mass concentration and aging in the parabolic anderson model with doubly exponential tails we study the non negative solution u u x t to the cauchy problem for the parabolic equation partial t u vardelta u xi u on mathbb z d times 0 infty with initial data u x 0 mathsf 1 0 x here vardelta is the discrete laplacian on mathbb z d and xi xi z z in mathbb z d is an i i d random field with doubly exponential upper tails we prove that for large t and with large probability most of the total mass u t sum x u x t of the solution resides in a bounded neighborhood of a site z t that achieves an optimal compromise between the local dirichlet eigenvalue of the anderson hamiltonian vardelta xi and the distance to the origin the processes t mapsto z t and t mapsto tfrac 1 t log u t are shown to converge in distribution under suitable scaling of space and time aging results for z t as well as for the solution to the parabolic problem are also established the proof uses the characterization of eigenvalue order statistics for vardelta xi in large sets recently proved by the first two authors

delocalising the parabolic anderson model through partial duplication of the potential the parabolic anderson model on mathbb z d with i i d potential is known to completely localise if the distribution of the potential is sufficiently heavy tailed at infinity in this paper we investigate a modification of the model in which the potential is partially duplicated in a symmetric way across a plane through the origin in the case of potential distribution with polynomial tail decay we exhibit a surprising phase transition in the model as the decay exponent varies for large values of the exponent the model completely localises as in the i i d case by contrast for small values of the exponent we show that the model may delocalise more precisely we show that there is an event of non negligible probability on which the solution has non negligible mass on two sites

multibody grouping from motion images we want to deduce from a sequence of noisy two dimensional images of a scene of several rigid bodies moving independently in three dimensions the number of bodies and the grouping of given feature points in the images to the bodies prior processing is assumed to have identified features or points common to all frames and the images are assumed to be created by orthographic projection i e perspective effects are minimal we describe a computationally inexpensive algorithm that can determine which points or features belong to which rigid body using the fact that with exact observations in orthographic projection points on a single body lie in a three or less dimensional linear manifold of frame space if there are enough observations and independent motions these manifolds can be viewed as a set linearly independent four or less dimensional subspaces we show that the row echelon canonical form provides direct information on the grouping of points to these subspaces treatment of the noise is the most difficult part of the problem this paper uses a statistical approach to estimate the grouping of points to subspaces in the presence of noise by computing which partition has the maximum likelihood the input data is assumed to be contaminated with independent gaussian noise the algorithm can base its estimates on a user supplied standard deviation of the noise or it can estimate the noise from the data the algorithm can also be used to estimate the probability of a user specified partition so that the hypothesis can be combined with others using bayesian statistics

optimal data collection for informative rankings expose well connected graphs given a graph where vertices represent alternatives and arcs represent pairwise comparison data the statistical ranking problem is to find a potential function defined on the vertices such that the gradient of the potential function agrees with the pairwise comparisons our goal in this paper is to develop a method for collecting data for which the least squares estimator for the ranking problem has maximal fisher information our approach based on experimental design is to view data collection as a bi level optimization problem where the inner problem is the ranking problem and the outer problem is to identify data which maximizes the informativeness of the ranking under certain assumptions the data collection problem decouples reducing to a problem of finding multigraphs with large algebraic connectivity this reduction of the data collection problem to graph theoretic questions is one of the primary contributions of this work as an application we study the yahoo movie user rating data set and demonstrate that the addition of a small number of well chosen pairwise comparisons can significantly increase the fisher informativeness of the ranking as another application we study the 2011 12 ncaa football schedule and propose schedules with the same number of games which are significantly more informative using spectral clustering methods to identify highly connected communities within the division we argue that the ncaa could improve its notoriously poor rankings by simply scheduling more out of conference games

a multibody factorization method for independently moving objects the structure from motion problem has been extensively studied in the field of computer vision yet the bulk of the existing work assumes that the scene contains only a single moving object the more realistic case where an unknown number of objects move in the scene has received little attention especially for its theoretical treatment in this paper we present a new method for separating and recovering the motion and shape of multiple independently moving objects in a sequence of images the method does not require prior knowledge of the number of objects nor is dependent on any grouping of features into an object at the image level for this purpose we introduce a mathematical construct of object shapes called the shape interaction matrix which is invariant to both the object motions and the selection of coordinate systems this invariant structure is computable solely from the observed trajectories of image features without grouping them into individual objects once the matrix is computed it allows for segmenting features into objects by the process of transforming it into a canonical form as well as recovering the shape and motion of each object the theory works under a broad set of projection models scaled orthography paraperspective and affine but they must be linear so it excludes projective cameras

intermittency in a catalytic random medium in this paper we study intermittency for the parabolic anderson equation u t u u where u d 0 is the diffusion constant is the discrete laplacian and d 0 is a space time random medium we focus on the case where is times the random medium that is obtained by running independent simple random walks with diffusion constant starting from a poisson random field with intensity throughout the paper we assume that 0 the solution of the equation describes the evolution of a reactant u under the influence of a catalyst we consider the annealed lyapunov exponents that is the exponential growth rates of the successive moments of u and show that they display an interesting dependence on the dimension d and on the parameters with qualitatively different intermittency behavior in d 1 2 in d 3 and in d 4 special attention is given to the asymptotics of these lyapunov exponents for 0 and

cut ratios and laplacian eigenvalues abstract we introduce local cut ratios as a sampling scheme for the cut ratio of a vertex subset in a graph we prove a bound for the sampling error as well as a two sided estimate for the local cut ratio in terms of laplacian eigenvalues we derive an inequality for the vertex cut ratio in terms of the edge cut ratio

the universality classes in the parabolic anderson model we discuss the long time behaviour of the parabolic anderson model the cauchy problem for the heat equation with random potential on mathbb z d we consider general i i d potentials and show that exactly four qualitatively different types of intermittent behaviour can occur these four universality classes depend on the upper tail of the potential distribution 1 tails at that are thicker than the double exponential tails 2 double exponential tails at studied by gartner and molchanov 3 a new class called almost bounded potentials and 4 potentials bounded from above studied by biskup and konig the new class 3 which contains both unbounded and bounded potentials is studied in both the annealed and the quenched setting we show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time the characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas respectively gaussian densities our analysis of class 3 relies on two large deviation results for the local times of continuous time simple random walk one of these results is proved by brydges and the first two authors in bhk04 and is also used here to correct a proof in bk01

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

on hadamard diagonalizable graphs of interest here is a characterization of the undirected graphs g such that the laplacian matrix associated with g can be diagonalized by some hadamard matrix many interesting and fundamental properties are presented for such graphs along with a partial characterization of the cographs that have this property

on graphs with algebraic connectivity equal to minimum edge density abstract given an undirected graph g v e and a nonempty subset x v the edge density of x is given by x v ex x v x where ex is the set of all edges with one end in x and the other end in v x it is known that the algebraic connectivity of g denoted by a g satisfies a g minx v x in this paper we study the graphs g for which equality holds in the above inequality

estimation of subspace arrangements with applications in modeling and segmenting mixed data recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high dimensional data that have very complicated structures from both geometric and statistical points of view such unsorted data are considered mixed as different parts of the data have significantly different structures which cannot be described by a single model in this paper we propose to use subspace arrangements a union of multiple subspaces for modeling mixed data each subspace in the arrangement is used to model just a homogeneous subset of the data thus multiple subspaces together can capture the heterogeneous structures within the data set in this paper we give a comprehensive introduction to a new approach for the estimation of subspace arrangements this is known as generalized principal component analysis gpca in particular we provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust even when the given data are corrupted by noise or contaminated with outliers this new method in many ways improves and generalizes extant methods for modeling or clustering mixed data there have been successful applications of this new method to many real world problems in computer vision image processing and system identification in this paper we will examine several of those representative applications this paper is intended to be expository in nature however in order that this may serve as a more complete reference for both theoreticians and practitioners we take the liberty of filling in several gaps between the theory and the practice in the existing literature

on the approximability of minimizing nonzero variables or unsatisfied relations in linear systems we investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning operations research and pattern recognition in the first class min ulr one wishes given a possibly infeasible system of linear relations to find a solution that violates as few relations as possible while satisfying all the others in the second class min rvls the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible for both min ulr and min rvls the four basic types of relational operators and are considered while min rvls with equations was mentioned to be np hard in garey and johnson 1979 we established in amaldi 1992 amaldi and kann 1995 that min ulr with equalities and inequalities are np hard even when restricted to homogeneous systems with bipolar coefficients the latter problems have been shown hard to approximate in arora et al 1993 in this paper we determine strong bounds on the approximability of various variants of min rvls and min ulr including constrained ones where the variables are restricted to take binary values or where some relations are mandatory while others are optional the various np hard versions turn out to have different approximability properties depending on the type of relations and the additional constraints but none of them can be approximated within any constant factor unless p np particular attention is devoted to two interesting special cases that occur in discriminant analysis and machine learning in particular we disprove a conjecture of van horn and martinez 1992 regarding the existence of a polynomial time algorithm to design linear classifiers or perceptrons that involve a close to minimum number of features

localization and selection in a mean field branching random walk in a random environment we consider a continuous time branching random walk on the finite set 1 2 ldots n with totally symmetric diffusion jumps and some site dependent i i d random birth rates which are unbounded we study this process as the time t and the space size n tend to infinity simultaneously in the classical law of large numbers setup for spatial branching models the growth of the population obeys an exponential limit law due to the localization of the overwhelming portion of particles in the record point of the medium this phenomenon is analyzed further the historical path in space of a typical particle picked at time t selection is of a rather simple and special nature and becomes in the limit singular in distribution to the path of the underlying mean field random walk in general the properties of the typical path depend on the relation in which t and n tend to infinity

motion segmentation in the presence of outlying incomplete or corrupted trajectories in this paper we study the problem of segmenting tracked feature point trajectories of multiple moving objects in an image sequence using the affine camera model this problem can be cast as the problem of segmenting samples drawn from multiple linear subspaces in practice due to limitations of the tracker occlusions and the presence of nonrigid objects in the scene the obtained motion trajectories may contain grossly mistracked features missing entries or corrupted entries in this paper we develop a robust subspace separation scheme that deals with these practical issues in a unified mathematical framework our methods draw strong connections between lossy compression rank minimization and sparse representation we test our methods extensively on the hopkins155 motion segmentation database and other motion sequences with outliers and missing data we compare the performance of our methods to state of the art motion segmentation methods based on expectation maximization and spectral clustering for data without outliers or missing information the results of our methods are on par with the state of the art results and in many cases exceed them in addition our methods give surprisingly good performance in the presence of the three types of pathological trajectories mentioned above all code and results are publicly available at http perception csl uiuc edu coding motion

recovery guarantees for exemplar based clustering for a certain class of distributions we prove that the linear programming relaxation of k medoids clustering a variant of k means clustering where means are replaced by exemplars from within the dataset distinguishes points drawn from nonoverlapping balls with high probability once the number of points drawn and the separation distance between any two balls are sufficiently large our results hold in the nontrivial regime where the separation distance is small enough that points drawn from different balls may be closer to each other than points drawn from the same ball in this case clustering by thresholding pairwise distances between points can fail we also exhibit numerical evidence of high probability recovery in a substantially more permissive regime

lambertian reflectance and linear subspaces we prove that the set of all lambertian reflectance functions the mapping from surface normals to intensities obtained with arbitrary distant light sources lies close to a 9d linear subspace this implies that in general the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace explaining prior empirical results we also provide a simple analytic characterization of this linear space we obtain these results by representing lighting using spherical harmonics and describing the effects of lambertian materials as the analog of a convolution these results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions we also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4d linear space we apply these algorithms to perform face recognition by finding the 3d model that best matches a 2d query image

geometric characterization of intermittency in the parabolic anderson model we consider the parabolic anderson problem tu u x u on r z d with localized initial condition u 0 x 0 x and random i i d potential under the assumption that the distribution of 0 has a double exponential or slightly heavier tail we prove the following geometric characterisation of intermittency with probability one as t 1 the overwhelming contribution to the total mass p x u t x comes from a slowly increasing number of islands which are located far from each other these islands are local regions of those high exceedances of the eld in a box of side length 2t log 2 t for which the local principal dirichlet eigenvalue of the random operator is close to the top of the spectrum in the box we also prove that the shape of in these regions is non random and that u t is close to the corresponding positive eigenfunction this is the geometric picture suggested by localization theory for the anderson hamiltonian

a geometric analysis of subspace clustering with outliers this paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes as is common in computer vision or unsupervised learning applications we do not know in advance how many subspaces there are nor do we have any information about their dimensions we develop a novel geometric analysis of an algorithm named sparse subspace clustering ssc 11 which signicantly broadens the range of problems where it is provably eective for instance we show that ssc can recover multiple subspaces each of dimension comparable to the ambient dimension we also prove that ssc can correctly cluster data points even when the subspaces of interest intersect further we develop an extension of ssc that succeeds when the data set is corrupted with possibly overwhelmingly many outliers underlying our analysis are clear geometric insights which may bear on other sparse recovery problems a numerical study complements our theoretical analysis and demonstrates the eectiveness of these methods

multiframe motion segmentation with missing data using powerfactorization and gpca we consider the problem of segmenting multiple rigid body motions from point correspondences in multiple affine views we cast this problem as a subspace clustering problem in which point trajectories associated with each motion live in a linear subspace of dimension two three or four our algorithm involves projecting all point trajectories onto a 5 dimensional subspace using the svd the powerfactorization method or ransac and fitting multiple linear subspaces representing different rigid body motions to the points in 5 using gpca unlike previous work our approach does not restrict the motion subspaces to be four dimensional and independent instead it deals gracefully with all the spectrum of possible affine motions from two dimensional and partially dependent to four dimensional and fully independent our algorithm can handle the case of missing data meaning that point tracks do not have to be visible in all images by using the powerfactorization method to project the data in addition our method can handle outlying trajectories by using ransac to perform the projection we compare our approach to other methods on a database of 167 motion sequences with full motions independent motions degenerate motions partially dependent motions missing data outliers etc on motion sequences with complete data our method achieves a misclassification error of less that 5 for two motions and 29 for three motions

low rank matrix completion in the presence of high coherence prevalent matrix completion methods capture only the low rank property which gives merely a constraint that the data points lie on some low dimensional subspace but generally ignore the extra structures beyond low rank that specify in more detail how the data points lie on the subspace whenever the data points are not uniformly distributed on the low dimensional subspace the row coherence of the target matrix to recover could be considerably high and accordingly prevalent methods might fail even if the target matrix is fairly low rank to relieve this challenge we suggest to consider a model termed low rank factor decomposition lrfd which imposes an additional restriction that the data points must be represented as linear compressive combinations of the bases in a given dictionary we show that lrfd can effectively mitigate the challenges of high row coherence provided that its dictionary is configured properly namely it is mathematically proven that if the dictionary is well conditioned and low rank then lrfd can weaken the dependence on the row coherence in particular if the dictionary itself is low rank then the dependence on the row coherence can be entirely removed subsequently we devise two practical algorithms to obtain proper dictionaries in unsupervised environments one uses the existing matrix completion methods to construct the dictionary in lrfd and the other tries to learn a proper dictionary from the data given experiments on randomly generated matrices and motion datasets show superior performance of our proposed algorithms

minimizing effective resistance of a graph the effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights the effective resistance comes up in many applications and fields in addition to electrical network analysis including for example markov chains and continuous time averaging networks in this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance i e the sum of the resistances between all pairs of nodes we show that this is a convex optimization problem and can be solved efficiently either numerically or in some cases analytically we show that optimal allocation of the edge weights can reduce the total effective resistance of the graph compared to uniform weights by a factor that grows unboundedly with the size of the graph we show that among all graphs with n nodes the path has the largest value of optimal total effective resistance and the complete graph has the least

segmentation of multivariate mixed data via lossy data coding and compression in this paper based on ideas from lossy data coding and compression we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of gaussian distributions which are allowed to be almost degenerate the goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data subject to a given distortion by analyzing the coding length rate of mixed data we formally establish some strong connections of data segmentation to many fundamental concepts in lossy data compression and rate distortion theory we show that a deterministic segmentation is approximately the asymptotically optimal solution for compressing mixed data we propose a very simple and effective algorithm that depends on a single parameter the allowable distortion at any given distortion the algorithm automatically determines the corresponding number and dimension of the groups and does not involve any parameter estimation simulation results reveal intriguing phase transition like behaviors of the number of segments when changing the level of distortion or the amount of outliers finally we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data

matrix completion by least square low rank and sparse self representations abstract conventional matrix completion methods are generally based on rank minimization these methods assume that the given matrix is of low rank and the data points are drawn from a single subspace of low dimensionality therefore they are not effective in completing matrices where the data are drawn from multiple subspaces in this paper we establish a novel matrix completion framework that is based on self representation specifically least square low rank and sparse self representations based matrix completion algorithms are provided the underlying idea is that one data point can be efficiently reconstructed by other data points belonging to a common subspace where the missing entries are recovered so as to fit the common subspace the proposed algorithms actually maximize the weighted correlations among the columns of a given matrix we prove that the proposed algorithms are approximations for rank minimization of the incomplete matrix in addition they are able to complete high rank or even full rank matrices when the data are drawn from multiple subspaces comparative studies are conducted on synthetic datasets natural image inpainting tasks temperature prediction task and collaborative filtering tasks the results show that the proposed algorithms often outperform other state of the art algorithms in various tasks

parabolic problems for the anderson model the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

matrix completion from noisy entries given a matrix m of low rank we consider the problem of reconstructing it from noisy observations of a small random subset of its entries the problem arises in a variety of applications from collaborative filtering the netflix problem to structure from motion and positioning we study a low complexity algorithm introduced by keshavan montanari and oh 2010 based on a combination of spectral techniques and manifold optimization that we call here optspace we prove performance guarantees that are order optimal in a number of circumstances

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

outlier robust matrix completion via ell _p minimization matrix completion refers to recovering a low rank matrix from only a subset of its possibly noisy entries and has a variety of important applications because many real world signals can be modeled by a n 1 times n 2 matrix with rank r ll min n 1 n 2 most existing techniques for matrix completion assume gaussian noise and thus they are not robust to outliers in this paper we devise two algorithms for robust matrix completion based on low rank matrix factorization and ell p norm minimization of the fitting error with 0 the first method tackles the low rank matrix factorization with missing data by iteratively solving n 1 n 2 linear ell p regression problems whereas the second applies the alternating direction method of multipliers admm in the ell p space at each iteration of the admm it requires performing a least squares ls matrix factorization and calculating the proximity operator of the p th power of the ell p norm the ls factorization is efficiently solved using linear ls regression while the proximity operator has closed form solution for p 1 or can be obtained by root finding of a scalar nonlinear equation for other values of p the two proposed algorithms have comparable recovery capability and computational complexity of mathcal o k omega r 2 where omega is the number of observed entries and k is a fixed constant of several hundreds to thousands and dimension independent it is demonstrated that they are superior to the singular value thresholding singular value projection and alternating projection schemes in terms of computational simplicity statistical accuracy and outlier robustness

block sparse recovery via convex optimization given a dictionary that consists of multiple blocks and a signal that lives in the range space of only a few blocks we study the problem of finding a block sparse representation of the signal i e a representation that uses the minimum number of blocks motivated by signal image processing and computer vision applications such as face recognition we consider the block sparse recovery problem in the case where the number of atoms in each block is arbitrary possibly much larger than the dimension of the underlying subspace to find a block sparse representation of a signal we propose two classes of nonconvex optimization programs which aim to minimize the number of nonzero coefficient blocks and the number of nonzero reconstructed vectors from the blocks respectively since both classes of problems are np hard we propose convex relaxations and derive conditions under which each class of the convex programs is equivalent to the original nonconvex formulation our conditions depend on the notions of mutual and cumulative subspace coherence of a dictionary which are natural generalizations of existing notions of mutual and cumulative coherence we evaluate the performance of the proposed convex programs through simulations as well as real experiments on face recognition we show that treating the face recognition problem as a block sparse recovery problem improves the state of the art results by 10 with only 25 of the training data

a structured sparse plus structured low rank framework for subspace clustering and completion recent advances in a low rank matrix completion have enabled the exact recovery of incomplete data drawn from a low dimensional subspace of a high dimensional observation space however in many applications the data are drawn from multiple low dimensional subspaces without knowing which point belongs to which subspace in such cases using a single low dimensional subspace to complete the data may lead to erroneous results because the complete data matrix need not be low rank in this paper we propose a structured sparse plus structured low rank text s 3 lr optimization framework for clustering and completing data drawn from a union of low dimensional subspaces the proposed text s 3 lr framework exploits the fact that each point in a union of subspaces can be expressed as a sparse linear combination of all other points and that the matrix of the points within each subspace is low rank this framework leads to a nonconvex optimization problem which we solve efficiently by using a combination of a linearized alternating direction method of multipliers and spectral clustering in addition we discuss the conditions that guarantee the exact matrix completion in a union of subspaces experiments on synthetic data motion segmentation data and cancer gene data validate the effectiveness of the proposed approach

multiscale hybrid linear models for lossy image representation in this paper we introduce a simple and efficient representation for natural images we view an image in either the spatial domain or the wavelet domain as a collection of vectors in a high dimensional space we then fit a piece wise linear model i e a union of affine subspaces to the vectors at each downsampling scale we call this a multiscale hybrid linear model for the image the model can be effectively estimated via a new algebraic method known as generalized principal component analysis gpca the hybrid and hierarchical structure of this model allows us to effectively extract and exploit multimodal correlations among the imagery data at different scales it conceptually and computationally remedies limitations of many existing image representation methods that are based on either a fixed linear transformation e g dct wavelets or an adaptive uni modal linear transformation e g pca or a multimodal model that uses only cluster means e g vq we will justify both quantitatively and experimentally why and how such a simple multiscale hybrid model is able to reduce simultaneously the model complexity and computational cost despite a small overhead of the model our careful and extensive experimental results show that this new model gives more compact representations for a wide variety of natural images under a wide range of signal to noise ratios than many existing methods including wavelets we also briefly address how the same hybrid linear modeling paradigm can be extended to be potentially useful for other applications such as image segmentation

robust recovery of subspace structures by low rank representation in this paper we address the subspace clustering problem given a set of data samples vectors approximately drawn from a union of multiple subspaces our goal is to cluster the samples into their respective subspaces and remove possible outliers as well to this end we propose a novel objective function named low rank representation lrr which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary it is shown that the convex program associated with lrr solves the subspace clustering problem in the following sense when the data is clean we prove that lrr exactly recovers the true subspace structures when the data are contaminated by outliers we prove that under certain conditions lrr can exactly recover the row space of the original data and detect the outlier as well for data corrupted by arbitrary sparse errors lrr can also approximately recover the row space with theoretical guarantees since the subspace membership is provably determined by the row space these further imply that lrr can perform robust subspace clustering and error correction in an efficient and effective way

generalized principal component analysis gpca this paper presents an algebro geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points we represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point when the number of subspaces is known we show that these polynomials can be estimated linearly from data hence subspace segmentation is reduced to classifying one point per subspace we select these points optimally from the data set by minimizing certain distance function thus dealing automatically with moderate noise in the data a basis for the complement of each subspace is then recovered by applying standard pca to the collection of derivatives normal vectors extensions of gpca that deal with data in a high dimensional space and with an unknown number of subspaces are also presented our experiments on low dimensional data show that gpca outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as k subspaces and expectation maximization we also present applications of gpca to computer vision problems such as face clustering temporal video segmentation and 3d motion segmentation from point correspondences in multiple affine views

parabolic problems for the anderson model i intermittency and related topics the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t 0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

on the stochastic heat equation with spatially colored random forcing we consider the stochastic heat equation of the following form t ut x lut x b ut x ut x t x for t 0 x r where l is the generator of a levy process and is a spatially colored temporally white gaussian noise we will be concerned mainly with the long term behavior of the mild solution to this stochastic pde for the most part we work under the assumptions that the initial data u0 is a bounded and measurable function and is nonconstant and lipschitz continuous in this case we find conditions under which the preceding stochastic pde admits a unique solution which is also weakly intermittent in addition we study the same equation in the case that lu is replaced by its massive dispersive analogue lu u where r and we describe accurately the effect of the parameter on the intermittence of the solution in the case that u is proportional to u the parabolic anderson model we also look at the linearized version of our stochastic pde that is the case when is identically equal to one any other constant works also in this case we study not only the existence and uniqueness of a solution but also the regularity of the solution when it exists and is unique

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

robust kernel low rank representation recently low rank representation lrr has shown promising performance in many real world applications such as face clustering however lrr may not achieve satisfactory results when dealing with the data from nonlinear subspaces since it is originally designed to handle the data from linear subspaces in the input space meanwhile the kernel based methods deal with the nonlinear data by mapping it from the original input space to a new feature space through a kernel induced mapping to effectively cope with the nonlinear data we first propose the kernelized version of lrr in the clean data case we also present a closed form solution for the resultant optimization problem moreover to handle corrupted data we propose the robust kernel lrr rklrr approach and develop an efficient optimization algorithm to solve it based on the alternating direction method in particular we show that both the subproblems in our optimization algorithm can be efficiently and exactly solved and it is guaranteed to obtain a globally optimal solution besides our proposed algorithm can also solve the original lrr problem which is a special case of our rklrr when using the linear kernel in addition based on our new optimization technique the kernelization of some variants of lrr can be similarly achieved comprehensive experiments on synthetic data sets and real world data sets clearly demonstrate the efficiency of our algorithm as well as the effectiveness of rklrr and the kernelization of two variants of lrr

on algebraic connectivity and spectral integral variations of graphs abstract let g be a simple connected graph and l g be the laplacian matrix of g let a g be the second smallest eigenvalue of l g an eigenvector of l g corresponding to the eigenvalue a g is called a fiedler vector of g let y be a fiedler vector of g a characteristic vertex is a vertex u of g such that y u 0 and such that there is a vertex w adjacent to u satisfying y w 0 a characteristic edge is an edge u v such that y u y v

localization for random walks among random obstacles in a single euclidean ball place an obstacle with probability 1 p independently at each vertex of mathbb z d and run a simple random walk before hitting one of the obstacles for d ge 2 and p strictly above the critical threshold for site percolation we condition on the environment where the origin is contained in an infinite connected component free of obstacles and we show that for environments with probability tending to one as n rightarrow infty there exists a unique discrete euclidean ball of volume d log 1 p n asymptotically such that the following holds conditioned on survival up to time n we have that at any time t in epsilon n n n for some epsilon n rightarrow n rightarrow infty 0 with probability tending to one the simple random walk is in this ball this work relies on and substantially improves a previous result of the authors on localization in a region of volume poly logarithmic in n for the same problem

spectral curvature clustering scc this paper presents novel techniques for improving the performance of a multi way spectral clustering framework govindu in proceedings of the 2005 ieee computer society conference on computer vision and pattern recognition cvpr 05 vol 1 pp 1150 1157 2005 chen and lerman 2007 preprint in the supplementary webpage for segmenting affine subspaces specifically it suggests an iterative sampling procedure to improve the uniform sampling strategy an automatic scheme of inferring the tuning parameter from the data a precise initialization procedure for k means as well as a simple strategy for isolating outliers the resulting algorithm spectral curvature clustering scc requires only linear storage and takes linear running time in the size of the data it is supported by theory which both justifies its successful performance and guides our practical choices we compare it with other existing methods on a few artificial instances of affine subspaces application of the algorithm to several real world problems is also discussed

high rank matrix completion this paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low rank subspaces this generalizesthe standardlow rankmatrix completion problem to situations in which the matrix rank can be quite high or even full rank since the columns belong to a union of subspaces this problem may also be viewed as a missing data version of the subspace clustering problem let x be an n n matrix whose complete columns lie in a union of at most k subspaces each of rank r 1 a constant depending on the usual incoherence conditions the geometrical arrangement of subspaces and the distribution of columns over the subspaces the result is illustrated with numerical experiments and an application to internet distance matrix completion and topology identification

hybrid linear modeling via local best fit flats we present a simple and fast geometric method for modeling data by a union of affine subspaces the method begins by forming a collection of local best fit affine subspaces i e subspaces approximating the data in local neighborhoods the correct sizes of the local neighborhoods are determined automatically by the jones 2 numbers we prove under certain geometric conditions that our method finds the optimal local neighborhoods the collection of subspaces is further processed by a greedy selection procedure or a spectral method to generate the final model we discuss applications to tracking based motion segmentation and clustering of faces under different illuminating conditions we give extensive experimental evidence demonstrating the state of the art accuracy and speed of the suggested algorithms on these problems and also on synthetic hybrid linear data as well as the mnist handwritten digits data and we demonstrate how to use our algorithms for fast determination of the number of affine subspaces

linear time subspace clustering via bipartite graph modeling we present a linear time subspace clustering approach that combines sparse representations and bipartite graph modeling the signals are modeled as drawn from a union of low dimensional subspaces and each signal is represented by a sparse combination of basis elements termed atoms which form the columns of a dictionary matrix the sparse representation coefficients are arranged in a sparse affinity matrix which defines a bipartite graph of two disjoint sets 1 atoms and 2 signals subspace clustering is obtained by applying low complexity spectral bipartite graph clustering that exploits the small number of atoms for complexity reduction the complexity of the proposed approach is linear in the number of signals thus it can rapidly cluster very large data collections performance evaluation of face clustering and temporal video segmentation demonstrates comparable clustering accuracies to state of the art at a significantly lower computational load

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

shape and motion from image streams under orthography a factorization method inferring scene geometry and camera motion from a stream of images is possible in principle but is an ill conditioned problem when the objects are distant with respect to their size we have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step an image stream can be represented by the 2fxp measurement matrix of the image coordinates of p points tracked through f frames we show that under orthographic projection this matrix is of rank 3 based on this observation the factorization method uses the singular value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively two of the three translation components are computed in a preprocessing stage the method can also handle and obtain a full solution from a partially filled in measurement matrix that may result from occlusions or tracking failures the method gives accurate results and does not introduce smoothing in either shape or motion we demonstrate this with a series of experiments on laboratory and outdoor image streams with and without occlusions

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

a two cities theorem for the parabolic anderson model the parabolic anderson problem is the cauchy problem for the heat equation partial derivative t u t z delta u t z xi z u t z on 0 infinity x z d with random potential xi z z is an element of z d we consider independent and identically distributed potentials such that the distribution function of z converges polynomially at infinity if u is initially localized in the origin that is if u 0 z 1 0 z we show that as time goes to infinity the solution is completely localized in two points almost surely and in one point with high probability we also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem

sparse subspace clustering algorithm theory and applications many real world problems deal with collections of high dimensional data such as images videos text and web documents dna microarray data and more often such high dimensional data lie close to low dimensional structures corresponding to several classes or categories to which the data belong in this paper we propose and study an algorithm called sparse subspace clustering to cluster data points that lie in a union of low dimensional subspaces the key idea is that among the infinitely many possible representations of a data point in terms of other points a sparse representation corresponds to selecting a few points from the same subspace this motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of the data into subspaces since solving the sparse optimization program is in general np hard we consider a convex relaxation and show that under appropriate conditions on the arrangement of the subspaces and the distribution of the data the proposed minimization program succeeds in recovering the desired sparse representations the proposed algorithm is efficient and can handle data points near the intersections of subspaces another key advantage of the proposed algorithm with respect to the state of the art is that it can deal directly with data nuisances such as noise sparse outlying entries and missing entries by incorporating the model of the data into the sparse optimization program we demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real world problems of motion segmentation and face clustering

extremal theory for spectrum of random discrete schrodinger operator ii distributions with heavy tails we study the asymptotic structure of the first k largest eigenvalues k v and the corresponding eigenfunctions k v of a finite volume anderson model discrete schrodinger operator mathcal h v kappa delta v xi cdot on the multidimensional lattice torus v increasing to the whole of lattice provided the distribution function f of i i d potential satisfies condition log 1 f t o t3 and some additional regularity conditions as t for z v denote by 0 z the principal eigenvalue of the single peak hamiltonian v z z in l2 v and let lambda 0 k v be the kth largest value of the sample 0 in v we first show that the eigenvalues k v are asymptotically close to lambda 0 k v we then prove extremal type limit theorems i e poisson statistics for the normalized eigenvalues k v bv av where the normalizing constants av 0 and bv are chosen the same as in the corresponding limit theorems for lambda 0 k v the eigenfunction k v is shown to be asymptotically completely localized as v at the sites zk v v defined by lambda 0 z k v lambda 0 k v proofs are based on the finite rank in particular rank one perturbation arguments for discrete schrodinger operator when potential peaks are sparse

exact asymptotics in a mean field model with random potential for a mean field operator with a random potential asymptotic properties of the eigenvalues and eigenfunctions are studied and applied to investigate the longerm behavior of the solutions of a corresponding large system of differential equations the total mass of the system is approximately concentrated in the record point of the random potential complete localization a more detailed inspection of the peaks shows that there is a phase transition only in the case of a moderate increase of time relatively to the growth of the space size the model behaves similarly to the system without diffusion but also in the non moderate case the asymptotic height of peaks can exactly be described

extrinsic methods for coding and dictionary learning on grassmann manifolds sparsity based representations have recently led to notable results in various visual recognition tasks in a separate line of research riemannian manifolds have been shown useful for dealing with features and models that do not lie in euclidean spaces with the aim of building a bridge between the two realms we address the problem of sparse coding and dictionary learning in grassmann manifolds i e the space of linear subspaces to this end we propose to embed grassmann manifolds into the space of symmetric matrices by an isometric mapping this in turn enables us to extend two sparse coding schemes to grassmann manifolds furthermore we propose an algorithm for learning a grassmann dictionary atom by atom lastly to handle non linearity in data we extend the proposed grassmann sparse coding and dictionary learning algorithms through embedding into higher dimensional hilbert spaces experiments on several classification tasks gender recognition gesture classification scene analysis face recognition action recognition and dynamic texture classification show that the proposed approaches achieve considerable improvements in discrimination accuracy in comparison to state of the art methods such as kernelized affine hull method and graph embedding grassmann discriminant analysis

complex hadamard diagonalisable graphs abstract in light of recent interest in hadamard diagonalisable graphs graphs whose laplacian matrix is diagonalisable by a hadamard matrix we generalise this notion from real to complex hadamard matrices we give some basic properties and methods of constructing such graphs we show that a large class of complex hadamard diagonalisable graphs have vertex sets forming an equitable partition and that the laplacian eigenvalues must be even integers we provide a number of examples and constructions of complex hadamard diagonalisable graphs including two special classes of graphs the cayley graphs over z r d and the non complete extended p sum neps we discuss necessary and sufficient conditions for laplacian fractional revival and perfect state transfer on continuous time quantum walks described by complex hadamard diagonalisable graphs and provide examples of such quantum state transfer