1,316 research outputs found
Non-Convex Multi-species Hopfield models
In this work we introduce a multi-species generalization of the Hopfield
model for associative memory, where neurons are divided into groups and both
inter-groups and intra-groups pair-wise interactions are considered, with
different intensities. Thus, this system contains two of the main ingredients
of modern Deep neural network architectures: Hebbian interactions to store
patterns of information and multiple layers coding different levels of
correlations. The model is completely solvable in the low-load regime with a
suitable generalization of the Hamilton-Jacobi technique, despite the
Hamiltonian can be a non-definite quadratic form of the magnetizations. The
family of multi-species Hopfield model includes, as special cases, the 3-layers
Restricted Boltzmann Machine (RBM) with Gaussian hidden layer and the
Bidirectional Associative Memory (BAM) model.Comment: This is a pre-print of an article published in J. Stat. Phy
Neural Networks retrieving Boolean patterns in a sea of Gaussian ones
Restricted Boltzmann Machines are key tools in Machine Learning and are
described by the energy function of bipartite spin-glasses. From a statistical
mechanical perspective, they share the same Gibbs measure of Hopfield networks
for associative memory. In this equivalence, weights in the former play as
patterns in the latter. As Boltzmann machines usually require real weights to
be trained with gradient descent like methods, while Hopfield networks
typically store binary patterns to be able to retrieve, the investigation of a
mixed Hebbian network, equipped with both real (e.g., Gaussian) and discrete
(e.g., Boolean) patterns naturally arises. We prove that, in the challenging
regime of a high storage of real patterns, where retrieval is forbidden, an
extra load of Boolean patterns can still be retrieved, as long as the ratio
among the overall load and the network size does not exceed a critical
threshold, that turns out to be the same of the standard
Amit-Gutfreund-Sompolinsky theory. Assuming replica symmetry, we study the case
of a low load of Boolean patterns combining the stochastic stability and
Hamilton-Jacobi interpolating techniques. The result can be extended to the
high load by a non rigorous but standard replica computation argument.Comment: 16 pages, 1 figur
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Design of exponential state estimators for neural networks with mixed time delays
This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Elsevier Ltd.In this Letter, the state estimation problem is dealt with for a class of recurrent neural networks (RNNs) with mixed discrete and distributed delays. The activation functions are assumed to be neither monotonic, nor differentiable, nor bounded. We aim at designing a state estimator to estimate the neuron states, through available output measurements, such that the dynamics of the estimation error is globally exponentially stable in the presence of mixed time delays. By using the Laypunov–Krasovskii functional, a linear matrix inequality (LMI) approach is developed to establish sufficient conditions to guarantee the existence of the state estimators. We show that both the existence conditions and the explicit expression of the desired estimator can be characterized in terms of the solution to an LMI. A simulation example is exploited to show the usefulness of the derived LMI-based stability conditions.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Nuffield Foundation of the UK under Grant NAL/00630/G, the Alexander von Humboldt Foundation of Germany, the Natural Science Foundation of Jiangsu Education Committee of China under Grants 05KJB110154 and BK2006064, and the National Natural Science Foundation of China under Grants 10471119 and 10671172
Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument
We consider a new model for shunting inhibitory cellular neural networks,
retarded functional differential equations with piecewise constant argument.
The existence and exponential stability of almost periodic solutions are
investigated. An illustrative example is provided.Comment: 24 pages, 1 figur
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