281 research outputs found
Pattern-recalling processes in quantum Hopfield networks far from saturation
As a mathematical model of associative memories, the Hopfield model was now
well-established and a lot of studies to reveal the pattern-recalling process
have been done from various different approaches. As well-known, a single
neuron is itself an uncertain, noisy unit with a finite unnegligible error in
the input-output relation. To model the situation artificially, a kind of 'heat
bath' that surrounds neurons is introduced. The heat bath, which is a source of
noise, is specified by the 'temperature'. Several studies concerning the
pattern-recalling processes of the Hopfield model governed by the
Glauber-dynamics at finite temperature were already reported. However, we might
extend the 'thermal noise' to the quantum-mechanical variant. In this paper, in
terms of the stochastic process of quantum-mechanical Markov chain Monte Carlo
method (the quantum MCMC), we analytically derive macroscopically deterministic
equations of order parameters such as 'overlap' in a quantum-mechanical variant
of the Hopfield neural networks (let us call "quantum Hopfield model" or
"quantum Hopfield networks"). For the case in which non-extensive number of
patterns are embedded via asymmetric Hebbian connections, namely,
for the number of neuron ('far from saturation'), we evaluate
the recalling processes for one of the built-in patterns under the influence of
quantum-mechanical noise.Comment: 10 pages, 3 figures, using jpconf.cls, Proc. of Statphys-Kolkata VI
The path-integral analysis of an associative memory model storing an infinite number of finite limit cycles
It is shown that an exact solution of the transient dynamics of an
associative memory model storing an infinite number of limit cycles with l
finite steps by means of the path-integral analysis. Assuming the Maxwell
construction ansatz, we have succeeded in deriving the stationary state
equations of the order parameters from the macroscopic recursive equations with
respect to the finite-step sequence processing model which has retarded
self-interactions. We have also derived the stationary state equations by means
of the signal-to-noise analysis (SCSNA). The signal-to-noise analysis must
assume that crosstalk noise of an input to spins obeys a Gaussian distribution.
On the other hand, the path-integral method does not require such a Gaussian
approximation of crosstalk noise. We have found that both the signal-to-noise
analysis and the path-integral analysis give the completely same result with
respect to the stationary state in the case where the dynamics is
deterministic, when we assume the Maxwell construction ansatz.
We have shown the dependence of storage capacity (alpha_c) on the number of
patterns per one limit cycle (l). Storage capacity monotonously increases with
the number of steps, and converges to alpha_c=0.269 at l ~= 10. The original
properties of the finite-step sequence processing model appear as long as the
number of steps of the limit cycle has order l=O(1).Comment: 24 pages, 3 figure
Dynamical Probability Distribution Function of the SK Model at High Temperatures
The microscopic probability distribution function of the
Sherrington-Kirkpatrick (SK) model of spin glasses is calculated explicitly as
a function of time by a high-temperature expansion. The resulting formula to
the third order of the inverse temperature shows that an assumption made by
Coolen, Laughton and Sherrington in their recent theory of dynamics is
violated. Deviations of their theory from exact results are estimated
quantitatively. Our formula also yields explicit expressions of the time
dependence of various macroscopic physical quantities when the temperature is
suddenly changed within the high-temperature region.Comment: LaTeX, 6 pages, Figures upon request (here revised), To be published
in J. Phys. Soc. Jpn. 65 (1996) No.
Synapse efficiency diverges due to synaptic pruning following over-growth
In the development of the brain, it is known that synapses are pruned
following over-growth. This pruning following over-growth seems to be a
universal phenomenon that occurs in almost all areas -- visual cortex, motor
area, association area, and so on. It has been shown numerically that the
synapse efficiency is increased by systematic deletion. We discuss the synapse
efficiency to evaluate the effect of pruning following over-growth, and
analytically show that the synapse efficiency diverges as O(log c) at the limit
where connecting rate c is extremely small. Under a fixed synapse number
criterion, the optimal connecting rate, which maximize memory performance,
exists.Comment: 15 pages, 16 figure
Fundamental properties of Tsallis relative entropy
Fundamental properties for the Tsallis relative entropy in both classical and
quantum systems are studied. As one of our main results, we give the parametric
extension of the trace inequality between the quantum relative entropy and the
minus of the trace of the relative operator entropy given by Hiai and Petz. The
monotonicity of the quantum Tsallis relative entropy for the trace preserving
completely positive linear map is also shown without the assumption that the
density operators are invertible.
The generalized Tsallis relative entropy is defined and its subadditivity is
shown by its joint convexity. Moreover, the generalized Peierls-Bogoliubov
inequality is also proven
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
Associative memory storing an extensive number of patterns based on a network of oscillators with distributed natural frequencies in the presence of external white noise
We study associative memory based on temporal coding in which successful
retrieval is realized as an entrainment in a network of simple phase
oscillators with distributed natural frequencies under the influence of white
noise. The memory patterns are assumed to be given by uniformly distributed
random numbers on so that the patterns encode the phase differences
of the oscillators. To derive the macroscopic order parameter equations for the
network with an extensive number of stored patterns, we introduce the effective
transfer function by assuming the fixed-point equation of the form of the TAP
equation, which describes the time-averaged output as a function of the
effective time-averaged local field. Properties of the networks associated with
synchronization phenomena for a discrete symmetric natural frequency
distribution with three frequency components are studied based on the order
parameter equations, and are shown to be in good agreement with the results of
numerical simulations. Two types of retrieval states are found to occur with
respect to the degree of synchronization, when the size of the width of the
natural frequency distribution is changed.Comment: published in Phys. Rev.
Analysis of Bidirectional Associative Memory using SCSNA and Statistical Neurodynamics
Bidirectional associative memory (BAM) is a kind of an artificial neural
network used to memorize and retrieve heterogeneous pattern pairs. Many efforts
have been made to improve BAM from the the viewpoint of computer application,
and few theoretical studies have been done. We investigated the theoretical
characteristics of BAM using a framework of statistical-mechanical analysis. To
investigate the equilibrium state of BAM, we applied self-consistent signal to
noise analysis (SCSNA) and obtained a macroscopic parameter equations and
relative capacity. Moreover, to investigate not only the equilibrium state but
also the retrieval process of reaching the equilibrium state, we applied
statistical neurodynamics to the update rule of BAM and obtained evolution
equations for the macroscopic parameters. These evolution equations are
consistent with the results of SCSNA in the equilibrium state.Comment: 13 pages, 4 figure
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