main overlap dynamics for multistate neural networks the authors present explicit formulae for one step dynamics of retrieval pattern errors main overlap for the q state potts and q or 4 clock neural networks solution of the fixed point equations and critical values of the saturation parameters alpha c q in the one step approximation are considered in the potts case

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

generalization in a multi state neural network the generalization ability of an extremely dilute feedback neural network with multi state neurons is studied by means of a deterministic noiseless parallel dynamics the overlap with any one of a macroscopic number of binary full activity concepts is determined when the network is trained with examples of variable activity according to a hebbian learning algorithm that favours stable symmetric mixture states explicit results about the phase diagram and the generalization error are obtained for a network with three state neurons which remain inactive below a threshold it is shown that the generalization ability can be considerably enhanced either by training the network with low activity examples or by means of a moderate increase in

robustness against random dilution in attractor neural networks the authors study the robustness of attractor neural networks against random disruption of a fraction of the synaptic couplings for the maximally stable network msn the authors determine the effect of different degrees of dilution on the overall storage capacity the size of the basins of attraction and the attractor overlap comparison with corresponding results for the hopfield model indicates that although msn is more robust at low degrees of dilution the hopfield network becomes more robust at high dilution

structure of metastable states in the hopfield model an upper bound for the number of metastable states in the hopfield model is calculated as a function of the hamming fraction from an input pattern for all finite values of alpha the ratio of number of patterns to nodes the hamming fraction from the input pattern to the nearest metastable state is infinite when alpha 0 113 the bound also implies that there is a gap between a set of states close to the input pattern and another set centred around the hamming fraction 0 5 from it

dynamics of ising random bond models neural network and random anisotropy axis model a discrete time retrieval dynamics for a class of random bond models with infinite range interactions is studied in a unified picture the hopfield model of neural networks with the hebb learning rule is considered together with the ising random anisotropy axis model in the strong anisotropy limit the main overlap magnetization the residual overlap and its dispersion as well as the correlation between the gaussian component of the residual overlap and the initial value of the latter obtained in the first step of a recursion relation are used to infer the structure of the recursion for large times a crucial assumption is the strong stationarity of the gaussian component the dynamics are discussed for finite alpha p n the storage ratio in the neural network problem or the ratio of random axis components per site in the limit where both p and n go to infinity the long time behaviour of the theory is shown to yield the equilibrium solution of an earlier work in mean field theory for a tri modal distribution of random axis components explicit results for the basins of attraction of either a ferromagnetic or a spin glass phase are obtained as well as the relaxation time with a square root power law decay near saturation

thermodynamic properties of fully connected q ising neural networks the thermodynamic and retrieval properties of fully connected q ising networks are studied in the replica symmetric mean field approximation in particular capacity gain parameter and capacity temperature phase diagrams are derived for q 3 4 and q infinity and different distributions of the stored patterns furthermore the optimal gain function is determined in order to obtain the best performance where appropriate the results are compared with the diluted and layered versions of these models

onset of super retrieval phase and enhancement of the storage capacity in neural networks of nonmonotonic neurons analogue neural networks of associative memory with continuous time dynamics are studied for nonmonotonic transfer functions using the method of self consistent signal to noise analysis the hebb learning rule with unbiased random patterns is assumed for the synaptic couplings a novel phenomenon is found to occur as a result of a phase transition concerning the property of the local field distribution in retrieval states of the newly found phase which the authors refer to as the super retrieval phase noise in the local field vanishes and the memory retrieval without errors ensures even for an extensive number of memory patterns stored under the local learning rule the storage capacity is obtained as a function of the parameter representing the degree of nonmonotonicity of the transfer functions with the result that an enhancement of the storage capacity can also occur

an improved bonferroni inequality and applications summary we present an improved bonferroni inequality which gives an upper bound for the probability of the union of an arbitrary sequence of events the bound is constructed in terms of the joint probability of pairs of events which are represented by edges on a graph examples of applications to periodicity location shift detection kolmogorovsmirnov tests and outlier detection are given

categorization in a layered neural network abstract the layered feedforward network of domany meir and kinzel is extended to investigate the ability to recognize any one of a macroscopic number of concepts ancestors when the network is trained through a finite number of examples descendents of each concept by means of a generalized hebbian learning rule between cells on two consecutive layers learning curves describing the generalization error are obtained as well as phase diagrams and basins of attraction that exhibit the possible coexistence of a generalization phase a retrieval phase of examples and a spin glass phase

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

categorization ability in a biologically motivated neural network abstract in this work we study the categorization properties of an extreme and asymmetrically diluted version of the hopfield model for associative memory when the effect of the refractory period of the neurons is taken into account in the dynamics of the system the simplest way of modeling these refractory periods is by means of a time dependent threshold that acts only on those neurons that emit a signal and favors them to be at rest during a given time interval the dynamic equations are derived in the limit of extreme dilution using an approach that explicitly preserves the dependence of the system on its whole history the categorization error is analyzed for different values of the parameters in particular we confront our analytical results with numerical simulations for the noiseless case t 0 when the number of examples or their correlations increases the system always categorizes independently of the amplitude of the potential that mimics the effect of the refractory period

on the parallel dynamics for the little hopfield model we propose a method algorithm for calculation of the explicit formulas for evolution of the main and the residual overlaps it allows us to confirm the gardner derrida mottishaw second step formula for the main overlap and to go beyond to the next steps we discuss the dynamical status of the amit gutfreund sompolinsky formula for the main overlap and some computersimulation results

in search of an optimal dilution algorithm for feedforward networks in a recent paper the authors presented a dilution algorithm that yields a storage capacity per synapse alpha eff larger than 2 in this letter two new algorithms with even higher alpha eff values are introduced they study a one step dilution algorithm where the perceptron is used to select a fraction of the couplings to be removed for the remaining bonds the perceptron of optimal stability is relearned they further compare simulation data from an iterative version of the one step dilution algorithm with phase space volume calculations

learning and forgetting on asymmetric diluted neural networks it is possible to construct diluted asymmetric models of neural networks for which the dynamics can be calculated exactly we test several learning schemes in particular models for which the values of the synapses remain bounded and depend on the history our analytical results on the relative efficiencies of the various learning schemes are qualitatively similar to the corresponding ones obtained numerically on fully connected symmetric networks

linear rank one preservers between spaces of matrices with zero trace abstract let x y be a pair of vector spaces over a field f associated with a bilinear form such that x y 0 for all y in y implies that x 0 let x y 0 be the subspace of x y spanned by all decomposable elements x y with x y 0 let u v be any two vector spaces over f in this note we study linear mappings from x y 0 to u v that send nonzero decomposable elements to nonzero decomposable elements and some of its consequences

categorization in a hopfield network trained with weighted examples i finite number of concepts abstract we consider the categorization problem in a hopfield network with a finite number of concepts and trained with s examples of weight 1 s we find that the retrieval capacity of an example with weight 1 and the corresponding categorization error depends also on the arithmetic mean m 1 s 1 2 s of the other weights for 1 m 1 we find that the line of first order transitions between the retrieval and categorization phases ends at a critical point in the s t plane when two solutions are present the global minimum of the free energy corresponds to the solution with the highest weight

generalization in a hopfield network with noise the generalization ability of the hopfield model of neural networks trained with examples is studied in mean field theory in the presence of synaptic noise although the latter improves the generalization ability for a finite number of concepts it does not for a macroscopic number of them nevertheless the network performance is still robust against synaptic noise numerical simulations are performed to verify the mean field theory predictions

feed forward chains of recurrent attractor neural networks near saturation we perform a stationary state replica analysis for a layered network of ising spin neurons with recurrent hebbian interactions within each layer in combination with strictly feed forward hebbian interactions between successive layers this model interpolates between the fully recurrent and symmetric attractor network studied by amit et al and the strictly feed forward attractor network studied by domany et al due to the absence of detailed balance it is as yet solvable only in the zero temperature limit the built in competition between two qualitatively different modes of operation feed forward ergodic within layers versus recurrent non ergodic within layers is found to induce interesting phase transitions

an improved bonferroni procedure for multiple tests of significance summary a modification of the bonferroni procedure for testing multiple hypotheses is presented the method based on the ordered p values of the individual tests is less conservative than the classical bonferroni procedure but is still simple to apply a simulation study shows that the probability of a type i error of the procedure does not exceed the nominal significance level a for a variety of multivariate normal and multivariate gamma test statistics for independent tests the procedure has type i error probability equal to a the method appears particularly advantageous over the classical bonferroni procedure when several highly correlated test statistics are involved

two state representations of three state neural networks the authors investigate to what extent the dynamical behaviour of nets consisting of three state neurons can be realised in nets using two state neurons they find that the generalisation from a two state neuron to a three state neuron cannot introduce any new behaviour in the deterministic case but can do so for noisy nets even in the zero noise limit for those noisy three state nets which have a hamiltonian description and a two state representation they investigate the relationship between the three state hamiltonian and the two state hamiltonian and find an interesting level splitting phenomenon

retrieval and chaos in layered q ising neural networks using a probabilistic approach we study the parallel dynamics of theq ising layered networks for arbitraryq by introducing auxiliary thermal fields we express the stochastic dynamics within the gain function formulation of the deterministic dynamics evolution equations are derived for arbitraryq at both zero and finite temperatures an explicit analysis of the fixed point equations is carried out for bothq 3 andq the retrieval properties are discussed in terms of the gain parameter the storage capacity and the temperature using the time evolution of the distance between two network configurations we investigate the possibility of microscopic chaos chaotic behavior is always present for arbitrary finiteq however in the limitq the existence of chaos depends on the parameters of the system

on multivariate runs tests for randomness this paper proposes several extensions of the concept of runs to the multivariate setup and studies the resulting tests of multivariate randomness against serial dependence two types of multivariate runs are defined i an elliptical extension of the spherical runs proposed by marden 1999 and ii an original concept of matrix valued runs the resulting runs tests themselves exist in various versions one of which is a function of the number of data based hyperplanes separating pairs of observations only all proposed multivariate runs tests are affine invariant and highly robust in particular they allow for heteroscedasticity and do not require any moment assumption their limiting distributions are derived under the null hypothesis and under sequences of local vector arma alternatives asymptotic relative efficiencies with respect to gaussian portmanteau tests are computed and show that while marden type runs tests suffer severe consistency problems tests based on matrix valued runs perform u

on the overlap dynamics of multi state neural networks with a finite number of patterns neural networks with multi state neurons are studied in the case of low loading for symmetric couplings satisfying a certain positivity condition a lyapunov function is shown to exist in the space of overlaps between the instantaneous microscopic state of the system and the learned patterns furthermore an algorithm is derived for zero temperature to determine all the fixed points as an illustration the three state model is worked out explicitly for hebbian couplings for finite temperature the time evolution of the overlap is studied for couplings which need not be symmetric the stability properties are discussed in detail for the three state model for asymmetric couplings limit cycle behaviour is shown to be possible

the roles of stability and symmetry in the dynamics of neural networks the authors study the retrieval phase of spin glass like neural networks considering that the dynamics should depend only on gauge invariant quantities they propose that two such parameters characterising the symmetry of the neural net s connections and the stabilities of the patterns are responsible for most of the dynamical effects this is supported by a numerical study of the shape of the basins of attraction for a one pattern neural network model the effects of stability and symmetry on the short time dynamics of this model are studied analytically and the full dynamics for vanishing symmetry is shown to be exactly solvable

parallel dynamics of the neural network with the pseudoinverse coupling matrix the authors investigate the parallel dynamics of the neural network with the pseudoinverse coupling matrix based on an exact dynamic theory they develop an approximate treatment for long timescales the temporal development of the overlap of correlation functions and of the remanent magnetization are investigated they present results for deterministic and stochastic dynamics large scale numerical simulations supplement their analytical findings

the three state layered neural network with finite dilution abstract the dynamics and the stationary states of an exactly solvable three state layered feed forward neural network model with asymmetric synaptic connections finite dilution and low pattern activity are studied in extension of a recent work on a recurrent network detailed phase diagrams are obtained for the stationary states and for the time evolution of the retrieval overlap with a single pattern it is shown that in spite of instabilities for low thresholds there is a gradual improvement in network performance with increasing threshold up to an optimal stage the robustness to synaptic noise is checked and the effects of dilution and of variable threshold on the information content of the network are also established

learning grey toned patterns in neural networks the problem of learning multi state patterns in neural networks is investigated an analysis of the space of couplings gardner approach yields the distribution of local fields the critical storage capacity alpha c and the minimum number of errors for an overloaded network for noisy local fields the classification error is minimized if the local fields of the patterns are allowed to lie in intervals of finite width a fast converging adaptive learning algorithm is presented which finds the coupling matrix of optimal stability

feed forward chains of recurrent attractor neural networks with finite dilution near saturation abstract a stationary state replica analysis for a dual neural network model that interpolates between a fully recurrent symmetric attractor network and a strictly feed forward layered network studied by coolen and viana is extended in this work to account for finite dilution of the recurrent hebbian interactions between binary ising units within each layer gradual dilution is found to suppress part of the phase transitions that arise from the competition between recurrent and feed forward operation modes of the network despite that a long chain of layers still exhibits a relatively good performance under finite dilution for a balanced ratio between inter layer and intra layer interactions

categorization in the pseudo inverse neural network we investigate analytically the emergence of the categorization ability in the pseudo inverse attractor neural network more pointedly we consider the problem of learning an extensive number of concepts by storing a finite number of examples s of each concept we find that there is a critical value beyond which the categorization error as measured by the average fraction of unstable sites in the concepts decreases monotonically with s

spectral preservers and approximate spectral preservers on operator algebras abstract let a and b be unital primitive banach algebras with minimal idempotents we prove that every surjective spectral isometry from a onto b is of the form where c with 1 and is either an isomorphism or an anti isomorphism from a onto b as an application we show that for all banach spaces x and y the spectral nearisometries and the approximate spectrum preserving maps from l x onto l y are perturbations of actual spectral isometries and spectrum preserving maps respectively

storing an extensive number of grey toned patterns in a neural network using multistate neurons grey toned patterns are pictures composed of pixels of several shades of grey the ability of neural networks using multistate neurons to store such patterns is systematically investigated if conventional generalizations of hopfield networks using analogue or soft neurons are considered it is impossible to stabilize these grey tones nevertheless it is shown that it can be done with networks that use neurons which have only a discrete set of possible activities this is demonstrated for the pseudo inverse rule for the synaptic couplings where only the stability of the patterns shrinks with increasing number q of grey tones one wants to store if the patterns are uncorrelated one can use the hebb rule and in this case the mean field theory is presented applying this rule the storage capacity decreases as q 2 with the number of grey tones

on the parallel dynamics of the q state potts and q ising neural networks using a probabilistic approach the parallel dynamics of theq state potts andq ising neural networks are studied at zero and at nonzero temperatures evolution equations are derived for the first time step and arbitraryq these formulas constitute recursion relations for the exact parallel dynamics of the extremely diluted asymmetric versions of these networks an explicit analysis including dynamical capacity temperature diagrams and the temperature dependence of the overlap is carried out forq 3 both types of models are compared

on the parallel dynamics of the diluted clock neural network abstract we present explicit formulae for one step evolution of the overlaps in a clock neural network for the parallel dynamics these formulae constitute recursion relations for the exact dynamics in the case of extreme dilution the fixed point equation for this exactly soluble model is investigated for varying temperature t capacity parameter and the number q of the clock states

information processing in three state neural networks we introduce networks of three state neurons where the additional state embodies the absence of information their dynamical behavior is studied from the standpoint of information processing these networks display strong pattern completion capabilities moreover inference naturally occurs between coherent patterns

neural networks a biased overview an overview of recent activity in the field of neural networks is presented the long range aim of this research is to understand how the brain works first some of the problems are stated and terminology defined then an attempt is made to explain why physicists are drawn to the field and their main potential contribution in particular in recent years some interesting models have been introduced by physicists a small subset of these models is described with particular emphasis on those that are analytically soluble finally a brief review of the history and recent developments of single and multilayer perceptrons is given bringing the situation up to date regarding the central immediate problem of the field search for a learning algorithm that has an associated convergence theorem

parallel dynamics of fully connected q ising neural networks using a probabilistic approach the parallel dynamics of fully connected q ising neural networks is studied for arbitrary q a novel recursive scheme is set up to determine the time evolution of the order parameters through the evolution of the distribution of the local field taking into account all feedback correlations in contrast to extremely diluted and layered network architectures the local field is no longer normally distributed but contains a discrete part as an illustrative example an explicit analysis is carried out for the first four time steps for the case of the q 2 and q 3 model the results are compared with extensive numerical simulations and excellent agreement is found finally equilibrium fixed point equations are derived and compared with the thermodynamic approach based upon the replica symmetric mean field approximation

predicting the future advantages of semilocal units in investigating gaussian radial basis function rbf networks for their ability to model nonlinear time series we have found that while rbf networks are much faster than standard sigmoid unit backpropagation for low dimensional problems their advantages diminish in high dimensional input spaces this is particularly troublesome if the input space contains irrelevant variables we suggest that this limitation is due to the localized nature of rbfs to gain the advantages of the highly nonlocal sigmoids and the speed advantages of rbfs we propose a particular class of semilocal activation functions that is a natural interpolation between these two families we present evidence that networks using these gaussian bar units avoid the slow learning problem of sigmoid unit networks and very importantly are more accurate than rbf networks in the presence of irrelevant inputs on the mackey glass and coupled lattice map problems the speedup over sigmoid networks is so dramatic that the difference in training time between rbf and gaussian bar networks is minor gaussian bar architectures that superpose composed gaussians gaussians of gaussians to approximate the unknown function have the best performance we postulate that an interesing behavior displayed by gaussian bar functions under gradient descent dynamics which we call automatic connection pruning is an important factor in the success of this representation

additive idempotence preservers abstract let a be a local matrix algebra over a field of characteristic different from 2 b an arbitrary algebra and x y banach spaces additive mappings a b which preserve idempotents are characterized the result is then used in classifying additive surjections b x b y which preserve idempotents and do not annihilate finite rank operators

a distribution free test for symmetry based on a runs statistic abstract i present a simple test based on a runs statistic for symmetry of a continuous distribution about a known median the statistic has a binomial sampling distribution and desirable invariance properties monte carlo studies demonstrate that for a wide variety of alternative asymmetric distributions the test is more powerful than tests proposed by butler 1969 rothman and woodroofe 1972 or hill and rao 1977

optimal capacity of graded response perceptrons a replica symmetry breaking solution optimal capacities of perceptrons with graded input output relations are studied within the first step replica symmetry breaking gardner approach input data errors and a limited output precision are allowed in particular the role of non monotonicity in the input output relations on the breaking and on the overall performance is determined

a maximally selected test of symmetry about zero the problem of testing symmetry about zero has a long and rich history in the statistical literature we introduce a new test that sequentially discards observations whose absolute value is below increasing thresholds defined by the data mcnemar s statistic is obtained at each threshold and the largest is used as the test statistic we obtain the exact distribution of this maximally selected mcnemar and provide tables of critical values and a program for computing p values power is compared with the t test the wilcoxon signed rank test and the sign test the new test mm is slightly less powerful than the t test and wilcoxon signed rank test for symmetric normal distributions with nonzero medians and substantially more powerful than all three tests for asymmetric mixtures of normal random variables with or without zero medians the motivation for this test derives from the need to appraise the safety profile of new medications if pre and post safety measures are obtained then under the null hypothesis the variables are exchangeable and the distribution of their difference is symmetric about a zero median large pre post differences are the major concern of a safety assessment the discarded small observations are not particularly relevant to safety and can reduce power to detect important asymmetry the new test was utilized on data from an on road driving study performed to determine if a hypnotic a drug used to promote sleep has next day residual effects

retrieval and chaos in extremely diluted q ising neural networks using a probabilistic approach the deterministic and the stochastic parallel dynamics of aq ising neural network are studied at finiteq and in the limitq exact evolution equations are presented for the first time step these formulas constitute recursion relations for the parallel dynamics of the extremely diluted asymmetric versions of these networks an explicit analysis of the retrieval properties is carried out in terms of the gain parameter the loading capacity and the temperature the results for theq network are compared with those for theq 3 andq 4 models possible chaotic microscopic behavior is studied using the time evolution of the distance between two network configurations for arbitrary finiteq the retrieval regime is always chaotic in the limitq the network exhibits a dynamical transition toward chaos

nonlinear neural networks i general theory a neural network is called nonlinear if the introduction of new data into the synaptic efficacies has to be performed through anonlinear operation the original hopfield model is linear whereas for instance clipped synapses constitute a nonlinear model here a general theory is presented to obtain the statistical mechanics of a neural network with finitely many patterns and arbitrary symmetric nonlinearity the problem is reduced to minimizing a free energy functional over all solutions of a fixed point equation with synaptic kernelq the case of clipped synapses with bimodal and gaussian probability distribution is analyzed in detail to this end a simple theory is developed that gives the spectrum ofq and thereby all the solutions that bifurcate from the high temperature phase

layered neural networks with non monotonic transfer functions abstract we investigate storage capacity and generalization ability for two types of fully connected layered neural networks with non monotonic transfer functions random patterns are embedded into the networks by a hebbian learning rule one of them is a layered network in which a non monotonic transfer function of even layers is different from that of odd layers the other is a layered network with intra layer connections in which the non monotonic transfer function of inter layer is different from that of intra layer and inter layered neurons and intra layered neurons are updated alternately we derive recursion relations for order parameters for those layered networks by the signal to noise ratio method we clarify that the storage capacity and the generalization ability for those layered networks are enhanced in comparison with those with a conventional monotonic transfer function when non monotonicity of the transfer functions is selected optimally we also point out that some chaotic behavior appears in the order parameters for the layered networks when non monotonicity of the transfer functions increases

linear additive preservers of rank 2 on spaces of alternate matrices over fields abstract let kn f be the linear space of all n n alternate matrices over a field f and let k n 2 f be its subset consisting of all rank 2 matrices an operator kn f kn f is said to be additive if a b a b for any a b kn f linear if is additive and aa af a for every a f and a kn f and a preserver of rank 2 on kn f if k n 2 f k n 2 f when n 4 we characterize all linear respectively additive preservers of rank 2 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself

a probabilistic approach to parallel dynamics for the little hopfield model presents new results on a probabilistic approach to parallel dynamics of the little hopfield model the authors propose a truncated auxiliary dynamics method to control a feedback noise in this symmetrical neural network with full connection it allows them to propose an ansatz for derivation of the explicit recurrence relations for the main and residual noisy overlap evolution for arbitrary discrete moment t

neural networks that use three state neurons the dynamics of a neural network which uses three state neurons 1 0 and 1 is solved exactly in the limit of non symmetric and highly dilute synapses recursion relations for the activity the fraction of non zero neurons and overlap of the network with a given pattern are derived which have three generic kinds of fixed points a retrieval fixed point a chaotic fixed point which corresponds to non zero activity but no overlap and a zero fixed point where all the neurons go to the 0 state as the non retrieval fixed points both have activities different from the retrieval fixed point one can easily tell whether a pattern has been recovered an analysis of which fixed points occur as a function of the thresholds and the storage ratio of the system yields remarkably rich phase diagrams optimising the threshold level can be very important especially when low activity patterns are stored a similar analysis can be applied to biased networks using two state 1 0 neurons finally one finds that mixture states which have an overlap with two patterns can be stabilised by a threshold in the networks using three state neurons this property allows larger higher activity memories to be naturally constructed out of smaller ones

synchronous versus sequential updating in the three state ising neural network with variable dilution abstract the q 3 state ising neural network with synchronous updating and variable dilution is discussed starting from the appropriate hamiltonians the thermodynamic and retrieval properties are examined using replica mean field theory the appearance and properties of two cycles are discussed capacity temperature phase diagrams are derived for several values of the pattern activity and different gradations of dilution and the information content is calculated it is found that the asymptotic behaviour is rather similar to that for sequential updating the retrieval region is enhanced marginally but the spin glass region is visibly enlarged only the presence of self coupling can enlarge the retrieval region substantially the dynamics of the network is studied for general q and both synchronous and sequential updating using an extension of the generating function technique the differences with the signal to noise approach are outlined typical flow diagrams for the q 3 overlap order parameter are presented

a finite size scaling investigation for q state hopfield models storage capacity and basins of attraction the storage capacity of a q state hopfield network is determined via finite size scaling for parallel dynamics and q or 8 the results are in good agreement with theoretical predictions by rieger the basins of attraction and other associative memory properties are discussed for q 4 6 a self controlling q state model with improved basins of attraction is proposed

order parameter evolution in a feedforward neural network we consider layered neural networks in which the weights are trained with the pseudoinverse rule to store a set of random patterns using many body diagrammatic techniques the evolution in the network can be described by the overlap order parameter m and the noise parameter delta looping effects are shown to be significant in contrast to a previous conjecture order parameter pairs corresponding to various input conditions are found to collapse on a universal curve

dilution and sparse coding in threshold linear nets the storage capacity of an autoassociative memory with extremely diluted connectivity and with threshold linear elementary units is studied in its dependence on the graded structure and on the sparseness of the coding scheme and on the form of the learning rule used as the coding becomes sparse more patterns can be stored and the difference in capacity measured for a given number of modifiable synapses per unit between fully connected and highly diluted systems vanishes graded non binary codings especially when used with learning rules nonlinear in their post synaptic factor further increase the number of patterns that can be stored by making their retrieved representation even sparser

data management of river water quality data a semi automatic procedure for data validation 1 monitoring networks typically generate large amounts of data before the data can be added to the database they have to be validated in this paper a semi automatic procedure is presented to validate river water quality data on the basis of historical data additive models are established to predict new observations and to construct prediction intervals pi s a new observation is accepted if it is located in the interval the coverage of the prediction intervals and its power to detect anomalous data are assessed in a simulation study the method is illustrated on two case studies in which the method detected abnormal nitrate concentrations in the water body provoked by a dry summer which was followed by an extreme winter period the case studies also show that similar to classical multivariate outlier detection tools the semi automatic procedure allows the detection of suspicious observations lying at the edges as well as observations lying at the center of the univariate distribution of the observations but without having to impose linear relationships typically associated with these classical methods

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

the weighted log rank class of permutation tests p values and confidence intervals using saddlepoint methods test statistics from the weighted log rank class are commonly used to compare treatment with control when there is right censoring this paper uses saddlepoint methods to determine mid p values from the null permutation distributions of tests from the weighted log rank class analytical saddlepoint computations replace the permutation simulations and provide mid p values that are virtually exact for all practical purposes the speed of these saddlepoint computations makes it practicable to invert the weighted log rank tests to determine nominal 95 confidence intervals for the treatment effect with right censored data such analytical inversions lead to permutation confidence intervals that are easily computed and virtually identical to the exact intervals that would normally require massive amounts of simulation

exact distribution of a maximally selected wilcoxon and a new hybrid test of symmetry recently a maximally selected normalized wilcoxon whose asymptotic distribution is a brownian bridge was proposed for testing symmetry of a distribution about zero the test sequentially discards observations whose absolute value is below increasing thresholds the wilcoxon is obtained at each threshold and the maximum is the test statistic we develop a recursive function for the exact distribution of a modification of the max wilcoxon test mw and provide critical values and a program for computing the p value for a sample a new hybrid test that combines the sign and mw tests is introduced the power of mw and the new hybrid test are compared with modarres and gastwirth s hybrid test mgh and the max mcnemar mm under the generalized lambda distributions gld family and two normal mixture models the mw and the new hybrid test outperform the mgh which is superior to the mm test in the gld family in one mixture model mm is the least powerful test and the remaining three are essentially equivalent in the second mixture model when the zero median assumption is nearly valid the mw test does well its performance degrades when this assumption is violated in the latter case the mm performs better than mw for the same degree of skewness because the mm simultaneously tests both symmetry and zero median data from a genetic study of monozygotic twins discordant for major depressive disorder is used to illustrate the new tests copyright 2014 john wiley sons ltd

asymptotic points for a test of symmetry about a specified median hill rao 1977 have discussed a family of statistics based on cramer von mises statistics for testing the symmetry of a continuous distribution about a specified median and have suggested one member of the family as a useful test statistic this statistic t is calculated as follows suppose that s x1 x4 is a random sample from an unspecified continuous distribution we wish to test the null hypothesis that the distribution is symmetric about the origin let z1 zn be the ordered absolute values i xj xj e s and let nk and pk k 1 n be the numbers of negative and positive values respectively in s such that i xji zk let s be the set of reciprocal values x1 x 1 and define zk n and pk analogously to zk nk and pk then

parallel dynamics of extremely diluted symmetric q ising neural networks the parallel dynamics of extremely diluted symmetric q ising neural networks is studied for arbitrary q using a probabilistic approach in spite of the extremely diluted architecture the feedback correlations arising from the symmetry prevent a closed form solution in contrast with the extremely diluted asymmetric model a recursive scheme is found determining the complete time evolution of the order parameters taking into account all feedback it is based upon the evolution of the distribution of the local field as in the fully connected model as an illustrative example an explicit analysis is carried out for the q 2 and q 3 model these results agree with and extend the partial results existing for q 2 for q 2 the analysis is entirely new finally equilibrium fixed point equations are derived and a capacity gain function diagram is obtained

pattern selectivity in neural networks as a means of understanding basin structures the authors study the problem of learning and retrieving for a pair of correlated patterns within an extensive number of uncorrelated patterns for networks where learning may be treated as an optimization process with respect to an arbitrary cost function this formulation is then applied to several specific examples where they study the pattern selectivity of these systems i e their ability to differentiate correlated patterns and investigate the process of basin shrinking with increasing loading levels they define and discuss several different retrieval phases whose existence depends on the competitive interplay of the loading level and the pattern correlation discussion of asymptotic retrieval is restricted to dilute asymmetric networks

mean field dynamics of sequence processing neural networks with finite connectivity abstract a recent dynamic mean field theory for sequence processing in fully connected neural networks of hopfield type is extended and analyzed here for a symmetrically diluted network with finite connectivity near saturation equations for the dynamics and the stationary states are obtained for the macroscopic observables and the precise equivalence is established with the single pattern retrieval problem in a layered feed forward network with finite connectivity

on the dynamics of analogue neurons with nonsigmoidal gain functions abstract the characteristic properties of the macroscopic retrieval dynamics of analogue neurons with hebbian coupling strengths are studied usign the shape of the gain function as a modeling parameter already at low loading a rich diversity in dynamical behaviour is observed covering the full range from point attractors to chaotic dynamics the attractors which are not of the fixed point type are interpreted as an intermediate phase between the well known retrieval states and the zero state enabling a waiting mode in the system using a probabilistic approach it is shown that these features persist in extremely and asymmetrically diluted systems at sufficiently low loading a number of generic examples are worked out in detail illustrating some properties of networks governed by nonmonotonic piecewise linear gain functions the critical storage level above which the chaotic behaviour is absent is numerically determined

additive mappings preserving operators of rank one abstract let b x be the algebra of all bounded linear operators on a nontrivial real or complex banach space and let f x be the subalgebra of all finite rank operators a characterization of additive mappings on f x which preserve operators of rank one or projections of rank one is given in the real case such mappings are automatically linear

a dilution algorithm for neural networks a dilution algorithm to enlarge the storage capacity per synapse alpha eff of neural networks is proposed the algorithm is a hybrid method where hebb s rule is used to select a fraction of couplings to be removed afterwards the perceptron of optimal stability for the remaining couplings is learned the authors present an analytical calculation and the results of the numerical simulations in comparison with the fully connected or the randomly diluted perceptron the effective storage capacity alpha eff is remarkably enlarged

layered neural networks exact solutions for the dynamics of layered feedforward neural networks are presented these networks are expected to respond to an input by going through a sequence of preassigned states on the various layers the family of networks considered has a variety of interlayer couplings linear and non linear hebbian hebbian with gaussian synaptic noise and with various kinds of dilution in addition the authors also solve the problem of layered networks with the pseudoinverse projector matrix of couplings in all cases the solutions take the form of layer to layer recursions for the mean overlap with a random key pattern and for the width of the embedding field distribution the dynamics is governed by the fixed points of these recursions for all cases non trivial domains of attraction of the memory states are found and graphically displayed

generalization and chaos in a layered neural network the generalization performance of a multi state and a graded response layered attractor neural network trained with examples of low activity is established exactly for monotonic and non monotonic input output functions complex behaviour is found which goes from fixed point attractors to chaos through a cascade of bifurcations depending on an appropriate threshold or cut off parameter the effect of the irregular behaviour on the generalization curves is explicitly demonstrated and phase diagrams for the recognition ratio of concepts in terms of the threshold cut off exhibit ordered generalization disordered paramagnetic or self sustained activity and chaotic phases

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

additive mappings that preserve rank one nilpotent operators let b x be the algebra of bounded operators on a complex infinite dimensional banach space x and f x be the subalgebra of all finite rank operators in b x a characterization of additive mappings on f x which preserve rank one nilpotent operators in both directions is given as applications of this result some additive preservers are described

some linear preserver problems onb h concerning rank and corank abstract as a continuation of the work of linear maps between operator algebras which preserve certain subsets of operators with finite rank or finite corank here we consider the problem inbetween that is we treat the question of preserving operators with infinite rank and infinite corank since as it turns out in this generality our preservers cannot be written in a nice form what we have got used to when dealing with linear preserver problems hence we restrict our attention to certain important classes of operators like idempotents or projections or partial isometries we conclude the paper with a result on the form of linear maps which preserve the left ideals in b h

partially connected models of neural networks a partially connected hopfield neural network model is studied under the restriction that w the ratio of connections per site to the size of the system remains finite as the size n to infinity with the connection structure at each site being the same the replica symmetric mean field theory equations for the order parameters are derived the zero temperature forms of these equations are then solved numerically for a few different local connectivity architectures showing phase transitions at different critical storage ratios alpha c where the states which the authors are trying to store in the network become discontinuously unstable they show that the information capacity per connection improves for partially connected systems

1 dimensional attractor neural networks we solve a class of attractor neural network models with a mixture of 1d nearest neighbour interactions and infinite range interactions which are both of a hebbian type form our solution is based on a combination of mean field methods transfer matrices and 1d random field techniques and is obtained both for boltzmann type equilibrium following sequential glauber dynamics and peretto type equilibrium following parallel dynamics competition between the alignment forces mediated via short range interactions and those mediated via infinite range ones is found to generate novel phenomena such as multiple locally stable pure states first order transitions between recall states 2 cycles and non recall states and domain formation leading to extremely long relaxation times we test our results against numerical simulations and simple benchmark cases and find excellent agreement

dynamics of three state excitable units on poisson vs power law random networks abstract the influence of the network topology on the dynamics of systems of coupled excitable units is studied numerically and demonstrates a lower dynamical variability for power law networks than for poisson ones this effect which reflects a robust collective excitable behavior is however lower than that observed for diffusion processes or network robustness instead the presence and number of triangles and larger loops in the networks appears as a parameter with strong influence on the considered dynamics