45 research outputs found

### A family of Schr\"odinger operators whose spectrum is an interval

By approximation, I show that the spectrum of the Schr\"odinger operator with
potential $V(n) = f(n\rho \pmod 1)$ for f continuous and $\rho > 0$, $\rho
\notin \N$ is an interval.Comment: Comm. Math. Phys. (to appear

### Retrieval Properties of Hopfield and Correlated Attractors in an Associative Memory Model

We examine a previouly introduced attractor neural network model that
explains the persistent activities of neurons in the anterior ventral temporal
cortex of the brain. In this model, the coexistence of several attractors
including correlated attractors was reported in the cases of finite and
infinite loading. In this paper, by means of a statistical mechanical method,
we study the statics and dynamics of the model in both finite and extensive
loading, mainly focusing on the retrieval properties of the Hopfield and
correlated attractors. In the extensive loading case, we derive the evolution
equations by the dynamical replica theory. We found several characteristic
temporal behaviours, both in the finite and extensive loading cases. The
theoretical results were confirmed by numerical simulations.Comment: 12 pages, 7 figure

### Storage capacity of correlated perceptrons

We consider an ensemble of $K$ single-layer perceptrons exposed to random
inputs and investigate the conditions under which the couplings of these
perceptrons can be chosen such that prescribed correlations between the outputs
occur. A general formalism is introduced using a multi-perceptron costfunction
that allows to determine the maximal number of random inputs as a function of
the desired values of the correlations. Replica-symmetric results for $K=2$ and
$K=3$ are compared with properties of two-layer networks of tree-structure and
fixed Boolean function between hidden units and output. The results show which
correlations in the hidden layer of multi-layer neural networks are crucial for
the value of the storage capacity.Comment: 16 pages, Latex2

### Realistic model of correlated disorder and Anderson localization

A conducting 1D line or 2D plane inside (or on the surface of) an insulator
is considered.Impurities displace the charges inside the insulator. This
results in a long-range fluctuating electric field acting on the conducting
line (plane). This field can be modeled by that of randomly distributed
electric dipoles. This model provides a random correlated potential with
decaying as 1/k . In the 1D case such correlations give essential
corrections to the localization length but do not destroy Anderson
localization

### Bi-stability of mixed states in neural network storing hierarchical patterns

We discuss the properties of equilibrium states in an autoassociative memory
model storing hierarchically correlated patterns (hereafter, hierarchical
patterns). We will show that symmetric mixed states (hereafter, mixed states)
are bi-stable on the associative memory model storing the hierarchical patterns
in a region of the ferromagnetic phase. This means that the first-order
transition occurs in this ferromagnetic phase. We treat these contents with a
statistical mechanical method (SCSNA) and by computer simulation. Finally, we
discuss a physiological implication of this model. Sugase et al. analyzed the
time-course of the information carried by the firing of face-responsive neurons
in the inferior temporal cortex. We also discuss the relation between the
theoretical results and the physiological experiments of Sugase et al.Comment: 18 pages, 6 figure

### Gradient descent learning in and out of equilibrium

Relations between the off thermal equilibrium dynamical process of on-line
learning and the thermally equilibrated off-line learning are studied for
potential gradient descent learning. The approach of Opper to study on-line
Bayesian algorithms is extended to potential based or maximum likelihood
learning. We look at the on-line learning algorithm that best approximates the
off-line algorithm in the sense of least Kullback-Leibler information loss. It
works by updating the weights along the gradient of an effective potential
different from the parent off-line potential. The interpretation of this off
equilibrium dynamics holds some similarities to the cavity approach of
Griniasty. We are able to analyze networks with non-smooth transfer functions
and transfer the smoothness requirement to the potential.Comment: 08 pages, submitted to the Journal of Physics

### Correlations between hidden units in multilayer neural networks and replica symmetry breaking

We consider feed-forward neural networks with one hidden layer, tree
architecture and a fixed hidden-to-output Boolean function. Focusing on the
saturation limit of the storage problem the influence of replica symmetry
breaking on the distribution of local fields at the hidden units is
investigated. These field distributions determine the probability for finding a
specific activation pattern of the hidden units as well as the corresponding
correlation coefficients and therefore quantify the division of labor among the
hidden units. We find that although modifying the storage capacity and the
distribution of local fields markedly replica symmetry breaking has only a
minor effect on the correlation coefficients. Detailed numerical results are
provided for the PARITY, COMMITTEE and AND machines with K=3 hidden units and
nonoverlapping receptive fields.Comment: 9 pages, 3 figures, RevTex, accepted for publication in Phys. Rev.

### Parisi Phase in a Neuron

Pattern storage by a single neuron is revisited. Generalizing Parisi's
framework for spin glasses we obtain a variational free energy functional for
the neuron. The solution is demonstrated at high temperature and large relative
number of examples, where several phases are identified by thermodynamical
stability analysis, two of them exhibiting spontaneous full replica symmetry
breaking. We give analytically the curved segments of the order parameter
function and in representative cases compute the free energy, the storage
error, and the entropy.Comment: 4 pages in prl twocolumn format + 3 Postscript figures. Submitted to
Physical Review Letter

### Metal-insulator transition in one-dimensional lattices with chaotic energy sequences

We study electronic transport through a one-dimensional array of sites by
using a tight binding Hamiltonian, whose site-energies are drawn from a chaotic
sequence. The correlation degree between these energies is controlled by a
parameter regulating the dynamic Lyapunov exponent measuring the degree of
chaos. We observe the effect of chaotic sequences on the localization length,
conductance, conductance distribution and wave function, finding evidence of a
Metal-Insulator Transition (MIT) at a critical degree of chaos. The
one-dimensional metallic phase is characterized by a Gaussian conductance
distribution and exhibits a peculiar non-selfaveraging.Comment: 5 pages, 5 figures (one figure replaced). Includes new results and a
few additional references. Improved style for publication. Accepted in
Physics Letters

### A canonical ensemble approach to graded-response perceptrons

Perceptrons with graded input-output relations and a limited output precision
are studied within the Gardner-Derrida canonical ensemble approach. Soft non-
negative error measures are introduced allowing for extended retrieval
properties. In particular, the performance of these systems for a linear and
quadratic error measure, corresponding to the perceptron respectively the
adaline learning algorithm, is compared with the performance for a rigid error
measure, simply counting the number of errors. Replica-symmetry-breaking
effects are evaluated.Comment: 26 pages, 10 ps figure