storage of sets of correlated data in neural network memories in most applications of neural network models one has to consider sets of correlated data the authors study the problem of the storage of associated patterns in neural network memories they consider two basic types of correlation semantic ones among the different patterns and spatial ones among the different sites of the network they apply gardner s program to evaluate optimal storage conditions for both kinds of correlation the spatial ones worsen generally speaking the storage properties of a simple perceptron while they improve them for the hopfield network in the case of semantic correlations they obtain bounds of the critical capacity for both kinds of networks the storage ratio of the perceptron may be significantly increased in this case

energy landscape of neural networks storing spatially correlated patterns hopfield like neural networks with spatially organized data are studied by a mean field theory the internal structure of the data is described by a matrix c whose elements cij are equal to the correlation between two pixels i and j of any input pattern the model considered here is described by the matrix c in which the pixel pixel correlation is the same for all pairs of pixels and is equal to lambda n the statistical properties of the model depend on three parameters the reduced number of the stored patterns alpha the temperature t and the reduced number of strength correlations of the pixels lambda the phase diagram in the space of parameters lambda and alpha at temperature t 0 is obtained the network can retrieve patterns at t 0 for alpha alpha c where alpha c approximately 0 14 as for the usual hopfield neural network but there is a new transition line above which a new local minimum of the free energy arises this minimum corresponds to a ferromagnetic ordering of the neurons there is another additional minimum between the next two lines that corresponds to mixed ordering we also find the region where the ferromagnetic state becomes the ground state of the system

critical storage capacity of the j 1 neural network for neural networks in which the coupling jij are allowed to take on the values jij 1 or jij 1 the authors determine numerically the critical storage capacity for random unbiased patterns as a function of the stability they use an exact enumeration scheme based on the gray code and a continuous distribution for the patterns to control finite size effects results are presented for n 25 they indicate an optimal storage capacity of c 0 82 n

almost maximal growth of the hilbert function abstract let a s j be a standard artinian graded algebra over the polynomial ring s a theorem of macaulay dictates the possible growth of the hilbert function of a from any degree to the next and if this growth is the maximal possible then strong consequences have been given by gotzmann it can be phrased in terms of the base locus of the linear system defined by the relevant component s of j if j is the artinian reduction of the ideal of a finite set of points in projective space then this maximal growth for a was shown by bigatti geramita and the second author to imply strong geometric consequences for the points we now suppose that the growth of the hilbert function is one less than maximal this again has not as strong consequences for the base locus defined by the relevant component and when j is the artinian reduction of the ideal of a finite set of points in projective space we prove that almost maximal growth again forces geometric consequences

properties of neural networks storing spatially correlated patterns the author studies the behaviour of a feedforward neural network supplied with spatially organized data this inner structure is taken into account by a matrix cij whose coefficients equal the average correlation between two pixels i and j of the input patterns the storage capacity alpha is computed as a function of the required stability and of the eigenvalues of c the author proposes a geometrical transformation allowing an intuitive interpretation of these results numerical simulations using real and binary patterns show a very good agreement with the theory finally the author analyses the synaptic couplings correlations resulting from the training of the network with structured patterns focusing on exponentially decreasing correlations one and two dimensions the author finds that they exhibit a mexican hat profile the excitatory centre size of which depends on alpha

content addressability and learning in neural networks the content addressability of patterns stored in ising spin neural network models with symmetric interactions is studied numerical results from simulations on the icl distributed array processor dap involving systems with up to 2048 neurons are presented behaviour consistent with finite size scaling characteristic of a first order phase transition is shown to be exhibited by the basins of attraction of the stored patterns both in the case of the hopfield model and for systems using a local iterative learning algorithm designed to optimise the basins of attraction estimates are obtained for the critical minimum overlaps which an input pattern must have with a stored pattern in order to successfully retrieve it

retrieval phase diagrams for attractor neural networks with optimal interactions the authors consider the retrieval properties of attractor neural networks whose synaptic matrices have been constructed to maximise the number of patterns which can be stored in a perceptron satisfying certain constraints retrieval is studied in the absence as well as in the presence of fast noise temperature the discussion is restricted to dilute networks for which dynamical equations for the overlaps are available when the patterns are stored with a prescribed lower limit on the stability parameter on every site the full stability of the perceptron ensures the existence of an attractor with perfect retrieval it is found that the curve of critical storage capacity alpha against temperature t as a line of first order transitions for high values of alpha and becomes second order for low alpha at a point of a tricritical nature the phase diagram is compared with the dilute hopfield model it is found that at high synaptic noise levels the diluted hopfield net stores more effectively than the network trained for optimal perceptron storage when a given fraction of sites is allowed to violate the stability bound the solution of the perceptron learning problem does not ensure the existence of an attractor of finite overlap even in the absence of noise this case is studied separately for t 0 and for finite t

linear readout of object manifolds objects are represented in sensory systems by continuous manifolds due to sensitivity of neuronal responses to changes in physical features such as location orientation and intensity what makes certain sensory representations better suited for invariant decoding of objects by downstream networks we present a theory that characterizes the ability of a linear readout network the perceptron to classify objects from variable neural responses we show how the readout perceptron capacity depends on the dimensionality size and shape of the object manifolds in its input neural representation

learning from correlated examples in a perceptron using the replica method the authors investigate learning from correlated data in a perceptron both types of association spatial within each example and semantic among the different ones are considered for both boolean and continuous weights and output functions spatial correlations of data may significantly improve the learning properties whereas the semantic ones do not practically influence them

optimally adapted attractor neural networks in the presence of noise by adapting an attractor neural network to an appropriate training overlap the authors optimize its attractor overlap and subsequently the storage capacity when retrieval noise temperature is present in the system the training overlap is determined self consistently by the optimal attractor overlap the phase diagram of the optimal attractor overlap in the temperature storage space is found a novel co existence phase of strong and weak retrievers is present the maximum storage capacity deviates from the storage capacity of the maximally stable network on increasing temperature and in the high temperature regime t or 0 38 for gaussian noise the hopfield network yields the maximum storage capacity this analysis demonstrates the principles of specialization and adaptation in neural networks

finite size effects and bounds for perceptron models ablncl in lhis paper we consider two main aspsln of the binary perccptron problem the maximal capacity when random paltem are stored model a and its generalhation ability model b we have extended previous numerical estimates of critical capacilies and studied thermal properties of systems of small sizes to lest recent replica predictions we have also considered some simpler versions of thest models the discrete spherical vemions can be salved exactly using gardner s replica calculation for the spherical model and are shown to give a rigorous upper bound and lower bound on the capacities of models a and b respectively iby vemions of models a and b are soivcd in detail and provide information which is useful for interpreting the bnile size eeeciects present in the numerical studies of models a and b

on the first infinitesimal neighborhood of a linear configuration of points in p2 abstract we consider the following open questions fix a hilbert function h that occurs for a reduced zero dimensional subscheme of p 2 among all subschemes x with hilbert function h what are the possible hilbert functions and graded betti numbers for the first infinitesimal neighborhood z of x i e the double point scheme supported on x is there a minimum h min and maximum h max such function the numerical information encoded in h translates to a type vector which allows us to find unions of points on lines called linear configurations with hilbert function h we give necessary and sufficient conditions for the hilbert function and graded betti numbers of the first infinitesimal neighborhoods of all such linear configurations to be the same even for those h for which the hilbert functions or graded betti numbers of the resulting double point schemes are not uniquely determined we give one depending only on h that does occur we prove the existence of h max in general and discuss h min our methods include liaison techniques

on the uniformity of zero dimensional complete intersections abstract after showing that the general cayley bacharach conjecture formulated by d eisenbud m green and j harris 1996 6 is equivalent to a conjecture about the region of uniformity of a zero dimensional complete intersection we prove this conjecture in a number of special cases in particular after splitting the conjecture into several intervals we prove it for the first the last and part of the penultimate interval moreover we generalize the uniformity results of j hansen 2003 12 and l gold j little and h schenck 2005 9 to level schemes and apply them to obtain bounds for the minimal distance of generalized reed muller codes

hilbert functions and level algebras abstract the authors consider certain quotients a r i of the polynomial ring r k x1 xr over an arbitrary field k they first determine upper and lower bounds on the hilbert functions of any algebra having the form a r v where v is the largest ideal of r agreeing in degrees at least j with the ideal v generated by a vector subspace v rj of degree j forms these bounds extend the macaulay bounds theorem 1 4 a level artinian quotient of a r i has socle soc a 0 m m x1 xr in a single degree j the authors next determine the extremal hilbert functions hmax t j r and hmin t j r that occur for level graded artin algebra quotients of r having socle degree j and type t dimk soc a and they describe the extremal strata theorem 1 8 they next give a natural upper bound for the hilbert function of level algebra quotients of the coordinate ring o z of a punctual subscheme z p n n r 1 in terms of the hilbert function h z and j t this bound was known and known to be sharp in the case z is locally gorenstein and t 1 finally they show that there are no level algebras of hilbert function t 1 3 4 5 6 2 proposition 2 7 given that there are smooth punctual subschemes z of p 2 with hilbert function h z 1 3 4 5 6 6 this result shows that the sharpness of the natural upper bound in the case where z is gorenstein and t 1 does not extend to level algebra quotients of type 2 even when z is smooth the extremality results for the hilbert functions of level algebras and some counterexamples use the extremality theorem of f h s macaulay 1927 proc london math soc 26 531 555 and g gotzmann s 1978 math z 158 61 70 results on the hilbert scheme

phase transitions in the generalization behaviour of multilayer perceptrons ii the influence of noise we extend our study of phase transitions in the generalization behaviour of multilayer perceptrons with non overlapping receptive fields to the problem of the influence of noise concerning for example the input units and or the couplings between the input units and the hidden units of the second layer input noise or the final output unit output noise without output noise the output itself is given by a general permutation invariant boolean function of the outputs of the hidden units as a result we find that the phase transitions which we found in the deterministic case mostly persist in the presence of noise the influence of the noise on the position of the phase transition as well as on the behaviour in other regimes of the loading parameter can often be described by a simple rescaling of depending on the strength and type of the noise we then consider the problem of the optimal noise level for gibbsian and bayesian learning looking at replica symmetry breaking as well finally we consider the question of why learning with errors is useful at all

estimates of optimal storage conditions in neural network memories based on random matrix theory the authors formulate a method for estimating the critical conditions for storage of sets of data in neural network memory this variational method is based on random matrix theory and calculating the average spectrum of a matrix whose elements are given by overlaps of the stored patterns several generic cases of random overlap matrices are considered they investigate the cases of simply uncorrelated random patterns and spatially and semantically correlated ones they obtain bounds of the critical curve in the control parameters space which determine the stability of the stored data sets

the gotzmann coefficients of hilbert functions abstract in this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the hilbert function of the general hyperplane section of a variety green s hyperplane restriction theorem these geometric consequences improve some results in this direction first given by green and extend others by bigatti geramita and migliore other applications of our detailed investigation of how the hilbert polynomial is written as a sum of binomials are to conditions that must be satisfied by a polynomial if it is to be the hilbert polynomial of a non degenerate integral subscheme of p n a problem posed by r p stanley we also give some new restrictions on the hilbert function of a zero dimensional reduced scheme with the uniform position property

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

generalization from educated teachers the authors study the generalization ability of a perceptron as a function of the structural similarity between teacher and student to this end they consider a teacher perceptron b designed to implement a certain number of random input output mappings and a student perceptron bold j trained with input output examples generated by the teacher if the pattern sets of teacher and student overlap different combinations of teacher and student learning rules result in different generalization probabilities generalization can be facilitated or can be hindered depending on the combination of learning rules and the storage levels of teacher and student analytical calculations of the generalization ability for several combinations of teacher and student learning rules are in very good agreement with numerical simulations

storage of hierarchically correlated patterns the maximal storage capacity for hierarchically correlated patterns is calculated without using an explicit learning rule if the lowest level branching ratio tends to infinity strong correlations increase the storage capacity otherwise they decrease it significantly there is probably no enlargement of the typical basins of attraction due to the hierarchical organisation the author considers both two level hierarchies and those with infinitely many levels

on the identifiability of ternary forms abstract we describe a new method to determine the minimality and identifiability of a waring decomposition a of a specific form symmetric tensor t in three variables our method which is based on the hilbert function of a can distinguish between forms in the span of the veronese image of a which in general contains both identifiable and not identifiable points depending on the choice of coefficients in the decomposition this makes our method applicable for all values of the length r of the decomposition from 2 up to the generic rank a range which was not achievable before though the method in principle can handle all cases of specific ternary forms we introduce and describe it in details for forms of degree 8

on the genus and hartshorne rao module of projective curves in this paper optimal upper bounds for the genus and the dimension of the graded components of the hartshorne rao module of curves in projective n space are established this generalizes earlier work by hartshorne h and martin deschamps and perrin mdp special emphasis is put on curves in bf p 4 the first main result is a so called restriction theorem it says that a non degenerate curve of degree d geq 4 in bf p 4 over a field of characteristic zero has a non degenerate general hyperplane section if and only if it does not contain a planar curve of degree d 1 see th 1 3 then using methods of brodmann and nagel bounds for the genus and hartshorne rao module of curves in bf p n with non degenerate general hyperplane section are derived it is shown that these bounds are best possible in a very strict sense coupling these bounds with the restriction theorem gives the second main result for curves in bf p 4 then curves of maximal genus are investigated the betti numbers of their minimal free resolutions are computed and a description of all reduced curves of maximal genus in bf p n of degree geq n 2 is given finally all pairs d g of integers which really occur as the degree d and genus g of a non degenerate curve in bf p 4 are described

generalizing with perceptrons in the case of structured phase and pattern spaces we investigate the influence of different kinds of structures on the learning behaviour of a perceptron performing a classification task defined by a teacher rule the underlying pattern distribution is permitted to have spatial correlations the prior distribution for the teacher coupling vectors itself is assumed to be nonuniform thus classification tasks of quite different difficulty are included as learning algorithms we discuss hebbian gibbs and bayesian learning with different priors using methods from statistics and the replica formalism we find that the hebb rule is quite sensitive to the structure of the actual learning problem failing asymptotically in most cases in contrast the behaviour of the more sophisticated methods of gibbs and bayes learning is influenced by the spatial correlations only in an intermediate regime of where specifies the size of the training set in view of the bayesian case we show how enhanced prior knowledge improves the performance

engel elements in the homotopy lie algebra an engel element in a graded lie algebra l over a field k is an element x l such that ad x is locally nilpotent the linear span of these elements is a graded subspace e l the depth of l is the infinum possibly of the set of integers m for which extulm k ul 0 it is known that if l is the rational homotopy lie algebra of a simply connected space x or the homotopy lie algebra of a local noetherian ring a then the depth of l is bounded above respectively by the lusternik schnirelmann category of x and the embedding dimension of a theorem if l is concentrated in degrees 0 or in degrees 0 and if depth l m then there are at most m linearly independent engel elements of even degree idimke2i depthl

towards a theory of early visual processing we propose a theory of the early processing in the mammalian visual pathway the theory is formulated in the language of information theory and hypothesizes that the goal of this processing is to recode in order to reduce a generalized redundancy subject to a constraint that specifies the amount of average information preserved in the limit of no noise this theory becomes equivalent to barlow s redundancy reduction hypothesis but it leads to very different computational strategies when noise is present a tractable approach for finding the optimal encoding is to solve the problem in successive stages where at each stage the optimization is performed within a restricted class of transfer functions we explicitly find the solution for the class of encodings to which the parvocellular retinal processing belongs namely linear and nondivergent transformations the solution shows agreement with the experimentally observed transfer functions at all levels of signal to noise

phase transitions in the generalization behaviour of multilayer neural networks we study the generalization ability of multilayer neural networks with tree architecture for the case of random training sets the first layer consists of k spherical perceptrons with binary output a boolean function b computes the final output from the k values produced by the first layer we first calculate the learning behaviour of gibbs learning in the case of learnable rules where teacher and student have the same architecture and the boolean function b is permutation symmetric with respect to the hidden units in the asymptotic case of high loading alpha to infinity as usual alpha is the loading parameter we find that the generalization error vanishes in the same way for all b the reason being is that there are two effects cancelling each other in the opposite limit for small alpha we find qualitatively different behaviour i e some networks undergo a phase transition we show how these differences in the behaviour depend on certain characteristics of the boolean function b we then study the bayes algorithm which generalizes according to the majority decision of a certain ensemble of machines and find that the parity function plays a special role if the teacher is a parity machine there always exists a single student parity machine that generalizes as well as the bayes algorithm

on the capacity of neural networks with binary weights the authors study the critical capacity alpha c of multilayered networks with binary couplings they show that for any network presenting a tree like architecture after the first hidden layer no fixed internal representation is required using gardner s calculations they apply statistical mechanics to the simplest network with two layers of adaptive weights following the same approach as for the binary perceptron they find from the zero entropy point a critical capacity alpha c 0 92 they discuss the validity of this result in view of exhaustive search simulations on small networks

duality between learning machines a bridge between supervised and unsupervised learning we exhibit a duality between two perceptrons that allows us to compare the theoretical analysis of supervised and unsupervised learning tasks the first perceptron has one output and is asked to learn a classification of p patterns the second dual perceptron has p outputs and is asked to transmit as much information as possible on a distribution of inputs we show in particular that the maximum information that can be stored in the couplings for the supervised learning task is equal to the maximum information that can be transmitted by the dual perceptron

storage of correlated patterns in a perceptron we calculate the storage capacity of a perceptron for correlated gaussian patterns we find that the storage capacity alpha c can be less than 2 if similar patterns are mapped onto different outputs and vice versa as long as the patterns are in a general position we obtain in contrast to previous works that alpha c or 1 in agreement with cover s theorem numerical simulations confirm the results

the weak and strong lefschetz properties for artinian k algebras abstract let a i 0ai be a standard graded artinian k algebra where chark 0 then a has the weak lefschetz property if there is an element l of degree 1 such that the multiplication l a i a i 1 has maximal rank for every i and a has the strong lefschetz property if l d a i a i d has maximal rank for every i and d the main results obtained in this paper are the following 1 every height three complete intersection has the weak lefschetz property our method surprisingly uses rank two vector bundles on p 2 and the grauert mulich theorem 2 we give a complete characterization including a concrete construction of the hilbert functions that can occur for k algebras with the weak or strong lefschetz property and the characterization is the same one 3 we give a sharp bound on the graded betti numbers achieved by our construction of artinian k algebras with the weak or strong lefschetz property and fixed hilbert function this bound is again the same for both properties some hilbert functions in fact force the algebra to have the maximal betti numbers 4 every artinian ideal in k x y possesses the strong lefschetz property this is false in higher codimension

the radical of the homotopy lie algebra introduction let t be a simply connected cw complex of finite type its homotopy lie algebra is the graded rational lie algebra 7r 9 t 0 q equipped with the samelson product let a be a local noetherian ring with residue field k of arbitrary characteristic its homotopy lie algebra 7r a is the naturally defined graded lie algebra whose enveloping algebra is exta k k cf av these graded lie algebras have finite type i e are finite dimensional in each degree and are concentrated in strictly positive degrees thus for the sake of simplicity we shall use graded lie algebra to mean graded lie algebra of finite type concentrated in strictly positive degrees throughout this paper on the other hand these graded lie algebras are usually nonzero in infinitely many degrees thus while we may and do form the sum r of all the solvable ideals in such a graded lie algebra and call it the radical it may well happen that r itself is not solvable indeed this does happen in the topological context since by quillen s result q every graded rational lie algebra of finite type occurs as the homotopy lie algebra of a space by contrast in the case of local rings or of cw complexes with finite lusternik schnirelmann l s category serious restrictions apply to the homotopy lie algebra indeed our first main result is the remarkable

generalization from correlated sets of patterns in the perceptron generalization is a central aspect of learning theory here we proposea 13 a framework that explores an auxiliary task dependent notion of gener a 13 alization and attempts to quantitatively answer the following question a 13 given two sets of patterns with a given degree of dissimilarity how easilya 13 will a network be able to unify their interpretation this is quantifieda 13 by the volume of the configurations of synaptic weights that classify thea 13 two sets in a similar manner to show the applicability of our idea ina 13 a concrete setting we compute this quantity for the perceptron a sim a 13 ple binary classifier using the classical statistical physics approach in thea 13 replica symmetric ansatz in this case we show how an analytical ex a 13 pression measures the distance based capacity the maximum load ofa 13 patterns sustainable by the network at fixed dissimilarity between pat a 13 terns and fixed allowed number of errors this curve indicates that gen a 13 eralization is possible at any distance but with decreasing capacity wea 13 propose that a distance based definition of generalization may be usefula 13 in numerical experiments with real world neural networks and to explorea 13 computationally sub dominant sets of synaptic solutions

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

learning algorithms with optimal stability in neural networks to ensure large basins of attraction in spin glass like neural networks of two state elements xi imu or 1 the authors propose to study learning rules with optimal stability delta where delta is the largest number satisfying delta or sigma j jij xi jmu xi imu mu 1 p i 1 n where n is the number of neurons and p is the number of patterns they motivate this proposal and provide optimal stability learning rules for two different choices of normalisation for the synaptic matrix jij in addition numerical work is presented which gives the value of the optimal stability for random uncorrelated patterns

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

a new approach to recognition of deformed patterns abstract by introducing a distance between original and deformed patterns and an adequate cost function we derive a gradient dynamics for transformation functions that transform one pattern into another one in its simplest form the cost function is lent from an elastic medium but in further improvements the corresponding lame constants are chosen pattern dependent in the case of line patterns a smearing out by means of a gaussian distribution is introduced and made time dependent the numerical simulations show the feasibility of the present approach for instance for the recognition of handwritten characters or of faces showing different facial expressions

on the ability of the optimal perceptron to generalise a linearly separable boolean function is derived from a set of examples by a perceptron with optimal stability the probability to reconstruct a pattern which is not learnt is calculated analytically using the replica method

learning drifting concepts with neural networks the learning of time dependent concepts with a neural network is studied analytically and numerically the linearly separable target rule is represented by an n vector whose time dependence is modelled by a random or deterministic drift process a single layer network is trained online using different hebb like algorithms training is based on examples which are chosen randomly and according to a query strategy the evolution of the generalization error can be calculated exactly in the thermodynamic limit n to infinity the rule is never learnt perfectly but can be tracked within a certain error margin the generalization performance of various learning rules is compared and simulations confirm the analytic results

bayes optimal performance in a discrete space we study a simple model of unsupervised learning where the single symmetry breaking vector has binary components 1 we calculate exactly the bayes optimal performance of an estimator which is required to lie in the same discrete space we also show that except for very special cases such an estimator cannot be obtained by minimization of a class of variationally optimal potentials

optimal storage of invariant sets of patterns in neural network memories the authors investigate optimal conditions for the storage of invariant sets of patterns in neural network memories the sets of patterns that they consider are highly correlated non random and invariant with respect to some symmetry transformations they generalize gardner s method to study the fractional volume in the space of neural interactions that allow for storage of invariant patterns they demonstrate that optimal storage conditions correspond to non trivial relations between the number of stored patterns their stability and other symmetry specific characteristics of patterns

structured patterns retrieval using a metric attractor network application to fingerprint recognition abstract the ability of a metric attractor neural networks mann to learn structured patterns is analyzed in particular we consider collections of fingerprints which present some local features rather than being modeled by random patterns the network retrieval proved to be robust to varying the pattern activity the threshold strategy the topological arrangement of the connections and for several types of noisy configuration we found that the lower the fingerprint patterns activity is the higher the load ratio and retrieval quality are a simplified theoretical framework for the unbiased case is developed as a function of five parameters the load ratio the finiteness connectivity the density degree of the network randomness ratio and the spatial pattern correlation linked to the latter appears a new neural dynamics variable the spatial neural correlation the theory agrees quite well with the experimental results

capacity of neural networks with discrete synaptic couplings the authors study the optimal storage capacity of neural networks with discrete local constraints on the synaptic couplings jij models with such constraints include those with binary couplings jij or 1 or jij 0 1 quantised couplings with larger synaptic range e g jij or 1 l or 2 l or 1 and in the limit continuous couplings confined to the hypercube mod jij mod or 1 box confinement they find that the optimal storage capacity alpha kappa is best determined by the vanishing of a suitably defined entropy as calculated in the replica symmetric approximation they also extend their results to cases with biased memories and make contact with sparse coding models

generalization performance of bayes optimal classification algorithm for learning a perceptron the generalization error of the bayes optimal classification algorithm when learning a perceptron from noise free random training examples is calculated exactly using methods of statistical mechanics it is shown that if an assumption of replica symmetry is made then in the thermodynamic limit the error of the bayes optimal algorithm is less than the error of a canonical stochastic learning algorithm by a factor approaching ensuremath surd 2 as the ratio of the number of training examples to perceptron weights grows in addition it is shown that approximations to the generalization error of the bayes optimal algorithm can be achieved by learning algorithms that use a two layer neutral net to learn a perceptron

dilution in boolean perceptrons that learn from noisy examples we investigate the effect of dilution after learning on the generalization ability of single layer boolean perceptrons that learn from noisy examples we present a thorough comparison between the relative performances of several well known learning rules in particular we show that the effect of dilution is always deleterious and that the bayes algorithm always gives the best generalization performance

geometric consequences of extremal behavior in a theorem of macaulay f s macaulay gave necessary and sufficient conditions on the growth of a nonnegative integer valued function which determine when such a function can be the hilbert function of a standard graded k algebra we investigate some algebraic and geometric consequences which arise from the extremal cases of macaulay s theorem our work also builds on the fundamental work of g gotzmann our principal applications are to the study of hilbert functions of zeroschemes with uniformity conditions as a consequence we have new strong limitations on the possible hilbert functions of the points which arise as a general hyperplane section of an irreducible curve