measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

the phase transition of the directed polymer on disordered hierarchical lattices the directed polymer on disordered hierarchical lattices is studied using an exact renormalization scheme the phase transition is studied and a hyperscaling relation is derived the fixed distributions of the renormalized energies are obtained numerically the specific heat and associated exponents are computed from the fixed distributions

differential migdal kadanoff renormalization group for disordered systems phase transitions in quenched random systems are studied within the migdal kadanoff approximation exact differential recursion relations for the distribution of coupling are derived the q state potts model with random ferromagnetic couplings is studied in two and three dimensions in three dimensions there is a non perturbative random fixed point which exists when the specific heat exponent is either positive or slightly negative if the specific heat exponent is slightly negative a multicritical point separates strong and weak disorder regimes the analytic results are confirmed by numerical simulations of the recursion relations

a reflective newton method for minimizing a quadratic function subject to bounds on some of the variables we propose a new algorithm a reflective newton method for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables this method applies to a general indefinite quadratic function for which a local minimizer subject to bounds is required and is particularly suitable for the large scale problem our new method exhibits strong convergence properties global and quadratic convergence and appears to have significant practical potential strictly feasible points are generated experimental results on moderately large and sparse problems support the claim of practicality for large scale problems

critical behaviour of dyson s hierarchical model with a random field the critical behaviour of dyson s hierarchical ising model is studied in the presence of a random field simple scaling relations determine the exponents eta eta exactly for random fields with either short or long range correlations the schwartz soffer inequality eta or 2 eta is satisfied as an equality provided the random field correlations are not too long ranged for short range random fields the exponent v is calculated to order in lambda 4 3 where lambda is the range parameter of the dyson model the exponent v is also computed to o in 2 with in 2 lambda 3 for the hierarchical o n model in zero field the results reveal striking similarities with the exponents of the corresponding one dimensional model with long range interactions

weighted mean field theory for the random field ising model we consider the mean field theory of the random field ising model obtained by weighing the many solutions of the mean field equations with boltzmann like factors these solutions are found numerically in three dimensions and we observe critical behaviour arising from the weighted sum we estimate the exponents

critical behaviour of the random field ising model we present an analytic real space renormalization group calculation for the random field ising model we apply the migdal kadanoff approximation for the renormalization of a cubic cell in dimensions d introducing a new field partitioning scheme which allows us to treat the random field fluctuations in a coherent manner our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension in three dimensions the magnetization exponent and the schwartz soffer inequality is almost satisfied as an equality we expand analytically in further we show that and the magnitude of the inequality go to zero exponentially with we calculate the crossover exponent from pure to the random field system and find surprisingly good agreement with experimental values we find that satisfies the schwartz soffer inequality the susceptibility exponent of the pure system we expand in and find that the magnitude of the inequality varies exponentially in finally we find that dimensional reduction is satisfied to first order in with the reduced dimension

on the missing scaling relation in random field systems abstract a detailed proof is given that for random field systems the susceptibility and correlation function are related at the transition this yields an additional scaling relation 2

fixed point distributions of short range ising spin glasses on hierarchical lattices fixed point distributions for the couplings of ising spin glasses with nearest neighbor interactions on hierarchical lattices are investigated numerically hierarchical lattices within the migdal kadanoff family with fractal dimensions in the range 2 58 ensuremath le d ensuremath le 7 as well as a lattice of the wheatstone bridge family with fractal dimension d ensuremath approx 3 58 are considered three initial distributions for the couplings are analyzed namely the gaussian bimodal and uniform ones in all cases after a few iterations of the renormalization group procedure the associated probability distributions approached universal fixed shapes for hierarchical lattices of the migdal kadanoff family the fixed point distributions were well fitted either by stretched exponentials or by q gaussian distributions both fittings recover the expected gaussian limit as d ensuremath rightarrow ensuremath infty in the case of the wheatstone bridge lattice the best fit was found by means of a stretched exponential distribution

intrinsic dimensionality estimation with optimally topology preserving maps a new method for analyzing the intrinsic dimensionality id of low dimensional manifolds in high dimensional feature spaces is presented compared to a previous approach by fukunaga and olsen 1971 the method has only linear instead of cubic time complexity with respect to the dimensionality of the input space moreover it is less sensitive to noise than the former approach experiments include id estimation of synthetic data for comparison and illustration as well as id estimation of an image sequence

on local intrinsic dimension estimation and its applications in this paper we present multiple novel applications for local intrinsic dimension estimation there has been much work done on estimating the global dimension of a data set typically for the purposes of dimensionality reduction we show that by estimating dimension locally we are able to extend the uses of dimension estimation to many applications which are not possible with global dimension estimation additionally we show that local dimension estimation can be used to obtain a better global dimension estimate alleviating the negative bias that is common to all known dimension estimation algorithms we illustrate local dimension estimation s uses towards additional applications such as learning on statistical manifolds network anomaly detection clustering and image segmentation

inequalities for quantum relative entropy some logarithmic trace inequalities involving the notions of relative entropy are reobtained from a log majorization result the thermodynamic inequality is generalized and a chain of equivalent statements involving this inequality and the peierls bogoliubov inequality is obtained

polymers on disordered hierarchical lattices a nonlinear combination of random variables the problem of directed polymers on disordered hierarchical and hypercubic lattices is considered for the hierarchical lattices the problem can be reduced to the study of the stable laws for combining random variables in a nonlinear way we present the results of numerical simulations of two hierarchical lattices finding evidence of a phase transition in one case for a limiting case we extend the perturbation theory developed by derrida and griffiths to nonzero temperature and to higher order and use this approach to calculate thermal and geometrical properties overlaps of the model in this limit we obtain an interpolation formula allowing one to obtain the noninteger moments of the partition function from the integer moments we obtain bounds for the transition temperature for hierarchical and hypercubic lattices and some similarities between the problem on the two different types of lattice are discussed

extensions of inequalities for unitarily invariant norms via log majorization abstract a capital letter means n n matrix t is said to be positive definite denoted by t 0 if t is positive semidefinite and invertible we shall show the following central results via log majorization obtained by an order preserving operator inequality theorem if a 0 and b 0 then for 0 1 t 0 1 and r t a 1 t 2 a t b a 1 t 2 s log w 2 a r b s a w 2 holds for 1 r t 1 t 1 s 1 where w 1 s r 1 t s our result extends the following recent elegant inequality by matharu and aujlia let a b be positive definite and 0 1 then i 1 k j a 1 b i 1 k j a b 1 k n also some results associated with log majorization are shown

on an eigenvalue inequality involving the hadamard product abstract let a b be n n positive definite matrices then for 0 t 1 i k n i a b i k n i a t b a 1 t b i k n i a b k 1 n this gives a weighted extension of a result of ando 1 the case where the eigenvalues i are replaced with the singular values i is also considered

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

intrinsic dimension estimation of manifolds by incising balls dimensionality reduction is a very important tool in data mining intrinsic dimension of data sets is a key parameter for dimensionality reduction however finding the correct intrinsic dimension is a challenging task in this paper a new intrinsic dimension estimation method is presented the estimator is derived by finding the exponential relationship between the radius of an incising ball and the number of samples included in the ball the method is compared with the previous dimension estimation methods experiments have been conducted on synthetic and high dimensional image data sets and on data sets of the santa fe time series competition and the results show that the new method is accurate and robust

intrinsic dimension estimation the paper reviews state of the art of the methods of intrinsic dimension id estimation the paper defines the properties that an ideal id estimator should have the paper reviews under the above mentioned framework the major id estimation methods underlining their advances and the open problems dimensionality reduction methods are preprocessing techniques used for coping with high dimensionality they have the aim of projecting the original data set of dimensionality n without information loss onto a lower m dimensional submanifold since the value of m is unknown techniques that allow knowing in advance the value of m called intrinsic dimension id are quite useful the aim of the paper is to review state of the art of the methods of id estimation underlining the recent advances and the open problems

the whitney reduction network a method for computing autoassociative graphs this article introduces a new architecture and associated algorithms ideal for implementing the dimensionality reduction of an m dimensional manifold initially residing in an n dimensional euclidean space where n m motivated by whitney s embedding theorem the network is capable of training the identity mapping employing the idea of the graph of a function in theory a reduction to a dimension d that retains the differential structure of the original data may be achieved for some d 2m 1 to implement this network we propose the idea of a good projection which enhances the generalization capabilities of the network and an adaptive secant basis algorithm to achieve it the effect of noise on this procedure is also considered the approach is illustrated with several examples

the golden thompson trace inequality is complemented we prove a class of trace inequalities which complements the golden thompson inequality for example tr epa epb 2 p tr ea b holds for all p 0 when a and b are hermitian matrices and denotes the geometric mean we also prove related trace inequalities involving the logarithmic function namely p 1tr x log yp 2xpyp 2 tr x log x log y p 1tr x log xp 2ypxp 2 for all p 0 when x and y are nonnegative matrices these inequalities supply lower and upper bounds on the relative entropy

spectral inequalities for kubo ando operator means abstract eigenvalue and singular value inequalities involving kubo ando operator connections or means are established previous results from lemos and soares in 2018 are generalized or complemented some log majorizations are included as a consequence a refinement of an independent result by ando and by visick in 1995 on the eigenvalues of the hadamard product is derived some singular value inequalities by zou in 2017 are further extended

real space renormalization group analysis of the random field ising model we present an analytic migdal kadanoff renormalization group analysis of the random field ising model the renormalization flows to a zero temperature critical point from which we calculate independently three critical exponents in arbitrary dimension in three dimensions the magnetization exponent and the schwartz soffer inequality is almost satisfied as an equality expanding analytically in we find that and the distance from the upper bound of the equality go to zero exponentially with

intrinsic dimension estimation via nearest constrained subspace classifier we consider the problems of classification and intrinsic dimension estimation on image data a new subspace based classifier is proposed for supervised classification or intrinsic dimension estimation the distribution of the data in each class is modeled by a union of a finite number of affine subspaces of the feature space the affine subspaces have a common dimension which is assumed to be much less than the dimension of the feature space the subspaces are found using regression based on the 0 norm the proposed method is a generalisation of classical nn nearest neighbor nfl nearest feature line classifiers and has a close relationship to ns nearest subspace classifier the proposed classifier with an accurately estimated dimension parameter generally outperforms its competitors in terms of classification accuracy we also propose a fast version of the classifier using a neighborhood representation to reduce its computational complexity experiments on publicly available datasets corroborate these claims highlightsa new classifier called ncsc and its fast version called ncsc ii are proposed classifiers of nn nfl and ns are the special cases of ncsc ncsc ncsc ii can be used as intrinsic dimension estimator using well tuned intrinsic parameter ncsc ncsc ii outperforms nn nfl and ns

some log majorizations and an extension of a determinantal inequality abstract an eigenvalue inequality involving a matrix connection and its dual is established and some log majorization type results are obtained in particular some eigenvalues inequalities considered by f hiai and m lin 9 an associated conjecture and a singular values inequality by l zou 20 are revisited a reformulation of the inequality det a u b det a b for positive semidefinite matrices a b with u a unitary matrix that appears in the polar decomposition of b a is also extended using some known norm inequalities associated to furuta inequality and araki cordes inequality

some exact inequalities for the random field ϕ4 system exact lower bound on the correlation function and susceptibility are obtained in terms of the quantity r r0 12u 2 d ddl l l where l l are the exact averages further lower bounds are obtained on r relating it to an exactly soluble model

on the harris criterion for hierarchical lattices the authors discuss the condition necessary for the harris criteria to be valid on hierarchical lattices they prove that disorder is always relevant when the specific heat exponent alpha p is positive and show how to construct lattices for which disorder becomes relevant while alpha p is still negative

a self consistent ornstein zernike approximation for the random field ising model we extend the self consistent ornstein zernike approximation scoza first formulated in the context of liquid state theory to the study of the random field ising model within the replica formalism we treat the quenched random field just as another spin variable thereby avoiding the usual average over the random field distribution this allows us to study the influence of the distribution on the phase diagram in finite dimensions the thermodynamics and the correlation functions are obtained as solutions of a set a coupled partial differential equations with magnetization temperature and disorder strength as independent variables a preliminary analysis based on high temperature and 1 d series expansions shows that the theory can predict accurately the dependence of the critical temperature on disorder strength no sharp transition however occurs for d 4 for the bimodal distribution we find a tricritical point which moves to weaker fields as the dimension is reduced for the gaussian distribution a tricritical point may appear for d around 4

estimating the intrinsic dimension of data with a fractal based method in this paper the problem of estimating the intrinsic dimension of a data set is investigated a fractal based approach using the grassberger procaccia algorithm is proposed since the grassberger procaccia algorithm 1983 performs badly on sets of high dimensionality an empirical procedure that improves the original algorithm has been developed the procedure has been tested on data sets of known dimensionality and on time series of santa fe competition

modeling the manifolds of images of handwritten digits this paper describes two new methods for modeling the manifolds of digitized images of handwritten digits the models allow a priori information about the structure of the manifolds to be combined with empirical data accurate modeling of the manifolds allows digits to be discriminated using the relative probability densities under the alternative models one of the methods is grounded in principal components analysis the other in factor analysis both methods are based on locally linear low dimensional approximations to the underlying data manifold links with other methods that model the manifold are discussed

counting closed 2 manifolds in tubes in hypercubic lattices we consider closed 2 manifolds 2 manifolds without boundary embedded in tubes in the hypercubic lattice z d for orientable 2 manifolds with fixed genus we prove that the exponential growth rate is independent of the genus and we use this to prove a pattern theorem for manifolds with fixed genus we prove a similar theorem for the non orientable case for if the genus is not fixed then we prove a pattern theorem and use this to show that 2 manifolds with genus less than any fixed number are exponentially rare so the typical genus increases with the size of the manifold in four and higher dimensions we prove that orientable manifolds are exponentially rare and are dominated by non orientable manifolds in four dimensions all except exponentially few 2 manifolds both orientable and non orientable contain a local knotted ball pair

some inequalities for the matrix heron mean abstract in this paper we prove some norm inequalities for the matrix heron mean of two positive definite matrices we also prove a determinant inequality for the t power mean of two positive definite matrices as a consequence we obtain a determinant inequality for the matrix heron mean

an arithmetic geometric mean inequality for singular values and its applications abstract in this short note we give a new equivalent form of the arithmetic geometric mean inequality for singular values as applications of our result we give a new proof of an inequality due to bhatia and davis 1993 4 and we obtain a singular value inequality for matrix means which is similar to one proved by drury 2012 9 finally we present a log majorization inequality for singular values

some inequalities for unitarily invariant norm abstract we shall prove the inequalities a b a b aa bb 2 ab a b a b 4 ab for all n n complex matrices a b and all unitarily invariant norms if further a b are positive definite it is proved that j 1 k j a b j 1 k j a 1 b 1 k n 0 1 where denotes the operator means considered by kubo and ando and j x 1 j n denote the eigenvalues of x arranged in the decreasing order whenever these all are real a number of inequalities are obtained as applications

renormalisation group study of a disordered model we study the potts model on a diamond hierarchical lattice with random interactions using weak disorder expansions we calculate analytically the position and the exponents of the random fixed point which appears when the specific heat exponent ap of the pure system becomes positive at ap 0 we find how the logarithmic singularity is modified by the disorder lastly we suggest that this model should present griffiths like singularities

real space renormalization group approach to the random field ising model the authors discuss results from two types of real space renormalization group rsrg calculations applied to the random field ising model in three dimensions starting from a lattice of size l the rsrg is used to reduce the lattice to a size l 2 on which the trace is done exactly in this way thermodynamic properties such as the magnetization and susceptibility can be determined approximately they find that for a given size the susceptibility increases as the temperature t is reduced down to the transition temperature tc and becomes essentially independent of temperature below tc both in the vicinity of tc and at lower temperatures there are large sample to sample fluctuations in the susceptibility which grow with increasing system size they interpret these results in terms of the droplet theory of the transition