a cautionary example on the use of second order methods for analyzing point patterns a popular approach to the analysis of mapped planar point patterns is through secondorder methods where the object of attention is the variance of the number of points falling in a test set of a given size and shape or the behaviour of all distances between pairs of points in the pattern standard references for the analysis of spatial data are works by ripley 1981 ch 8 and diggle 1983 many of the methods for point pattern analysis given in the literature are second order in nature the analysis of mapped point patterns is important in a variety of biological applications ranging from ecology to physiology and some examples are given in these books by ripley and by diggle suppose that an observed point pattern can be considered as a realization of a random point process model which is stationary and isotropic of course this is a strong assumption which may not be justified in practice but it is nevertheless the assumption under which much of the existing point process methodology has been developed the property of the underlying random process elicited directly or indirectly by second order methods is the second moment cumulative function k t which satisfies the following properties under suitable regularity conditions see ripley 1977 p 150 i if x is the intensity of the process then xk t is the expected number of further points within distance t of a typical point of the process ii specifying k t for all t is equivalent to specifying the variance of the number of points falling in any given set the function k t defined by ripley 1977 is an edge corrected version of the empirical distribution function of all interpoint distances in the observed pattern and provides an approximately unbiased estimator of k t the estimate k t can be used to construct tests of the hypothesis that an observed pattern is consistent with the completely random poisson point process model and more importantly to quantify the apparent deviation

regression on manifolds estimation of the exterior derivative collinearity and near collinearity of predictors cause difficulties when doing regression in these cases variable selection becomes untenable because of mathematical issues concerning the existence and numerical stability of the regression coefficients and interpretation of the coefficients is ambiguous because gradients are not defined using a differential geometric interpretation in which the regression coefficients are interpreted as estimates of the exterior derivative of a function we develop a new method to do regression in the presence of collinearities our regularization scheme can improve estimation error and it can be easily modified to include lasso type regularization these estimators also have simple extensions to the large p small n context

group preference aggregation methods employed in ahp an evaluation and an intrinsic process for deriving members weightages abstract the analytic hierarchy process ahp is one of the popular and powerful techniques for decision making a detailed survey of the literature has revealed that there exists no formal evaluation of the group preference aggregation methods currently employed in ahp this paper provides such an evaluation using well established social choice axioms which govern the process of combining individual opinions to obtain a single group opinion the geometric mean method gmm and the weighted arithmetic mean method wamm are the two methods evaluated it is shown using counter examples that the gmm does not always satisfy the pareto optimality axiom which is one of the prominent and widely accepted social choice axioms this finding is significant as the gmm has been the most commonly used method in ahp for combining individual opinions to form a group opinion the other method viz wamm has satisfied all the axioms except the independence of irrelevant alternatives axiom in order to use the wamm one has to find the weightages importance to be assigned to the members of the group this is often a difficult task especially so if the group is large as in the case of public policy decisions and when judgements are elicited through the use of questionnaires these situations need an objective method to derive members weightages but only a few studies are available in the literature to address such a situation we propose a simple and intuitively appealing eigenvector based method to intrinsically determine the weightages for group members using their own subjective opinions the superiority of the proposed method over the previous methods is brought out in the paper

a cosine maximization method for the priority vector derivation in ahp abstract the derivation of a priority vector from a pair wise comparison matrix pcm is an important issue in the analytic hierarchy process ahp the existing methods for the priority vector derivation from pcm include eigenvector method ev weighted least squares method wls additive normalization method an logarithmic least squares method lls etc the derived priority vector should be as similar to each column vector of the pcm as possible if a pair wise comparison matrix pcm is not perfectly consistent therefore a cosine maximization method cm based on similarity measure is proposed which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a pcm an optimization model for the cm is proposed to derive the reliable priority vector using three numerical examples the cm is compared with the other prioritization methods based on two performance evaluation criteria euclidean distance and minimum violation the results show that the cm is flexible and efficient

strong law of large numbers for betti numbers in the thermodynamic regime we establish the strong law of large numbers for betti numbers of random cech complexes built on mathbb r n valued binomial point processes and related poisson point processes in the thermodynamic regime here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the lebesgue measure on mathbb r n and the case where it is supported on a c 1 compact manifold of dimension strictly less than n the strong law is proved under very mild assumption which only requires that the common probability density function belongs to l p spaces for all 1 le p infty

nonlinear manifold representations for functional data for functional data lying on an unknown nonlinear low dimensional space we study manifold learning and introduce the notions of manifold mean manifold modes of functional variation and of functional manifold components these constitute nonlinear representations of functional data that complement classical linear representations such as eigenfunctions and functional principal components our manifold learning procedures borrow ideas from existing nonlinear dimension reduction methods which we modify to address functional data settings in simulations and applications we study examples of functional data which lie on a manifold and validate the superior behavior of manifold mean and functional manifold components over traditional cross sectional mean and functional principal components we also include consistency proofs for our estimators under certain assumptions

chaos in the mixed even spin models we consider a disordered system obtained by coupling two mixed even spin models together the chaos problem is concerned with the behavior of the coupled system when the external parameters in the two models such as temperature disorder or external field are slightly different it is conjectured that the overlap between two independently sampled spin configurations from respectively the gibbs measures of the two models is essentially concentrated around a constant under the coupled gibbs measure using the extended guerra replica symmetry breaking bound together with a recent development of controlling the overlap using the ghirlanda guerra identities as well as a new family of identities we present rigorous results on chaos in temperature in addition chaos in disorder and in external field are addressed

geometric representations of random hypergraphs abstractwe introduce a novel parameterization of distributions on hypergraphs based on the geometry of points in rd the idea is to induce distributions on hypergraphs by placing priors on point configurations via spatial processes this specification is then used to infer conditional independence models or markov structure for multivariate distributions this approach results in a broader class of conditional independence models beyond standard graphical models factorizations that cannot be retrieved via a graph are possible inference of nondecomposable graphical models is possible without requiring decomposability or the need of gaussian assumptions this approach leads to new metropolis hastings markov chain monte carlo algorithms with both local and global moves in graph space generally offers greater control on the distribution of graph features than currently possible and naturally extends to hypergraphs we provide a comparative performance evaluation against state of the art approaches and

loss surface of xor artificial neural networks training an artificial neural network involves an optimization process over the landscape defined by the cost loss as a function of the network parameters we explore these landscapes using optimization tools developed for potential energy landscapes in molecular science the number of local minima and transition states saddle points of index one as well as the ratio of transition states to minima grow rapidly with the number of nodes in the network there is also a strong dependence on the regularization parameter with the landscape becoming more convex fewer minima as the regularization term increases we demonstrate that in our formulation stationary points for networks with n h hidden nodes including the minimal network required to fit the xor data are also stationary points for networks with n h 1 hidden nodes when all the weights involving the additional node are zero hence smaller networks trained on xor data are embedded in the landscapes of larger networks our results clarify certain aspects of the classification and sensitivity to perturbations in the input data of minima and saddle points for this system and may provide insight into dropout and network compression

approximate ultrametricity for random measures and applications to spin glasses in this paper we introduce a notion called approximate ultrametricity which encapsulates the phenomenology of a sequence of random probability measures having supports that behave like ultrametric spaces insofar as they decompose into nested balls we provide a sufficient condition for a sequence of random probability measures on the unit ball of an infinite dimensional separable hilbert space to admit such a decomposition whose elements we call clusters we also characterize the laws of the measures of the clusters by showing that they converge in law to the weights of a ruelle probability cascade these results apply to a large class of classical models in mean field spin glasses we illustrate the notion of approximate ultrametricity by proving a conjecture of talagrand regarding mixed p spin glasses that is known to imply a prediction of dotsenko franz mezard 2016 wiley periodicals inc

a tomography of the grem beyond the rem conjecture in a companion paper we proved that in a large class of gaussian disordered spin systems the local statistics of energy values near levels n 1 2 with 1 2 are described by a simple poisson process in this paper we address the issue as to whether this is optimal and what will happen if 1 2 we do this by analysing completely the gaussian generalised random energy models grem we show that the rem behaviour persists up to the level c n where c denotes the critical temperature we show that beyond this value the simple poisson process must be replaced by more and more complex mixed poisson point processes

on properties of parisi measures we investigate the structure of parisi measures the functional order parameters of mixed p spin models in mean field spin glasses in the absence of external field we prove that a parisi measure satisfies the following properties first at all temperatures the support of any parisi measure contains the origin if it contains an open interval then the measure has a smooth density on this interval next we give a criterion on temperature parameters for which a parisi measure is neither replica symmetric nor one replica symmetry breaking finally we show that in the sherrington kirkpatrick model slightly above the critical temperature the largest number in the support of a parisi measure is a jump discontinuity an analogue of these results is discussed in the spherical mixed p spin models as a tool to establish these facts and of independent interest we study functionals of the associated parisi pdes and derive regularity properties of their solutions

a multicriteria evaluation of methods for obtaining weights from ratio scale matrices abstract ratio scale matrices are commonly used as a vehicle to distinguish among competing alternatives or conflicting criteria the issue of extracting weights out of such matrices has attracted increasing attention in recent years and resulted in the development of several scaling methods this paper presents an attempt to evaluate some of the more known techniques in this area the evaluation procedure includes random generation of a large number of representative matrices which are used to evaluate the various scaling methods according to the specified criteria the performance of the different techniques is analyzed with respect to the proposed criteria and conclusions are drawn

the complexity of spherical p spin models a second moment approach recently auffinger ben arous and cern initiated the study of critical points of the hamiltonian in the spherical pure pp spin spin glass model and established connections between those and several notions from the physics literature denoting the number of critical values less than nunu by crtn u crtn u they computed the asymptotics of 1nlog ecrtn u 1nlog ecrtn u as nn the dimension of the sphere goes to we compute the asymptotics of the corresponding second moment and show that for p 3p 3 and sufficiently negative uu it matches the first moment e crtn u 2 e crtn u 2 1 as an immediate consequence we obtain that crtn u e crtn u 1crtn u e crtn u 1 in l2l2 and thus in probability for any uu for which ecrtn u ecrtn u does not tend to 00 we prove that the moments match on an exponential scale

the parisi formula in this chapter we prove the parisi formula which gives the limiting value of the free energy per site for the sherrington kirkpatrick model at each temperature starting with the famous result of guerra that the parisi formula provided a bound for this free energy we make full use of poisson dirichlet cascades we then use the same techniques to prove in a strong sense that in presence of an external field the overlaps are positive

on the path density of a gradient field we consider the problem of reliably finding filaments in point clouds realistic data sets often have numerous filaments of various sizes and shapes statistical techniques exist for finding one or a few filaments but these methods do not handle noisy data sets with many filaments other methods can be found in the astronomy literature but they do not have rigorous statistical guarantees we propose the following method starting at each data point we construct the steepest ascent path along a kernel density estimator we locate filaments by finding regions where these paths are highly concentrated formally we define the density of these paths and we construct a consistent estimator of this path density

heuristic aggregation of individual judgments in ahp group decision making using simulated annealing algorithm sa algorithm is used as aggregation procedure in ahp group decision making the group priority vector is obtained by minimization of the euclidean distance objective search for maximum consensus between individuals within the group proposed procedure is invariant to any prioritization method proposed procedure performs better or at least equally to several other procedures there are various aggregation procedures for obtaining a group priority vector within analytic hierarchy process ahp supported decision making processes this paper will introduce a heuristic aggregation procedure based on simulated annealing sa algorithm to be used for the purposes of obtaining a group priority vector at any node of an ahp hierarchy the proposed procedure performs its aggregation process by minimizing the group euclidean distance ged consensus measure across group weights and judgments and the group vector obtained in this way is invariant to any prioritization method in other words there is no need to have individual priority vectors as is required by some other aggregation procedures along with sa minimization of the ged the group rank reversal minimum violation criterion is implemented as a control mechanism as well as the consensus measure based on the ranks of alternatives the presented procedure is compared with several reported combinations of different prioritization methods and group aggregation procedures five examples from literature are used to show that the proposed procedure performs better or at least equally to several other well known combinations of prioritization and aggregation in ahp group decision making frameworks

on the overlap in the multiple spherical sk models in order to study certain questions concerning the distribution of the overlap in sherrington kirkpatrick type models such as the chaos and ultrametricity problems it seems natural to study the free energy of multiple systems with constrained overlaps one can write analogues of guerra s replica symmetry breaking bound for such systems but it is not at all obvious how to choose informative functional order parameters in these bounds we were able to make some progress for spherical pure p spin sk models where many computations can be made explicitly for pure 2 spin model we prove ultrametricity and chaos in an external field for the pure p spin model for even p 4 without an external field we describe two possible values of the overlap of two systems at different temperatures we also prove a somewhat unexpected result which shows that in the 2 spin model the support of the joint overlap distribution is not always witnessed at the level of the free energy and for example ultrametricity holds only in a weak sense

the geometric consistency index approximated thresholds abstract crawford and williams journal of mathematical psychology 29 1985 387 suggested for the row geometric mean method rgmm one of the most extended ahp s priorization procedure a measure of the inconsistency based on stochastic properties of a subjacent model in this paper we formalize this inconsistency measure hereafter called the geometric consistency index gci and provide the thresholds associated with it these thresholds allow us an interpretation of the inconsistency tolerance level analogous to that proposed by saaty multicriteria decision making the analytic hierarchy process new york 1980 for the consistency ratio cr used with the right eigenvector method in conventional ahp

on the energy landscape of the mixed even p spin model we investigate the energy landscape of the mixed even p spin model with ising spin configurations we show that for any given energy level between zero and the maximal energy with overwhelming probability there exist exponentially many distinct spin configurations such that their energies stay near this energy level furthermore their magnetizations and overlaps are concentrated around some fixed constants in particular at the level of maximal energy we prove that the hamiltonian exhibits exponentially many orthogonal peaks this improves the results of chatterjee disorder chaos and multiple valleys in spin glasses 2009 and ding et al ann probab 43 6 3468 3493 2015 where the former established a logarithmic size of the number of the orthogonal peaks while the latter proved a polynomial size our second main result obtains disorder chaos at zero temperature and at any external field as a byproduct this implies that the fluctuation of the maximal energy is superconcentrated when the external field vanishes and obeys a gaussian limit law when the external field is present

a central limit theorem for the euler characteristic of a gaussian excursion set we study the euler characteristic of an excursion set of a stationary isotropic gaussian random field x omega times mathbb r d to mathbb r let us fix a level u in r and let us consider the excursion set above u a t u t in t x t ge u where t is a bounded cube subset r d the aim of this paper is to establish a central limit theorem for the euler characteristic of a t u as t grows to r d as conjectured by r adler more than ten years ago the required assumption on x is c 3 regularity of the trajectories non degeneracy of the gaussian vector x t and derivatives at any fixed point t in r d as well as integrability on r d of the covariance function and its derivatives the fact that x is c 3 is stronger than geman s assumption traditionally used in dimension one nevertheless our result extends what is known in dimension one to higher dimension in that case the euler characteristic of a t u equals the number of up crossings of x at level u plus eventually one if x is above u at the left bound of the interval t

random matrices and complexity of spin glasses cern y abstract we give an asymptotic evaluation of the complexity of spherical p spin spin glass models via random matrix theory this study enables us to obtain detailed infor mation about the bottom of the energy landscape including the absolute minimum the ground state the other local minima and describe an interesting layered structure of the low critical values for the hamiltonians of these models we also show that our ap proach allows us to compute the related tap complexity and extend the results known in the physics literature as an independent tool we prove a ldp for the k th largest eigenvalue of the goe extending the results of bdg01 how many critical values of given index and below a given level does a typical random morse function have on a high dimensional manifold our work addresses this question in a very special case we look at certain natural random gaussian functions on the n dimensional sphere known as p spin spherical spin glass models we cannot yet answer the question above about the typical number but we can study thoroughly the mean number which we show is exponentially large in n we introduce a new identity based on the classical kac rice formula relating random matrix theory and the problem of counting these critical values using this identity and tools from random matrix theory we give an asymptotic evaluation of the complexity of these spherical spin glass models the complexity mentioned here is defined as the mean number of critical points of given index whose value is below or above a given level this includes the important question of counting the mean number of local minima below a given level and in particular the question of finding the ground state energy the minimal value of the hamiltonian we show that this question is directly related to the study of the edge of the spectrum of the gaussian orthogonal ensemble goe the question of computing the complexity of mean field spin glass models has recently been thoroughly studied in the physics literature see for example clr03 and the refer ences therein mainly for a different measure of the complexity i e the mean number of solutions to the thouless anderson palmer equations or tap complexity our approach to the complexity enables us to recover known results in the physics literature about tap complexity to compute the ground state energy when p is even and to describe an interesting layered structure of the low energy levels of the hamiltonians of these models which might prove useful for the study of the metastability of langevin dynamics for these models in longer time scales than those studied in bdg01 the paper is organised as follows in section 2 we give our main results in section 3 we prove two main formulas theorem 2 1 and 2 2 relating random matrix theory specif ically the goe and spherical spin glasses these formulas are direct consequences of the kac rice formula we learned the version needed here in the book at07 for another modern account see aw09 the main new ingredient is the fact that for spherical spin glass models the hessian of the hamiltonian at a critical point conditioned on the value of the hamiltonian is a symmetric gaussian random matrix with independent entries up to symmetry plus a diagonal matrix this implies in particular that it is possible to

replica symmetry breaking condition exposed by random matrix calculation of landscape complexity we start with a rather detailed general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random n 1 dimensional gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity then we employ random matrix methods to calculate the density of stationary points as well as minima of the associated energy surface this is used to show that for a generic smooth concave confining potentials the condition of the zero temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape and the mean number of minima are exponential in n for such systems the annealed complexity of minima vanishes cubically when approaching the critical confinement whereas the cumulative annealed complexity vanishes quadratically different behaviour reported in our earlier short communication fyodorov et al in jetp lett 85 261 2007 was due to non analyticity of the hard wall confinement potential finally for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points in particular we show that in the vicinity of critical confinement the saddle points with a positive annealed complexity must be close to minima as they must have a vanishing fraction of negative eigenvalues in the hessian

energy landscape for large average submatrix detection problems in gaussian random matrices the problem of finding large average submatrices of a real valued matrix arises in the exploratory analysis of data from a variety of disciplines ranging from genomics to social sciences in this paper we provide a detailed asymptotic analysis of large average submatrices of an n times n gaussian random matrix the first part of the paper addresses global maxima for fixed k we identify the average and the joint distribution of the k times k submatrix having largest average value as a dual result we establish that the size of the largest square sub matrix with average bigger than a fixed positive constant is with high probability equal to one of two consecutive integers that depend on the threshold and the matrix dimension n the second part of the paper addresses local maxima specifically we consider submatrices with dominant row and column sums that arise as the local optima of iterative search procedures for large average submatrices for fixed k we identify the limiting average value and joint distribution of a k times k submatrix conditioned to be a local maxima in order to understand the density of such local optima and explain the quick convergence of such iterative procedures we analyze the number l n k of local maxima beginning with exact asymptotic expressions for the mean and fluctuation behavior of l n k for fixed k the mean of l n k is theta n k log n k 1 2 while the standard deviation is theta n 2k 2 k 1 log n k 2 k 1 our principal result is a gaussian central limit theorem for l n k that is based on a new variant of stein s method

local energy statistics in disordered systems a proof of the local rem conjecture recently bauke and mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model here we give necessary conditions for this hypothesis to be true which we show to be satisfied in wide classes of examples short range spin glasses and mean field spin glasses of the sk type we also show that under certain conditions the conjecture holds even if energy levels that grow moderately with the volume of the system are considered

the topology of probability distributions on manifolds let mathcal p be a set of n random points in mathbb r d generated from a probability measure on a m dimensional manifold mathcal m subset mathbb r d in this paper we study the homology of mathcal u mathcal p r the union of d dimensional balls of radius r around mathcal p as n rightarrow infty and r rightarrow 0 in addition we study the critical points of d mathcal p the distance function from the set mathcal p these two objects are known to be related via morse theory we present limit theorems for the betti numbers of mathcal u mathcal p r as well as for number of critical points of index k for d mathcal p depending on how fast r decays to zero as n grows these two objects exhibit different types of limiting behavior in one particular case n r m ge c log n we show that the betti numbers of mathcal u mathcal p r perfectly recover the betti numbers of the original manifold mathcal m a result which is of significant interest in topological manifold learning

free energy and complexity of spherical bipartite models we investigate both free energy and complexity of the spherical bipartite spin glass model we first prove a variational formula in high temperature for the limiting free energy based on the well known crisanti sommers representation of the mixed p spin spherical model next we show that the mean number of local minima at low levels of energy is exponentially large in the size of the system and we derive a bound on the location of the ground state energy

low temperature asymptotics of spherical mean field spin glasses in this paper we study the low temperature limit of the spherical crisanti sommers variational problem we identify the gamma limit of the crisanti sommers functionals thereby establishing a rigorous variational problem for the ground state energy of spherical mixed p spin glasses as an application we compute moderate deviations of the corresponding minimizers in the low temperature limit in particular for a large class of models this yields moderate deviations for the overlap distribution as well as providing sharp interpolation estimates between models we then analyze the ground state energy problem we show that this variational problem is dual to an obstacle type problem this duality is at the heart of our analysis we present the regularity theory of the optimizers of the primal and dual problems this culminates in a simple method for constructing a finite dimensional space in which these optimizers live for any model as a consequence of these results we unify independent predictions of crisanti leuzzi and auffinger ben arous regarding the one step replica symmetry breaking 1rsb phase in this limit we find that the positive replicon eigenvalue and pure like conditions are together necessary for optimality but that neither are themselves sufficient answering a question of auffinger and ben arous in the negative we end by proving that these conditions completely characterize the 1rsb phase in 2 p spin models

local minima in disordered mean field ferromagnets we consider the complexity of random ferromagnetic landscapes on the hypercube pm 1 n given by ising models on the complete graph with i i d non negative edge weights this includes in particular the case of bernoulli disorder corresponding to the ising model on a dense random graph mathcal g n p previous results had shown that with high probability as n rightarrow infty the gradient search energy lowering algorithm initialized uniformly at random converges to one of the homogeneous global minima all plus or all minus here we devise two modified algorithms tailored to explore the landscape at near zero magnetizations where the effect of the ferromagnetic drift is minimized with these we numerically verify the landscape complexity of random ferromagnets finding a diverging number of 1 spin flip stable local minima as n rightarrow infty we then investigate some of the properties of these local minima e g typical energy and magnetization and compare to the situation where the edge weights are drawn from a heavy tailed distribution

on the optimization landscape of tensor decompositions non convex optimization with local search heuristics has been widely used in machine learning achieving many state of art results it becomes increasingly important to understand why they can work for these np hard problems on typical data the landscape of many objective functions in learning has been conjectured to have the geometric property that all local optima are approximately global optima and thus they can be solved efficiently by local search algorithms however establishing such property can be very difficult in this paper we analyze the optimization landscape of the random over complete tensor decomposition problem which has many applications in unsupervised learning especially in learning latent variable models in practice it can be efficiently solved by gradient ascent on a non convex objective we show that for any small constant varepsilon 0 among the set of points with function values 1 varepsilon factor larger than the expectation of the function all the local maxima are approximate global maxima previously the best known result only characterizes the geometry in small neighborhoods around the true components our result implies that even with an initialization that is barely better than the random guess the gradient ascent algorithm is guaranteed to solve this problem however achieving such a initialization with random guess would still require super polynomial number of attempts our main technique uses kac rice formula and random matrix theory to our best knowledge this is the first time when kac rice formula is successfully applied to counting the number of local optima of a highly structured random polynomial with dependent coefficients

random fields of multivariate test statistics with applications to shape analysis our data are random fields of multivariate gaussian observations and we fit a multivariate linear model with common design matrix at each point we are interested in detecting those points where some of the coefficients are nonzero using classical multivariate statistics evaluated at each point the problem is to find the p value of the maximum of such a random field of test statistics we approximate this by the expected euler characteristic of the excursion set our main result is a very simple method for calculating this which not only gives us the previous result of cao and worsley ann statist 27 1999 925 942 for hotelling s t 2 but also random fields of roy s maximum root maximum canonical correlations ann appl probab 9 1999 1021 1057 multilinear forms ann statist 29 2001 328 371 x 2 statist probab lett 32 1997 367 376 ann statist 25 1997 2368 2387 and x 2 scale space adv in appl probab 33 2001 773 793 the trick involves approaching the problem from the point of view of roy s union intersection principle the results are applied to a problem in shape analysis where we look for brain damage due to nonmissile trauma

fractal dimension and the persistent homology of random geometric complexes abstract we prove that the fractal dimension of a metric space equipped with an ahlfors regular measure can be recovered from the persistent homology of random samples our main result is that if x 1 x n are i i d samples from a d ahlfors regular measure on a metric space and e 0 x 1 x n denotes the weight of the minimum spanning tree on x 1 x n e 0 x 1 x n e t x 1 x n e then there exist constants 0 c 1 c 2 so that c 1 n d d e 0 x 1 x n c 2 with high probability as n in particular d can be recovered from the limit log e 0 x 1 x n log n d d this is a generalization of a result of steele 62 from the non singular case to the fractal setting we also construct an example of an ahlfors regular measure for which the limit lim n n d d e 0 x 1 x n does not exist with high probability and prove analogous results for weighted sums defined in terms of higher dimensional persistent homology

symbols of truncated toeplitz operators abstract we consider three topics connected with coinvariant subspaces of the backward shift operator in hardy spaces h p properties of truncated toeplitz operators carleson type embedding theorems for the coinvariant subspaces factorizations of pseudocontinuable functions from h 1 these problems turn out to be closely connected and even in a sense equivalent the new approach based on the factorizations allows us to answer a number of challenging questions about truncated toeplitz operators posed by d sarason

bernstein type inequalities for shift coinvariant subspaces and their applications to carleson embeddings abstract we study weighted norm inequalities for the derivatives bernstein type inequalities in the shift coinvariant subspaces k p of the hardy class h p in the upper half plane it is shown that the differentiation operator acts from k p to certain spaces of the form l p w where the weight w x depends on the density of the spectrum of near the point x of the real line we discuss an application of the bernstein type inequalities to the problems of the description of measures for which k p l p and of compactness of such embeddings new versions of carleson type embedding theorems are obtained generalizing the theorems due to w s cohn and a l volberg s r treil

a new fuzzy programming method to derive the priority vector from an interval reciprocal comparison matrix in this paper we discuss the interval reciprocal comparison matrices and suggest a new fuzzy programming method nfpm to derive the priority vector from an interval reciprocal comparison matrix based on mikhailov s membership function we first give a membership function which is used to measure the decision maker s satisfaction degree for the priority vector derived from each interval constraint then by using the membership function we propose a new fuzzy programming method to get the most satisfaction degree of each priority vector the nfpm has many prominent characteristics especially it is capable of deriving the priority vector from a consistent or inconsistent interval reciprocal comparison matrix moreover the nfpm can overcome some drawbacks of mikhailov s fuzzy programming method fpm for inconsistent interval reciprocal comparison matrices finally several examples are given to illustrate the applicability and advantage of the proposed method

large deviations for wigner s law and voiculescu s non commutative entropy we study the spectral measure of gaussian wigner s matrices and prove that it satisfies a large deviation principle we show that the good rate function which governs this principle achieves its minimum value at wigner s semicircular law which entails the convergence of the spectral measure to the semicircular law as a conclusion we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about voiculescu s non commutative entropy

a statistical comparative study of different similarity measures of consensus in group decision making an essential aim in group decision making gdm problems is to achieve a high level of consensus among experts consensus is defined as general or widespread agreement and it is usually modelled mathematically via a similarity function measuring how close experts opinions or preferences are similarity functions are defined based on the use of a metric describing the distance between experts opinions or preferences in the literature different metrics or distance functions have been proposed to implement in consensus models but no study has been carried out to analyse the influence the use of different distance functions can have in the gdm process this paper presents a comparative study of the effect of the application of some different distance functions for measuring consensus in gdm by using the nonparametric wilcoxon matched pairs signed ranks test it is concluded that different distance functions can produce significantly different results moreover it is also shown that their application also has a significant effect on the speed of achieving consensus finally these results are analysed and used to derive decision support rules based on a convergent criterion that can be used to control the convergence speed of the consensus process using the compared distance functions

the extremal process of critical points of the pure p spin spherical spin glass model recently sharp results concerning the critical points of the hamiltonian of the p spin spherical spin glass model have been obtained by means of moments computations in particular these moments computations allow for the evaluation of the leading term of the ground state i e of the global minimum in this paper we study the extremal point process of critical points that is the point process associated to all critical values in the vicinity of the ground state we show that the latter converges in distribution to a poisson point process of exponential intensity in particular we identify the correct centering of the ground state and prove the convergence in distribution of the centered minimum to a minus gumbel variable these results are identical to what one obtains for a sequence of i i d variables correctly normalized namely we show that the model is in the universality class of rem

group aggregation of pairwise comparisons using multi objective optimization in group decision making multiple decision makers dms aim to reach a consensus ranking of alternatives in a decision problem the differing expertise experience and potentially conflicting interests of the dms will result in the need for some form of conciliation to achieve consensus pairwise comparisons are commonly used to elicit values of preference of a dm the aggregation of the preferences of multiple dms must additionally consider potential conflict between dms and how these conflicts may result in a need for compromise to reach group consensus we present an approach to aggregating the preferences of multiple dms utilizing multi objective optimization to derive and highlight underlying conflict between the dms when seeking to achieve consensus extracting knowledge of conflict facilitates both traceability and transparency of the trade offs involved when reaching a group consensus further the approach incorporates inconsistency reduction during the aggregation process to seek to diminish adverse effects upon decision outcomes the approach can determine a single final solution based on either global compromise information or through utilizing weights of importance of the dms within multi criteria decision making we present a case study within the analytical hierarchy process from which we derive a richer final ranking of the decision alternatives

complexity of random smooth functions on the high dimensional sphere we analyze the landscape of general smooth gaussian functions on the sphere in dimension n when n is large we give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean euler characteristic of level sets we then find two possible scenarios for the bottom landscape one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large we end the paper by discussing how these results can be interpreted in the language of spin glasses models

a comparative study of the numerical scales and the prioritization methods in ahp abstract in the analytic hierarchy process ahp a decision maker first gives linguistic pairwise comparisons then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them and finally derives a priority vector from the numerical pairwise comparisons in particular the validity of this decision making tool relies on the choice of numerical scale and the design of prioritization method by introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices lpcms and by defining the deviation measures of lpcms we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods using these performance measure algorithms we compare the most common numerical scales the saaty scale the geometrical scale the ma zheng scale and the salo hamalainen scale and the prioritization methods the eigenvalue method and the logarithmic least squares method in addition we also discuss the parameter of the geometrical scale develop a new prioritization method and construct an optimization model to select the appropriate numerical scales for the ahp decision makers the findings in this paper can help the ahp decision makers select suitable numerical scales and prioritization methods

tilted euler characteristic densities for central limit random fields with application to bubbles local increases in the mean of a random field are detected con servatively by thresholding a field of test statistics at a level u chosen to control the tail probability or p value of its maximum this p value is approximated by the expected euler characteristic ec of the ex cursion set of the test statistic field above u denoted e au under isotropy one can use the expansion e au p k vkk u where vk is an intrinsic volume of the parameter space andk is an ec density of the field ec densities are available for a number of pro cesses mainly those constructed from multivariate gaussian fields via smooth functions using saddlepoint methods we derive an ex pansion fork u for fields which are only approximately gaussian but for which higher order cumulants are available we focus on linear combinations of n independent non gaussian fields whence a central limit theorem is in force the threshold u is allowed to grow with the sample size n in which case our expression has a smaller relative asymptotic error than the gaussian ec density several illustrative examples including an application to bubbles data accompany the theory

free energy of the spherical mean field model we compute at any temperature the free energy of the multi p spin spherical model when only terms for p even are considered

bounded symbols and reproducing kernel thesis for truncated toeplitz operators compressions of toeplitz operators to coinvariant subspaces of h2 are called truncated toeplitz operators we study two questions related to these operators the first raised by sarason is whether boundedness of the operator implies the existence of a bounded symbol the second is the reproducing kernel thesis we show that in general the answer to the first question is negative and we exhibit some classes of spaces for which the answers to both questions are positive

the parisi ultrametricity conjecture in this paper we prove that the support of a random measure on the unit ball of a separable hilbert space that satises the ghirlanda guerra identities must be ultrametric with probability one this implies the parisi ultrametricity conjecture in mean eld spin glass models such as the sherrington kirkpatrick and mixed p spin models for which gibbs measures are known to satisfy the ghirlanda guerra identities in the thermodynamic limit

distance functions and geodesics on submanifolds of r d and point clouds a theoretical and computational framework for computing intrinsic distance functions and geodesics on submanifolds of bbb r d given by point clouds is introduced and developed in this paper

bi criteria evolution strategy in estimating weights from the ahp ratio scale matrices abstract the problem of deriving weights from ratio scale matrices in an analytic hierarchy process ahp is addressed by researchers worldwide there are various ways to solve the problem that are generally grouped into simple matrix and optimization methods all methods have received criticism regarding the accuracy of derived weights and different criteria are in use to compare the weights obtained from different methods because the set of pareto non dominated solutions weights is unknown and for inconsistent matrices is indefinite a bi criterion optimization approach is proposed for manipulating such matrices the problem specific evolution strategy algorithm esa is implemented for a robust stochastic search over a feasible indefinite solution space the fitness function is defined as a scalar vector function composed of the common error measure i e the euclidean distance and a minimum violation error that accounts for no violation of the rank ordering the encoding scheme and other components of the search engine are adjusted to preserve the imposed constraints related to the required normalized values of the weights the solutions generated by the proposed approach are compared with solutions obtained by five well known prioritization techniques for three judgment matrices taken from the literature in these and other test applications the prioritization method that uses the entitled weights estimation by evolution strategy algorithm weesa appears to be superior to other methods if only two the most commonly used methods are applied the euclidean distance and minimum violation exclusion criteria

aging of spherical spin glasses sompolinski and zippelius 1981 propose the study of dynamical systems whose invariant measures are the gibbs measures for hard to analyze statistical physics models of interest in the course of doing so physicists often report of an aging phenomenon for example aging is expected to happen for the sherrington kirkpatrick model a disordered mean field model with a very complex phase transition in equilibrium at low temperature we shall study the langevin dynamics for a simplified spherical version of this model the induced rotational symmetry of the spherical model reduces the dynamics in question to an n dimensional coupled system of ornstein uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices we obtain the limiting dynamics for n approaching infinity and by analyzing its long time behavior explain what is aging mathematically speaking what causes this phenomenon and what is its relationship with the phase transition of the corresponding equilibrium invariant measures

spectral clustering and the high dimensional stochastic blockmodel networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors social ne tworks representing people who communicate with each other are one example communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks spectral clustering is a popular and computationally feasi ble method to discover these communities the stochastic block model holland et al 1983 is a social network model with well defined communities each node is a member of one community for a network generated from the stochastic block model we bound the number of nodes misclus tered by spectral clustering the asymptotic results in th is paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes hence the name high dimensional in order to study spectral clustering under the stochastic block model we first show that under the more general latent space model the eigenvectors of the normalized graph laplacian asymptotically converge to the eigenvectors of a population normal ized graph laplacian aside from the implication for spectral clustering this provides insight into a graph visualization technique our method of studying the eigenvectors of random matrices is original ams 2000 subject classifications primary 62h30 62h25 secondary 60b20

a topological view of unsupervised learning from noisy data in this paper we take a topological view of unsupervised learning from this point of view clustering may be interpreted as trying to find the number of connected components of any underlying geometrically structured probability distribution in a certain sense that we will make precise we construct a geometrically structured probability distribution that seems appropriate for modeling data in very high dimensions a special case of our construction is the mixture of gaussians where there is gaussian noise concentrated around a finite set of points the means more generally we consider gaussian noise concentrated around a low dimensional manifold and discuss how to recover the homology of this underlying geometric core from data that do not lie on it we show that if the variance of the gaussian noise is small in a certain sense then the homology can be learned with high confidence by an algorithm that has a weak linear dependence on the ambient dimension our algorithm has a natural interpretation as a spectral learning algorithm using a combinatorial laplacian of a suitable data derived simplicial complex

distance based consensus models for fuzzy and multiplicative preference relations this paper proposes a distance based consensus model for fuzzy preference relations where the weights of fuzzy preference relations are automatically determined two indices an individual to group consensus index ici and a group consensus index gci are introduced an iterative consensus reaching algorithm is presented and the process terminates until both the ici and gci are controlled within predefined thresholds the model and algorithm are then extended to handle multiplicative preference relations finally two examples are illustrated and comparative analyses demonstrate the effectiveness of the proposed methods

group consensus algorithms based on preference relations in many group decision making situations decision makers preferences for alternatives are expressed in preference relations including fuzzy preference relations and multiplicative preference relations an important step in the process of aggregating preference relations is to determine the importance weight of each preference relation in this paper we develop a number of goal programming models and quadratic programming models based on the idea of maximizing group consensus our models can be used to derive the importance weights of fuzzy preference relations and multiplicative preference relations we further develop iterative algorithms for reaching acceptable levels of consensus in group decision making based on fuzzy preference relations or multiplicative preference relations finally we include an illustrative example

random geometric complexes in the thermodynamic regime we consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices in particular we study the betti numbers of these complexes as the number of vertices becomes large obtaining limit theorems for means strong laws concentration inequalities and central limit theorems as opposed to most prior papers treating random complexes the limit with which we work is in the so called thermodynamic regime which includes the percolation threshold in which the complexes become very large and complicated with complex homology characterised by diverging betti numbers the proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of mayer vietoris exact sequences the mayer vietoris arguments are crucial since homology in general and betti numbers in particular are global rather than local phenomena and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality

estimating the number of peaks in a random field using the hadwiger characteristic of excursion sets with applications to medical images excursion sets has been studied by adler who defined the ig integral geometry characteristic of excursion sets as a multidimensional analogue of the number of upcrossings of the threshold by a unidimensional process the ig characteristic equals the euler characteristic of an excursion set provided that the set does not touch the boundary of the volume and adler found the expected ig characteristic for a stationary random field inside a fixed volume worsley evans marrett and neelin used the ig characteristic as an estimator of the number of regions of activation of positron emission tomography pet images of blood flow in the brain unfortunately the ig characteristic is only defined on intervals it is not invariant under rotations and it can fail to count connected regions that touch the boundary this is important since activation often occurs in the cortical regions near the boundary of the brain in this paper we study the hadwiger characteristic which is defined on arbitrary sets is invariant under rotations and does count connected regions whether they touch the boundary or not our main result is a simple expression for the expected hadwiger characteristic for an isotropic stationary random field in two and three dimensions and on a smooth surface embedded in three dimensions results are applied to pet studies of pain perception and word recognition

on the energy landscape of spherical spin glasses abstract we investigate the energy landscape of the spherical mixed even p spin model near its maximum energy we relate the distance between pairs of near maxima to the support of the parisi measure at zero temperature we then provide an algebraic relation that characterizes one step replica symmetric breaking parisi measures for these measures we show that any two nonparallel spin configurations around the maximum energy are asymptotically orthogonal to each other in sharp contrast we study models with full replica symmetry breaking and show that all possible values of the asymptotic distance are attained near the maximum energy

the geometry of the gibbs measure of pure spherical spin glasses we analyze the statics for pure p spin spherical spin glass models with p ge 3 at low enough temperature with f n beta denoting the free energy we compute the logarithmic second order term of nf n beta and prove that for an appropriate centering c n beta nf n beta c n beta is a tight sequence we further establish the absence of temperature chaos those results follow from the following geometric picture we prove for the gibbs measure of interest by itself asymptotically the measure splits into infinitesimal spherical bands centered at deep minima playing the role of pure states for the pure models the latter makes precise the picture of many valleys separated by high mountains and significant parts of the tap analysis from the physics literature

duality theorems for coinvariant subspaces of h1 abstract let be an inner function satisfying the connected level set condition of b cohn and let k 1 be the shift coinvariant subspace of the hardy space h 1 generated by we describe the dual space to k 1 in terms of a bounded mean oscillation with respect to the clark measure of namely we prove that k 1 z h 1 bmo the result yields a two sided estimate for the operator norm of a finite hankel matrix of size n n via bmo 2 n norm of its standard symbol where 2 n is the haar measure on the group c 2 n 1