scaled rotation regularization abstract a new regularization method a scaled rotation is proposed and compared with the standard linear regularized discriminant analysis a sense of the method consists in the singular value decomposition s tdt of a sample covariance matrix s and a use of the following representation of an inverse of the covariance matrix s 1 t d i 1 t for certain data structures the scaled rotation helps to reduce the generalization error in small learning set and high dimensionality cases efficacy of the scaled rotation increases if one transforms the data by y d i 1 2 t x and uses an optimally stopped single layer perceptron classifier afterwards

birational geometry of quartic 3 folds ii the importance of being factorial the paper explores the birational geometry of terminal quartic 3 folds in doing this i develop a new approach to study maximal singularities with positive dimensional centers this allows to determine the pliability of a open image in new window factorial quartic with ordinary double points and it shows the importance of open image in new window factoriality in the context of birational geometry of uniruled 3 folds

birational rigidity of complete intersections we prove that every smooth complete intersection x x d 1 ldots d s subset mathbb p sum i 1 s d i defined by s hypersurfaces of degree d 1 ldots d s is birationally superrigid if 5s 1 le frac 2 sum i 1 s d i 1 sqrt prod i 1 s d i in particular x is non rational and mathrm bir x mathrm aut x we also prove birational superrigidity of singular complete intersections with similar numerical condition these extend the results proved by tommaso de fernex

zeta function regularization the multiplicative anomaly and the wodzicki residue the multiplicative anomaly associated with the zeta function regularized determinant is computed for the laplace type operators v1 v1 and v2 v2 with v1 v2 constant in a d dimensional compact smooth manifold md making use of several results due to wodzicki and by direct calculations in some explicit examples it is found that the multiplicative anomaly is vanishing for d odd and for d 2 an application to the one loop effective potential of the o 2 self interacting scalar model is outlined

spectral density of sparse sample covariance matrices applying the replica method of statistical mechanics we evaluate the eigenvalue density of the large random matrix sample covariance matrix of the form j a t a where a is an m n real sparse random matrix the difference from a dense random matrix is the most significant in the tail region of the spectrum we compare the results of several approximation schemes focusing on the behaviour in the tail region

inferring hidden states in langevin dynamics on large networks average case performance we present average performance results for dynamical inference problems in large networks where a set of nodes is hidden while the time trajectories of the others are observed examples of this scenario can occur in signal transduction and gene regulation networks we focus on the linear stochastic dynamics of continuous variables interacting via random gaussian couplings of generic symmetry we analyze the inference error given by the variance of the posterior distribution over hidden paths in the thermodynamic limit and as a function of the system parameters and the ratio ensuremath alpha between the number of hidden and observed nodes by applying kalman filter recursions we find that the posterior dynamics is governed by an effective drift that incorporates the effect of the observations we present two approaches for characterizing the posterior variance that allow us to tackle respectively equilibrium and nonequilibrium dynamics the first appeals to random matrix theory and reveals average spectral properties of the inference error and typical posterior relaxation times the second is based on dynamical functionals and yields the inference error as the solution of an algebraic equation

birational automorphisms of fano hypersurfaces v v y v 0 between smooth three dimensional quartics v v4 v 0 v 0 4 in p is a projective isomorphism this is the original wording of their theorem in fact the arguments of im give a much stronger result namely they actually prove that smooth quartics are birationally superrigid the concept of birational rigidity is not visual being rather technical it makes little impression it is immediate implications of the property of being birationally super rigid that are really impressive for a projective variety x smooth in codimension 1 and a linear system jdj free from xed components we de ne the threshold of canonical adjunction or brie y just threshold of the pair x d setting

global and local aspects of spectral actions the principal object in noncommutative geometry is the spectral triple consisting of an algebra a hilbert space and a dirac operator field theories are incorporated in this approach by the spectral action principle which sets the field theory action to where f is a real function such that the trace exists and is a cutoff scale in the low energy weak field limit the spectral action reproduces reasonably well the known physics including the standard model however not much is known about the spectral action beyond the low energy approximation in this paper after an extensive introduction to spectral triples and spectral actions we study various expansions of the spectral actions exemplified by the heat kernel we derive the convergence criteria for a commutative spectral triple we compute the heat kernel on the torus up to the second order in gauge connection and consider limiting cases this article is part of a special issue of journal of physics a mathematical and theoretical in honour of stuart dowker s 75th birthday devoted to applications of zeta functions and other spectral functions in mathematics and physics

the spectral action for moyal planes extending a result of vassilevich we obtain the asymptotic expansion for the trace of a spatially regularized heat operator l f e t where is a generalized laplacian defined with moyal products and l f is moyal left multiplication the moyal planes corresponding to any skewsymmetric matrix being spectral triples the spectral action introduced in noncommutative geometry by chamseddine and connes is computed this result generalizes the connes lott action previously computed by gayral for symplectic

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

spectral action for torsion with and without boundaries we derive a commutative spectral triple and study the spectral action for a rather general geometric setting which includes the skew symmetric torsion and the chiral bag conditions on the boundary the spectral action splits into bulk and boundary parts in the bulk we clarify certain issues of the previous calculations show that many terms in fact cancel out and demonstrate that this cancellation is a result of the chiral symmetry of spectral action on the boundary we calculate several leading terms in the expansion of spectral action in four dimensions for vanishing chiral parameter of the boundary conditions and show that 0 is a critical point of the action in any dimension and at all orders of the expansion

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

spectral triples and manifolds with boundary abstract we investigate manifolds with boundary in noncommutative geometry spectral triples associated to a symmetric differential operator and a local boundary condition are constructed we show that there is no tadpole for classical dirac operators with a chiral boundary condition on spin manifolds

accuracy of pseudo inverse covariance learning a random matrix theory analysis for many learning problems estimates of the inverse population covariance are required and often obtained by inverting the sample covariance matrix increasingly for modern scientific data sets the number of sample points is less than the number of features and so the sample covariance is not invertible in such circumstances the moore penrose pseudo inverse sample covariance matrix constructed from the eigenvectors corresponding to nonzero sample covariance eigenvalues is often used as an approximation to the inverse population covariance matrix the reconstruction error of the pseudo inverse sample covariance matrix in estimating the true inverse covariance can be quantified via the frobenius norm of the difference between the two the reconstruction error is dominated by the smallest nonzero sample covariance eigenvalues and diverges as the sample size becomes comparable to the number of features for high dimensional data we use random matrix theory techniques and results to study the reconstruction error for a wide class of population covariance matrices we also show how bagging and random subspace methods can result in a reduction in the reconstruction error and can be combined to improve the accuracy of classifiers that utilize the pseudo inverse sample covariance matrix we test our analysis on both simulated and benchmark data sets

perturbations of gibbs semigroups we present the analytic perturbation theory for gibbs semigroups in the case when perturbations of generators are relatively bounded analyticity with respect to perturbation and semigroup parameters in the tr norm topology is proved and the corresponding domains are described

first order tree type dependence between variables and classification performance structuralization of the covariance matrix reduces the number of parameters to be estimated from the training data and does not affect an increase in the generalization error asymptotically as both the number of dimensions and training sample size grow a method to benefit from approximately correct assumptions about the first order tree dependence between components of the feature vector is proposed we use a structured estimate of the covariance matrix to decorrelate and scale the data and to train a single layer perceptron in the transformed feature space we show that training the perceptron can reduce negative effects of inexact a priori information experiments performed with 13 artificial and 10 real world data sets show that the first order tree type dependence model is the most preferable one out of two dozen of the covariance matrix structures investigated

heat kernel and number theory on nc torus the heat trace asymptotics on the noncommutative torus where generalized laplacians are made out of left and right regular representations is fully determined it turns out that this question is very sensitive to the number theoretical aspect of the deformation parameters the central condition we use is of a diophantine type more generally the importance of number theory is made explicit in a few examples we apply the results to the spectral action computation and revisit the uv ir mixing phenomenon for a scalar theory although we find non local counterterms in the nc phi 4 theory on mathbb t 4 we show that this theory can be made renormalizable at least at one loop and maybe even beyond

renormalization of the asymptotically expanded yang mills spectral action we study renormalizability aspects of the spectral action for the yang mills system on a flat 4 dimensional background manifold focusing on its asymptotic expansion interpreting the latter as a higher derivative gauge theory a power counting argument shows that it is superrenormalizable we determine the counterterms at one loop using zeta function regularization in a background field gauge and establish their gauge invariance consequently the corresponding field theory can be renormalized by a simple shift of the spectral function appearing in the spectral action

distribution of eigenvalues of ensembles of asymmetrically diluted hopfield matrices using grassmann variables and an analogy with two dimensional electrostatics we obtain the average eigenvalue distribution of ensembles of asymmetrically diluted hopfield matrices in the limit we found that in the limit of strong dilution the distribution is uniform in a circle in the complex plane

the cesaro behaviour of distributions a theory of summability for generalized functions of one real variable is presented this theory generalizes the classical theory of summability of series and integrals the relationship between the average behaviour of generalized functions according to this summability theory and the distributional behaviour is studied several applications are given

inversion of adjunction for local complete intersection varieties we extend inversion of adjunction and semicontinity of minimal log discrepancies from the case of a smooth ambient variety to the case of local complete intersection varieties

dynamics of batch training in a perceptron early stopping and weight decay are studied in a linear perceptron using a new simplified approach to the dynamics in the thermodynamical limit the approach is directly deduced from the gradient descent weight update it allows an exact description of the dynamics of the batch training process the results are compared with a recent study of early stopping and weight decay based on the equilibrium statistical mechanics approach it is shown that the equilibrium results for early stopping are good approximations and are exact for weight decay furthermore in the dynamical approach it is possible to determine the necessary number of training steps to fulfil certain termination conditions it can be shown that asymptotically i e if the number of examples is large only two batch steps are required to reach the optimal convergence if the learning rate is optimally chosen

eigenvalues of covariance matrices application to neural network learning the learing time of a simple neural network model is obtained through an analytic computation of the eigenvalue spectrum for the hessian matrix which describes the second order properties of the objective function in the space of coupling coefficients the results are generic for symmetric matrices obtained by summing outer products of random vectors the form of the eigenvalue distribution suggests new techniques for accelerating the learning process and provides a theoretical justification for the choice of centered versus biased state variables

on gravity torsion and the spectral action principle abstract we consider compact riemannian spin manifolds without boundary equipped with orthogonal connections we investigate the induced dirac operators and the associated commutative spectral triples in case of dimension four and totally anti symmetric torsion we compute the chamseddine connes spectral action deduce the equations of motions and discuss critical points

moyal planes are spectral triples axioms for nonunital spectral triples extending those introduced in the unital case by connes are proposed as a guide and for the sake of their importance in noncommutative quantum field theory the spaces r2n endowed with moyal products are intensively investigated some physical applications such as the construction of noncommutative wick monomials and the computation of the connes lott functional action are given for these noncommutative hyperplanes