Reimer Kühn

Hebbian learning reconsidered: Representation of static and dynamic objects in associative neural nets

According to Hebbs postulate for learning, information presented to a neural net during a learning session is stored in the synaptic efficacies. Long-term potentiation occurs only if the postsynaptic neuron becomes active in a time window set up by the presynaptic one. We carefully interpret and mathematically implement the Hebb rule so as to handle both stationary and dynamic objects such as single patterns and cycles. Since the natural dynamics contains a rather broad distribution of delays, the key idea is to incorporate these delays in the learning session. As theory and numerical simulation show, the resulting procedure is surprisingly robust and faithful. It also turns out that pure Hebbian learning is by selection: the network produces synaptic representations that are selected according to their resonance with the input percepts.

Functional correlation approach to operational risk in banking organizations

A Value-at-Risk-based model is proposed to compute the adequate equity capital necessary to cover potential losses due to operational risks, such as human and system process failures, in banking organizations. Exploring the analogy to a lattice gas model from physics, correlations between sequential failures are modeled by as functionally defined, heterogeneous couplings between mutually supportive processes. In contrast to traditional risk models for market and credit risk, where correlations are described as equal-time-correlations by a covariance matrix, the dynamics of the model shows collective phenomena such as bursts and avalanches of process failures.

The Hebb Rule: Storing Static and Dynamic Objects in an Associative Neural Network

The Hebb rule (Hebb, 1949) indicates how information presented to a neural network during a learning session is stored in the synapses, local elements which act as mediators between neurons. In this paper we demonstrate that the Hebb rule can be used to handle both stationary and dynamic objects such as single patterns and cycles. The two main ideas are: a) a broad distribution of delays as they occur in the natural dynamics and b) incorporation of the very same delays during the learning session. Our work shows that the resulting procedure is robust and faithful.

Collective phenomena in neural networks

In this paper we review some central notions of the theory of neural networks. In so doing we concentrate on collective aspects of the dynamics of large networks. The neurons are usually taken to be formal but this is not a necessary requirement for the central notions to be applicable. Formal neurons just make the theory simpler.

Spectra of sparse random matrices

We compute the spectral density for ensembles of of sparse symmetric random matrices using replica, managing to circumvent difficulties that have been encountered in earlier approaches along the lines first suggested in a seminal paper by Rodgers and Bray. Due attention is payed to the issue of localization. Our approach is not restricted to matrices defined on graphs with Poissonian degree distribution. Matrices defined on regular random graphs or on scale-free graphs, are easily handled. We also look at matrices with row constraints such as discrete graph Laplacians. Our approach naturally allows to unfold the total density of states into contributions coming from vertices of different local coordination.

Cavity approach to the spectral density of sparse symmetric random matrices.

The spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: matrices whose associated graphs are locally treelike, and sparse covariance matrices. We derive a closed set of equations from which the density of eigenvalues can be efficiently calculated. Within this approach, the Wigner semicircle law for Gaussian matrices and the Marcenko-Pastur law for covariance matrices are recovered easily. Our results are compared with numerical diagonalization, showing excellent agreement.

Critical behavior of the randomly spin diluted 2D Ising model: A grand ensemble approach.

The critical behaviour of the randomly spin-diluted Ising model in two space dimensions is investigated by a new method which combines a grand ensemble approach to disordered systems proposed by Morita with the phenomenological renormalization group scheme of Nightingale. Accurate approximations for the phase diagram and for the connectivity length exponent of the percolation transition are obtained. Our results suggest that the thermal phase transition of the disordered system might be different from that of the pure system: we observe a continuous variation of critical exponents with the density

Derivatives and credit contagion in interconnected networks

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Encoding and Decoding of Patterns which are Correlated in Space and Time

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Equilibrium ensemble approach to disordered systems I: general theory, exact results

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