stationary distribution and cover time of sparse directed configuration models we consider sparse digraphs generated by the configuration model with given in degree and out degree sequences we establish that with high probability the cover time is linear up to a poly logarithmic correction for a large class of degree sequences we determine the exponent gamma ge 1 of the logarithm and show that the cover time grows as n log gamma n where n is the number of vertices the results are obtained by analysing the extremal values of the stationary distribution of the digraph in particular we show that the stationary distribution pi is uniform up to a poly logarithmic factor and that for a large class of degree sequences the minimal values of pi have the form frac 1 n log 1 gamma n while the maximal values of pi behave as frac 1 n log 1 kappa n for some other exponent kappa in 0 1 in passing we prove tight bounds on the diameter of the digraphs and show that the latter coincides with the typical distance between two vertices

generalized iterative relief for supervised distance metric learning the relief algorithm is a popular approach for feature weighting many extensions of the relief algorithm are developed and i relief is one of the famous extensions in this paper i relief is generalized for supervised distance metric learning to yield a mahananobis distance function the proposed approach is justified by showing that the objective function of the generalized i relief is closely related to the expected leave one out nearest neighbor classification rate in addition the relationships among the generalized i relief the neighbourhood components analysis and graph embedding are also pointed out experimental results on various data sets all demonstrate the superiority of the proposed approach

field theory of bicritical and tetracritical points i statics we calculate the static critical behavior of systems of o n plus sign in circle o n perpendicular symmetry by the renormalization group method within the minimal subtraction scheme in two loop order summation methods lead to fixed points describing multicritical behavior their stability border lines in the space of the order parameter components n and n perpendicular and spatial dimension d are calculated the essential features obtained already in two loop order for the interesting case of an antiferromagnet in a magnetic field n 1 n perpendicular 2 are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points leading to very small transient exponents we are also able to calculate the flow of static couplings which allows us to consider the attraction region depending on the nonuniversal background parameters the existence of different multicritical behavior bicritical or tetracritical is possible including a triple point

copolymer networks and stars scaling exponents we explore and calculate the rich scaling behavior of copolymer networks in solution by renormalization group methods we establish a field theoretic description in terms of composite operators our third order resummation of the spectrum of scaling dimensions brings about remarkable features the special convexity properties of the spectra allow for a multifractal interpretation while preserving stability of the theory this behavior could not be found for power of field operators of usual ensuremath varphi 4 field theory the two dimensional 2d limit of the mutually avoiding walk star apparently corresponds to results of a conformal kac series such a classification seems not possible for the 2d limit of other copolymer stars we furthermore provide a consistency check of two complementary renormalization schemes ensuremath varepsilon expansion and renormalization at fixed dimension calculating a large collection of independent exponents in both approaches

iterative relief for feature weighting algorithms theories and applications relief is considered one of the most successful algorithms for assessing the quality of features in this paper we propose a set of new feature weighting algorithms that perform significantly better than relief without introducing a large increase in computational complexity our work starts from a mathematical interpretation of the seemingly heuristic relief algorithm as an online method solving a convex optimization problem with a margin based objective function this interpretation explains the success of relief in real application and enables us to identify and address its following weaknesses relief makes an implicit assumption that the nearest neighbors found in the original feature space are the ones in the weighted space and relief lacks a mechanism to deal with outlier data we propose an iterative relief i relief algorithm to alleviate the deficiencies of relief by exploring the framework of the expectation maximization algorithm we extend i relief to multiclass settings by using a new multiclass margin definition to reduce computational costs an online learning algorithm is also developed convergence analysis of the proposed algorithms is presented the results of large scale experiments on the uci and microarray data sets are reported which demonstrate the effectiveness of the proposed algorithms and verify the presented theoretical results

a boosting approach for supervised mahalanobis distance metric learning determining a proper distance metric is often a crucial step for machine learning in this paper a boosting algorithm is proposed to learn a mahalanobis distance metric similar to most boosting algorithms the proposed algorithm improves a loss function iteratively in particular the loss function is defined in terms of hypothesis margins and a metric matrix base learner specific to the boosting framework is also proposed experimental results show that the proposed approach can yield effective mahalanobis distance metrics for a variety of data sets and demonstrate the feasibility of the proposed approach

an optimal global nearest neighbor metric a quadratic metric dao x y x y t ao x y is proposed which minimizes the mean squared error between the nearest neighbor asymptotic risk and the finite sample risk under linearity assumptions a heuristic argument is given which indicates that this metric produces lower mean squared error than the euclidean metric a nonparametric estimate of ao is developed if samples appear to come from a gaussian mixture an alternative parametrically directed distance measure is suggested for nearness decisions within a limited region of space examples of some two class gaussian mixture distributions are included

local discriminative distance metrics ensemble learning the ultimate goal of distance metric learning is to incorporate abundant discriminative information to keep all data samples in the same class close and those from different classes separated local distance metric methods can preserve discriminative information by considering the neighborhood influence in this paper we propose a new local discriminative distance metrics lddm algorithm to learn multiple distance metrics from each training sample a focal sample and in the vicinity of that focal sample focal vicinity to optimize local compactness and local separability those locally learned distance metrics are used to build local classifiers which are aligned in a probabilistic framework via ensemble learning theoretical analysis proves the convergence rate bound the generalization bound of the local distance metrics and the final ensemble classifier we extensively evaluate lddm using synthetic datasets and large benchmark uci datasets

graph embedding and extensions a general framework for dimensionality reduction a large family of algorithms supervised or unsupervised stemming from statistics or geometry theory has been designed to provide different solutions to the problem of dimensionality reduction despite the different motivations of these algorithms we present in this paper a general formulation known as graph embedding to unify them within a common framework in graph embedding each algorithm can be considered as the direct graph embedding or its linear kernel tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided furthermore the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms by utilizing this framework as a tool we propose a new supervised dimensionality reduction algorithm called marginal fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class while the penalty graph connects the marginal points and characterizes the interclass separability we show that mfa effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions real face recognition experiments show the superiority of our proposed mfa in comparison to lda also for corresponding kernel and tensor extensions

critical exponents of manhattan hamiltonian walks in two dimensions from potts and o n models we consider a set of hamiltonian circuits filling a manhattan lattice i e a square lattice with alternating traffic regulation we show that the generating function with fugacityz of this set is identical to the critical partition function of aq state potts model on an unoriented square lattice withq1 2 z the set of critical exponents governing correlations of hamiltonian circuits is derived using a coulomb gas technique these exponents are also found to be those of an o n vector model in the low temperature phase withn q1 2 z the critical exponents in the limitz 0 are then those of spanning trees q 0 and of dense polymers n 0 t tc corresponding to a conformal theory with central chargec 2 this shows that the manhattan orientation and the hamiltonian constraint of filling all the lattice are irrelevant for the infrared critical properties of hamiltonian walks

two dimensional copolymers and multifractality comparing perturbative expansions monte carlo simulations and exact results we analyze the scaling laws for a set of two different species of long flexible polymer chains joined together at one of their extremities copolymer stars in space dimension d 2 we use a formerly constructed field theoretic description and compare our perturbative results for the scaling exponents with recent conjectures for exact conformal scaling dimensions derived by a conformal invariance technique in the context of d 2 quantum gravity a simple monte carlo simulation brings about reasonable agreement with both approaches we analyze the remarkable multifractal properties of the spectrum of scaling exponents

discriminant adaptive nearest neighbor classification nearest neighbour classification expects the class conditional probabilities to be locally constant and suffers from bias in high dimensions we propose a locally adaptive form of nearest neighbour classification to try to ameliorate this curse of dimensionality we use a local linear discriminant analysis to estimate an effective metric for computing neighbourhoods we determine the local decision boundaries from centroid information and then shrink neighbourhoods in directions orthogonal to these local decision boundaries and elongate them parallel to the boundaries thereafter any neighbourhood based classifier can be employed using the modified neighbourhoods the posterior probabilities tend to be more homogeneous in the modified neighbourhoods we also propose a method for global dimension reduction that combines local dimension information in a number of examples the methods demonstrate the potential for substantial improvements over nearest neighbour classification

higher conformal multifractality we derive from conformal invariance and quantum gravity the multifractal spectrum f of the harmonic measure i e electrostatic potential or diffusion field near any conformally invariant fractal in two dimensions it gives the hausdorff dimension of the set of points where the potential varies with distance r to the fractal frontier as r first examples are a random walk i e a brownian motion a self avoiding walk or a critical percolation cluster the generalized dimensions d n as well as the multifractal functions f are derived and are all identical for these three cases the external frontiers of a brownian motion and of a percolation cluster are thus identical to a self avoiding walk in the scaling limit the multifractal mf function f c of the electrostatic potential near any conformally invariant fractal boundary like a critical o n loop or a q state potts cluster is given as a function of the central charge c of the associated conformal field theory the dimensions dep of the external perimeter and dh of the hull of a critical scaling curve or cluster obey the superuniversal duality equation d text ep 1 d text d 1 tfrac 1 4 finally for a conformally invariant scaling curve which is simple i e without double points we derive higher multifractal functions like the universal function f2 which gives the hausdorff dimension of the points where the potential varies jointly with distance r as r on one side of the curve and as r on the other the general case of the potential distribution between the branches of a star made of an arbitrary number of scaling paths is also treated the results apply to critical o n loops potts clusters and to the sle process we present a duality between external perimeters of potts clusters and o n loops at their critical point as well as the corresponding duality in the sle process for 16

random walks polymers percolation and quantum gravity in two dimensions we consider l planar random walks or brownian motions of large length t starting at neighboring points and the probability pl t t l that their paths do not intersect by a 2d quantum gravity method i e a non linear map onto a random riemann surface the former conjecture that l 124 4l2 1 is established this also applies to the half plane where l l3 1 2l as well as to non intersection exponents of unions of paths mandelbrot s conjecture for the hausdorff dimension dh 43 of the frontier of a brownian path follows from dh 2 32 we then consider in two dimensions the most general star shaped copolymer mixing random rw or self avoiding walks saw with specific mutual avoidance interactions thereof its exact conformal scaling dimensions in the plane are derived the harmonic measures or electrostatic potential or diffusion field near a rw or a saw or near a critical percolation cluster are also considered their moments exhibit a multifractal spectrum the generalized dimensions d n as well as the function are derived and are shown to be all identical these are examples of exact conformal multifractality

nonintersection exponents for brownian paths ii estimates and applications to a random fractal let x and y be independent two dimensional brownian motions x 0 0 0 y 0 e 0 and let p e p x 0 1 y 0 1 o q e y 0 1 does not contain a closed loop around 0 asymptotic estimates when e 0 of p e q e and some related probabilities are given let f be the boundary of the unbounded connected component of r 2 z 0 1 where z t x t tx 1 for t 0 1 then f is a closed jordan arc and the hausdorff dimension of f is less or equal to 3 2 1 4 2

dimensionality reduction of multimodal labeled data by local fisher discriminant analysis reducing the dimensionality of data without losing intrinsic information is an important preprocessing step in high dimensional data analysis fisher discriminant analysis fda is a traditional technique for supervised dimensionality reduction but it tends to give undesired results if samples in a class are multimodal an unsupervised dimensionality reduction method called locality preserving projection lpp can work well with multimodal data due to its locality preserving property however since lpp does not take the label information into account it is not necessarily useful in supervised learning scenarios in this paper we propose a new linear supervised dimensionality reduction method called local fisher discriminant analysis lfda which effectively combines the ideas of fda and lpp lfda has an analytic form of the embedding transformation and the solution can be easily computed just by solving a generalized eigenvalue problem we demonstrate the practical usefulness and high scalability of the lfda method in data visualization and classification tasks through extensive simulation studies we also show that lfda can be extended to non linear dimensionality reduction scenarios by applying the kernel trick

multifractality of brownian motion near absorbing polymers we characterize the multifractal behavior of brownian motion in the vicinity of an absorbing star polymer we map the problem to an o m symmetric phi 4 field theory relating higher moments of the laplacian field of brownian motion to corresponding composite operators the resulting spectra of scaling dimensions of these operators display the convexity properties that are necessarily found for multifractal scaling but unusual for power of field operators in field theory using a field theoretic renormalization group approach we obtain the multifractal spectrum for absorption at the core of a polymer star as an asymptotic series we evaluate these series using resummation techniques

random walk on sparse random digraphs a finite ergodic markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale originally discovered in the context of card shuffling aldous and diaconis in am math mon 93 333 348 1986 this remarkable phenomenon is now rigorously established for many reversible chains here we consider the non reversible case of random walks on sparse directed graphs for which even the equilibrium measure is far from being understood we work under the configuration model allowing both the in degrees and the out degrees to be freely specified we establish the cutoff phenomenon determine its precise window and prove that the cutoff profile approaches a universal shape we also provide a detailed description of the equilibrium measure

distance metric learning for large margin nearest neighbor classification the accuracy of k nearest neighbor knn classification depends significantly on the metric used to compute distances between different examples in this paper we show how to learn a mahalanobis distance metric for knn classification from labeled examples the mahalanobis metric can equivalently be viewed as a global linear transformation of the input space that precedes knn classification using euclidean distances in our approach the metric is trained with the goal that the k nearest neighbors always belong to the same class while examples from different classes are separated by a large margin as in support vector machines svms the margin criterion leads to a convex optimization based on the hinge loss unlike learning in svms however our approach requires no modification or extension for problems in multiway as opposed to binary classification in our framework the mahalanobis distance metric is obtained as the solution to a semidefinite program on several data sets of varying size and difficulty we find that metrics trained in this way lead to significant improvements in knn classification sometimes these results can be further improved by clustering the training examples and learning an individual metric within each cluster we show how to learn and combine these local metrics in a globally integrated manner

granger causality driven ahp for feature weighted knn the knn algorithm remains a popular choice for pattern classification till date due to its non parametric nature easy implementation and the fact that its classification error is bounded by twice the bayes error in this paper we show that the performance of the knn classifier improves significantly from the use of training class wise group statistics based two criteria during pairwise comparison of features in a given dataset granger causality is employed to assign preferences to each criteria analytic hierarchy process ahp is applied to obtain weights for different features from the two criteria and their preferences finally these weights are used to build a weighted distance function for the knn classification comprehensive experimentation on fifteen benchmark datasets of the uci machine learning repository clearly reveals the supremacy of the proposed granger causality driven ahp induced knn algorithm over the knn method with many different distance metrics and with various feature selection strategies in addition the proposed method is also shown to perform well on high dimensional face and hand writing recognition datasets highlightsfeature weighting for knn by a multi criteria based decision analysis tool called ahp automated weight assignment in criteria matrix of ahp using group statistics criteria preference selection in ahp with granger causality superior classification performance over knn with many other feature weighting selection methods

universality classes of the three dimensional mn vector model we study the conditions under which the critical behaviour of the three dimensional mn vector model does not belong to the spherically symmetrical universality class in the calculations we rely on the field theoretical re normalization group approach in different regularization schemes adjusted by resummation and extended analysis of the series for renormalization group functions we address the question why the renormalization group perturbation theory expansions available for the model with a record six loop accuracy have not allowed so far for a definite answer about the universality class for certain particular values of dimensions m n we show that an analysis based on the marginal dimensions rather than on the stability exponents leads to the robust results about the phase diagram of the model

on the criticality of frustrated spin systems with noncollinear order we analyse the universal features of the critical behaviour of frustrated spin systems with noncollinear order by means of the field theoretical renormalization group approach we study the 3d model of a frustrated magnet and obtain pseudo e expansions for its universal order parameter marginal dimensions these dimensions govern accessibility of the renormalization group transformation fixed points and hence define the scenario of the phase transition

cutoff at the entropic time for sparse markov chains we study convergence to equilibrium for a large class of markov chains in random environment the chains are sparse in the sense that in every row of the transition matrix p the mass is essentially concentrated on few entries moreover the random environment is such that rows of p are independent and such that the entries are exchangeable within each row this includes various models of random walks on sparse random directed graphs the models are generally non reversible and the equilibrium distribution is itself unknown in this general setting we establish the cutoff phenomenon for the total variation distance to equilibrium with mixing time given by the logarithm of the number of states times the inverse of the average row entropy of p as an application we consider the case where the rows of p are i i d random vectors in the domain of attraction of a poisson dirichlet law with index 0 1 our main results are based on a detailed analysis of the weight of the trajectory followed by the walker this approach offers an interpretation of cutoff as an instance of the concentration of measure phenomenon

acquiring linear subspaces for face recognition under variable lighting previous work has demonstrated that the image variation of many objects human faces in particular under variable lighting can be effectively modeled by low dimensional linear spaces even when there are multiple light sources and shadowing basis images spanning this space are usually obtained in one of three ways a large set of images of the object under different lighting conditions is acquired and principal component analysis pca is used to estimate a subspace alternatively synthetic images are rendered from a 3d model perhaps reconstructed from images under point sources and again pca is used to estimate a subspace finally images rendered from a 3d model under diffuse lighting based on spherical harmonics are directly used as basis images in this paper we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low dimensional linear space and that this subspace is close to those acquired by the other methods more specifically there exist configurations of k point light source directions with k typically ranging from 5 to 9 such that by taking k images of an object under these single sources the resulting subspace is an effective representation for recognition under a wide range of lighting conditions since the subspace is generated directly from real images potentially complex and or brittle intermediate steps such as 3d reconstruction can be completely avoided nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse harmonic light fields we validate the use of subspaces constructed in this fashion within the context of face recognition

supervised distance metric learning through maximization of the jeffrey divergence over the past decades distance metric learning has attracted a lot of interest in machine learning and related fields in this work we propose an optimization framework for distance metric learning via linear transformations by maximizing the jeffrey divergence between two multivariate gaussian distributions derived from local pairwise constraints in our method the distance metric is trained on positive and negative difference spaces which are built from the neighborhood of each training instance so that the local discriminative information is preserved we show how to solve this problem with a closed form solution rather than using tedious optimization procedures the solution is easy to implement and tractable for large scale problems experimental results are presented for both a linear and a kernelized version of the proposed method for k nearest neighbors classification we obtain classification accuracies superior to the state of the art distance metric learning methods in several cases while being competitive in others highlightswe propose a novel distance metric learning method dmlmj for classification dmlmj is simple to implement and it can be solved analytically we extend dmlmj into a kernelized version to tackle non linear problems experiments on several data sets show the effectiveness of the proposed method

on the time taken by random walks on finite groups to visit every state consider a random walk on a finite group g suppose that the time taken for the transition probabilities to approach the uniform distribution is small compared with g then the time taken for the random walk to visit every state is approximately r g log g where r is the mean number of returns to the initial state in the short term

stein s method for concentration inequalities we introduce a version of stein s method for proving concentration and moment inequalities in problems with dependence simple illustrative examples from combinatorics physics and mathematical statistics are provided