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

Retrieval behavior and thermodynamic properties of symmetrically diluted Q-Ising neural networks

The retrieval behavior and thermodynamic properties of symmetrically diluted Q-Ising neural networks are derived and studied in replica-symmetric mean-field theory generalizing earlier works on either the fully connected or the symmetrical extremely diluted network. Capacity-gain parameter phase diagrams are obtained for the Q=3, Q=4 and $Q=\infty$ state networks with uniformly distributed patterns of low activity in order to search for the effects of a gradual dilution of the synapses. It is shown that enlarged regions of continuous changeover into a region of optimal performance are obtained for finite stochastic noise and small but finite connectivity. The de Almeida-Thouless lines of stability are obtained for arbitrary connectivity, and the resulting phase diagrams are used to draw conclusions on the behavior of symmetrically diluted networks with other pattern distributions of either high or low activity.Comment: 21 pages, revte

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

Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples

Optimally adapted multi-state neural networks trained with noise

The principle of adaptation in a noisy retrieval environment is extended here to a diluted attractor neural network of Q-state neurons trained with noisy data. The network is adapted to an appropriate noisy training overlap and training activity which are determined self-consistently by the optimized retrieval attractor overlap and activity. The optimized storage capacity and the corresponding retriever overlap are considerably enhanced by an adequate threshold in the states. Explicit results for improved optimal performance and new retriever phase diagrams are obtained for Q=3 and Q=4, with coexisting phases over a wide range of thresholds. Most of the interesting results are stable to replica-symmetry-breaking fluctuations.Comment: 22 pages, 5 figures, accepted for publication in PR

Threshold-induced phase transitions in perceptrons

Error rates of a Boolean perceptron with threshold and either spherical or Ising constraint on the weight vector are calculated for storing patterns from biased input and output distributions derived within a one-step replica symmetry breaking (RSB) treatment. For unbiased output distribution and non-zero stability of the patterns, we find a critical load, α p, above which two solutions to the saddlepoint equations appear; one with higher free energy and zero threshold and a dominant solution with non-zero threshold. We examine this second-order phase transition and the dependence of α p on the required pattern stability, κ, for both one-step RSB and replica symmetry (RS) in the spherical case and for one-step RSB in the Ising case

Synthesis of reaction-adapted zeolites as methanol-to-olefins catalysts with mimics of reaction intermediates as organic structure-directing agents

[EN] Catalysis with enzymes and zeolites have in common the presence of well-defined single active sites and pockets/cavities where the reaction transition states can be stabilized by longer-range interactions. We show here that for a complex reaction, such as the conversion of methanol-to-olefins (MTO), it is possible to synthesize reaction-adapted zeolites by using mimics of the key molecular species involved in the MTO mechanism. Effort has focused on the intermediates of the paring mechanism because the paring is less favoured energetically than the side-chain route. All the organic structure-directing agents based on intermediate mimics crystallize cage-based small-pore zeolitic materials, all of them capable of performing the MTO reaction. 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