J. van Mourik

Stability properties of potts neural networks with biased patterns and low loading

The q-state Potts glass model of neural networks is extended to include biased patterns. For a finite number of such patterns, the existence and stability properties of the Mattis states and symmetric states are discussed in detail as a function of the bias. Analytic results are presented for all q at zero temperature. For finite temperatures numerical results are obtained for q=3 and two classes of representative bias parameters. A comparison is made with the Hopfield model.

Average and reliability error exponents in low-density parity-check codes

We present a theoretical method for a direct evaluation of the average and reliability error exponents in low-density parity-check error-correcting codes using methods of statistical physics. Results for the binary symmetric channel are presented for codes of both finite and infinite connectivity.

Swollen-collapsed transition in random hetero-polymers

Abstract:A lattice model of a hetero-polymer with random hydrophilic-hydrophobic charges interacting with the solvent is introduced, whose continuum counterpart has been proposed by Garel, Leibler and Orland [#!GLO!#]. The transfer matrix technique is used to study various constrained annealed systems which approximate at various degrees of accuracy the original quenched model. For highly hydrophobic chains an ordinary -point transition is found from a high temperature swollen phase to a low temperature compact phase. Depending on the type of constrained averages, at very low temperatures a swollen phase or a coexistence between compact and swollen phases are found. The results are carefully compared with the corresponding ones obtained in the continuum limit, and various improvements in the original calculations are discussed.

The Optimal Storage Capacity for a Neural Network with Multi-State Neurons

Using the Gardner approach, the optimal storage capacity is discussed for a network with multi-stage neurons and a spherical constraint on the couplings. It is found that this capacity decreases with the number of states and the corresponding separation of the plateaus of the gain function. An analysis of the validity of the replica-symmetric approximation is made.

Cluster derivation of Parisi's RSB solution for disordered systems

We propose a general scheme in which disordered systems are allowed to sacrifice energy equi-partitioning and separate into a hierarchy of ergodic sub-systems (clusters) with different characteristic timescales and temperatures. The details of the break-up follow from the requirement of stationarity of the entropy of the slower cluster, at every level in the hierarchy. We apply our ideas to the Sherrington-Kirkpatrick model, and show how the Parisi solution can be derived quantitatively from plausible physical principles. Our approach gives new insight into the physics behind Parisis solution and its relations with other theories, numerical experiments, and short-range models.

From shrinking to percolation in an optimization model

A model of noise reduction for signal processing and other optimization tasks is introduced. Each noise source puts a symmetric constraint on the space of the signal vector within a tolerance bound. When the number of noise sources increases, sequences of transitions take place, causing the solution space to vanish. We find that the transition from an extended solution space to a shrunk space is retarded because of the symmetry of the constraints, in contrast with the analogous problem of pattern storage. For low tolerance, the solution space vanishes by volume reduction, whereas for high tolerance, the vanishing becomes more and more like percolation.

Capacity of diluted multi-state neural networks

The optimal storage capacity is studied for diluted networks with multi-state neurons and continuous respectively discrete couplings, within the replica-symmetric Gardner approach. The Gardner-Derrida line, the de Almeida-Thouless line and the zero-entropy line are compared and the validity of the replica-symmetric approximation is discussed in detail. The distribution of the synaptic couplings is determined. The results are analysed in terms of the number of states of the neuron, the distribution of the stored patterns, the amount of dilution and the number of discrete values for the couplings.

Magnetization enumerator for LDPC codes - a statistical physics approach

We propose a method based on the magnetization enumerator to determine the critical noise level for Gallager type low density parity check error correcting codes (LDPC). Our method provides an appealingly simple interpretation to the relation between different decoding schemes, and provides more optimistic critical noise levels than those reported in the information theory literature.

The Influence of Memory in a Threshold Model for Distributed Task Assignment

A nature inspired decentralized multi-agent algorithm is proposed to solve a problem of distributed task selection in which cities produce and store batches of different mail types. Agents must collect and process the mail batches, without a priori knowledge of the available mail at the cities or inter-agent communication. In order to process a different mail type than the previous one, agents must undergo a change-over during which it remains inactive.We propose a threshold based algorithm in order to maximize the overall efficiency (the average amount of mail collected). We show that memory, i.e. the possibility for agents to develop preferences for certain cities, not only leads to emergent cooperation between agents, but also to a significant increase in efficiency (above the theoretical upper limit for any memoryless algorithm), and we systematically investigate the influence of the various model parameters. Finally, we demonstrate the flexibility of the algorithm to changes in circumstances, and its excellent scalability.

Cluster derivation of the Parisi scheme for disordered systems

We propose a general quantitative scheme in which systems are given the freedom to sacrifice energy equi-partitioning on the relevant time-scales of observation, and have phase transitions by separating autonomously into ergodic sub-systems (clusters) with different characteristic time-scales and temperatures. The details of the break-up follow uniquely from the requirement of zero entropy for the slower cluster. Complex systems, such as the Sherrington-Kirkpatrick model, are found to minimize their free energy by spontaneously decomposing into a hierarchy of ergodically equilibrating degrees of freedom at different (effective) temperatures. This leads exactly and uniquely to Parisi’s replica symmetry breaking scheme. Our approach, which is somewhat akin to an earlier one by Sompolinsky, gives new insight into the physical interpretation of the Parisi scheme and its relations with other approaches, numerical experiments, and short range models. Furthermore, our approach shows that the Parisi scheme can be...