1 figur
Shock Profiles for the Asymmetric Simple Exclusion Process in One Dimension
The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice
is a system of particles which jump at rates and (here ) to
adjacent empty sites on their right and left respectively. The system is
described on suitable macroscopic spatial and temporal scales by the inviscid
Burgers' equation; the latter has shock solutions with a discontinuous jump
from left density to right density , , which
travel with velocity . In the microscopic system we
may track the shock position by introducing a second class particle, which is
attracted to and travels with the shock. In this paper we obtain the time
invariant measure for this shock solution in the ASEP, as seen from such a
particle. The mean density at lattice site , measured from this particle,
approaches at an exponential rate as , with a
characteristic length which becomes independent of when
. For a special value of the
asymmetry, given by , the measure is
Bernoulli, with density on the left and on the right. In the
weakly asymmetric limit, , the microscopic width of the shock
diverges as . The stationary measure is then essentially a
superposition of Bernoulli measures, corresponding to a convolution of a
density profile described by the viscous Burgers equation with a well-defined
distribution for the location of the second class particle.Comment: 34 pages, LaTeX, 2 figures are included in the LaTeX file. Email:
[email protected], [email protected], [email protected]
SCHR\"Odinger Invariance and Strongly Anisotropic Critical Systems
The extension of strongly anisotropic or dynamical scaling to local scale
invariance is investigated. For the special case of an anisotropy or dynamical
exponent , the group of local scale transformation considered is
the Schr\"odinger group, which can be obtained as the non-relativistic limit of
the conformal group. The requirement of Schr\"odinger invariance determines the
two-point function in the bulk and reduces the three-point function to a
scaling form of a single variable. Scaling forms are also derived for the
two-point function close to a free surface which can be either space-like or
time-like. These results are reproduced in several exactly solvable statistical
systems, namely the kinetic Ising model with Glauber dynamics, lattice
diffusion, Lifshitz points in the spherical model and critical dynamics of the
spherical model with a non-conserved order parameter. For generic values of
, evidence from higher order Lifshitz points in the spherical model and
from directed percolation suggests a simple scaling form of the two-point
function.Comment: Latex 44 pp. (no figs), Oxford preprint OUTP-93-33S, Geneva preprint
UGVA-DPT 1993/09-83