nonuniqueness of nematic liquid crystal flows in dimension three abstract for suitable initial and boundary data we construct infinitely many weak solutions to the nematic liquid crystal flows in dimension three these solutions are in the axisymmetric class with bounded energy and backward bubbling at a large time

consistency of spectral clustering consistency is a key property of all statistical procedures analyzing randomly sampled data surprisingly despite decades of work little is known about consistency of most clustering algorithms in this paper we investigate consistency of the popular family of spectral clustering algorithms which clusters the data with the help of eigenvectors of graph laplacian matrices we develop new methods to establish that for increasing sample size those eigenvectors converge to the eigenvectors of certain limit operators as a result we can prove that one of the two major classes of spectral clustering normalized clustering converges under very general conditions while the other unnormalized clustering is only consistent under strong additional assumptions which are not always satisfied in real data we conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering

manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

on the probability of staying above a wall for the 2 1 dimensional sos model at low temperature we obtain sharp asymptotics for the probability that the 2 1 dimensional discrete sos interface at low temperature is positive in a large region for a square region lambda both under the infinite volume measure and under the measure with zero boundary conditions around lambda this probability turns out to behave like exp tau beta 0 l log l with tau beta 0 the surface tension at zero tilt also called step free energy and l the box side this behavior is qualitatively different from the one found for continuous height massless gradient interface models bolthausen et al commun math phys 170 2 417 443 1995 deuschel et al stochastic process appl 89 2 333 354 2000

unified framework for representing and ranking in the database retrieval and nearest neighbor classification tasks the two basic problems are to represent the query and database objects and to learn the ranking scores of the database objects to the query many studies have been conducted for the representation learning and the ranking score learning problems however they are always learned independently from each other in this paper we argue that there are some inner relationships between the representation and ranking of database objects and try to investigate their relationships by learning them in a unified way to this end we proposed the unified framework for representation and ranking ur2 of objects for the database retrieval and nearest neighbor classification tasks the learning of representation parameter and the ranking scores are modeled within one single unified objective function the objective function is optimized alternately with regard to representation parameter and the ranking scores based on the optimization results iterative algorithms are developed to learn the representation parameter and the ranking scores on a unified way moreover with two different formulas of representation feature selection and subspace learning we give two versions of ur2 the proposed algorithms are tested on two challenging tasks mri image based brain tumor retrieval and nearest neighbor classification based protein identification the experiments show the advantage of the proposed unified framework over the state of the art independent representation and ranking methods highlightsobject representation and ranking are two basic problems for database retrieval they are always studies separately we proposed the first unified framework for both representation and ranking experimental results show its advantage over independent representation or ranking methods

nonuniqueness for the heat flow of harmonic maps on the disk we prove that for suitable initial data the heat flow of harmonic maps from the disk to the sphere admits infinitely many solutions characterised by backward bubbling at some arbitrarily large time all having uniformly bounded energy

spectral embedded clustering a framework for in sample and out of sample spectral clustering spectral clustering sc methods have been successfully applied to many real world applications the success of these sc methods is largely based on the manifold assumption namely that two nearby data points in the high density region of a low dimensional data manifold have the same cluster label however such an assumption might not always hold on high dimensional data when the data do not exhibit a clear low dimensional manifold structure e g high dimensional and sparse data the clustering performance of sc will be degraded and become even worse than k means clustering in this paper motivated by the observation that the true cluster assignment matrix for high dimensional data can be always embedded in a linear space spanned by the data we propose the spectral embedded clustering sec framework in which a linearity regularization is explicitly added into the objective function of sc methods more importantly the proposed sec framework can naturally deal with out of sample data we also present a new laplacian matrix constructed from a local regression of each pattern and incorporate it into our sec framework to capture both local and global discriminative information for clustering comprehensive experiments on eight real world high dimensional datasets demonstrate the effectiveness and advantages of our sec framework over existing sc methods and k means based clustering methods our sec framework significantly outperforms sc using the nystrom algorithm on unseen data

random walks in cones we study the asymptotic behavior of a multidimensional random walk in a general cone we find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone the main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments for the proof of tail asymptotics and integral limit theorems we use a strong approximation of random walks by brownian motion for the proof of local limit theorems we suggest a rather simple approach which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks we also discuss some possible applications of our results to ordered random walks and lattice path enumeration

discriminative semi supervised feature selection via manifold regularization feature selection has attracted a huge amount of interest in both research and application communities of data mining we consider the problem of semi supervised feature selection where we are given a small amount of labeled examples and a large amount of unlabeled examples since a small number of labeled samples are usually insufficient for identifying the relevant features the critical problem arising from semi supervised feature selection is how to take advantage of the information underneath the unlabeled data to address this problem we propose a novel discriminative semi supervised feature selection method based on the idea of manifold regularization the proposed approach selects features through maximizing the classification margin between different classes and simultaneously exploiting the geometry of the probability distribution that generates both labeled and unlabeled data in comparison with previous semi supervised feature selection algorithms our proposed semi supervised feature selection method is an embedded feature selection method and is able to find more discriminative features we formulate the proposed feature selection method into a convex concave optimization problem where the saddle point corresponds to the optimal solution to find the optimal solution the level method a fairly recent optimization method is employed we also present a theoretic proof of the convergence rate for the application of the level method to our problem empirical evaluation on several benchmark data sets demonstrates the effectiveness of the proposed semi supervised feature selection method

entropic repulsion of an interface in an external field we consider an interface above an attractive hard wall in the complete wetting regime and submitted to the action of an external increasing convex potential and study its delocalization as the intensity of this potential vanishes our main motivation is the analysis of critical prewetting which corresponds to the choice of a linear external potential we also present partial results on critical prewetting in the two dimensional ising model as well as a few weak results on pathwise estimates for the pure wetting problem for effective interface models

active learning based on locally linear reconstruction we consider the active learning problem which aims to select the most representative points out of many existing active learning techniques optimum experimental design oed has received considerable attention recently the typical oed criteria minimize the variance of the parameter estimates or predicted value however these methods see only global euclidean structure while the local manifold structure is ignored for example i optimal design selects those data points such that other data points can be best approximated by linear combinations of all the selected points in this paper we propose a novel active learning algorithm which takes into account the local structure of the data space that is each data point should be approximated by the linear combination of only its neighbors given the local reconstruction coefficients for every data point and the coordinates of the selected points a transductive learning algorithm called locally linear reconstruction llr is proposed to reconstruct every other point the most representative points are thus defined as those whose coordinates can be used to best reconstruct the whole data set the sequential and convex optimization schemes are also introduced to solve the optimization problem the experimental results have demonstrated the effectiveness of our proposed method

network unfolding map by vertex edge dynamics modeling the emergence of collective dynamics in neural networks is a mechanism of the animal and human brain for information processing in this paper we develop a computational technique using distributed processing elements in a complex network which are called particles to solve semisupervised learning problems three actions govern the particles dynamics generation walking and absorption labeled vertices generate new particles that compete against rival particles for edge domination active particles randomly walk in the network until they are absorbed by either a rival vertex or an edge currently dominated by rival particles the result from the model evolution consists of sets of edges arranged by the label dominance each set tends to form a connected subnetwork to represent a data class although the intrinsic dynamics of the model is a stochastic one we prove that there exists a deterministic version with largely reduced computational complexity specifically with linear growth furthermore the edge domination process corresponds to an unfolding map in such way that edges stretch and shrink according to the vertex edge dynamics consequently the unfolding effect summarizes the relevant relationships between vertices and the uncovered data classes the proposed model captures important details of connectivity patterns over the vertex edge dynamics evolution in contrast to the previous approaches which focused on only vertex or only edge dynamics computer simulations reveal that the new model can identify nonlinear features in both real and artificial data including boundaries between distinct classes and overlapping structures of data

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

an invariance principle to ferrari spohn diffusions we prove an invariance principle for a class of tilted 1 1 dimensional sos models or equivalently for a class of tilted random walk bridges in mathbb z the limiting objects are stationary reversible ergodic diffusions with drifts given by the logarithmic derivatives of the ground states of associated singular sturm liouville operators in the case of a linear area tilt we recover the ferrari spohn diffusion with log airy drift which was derived in ferrari and spohn ann probab 33 4 1302 1325 2005 in the context of brownian motions conditioned to stay above circular and parabolic barriers

new semi supervised classification method based on modified cluster assumption the cluster assumption which assumes that similar instances should share the same label is a basic assumption in semi supervised classification learning and has been found very useful in many successful semi supervised classification methods it is rarely noticed that when the cluster assumption is adopted there is an implicit assumption that every instance should have a crisp class label assignment in real applications however there are cases where it is difficult to tell that an instance definitely belongs to one class and does not belong to other neighboring classes in such cases it is more adequate to assume that similar instances should share similar label memberships rather than sharing a crisp label assignment here label memberships can be represented as a vector where each element corresponds to a class and the value at the element expresses the likelihood of the concerned instance belonging to the class by adopting this modified cluster assumption in this paper we propose a new semi supervised classification method that is semi supervised classification based on class membership ssccm specifically we try to solve the decision function and adequate label memberships for instances simultaneously and constrain that an instance and its local weighted mean lwm share the same label membership vector where the lwm is a robust image of the instance constructed by calculating the weighted mean of its neighboring instances we formulate the problem in a unified objective function for the labeled unlabeled data and their lwms based on the square loss function and take an alternating iterative strategy to solve it in which each step generates a closed form solution and the convergence is guaranteed the solution will provide both the decision function and the label membership function for classification their classification results can verify each other and the reliability of semi supervised classification learning might be enhanced by checking the consistency between those two predictions experiments show that ssccm obtains encouraging results compared to state of the art semi supervised classification methods

harmonic map heat flow with rough boundary data abstract let b1 be the unit open disk in r2 and m be a closed riemannian manifold in this note we first prove the uniqueness for weak solutions of the harmonic map heat flow in h1 0 t b1 m whose energy is non increasing in time given initial data u0 h b1 m and boundary data u0 b1 previously this uniqueness result was obtained by riviere when m is the round sphere and the energy of initial data is small and freire when m is an arbitrary closed riemannian manifold given that u0 h b1 m and u0 b1 h b1 the point of our uniqueness result is that no boundary regularity assumption is needed second we prove the exponential convergence of the harmonic map heat flow assuming that energy is small at all times

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

constrained brownian motion fluctuations away from circular and parabolic barriers motivated by the polynuclear growth model we consider a brownian bridge b t with b t 0 conditioned to stay above the semicircle ct t p t 2 t2 in the limit of large t the fluctuation scale of b t ct t is t 1 3 and its time correlation scale is t 2 3 we prove that in the sense of weak convergence of path measures the conditioned brownian bridge when properly rescaled converges to a stationary diffusion process with a drift explicitly given in terms of airy functions the dependence on the reference point t t 2 1 1 is only through the second derivative of ct t at t t we also prove a corresponding result where instead of the semicircle the barrier is a parabola of height t 1 2 the fluctuation scale is then t 2 3 more general conditioning shapes are briefly discussed 1 introduction and main results we consider the brownian bridge b t over the time interval t t t 0 b t b t 0 conditioned to lie above the semicircle ct t t 2 t 2 let b t be the conditioned brownian bridge and let xt t b t ct t be the deviation of b t away from ct t see figure 1 clearly xt t 0 xt t xt t 0 and the path measure of the process is defined on c t t r c t t the space of continuous functions over the interval t t equipped with the supremum norm the issue is to understand the statistical properties of xt t for large t a well studied special case is when ct t is replaced by the function zero the brownian bridge is then constrained to stay positive a stochastic process known as brownian excursion in the limit of large t it converges to the three dimensional bessel process time dependent barriers like the circle seem to be hardly studied an exception is the parabola gt 2 t t 2 t 2 for

from local similarities to global coding a framework for coding applications feature coding has received great attention in recent years as a building block of many image processing algorithms in particular the importance of the locality assumption in coding approaches has been studied in many previous works we review this assumption and claim that using the similarity of data points to a more global set of anchor points does not necessarily weaken the coding method as long as the underlying structure of the anchor points is considered we propose to capture the underlying structure by assuming a random walker over the anchor points we also show that our method is a fast approximation to the diffusion map kernel experiments on various data sets show that with a knowledge of the underlying structure of anchor points different state of the art coding algorithms may boost their performance in different learning tasks by utilizing the proposed method

dyson ferrari spohn diffusions and ordered walks under area tilts we consider families of non colliding random walks above a hard wall which are subject to a self potential of tilted area type we view such ensembles as effective models for the level lines of a class of 2 1 dimensional discrete height random surfaces in statistical mechanics we prove that under rather general assumptions on the step distribution and on the self potential such walks converge under appropriate rescaling to non intersecting ferrari spohn diffusions associated with limiting sturm liouville operators in particular the limiting invariant measures are given by the squares of the corresponding slater determinants

a variance minimization criterion to feature selection using laplacian regularization in many information processing tasks one is often confronted with very high dimensional data feature selection techniques are designed to find the meaningful feature subset of the original features which can facilitate clustering classification and retrieval in this paper we consider the feature selection problem in unsupervised learning scenarios which is particularly difficult due to the absence of class labels that would guide the search for relevant information based on laplacian regularized least squares which finds a smooth function on the data manifold and minimizes the empirical loss we propose two novel feature selection algorithms which aim to minimize the expected prediction error of the regularized regression model specifically we select those features such that the size of the parameter covariance matrix of the regularized regression model is minimized motivated from experimental design we use trace and determinant operators to measure the size of the covariance matrix efficient computational schemes are also introduced to solve the corresponding optimization problems extensive experimental results over various real life data sets have demonstrated the superiority of the proposed algorithms