an almost sure large deviation principle for the hopfield model we prove a large deviation principle for the finite dimensional marginals of the gibbs distribution of the macroscopic overlap parameters in the hopfield model in the case where the number of random patterns m as a function of the system size n satisfies lim sup m n n 0 in this case the rate function is independent of the disorder for almost all realizations of the patterns

deformed preprojective algebras of generalized dynkin type we introduce the class of deformed preprojective algebras of generalized dynkin graphs an n 1 d n n 4 eg e 7 eg and l n n 1 and prove that it coincides with the class of all basic connected finite dimensional self injective algebras for which the inverse nakayama shift v 1 s of every non projective simple module s is isomorphic to its third syzygy 3 s

hyper sparse optimal aggregation given a finite set f of functions and a learning sample the aim of an aggregation procedure is to have a risk as close as possible to risk of the best function in f up to now optimal aggregation procedures are convex combinations of every elements of f in this paper we prove that optimal aggregation procedures combining only two functions in f exist such algorithms are of particular interest when f contains many irrelevant functions that should not appear in the aggregation procedure since selectors are suboptimal aggregation procedures this proves that two is the minimal number of elements of f required for the construction of an optimal aggregation procedure in every situations then we perform a numerical study for the problem of selection of the regularization parameters of the lasso and the elastic net estimators we compare on simulated examples our aggregation algorithms to aggregation with exponential weights to mallow s cp and to cross validation selection procedures

resolutions of mesh algebras periodicity and calabi yau dimensions a triangulated category mathcal t sigma is said to be calabi yau of dimension d if d is a serre functor we determine which stable module categories of self injective algebras of finite type are calabi yau and compute their calabi yau dimensions correcting errors in previous work we first show that the calabi yau property of mod can be detected in the minimal projective resolution of the stable auslander algebra of over its enveloping algebra we then describe the beginning of such a minimal resolution for any mesh algebra of a stable translation quiver and apply covering theory to relate these minimal resolutions to those of the generalized preprojective algebras of dynkin graphs for representation finite self injective algebras of torsion order t 1 we obtain a complete description of their stable calabi yau properties but only partial results for those algebras of torsion order t 2 we also obtain some new information about the periods of the representation finite self injective algebras of torsion order t 1 finally we describe how these questions can also be approached by realizing the stable categories of representation finite self injective algebras as orbit categories of the bounded derived categories of hereditary algebras and illustrate this technique with several explicit computations that our previous methods left unsettled

gibbs states of the hopfield model with extensively many patterns we consider the hopfield model withm n n patterns wheren is the number of neurons we show that if is sufficiently small and the temperature sufficiently low then there exist disjoint gibbs states for each of the stored patterns almost surely with respect to the distribution of the random patterns this solves a provlem left open in previous work the key new ingredient is a self averaging result on the free energy functional this result has considerable additional interest and some consequences are discussed a similar result for the free energy of the sherrington kirkpatrick model is also given

quiver varieties and lusztig s algebra abstract we study preprojective algebras of graphs and their relationship to module categories over representations of quantum sl 2 as an application ade quiver varieties of nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by lusztig

moderate deviations for the overlap parameter in the hopfield model we derive moderate deviation principles for the overlap parameter in the hopfield model of spin glasses and neural networks if the inverse temperature is different from the critical inverse temperature c 1 and the number of patterns m n satisfies m n n 0 the overlap parameter multiplied by n 1 2 1 obeys a moderate deviation principle with speed n1 2 and a quadratic rate function i e the gaussian limit for 1 2 remains visible on the moderate deviation scale at the critical temperature we need to multiply the overlap parameter by n 1 4 1 if then m n satisfies m n 6 log n m n 2n4 log n n 0 the rescaled overlap parameter obeys a moderate deviation principle with speed n1 4 and a rate function that is basically a fourth power the random term occurring in the central limit theorem for the overlap at c 1 is no longer present on a moderate deviation scale if the scaling is even closer to n1 4 e g if we multiply the overlap parameter by n1 4 log log n the moderate deviation principle breaks down the case of variable temperature converging to one is also considered if n converges to c fast enough i e faster than open image in new window the non gaussian rate function persists whereas for n converging to one slower than open image in new window the moderate deviations principle is given by the gaussian rate at the borderline the moderate deviation rate function is the one at criticality plus an additional gaussian term

the symmetry period and calabi yau dimension of finite dimensional mesh algebras abstract within the class of finite dimensional mesh algebras we identify those which are symmetric and those whose stable module category is calabi yau we also give in combinatorial terms explicit formulas for the period of any such algebra where is the syzygy functor and for the calabi yau frobenius and the stable calabi yau dimensions when they are defined

gibbs states of the hopfield model in the regime of perfect memory we study the thermodynamic properties of the hopfield model of an autoassociative memory ifn denotes the number of neurons andm n the number of stored patterns we prove the following results ifm n 0 asn then there exists an infinite number of infinite volume gibbs measures for all temperaturest 1 concentrated on spin configurations that have overlap with exactly one specific pattern moreover the measures induced on the overlap parameters are dirac measures concentrated on a single point and the gibbs measures on spin configurations are products of bernoulli measures ifm n asn for small enough we show that for temperaturest smaller than somet 1 the induced measures can have support only on a disjoint union of balls around the previous points but we cannot construct the infinite volume measures through convergent sequences of measures

from ɛ entropy to kl entropy analysis of minimum information complexity density estimation we consider an extension of e entropy to a kl divergence based complexity measure for randomized density estimation methods based on this extension we develop a general information theoretical inequality that measures the statistical complexity of some deterministic and randomized density estimators consequences of the new inequality will be presented in particular we show that this technique can lead to improvements of some classical results concerning the convergence of minimum description length and bayesian posterior distributions moreover we are able to derive clean finite sample convergence bounds that are not obtainable using previous approaches

noncommutative geometry and quiver algebras abstract we develop a new framework for noncommutative differential geometry based on double derivations this leads to the notion of moment map and of hamiltonian reduction in noncommutative symplectic geometry for any smooth associative algebra b we define its noncommutative cotangent bundle t b which is a basic example of noncommutative symplectic manifold applying hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras b that includes preprojective algebras associated with quivers our formalism of noncommutative hamiltonian reduction provides the space with a lie algebra structure analogous to the poisson bracket on the zero fiber of the moment map in the special case where is the preprojective algebra associated with a quiver of non dynkin type we give a complete description of the gerstenhaber algebra structure on the hochschild cohomology of in terms of the lie algebra

adaptive regression by mixing adaptation over different procedures is of practical importance different procedures perform well under different conditions in many practical situations it is rather hard to assess which conditions are approximately satisfied so as to identify the best procedure for the data at hand thus automatic adaptation over various scenarios is desirable a practically feasible method named adaptive regression by mixing arm is proposed to convexly combine general candidate regression procedures under mild conditions the resulting estimator is theoretically shown to perform optimally in rates of convergence without knowing which of the original procedures work the best simulations are conducted in several settings including comparing a parametric model with nonparametric alternatives comparing a neural network with a projection pursuit in multidimensional regression and combining bandwidths in kernel regression the results clearly support the theoretical property of arm the arm algorithm assigns we

the product in the hochschild cohomology ring of preprojective algebras of dynkin quivers abstract in this paper we compute the cup product structure of the preprojective algebra dynkin quivers of type d and e over a field of characteristic zero this is a continuation of the work done in p etingof c eu hochschild and cyclic homology of preprojective algebras of ade quivers arxiv math ag 0609006 where the additive structure of the hochschild cohomology together with its grading was computed together with the results in k erdmann n snashall on hochschild cohomology of preprojective algebras i j algebra 205 2 1998 391 412 ii j algebra 205 2 1998 413 434 where the a case was studied this yields a complete description of the product in the hochschild cohomology of ade quivers over a field of characteristic zero

bv structure on hochschild cohomology of the group ring of quaternion group of order eight in characteristic two abstract let k be an algebraically closed field of characteristic two and let q 8 be the quaternion group of order 8 we determine the gerstenhaber lie algebra structure and the batalin vilkovisky structure on the hochschild cohomology ring of the group algebra k q 8

the hochschild cohomology ring of a frobenius algebra with semisimple nakayama automorphism is a batalin vilkovisky algebra abstract in analogy with a recent result of n kowalzig and u krahmer for twisted calabi yau algebras we show that the hochschild cohomology ring of a frobenius algebra with semisimple nakayama automorphism is a batalin vilkovisky algebra thus generalizing a result of t tradler for finite dimensional symmetric algebras we give a criterion to determine when a frobenius algebra given by quiver with relations has semisimple nakayama automorphism and apply it to some known classes of tame frobenius algebras we also provide ample examples including quantum complete intersections finite dimensional hopf algebras defined over an algebraically closed field of characteristic zero and the koszul duals of koszul artin schelter regular algebras of dimension three

replica symmetry breaking and exponential inequalities for the sherrington kirkpatrick model we provide an extremely accurate picture of the sherrington kirkpatrick model in three cases for high temperature for large external field and for any temperature greater than or equal to 1 and sufficiently small external field we describe the system at the level of the central limit theorem or as physicists would say at the level of fluctuations around the mean field we also obtain much more detailed information in the form of exponential inequalities that express a uniform control over higher order moments we give a complete rigorous proof that at the generic point of the predicted low temperature region there is replica symmetry breaking in the sense that the system is unstable with respect to an infinitesimal coupling between two replicas

central extensions of preprojective algebras the quantum heisenberg algebra and 2 dimensional complex reflection groups preprojective algebras of quivers were introduced in 1979 by gelfand and ponomarev gp because for quivers of finite ade type they are models for indecomposable representations they contain each indecomposable exactly once twenty years later these algebras and their deformed versions introduced in cbh for arbitrary quivers became a subject of intense interest since their representation varieties called quiver varieties played an important role in geometric representation theory ironically it is exactly for quivers of finite ade type that preprojective algebras fail to have good properties they are not koszul and their deformed versions are not flat

metastates in disordered mean field models ii the superstates we continue to investigate the size dependence of disordered mean field models with finite local spin space in more detail illustrating the concept of superstates as recently proposed by bovier and gayrard we discuss various notions of convergence for the behavior of the paths t tn t 0 1 in the thermodynamic limit n here n is the gibbs measure in the finite volume 1 n and is the disorder variable in particular we prove refined convergence statements in our concrete examples the hopfield model with finitely many patterns having continuous paths and the curie weiss random field ising model having singular paths

ordered overlaps in disordered mean field models let m n be a sequence of integers with m as n and m o n for bounded i i d r v i k and bounded i i d r v i we study the large deviation of the family of ordered scalar products x k n 1 i 1 n i i k k m under the distribution conditioned on the i k s to get a full large deviation principle it is necessary to specify also the total norm k m x k 2 1 2 which turns to be associated with some extra gaussian distribution our results apply to disordered mean field systems including generalized hopfield models in the regime of a sublinear number of patterns we build also a class of examples where this norm is the crucial order parameter

self averaging in a class of generalized hopfield models we prove the almost sure convergence to zero of the fluctuations of the free energy in a class of disordered mean field spin systems that generalize the hopfield model in two ways i multi spin interactions are permitted and ii the random variables xi imu describing the patterns can have arbitrary distributions with mean zero and finite 4 epsilon th moments the number of patterns m is allowed to be an arbitrary multiple of the system size this generalizes a previous result of bovier gayrard and picco for the standard hopfield model and improves a result of feng and tirozzi that required m to be a finite constant note that the convergence of the mean of the free energy is not proven

multiple critical behavior of probabilistic limit theorems in the neighborhood of a tricritical point we derive probabilistic limit theorems that reveal the intricate structure of the phase transitions in a mean field version of the blume emery griffiths model phys rev a4 1971 1071 1077 these probabilistic limit theorems consist of scaling limits for the total spin and moderate deviation principles mdps for the total spin the model under study is defined by a probability distribution that depends on the parameters n and k which represent respectively the number of spins the inverse temperature and the interaction strength the intricate structure of the phase transitions is revealed by the existence of 18 scaling limits and 18 mdps for the total spin these limit results are obtained as k converges along appropriate sequences n kn to points belonging to various subsets of the phase diagram which include a curve of second order points and a tricritical point the forms of the limiting densities in the scaling limits and of the rate functions in the mdps reflect the influence of one or more sets that lie in neighborhoods of the critical points and the tricritical point of all the scaling limits the structure of those near the tricritical point is by far the most complex exhibiting new types of critical behavior when observed in a limit theorem phase diagram in the space of the two parameters that parametrize the scaling limits

on hochschild cohomology of preprojective algebras i abstract we study the hochschild cohomology of a finite dimensional preprojective algebra this is periodic by a result of a schofield we determine the ring structure of the hochschild cohomology ring given by the yoneda product as a result we obtain an explicit presentation by generators and relations

model selection an integral part of inference we argue that model selection uncertainty should be fully incorporated into statistical inference whenever estimation is sensitive to model choice and that choice is made with reference to the data we consider different philosophies for achieving this goal and suggest strategies for data analysis we illustrate our methods through three examples the first is a poisson regression of bird counts in which a choice is to be made between inclusion of one or both of two covariates the second is a line transect data set for which different models yield substantially different estimates of abundance the third is a simulated example in which truth is known

a free energy bound for the hopfield model the author gives a simple upper and lower bound on the free energy density of the hopfield model of size n with p stored patterns in the limit where n and p tend to infinity with p n to alpha 1 the two bounds coincide for alpha 0

on central extensions of preprojective algebras we show that the hilbert polynomial p t of the trace space a a a of the centrally extended preprojective algebra a of an ade quiver is equal to the hilbert series of the maximal nilpotent subalgebra of the corresponding simple lie algebra under the principal gradation this implies that the hilbert polynomial of the center of a is t2h 4p 1 t where h is the coxeter number

lower bounds and aggregation in density estimation in this paper we prove the optimality of an aggregation procedure we prove lower bounds for aggregation of model selection type of m density estimators for the kullback leibler divergence kl the hellinger s distance and the l1 distance the lower bound with respect to the kl distance can be achieved by the on line type estimate suggested among others by yang 2000a combining these results we state that log m n is an optimal rate of aggregation in the sense of tsybakov 2003 where n is the sample size

rigorous results for the hopfield model with many patterns we perform a thorough investigation of the main aspects of the hopfield model with many patterns advances are made toward the validity of the replica symmetric solution strong evidence of the validity of this solution is given over the entire domain where this validity is conjectured complete proof is given in a subregion that contains strictly the ergodic region

large deviation principles for the hopfield model and the kac hopfield model we study the kac version of the hopfield model and prove a lebowitz penrose theorem for the distribution of the overlap parameters at the same time we prove a large deviation principle for the standard hopfield model with infinitely many patterns

thermodynamic limit of the q state potts hopfield model with infinitely many patterns we prove the almost sure convergence of the free energy and of the overlap order parameters in aq state version of the hopfield neural network model we compute explicitly these limits for all temperatures different from some critical value the number of stored patterns is allowed to grow with the size of the systemn like lnq lnn we study the limiting behavior of the extremal states of the model that are the measures induced on the gibbs measures by the overlap parameters

some rigorous results on the hopfield neural network model we analyze the thermal equilibrium distribution of 2p mean field variables for the hopfield model withp stored patterns in the case where 2p is small compared to the number of spins in particular we give a full description of the free energy density in the thermodynamic limit and of the so called symmetric solutions for the mean field equations

dualising complexes and twisted hochschild co homology for noetherian hopf algebras we show that many noetherian hopf algebras a have a rigid dualising complex r with r a1 d here d is the injective dimension of the algebra and is a certain k algebra automorphism of a unique up to an inner automorphism in honour of the finite dimensional theory which is hereby generalised we call the nakayama automorphism of a we prove that s2 where s is the antipode of a and is the left winding automorphism of a determined by the left integral of a the hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are shown to be non zero in the top dimension d when a is an artin schelter regular noetherian hopf algebra of global dimension d twisted poincare duality holds in this setting as is deduced from a theorem of van den bergh calculating for a using also the opposite coalgebra structure we determine a formula for s4 generalising a 1976 formula of radford for a finite dimensional applications of the results to the cases where a is pi an enveloping algebra a quantum group a quantised function algebra and a group algebra are outlined

the replica symmetric solution without replica trick for the hopfield model we derive the saddle point equations for the order parameters of the hopfield model in the case of replica symmetry without using the replica trick but assuming that the edwards anderson parameter is a self averaging quantity

information theory and mixing least squares regressions for gaussian regression we develop and analyze methods for combining estimators from various models for squared error loss an unbiased estimator of the risk of the mixture of general estimators is developed special attention is given to the case that the component estimators are least squares projections into arbitrary linear subspaces such as those spanned by subsets of explanatory variables in a given design we relate the unbiased estimate of the risk of the mixture estimator to estimates of the risks achieved by the components this results in simple and accurate bounds on the risk and its estimate in the form of sharp and exact oracle inequalities that is without advance knowledge of which model is best the resulting performance is comparable to or perhaps even superior to what is achieved by the best of the individual models furthermore in the case that the unknown parameter has a sparse representation our mixture estimator adapts to the underlying sparsity simulations show that the performance of these mixture estimators is better than that of a related model selection estimator which picks a model with the highest weight also the connection between our mixtures with bayes procedures is discussed

topological conformal field theories and calabi yau categories abstract this is the first of two papers which construct a purely algebraic counterpart to the theory of gromov witten invariants at all genera these gromov witten type invariants depend on a calabi yau a category which plays the role of the target in ordinary gromov witten theory when we use an appropriate a version of the derived category of coherent sheaves on a calabi yau variety this constructs the b model at all genera when the fukaya category of a compact symplectic manifold x is used it is shown under certain assumptions that the usual gromov witten invariants are recovered the assumptions are that open closed gromov witten theory can be constructed for x and that the natural map from the hochschild homology of the fukaya category of x to the ordinary homology of x is an isomorphism

sequential procedures for aggregating arbitrary estimators of a conditional mean in this correspondence a sequential procedure for aggregating linear combinations of a finite family of regression estimates is described and analyzed particular attention is given to linear combinations having coefficients in the generalized simplex the procedure is based on exponential weighting and has a computationally tractable approximation analysis of the procedure is based in part on techniques from the sequential prediction of nonrandom sequences here these techniques are applied in a stochastic setting to obtain cumulative loss bounds for the aggregation procedure from the cumulative loss bounds we derive an oracle inequality for the aggregate estimator for an unbounded response having a suitable moment generating function the inequality shows that the risk of the aggregate estimator is less than the risk of the best candidate linear combination in the generalized simplex plus a complexity term that depends on the size of the coefficient set the inequality readily yields convergence rates for aggregation over the unit simplex that are within logarithmic factors of known minimax bounds some preliminary results on model selection are also presented

higman ideal stable hochschild homology and auslander reiten conjecture let a and b be two finite dimensional algebras over an algebraically closed field related to each other by a stable equivalence of morita type we prove that a and b have the same number of isomorphism classes of simple modules if and only if their 0 degree hochschild homology groups hh0 a and hh0 b have the same dimension the first of these two equivalent conditions is claimed by the auslander reiten conjecture for symmetric algebras we will show that the auslander reiten conjecture is equivalent to other dimension equalities involving the centers and the projective centers of a and b this motivates our detailed study of the projective center which now appears to contain the main obstruction to proving the auslander reiten conjecture for symmetric algebras as a by product we get several new invariants of stable equivalences of morita type

calabi yau frobenius algebras abstract we define calabi yau frobenius algebras over arbitrary base commutative rings we define a hochschild analogue of tate cohomology and show that this stable hochschild cohomology of periodic cy frobenius algebras has a batalin vilkovisky and frobenius algebra structure such algebras include centrally extended preprojective algebras of generalized dynkin quivers and group algebras of classical periodic groups we use this theory to compute for the first time the hochschild cohomology of many algebras related to quivers and to simplify the description of known results furthermore we compute the maps on cohomology from extended dynkin preprojective algebras to the dynkin ones which relates our cy property for frobenius algebras to that of ginzburg for algebras of finite hochschild dimension

graph degree sequence solely determines the expected hopfield network pattern stability we analyze the effect of network topology on the pattern stability of the hopfield neural network in the case of general graphs the patterns are randomly selected from a uniform distribution we start the hopfield procedure from some pattern v an error in an entry e of v is the situation where if the procedure is started at e the value of e flips such an entry is an instability point note that we disregard the value at e by the end of the procedure as well as what happens if we start the procedure from another pattern or another entry of v we measure the instability of the system by the expected total number of instability points of all the patterns our main result is that the instability of the system does not depend on the exact topology of the underlying graph but rather only on its degree sequence moreover for a large number of nodes the instability can be approximated by where is the standard normal distribution function and are the degrees of the nodes

large deviation estimates for a conditional probability distribution applications to random interaction gibbs measures let x i y i d be independent identically distributed random variables with arbitrary distribution we show that for almost every y i i the conditional law of the empirical field given y i i satisfies to large deviation inequalities this applies to the study of gibbs measures with random interaction in the case of some mean field models as well as of short range summable interaction we show that the pressure is nonrandom and is given by a variational formula these random gibbs measures have the same large deviation rate which does not depend on the particular realization of the interaction their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula

mixing strategies for density estimation general results on adaptive density estimation are obtained with respect to any countable collection of estimation strategies under kullback leibler and squared l 2 losses it is shown that without knowing which strategy works best for the underlying density a single strategy can be constructed by mixing the proposed ones to be adaptive in terms of statistical risks a consequence is that under some mild conditions an asymptotically minimax rate adaptive estimator exists for a given countable collection of density classes that is a single estimator can be constructed to be simultaneously minimax rate optimal for all the function classes being considered a demonstration is given for high dimensional density estimation on 0 1 d where the constructed estimator adapts to smoothness and interaction order over some piecewise besov classes and is consistent for all the densities with finite entropy

rigorous results on the thermodynamics of the dilute hopfield model we study the hopfield model of an autoassociative memory on a random graph onn vertices where the probability of two vertices being joined by a link isp n assuming thatp n goes to zero more slowly thano 1 n we prove the following results 1 if the number of stored patternsm n is small enough such thatm n np n 0 asn then the free energy of this model converges upon proper rescaling to that of the standard curie weiss model for almost all choices of the random graph and the random patterns 2 if in additionm n ln n ln 2 we prove that there exists fort 1 a gibbs measure associated to each original pattern whereas for higher temperatures the gibbs measure is unique the basic technical result in the proofs is a uniform bound on the difference between the hamiltonian on a random graph and its mean value