a note on self adjoint extensions of the laplacian on weighted graphs abstract we study the uniqueness of self adjoint and markovian extensions of the laplacian on weighted graphs we first show that for locally finite graphs and a certain family of metrics completeness of the graph implies uniqueness of these extensions moreover in the case when the graph is not metrically complete and the cauchy boundary has finite capacity we characterize the uniqueness of the markovian extensions

damage detection from changes in curvature mode shapes abstract a damage in a structure alters its dynamic characteristics the change is characterized by changes in the eigenparameters i e natural frequency damping values and the mode shapes associated with each natural frequency considerable effort has been spent in obtaining a relationship between the changes in eigenparameters the damage location and the damage size most of the emphasis has been on using the changes in the natural frequencies and the damping values to determine the location and the size of the damage in this paper a new parameter called curvature mode shape is investigated as a possible candidate for identifying and locating damage in a structure by using a cantilever and a simply supported analytical beam model it is shown here that the absolute changes in the curvature mode shapes are localized in the region of damage and hence can be used to detect damage in a structure the changes in the curvature mode shapes increase with increasing size of damage this information can be used to obtain the amount of damage in the structure finite element analysis was used to obtain the displacement mode shapes of the two models by using a central difference approximation curvature mode shapes were then calculated from the displacement mode shapes

essential self adjointness of the graph laplacian we study the operator theory associated with such infinite graphs g as occur in electrical networks in fractals in statistical mechanics and even in internet search engines our emphasis is on the determination of spectral data for a natural laplace operator associated with the graph in question this operator will depend not only on g but also on a prescribed positive real valued function c defined on the edges in g in electrical network models this function c will determine a conductance number for each edge we show that the corresponding laplace operator is automatically essential self adjoint by this we mean that is defined on the dense subspace d of all the real valued functions on the set of vertices g0 with finite support in the hilbert space l2 g0 the conclusion is that the closure of the operator is self adjoint in l2 g0 and so in particular that it has a unique spectral resolution determined by a projection valued measure on the borel subsets of the infinite half line we

eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

efficient high order matching we present a computational approach to high order matching of data sets in rd those are matchings based on data affinity measures that score the matching of more than two pairs of points at a time high order affinities are represented by tensors and the matching is then given by a rank one approximation of the affinity tensor and a corresponding discretization our approach is rigorously justified by extending zass and shashua s hypergraph matching to high order spectral matching this paves the way for a computationally efficient dual marginalization spectral matching scheme we also show that based on the spectral properties of random matrices affinity tensors can be randomly sparsified while retaining the matching accuracy our contributions are experimentally validated by applying them to synthetic as well as real data sets

infinite networks and variation of conductance functions in discrete laplacians for a given infinite connected graph g v e and an arbitrary but fixed conductance function c we study an associated graph laplacian c it is a generalized difference operator where the differences are measured across the edges e in g and the conductance function c represents the corresponding coefficients the graph laplacian a key tool in the study of infinite networks acts in an energy hilbert space e computed from c using a certain parseval frame we study the spectral theoretic properties of graph laplacians in fact for fixed c there are two versions of the graph laplacian one defined naturally in the l2 space of v and the other in e the first is automatically selfadjoint but the second involves a krein extension we prove that as sets the two spectra are the same aside from the point 0 the point zero may be in the spectrum of the second but not the first we further study the fine structure of the respective spectra as the conductance function varies showing now how the spect

a cross product approach for low rank approximations of large matrices abstract in this paper we present a new type of restarted krylov method for calculating low rank approximations of large matrices in contrast to former krylov methods our approach does not apply the lanczos algorithm or lanczos bidiagonalization this simplifies the basic iteration and allows the introduction of several innovations an advantage of the proposed method is that it requires a minimal amount of computer storage the krylov matrix which is built at each iteration is using an improved starting vector and a new three term recurrence relation these modifications lead to fast rate of convergence numerical experiments illustrate the usefulness of the proposed approach

sampling from large matrices an approach through geometric functional analysis we study random submatrices of a large matrix a we show how to approximately compute a from its random submatrix of the smallest possible size o rlog r with a small error in the spectral norm where r e vav2f vav22 is the numerical rank of a the numerical rank is always bounded by and is a stable relaxation of the rank of a this yields an asymptotically optimal guarantee in an algorithm for computing low rank approximations of a we also prove asymptotically optimal estimates on the spectral norm and the cut norm of random submatrices of a the result for the cut norm yields a slight improvement on the best known sample complexity for an approximation algorithm for max 2csp problems we use methods of probability in banach spaces in particular the law of large numbers for operator valued random variables

an experimentally validated damage detection theory in smart structures abstract a method for the non destructive detection and location of damage using parameterized partial differential equations and galerkin approximation techniques is presented damages in a structure cause changes in the physical coefficients of mass density elastic modulus and damping coefficients this paper examines the use of beam like structure with piezoceramic sensors and actuators to perform identification of those physical parameters and hence to detect the damage the method casts the inverse problem as an optimization problem the iterative method is based on enhanced least square error minimization experimental results are presented from tests on cantilevered aluminum beams damaged at different locations and with damage of different dimensions it is demonstrated that the method can sense the presence of damage and locate and characterize the damage to a satisfactory precision

fast monte carlo algorithms for matrices iii computing a compressed approximate matrix decomposition in many applications the data consist of or may be naturally formulated as an m times n matrix a which may be stored on disk but which is too large to be read into random access memory ram or to practically perform superlinear polynomial time computations on it two algorithms are presented which when given an m times n matrix a compute approximations to a which are the product of three smaller matrices c u and r each of which may be computed rapidly let a cur be the computed approximate decomposition both algorithms have provable bounds for the error matrix a a in the first algorithm c columns of a and r rows of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling and the r times n matrix r consists of those r rows of a also after appropriate rescaling then the c times r matrix u may be calculated from c and r for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively it is proven that left a a right xi le min d mathrm rank d le k left a d right xi poly k 1 c left a right f holds in expectation and with high probability for both xi 2 f and for all k 1 ldots mbox rank a thus by appropriate choice of k left a a right 2 le epsilon left a right f also holds in expectation and with high probability this algorithm may be implemented without storing the matrix a in ram provided it can make two passes over the matrix stored in external memory and use o m n additional ram assuming that c and r are constants independent of the size of the input the second algorithm is similar except that it approximates the matrix c by randomly sampling a constant number of rows of c thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a f both algorithms take time which is a low degree polynomial in k 1 epsilon and 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n the proofs for the error bounds make important use of matrix perturbation theory and previous work on approximating matrix multiplication and computing low rank approximations to a matrix the probability distribution over columns and rows and the rescaling are crucial features of the algorithms and must be chosen judiciously

fast monte carlo algorithms for finding low rank approximations we consider the problem of approximating a given m n matrix a by another matrix of specified rank k which is smaller than m and n the singular value decomposition svd can be used to find the best such approximation however it takes time polynomial in m n which is prohibitive for some modern applications in this article we develop an algorithm that is qualitatively faster provided we may sample the entries of the matrix in accordance with a natural probability distribution in many applications such sampling can be done efficiently our main result is a randomized algorithm to find the description of a matrix d of rank at most k so that holds with probability at least 1 where v vf is the frobenius norm the algorithm takes time polynomial in k 1 e log 1 only and is independent of m and n in particular this implies that in constant time it can be determined if a given matrix of arbitrary size has a good low rank approximation

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

sampling methods for the nystrom method the nystrom method is an efficient technique to generate low rank matrix approximations and is used in several large scale learning applications a key aspect of this method is the procedure according to which columns are sampled from the original matrix in this work we explore the efficacy of a variety of fixed and adaptive sampling schemes we also propose a family of ensemble based sampling algorithms for the nystrom method we report results of extensive experiments that provide a detailed comparison of various fixed and adaptive sampling techniques and demonstrate the performance improvement associated with the ensemble nystrom method when used in conjunction with either fixed or adaptive sampling schemes corroborating these empirical findings we present a theoretical analysis of the nystrom method providing novel error bounds guaranteeing a better convergence rate of the ensemble nystrom method in comparison to the standard nystrom method

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

the problem of deficiency indices for discrete schrodinger operators on locally finite graphs the number of self adjoint extensions of a symmetric operator acting on a complex hilbert space is characterized by its deficiency indices given a locally finite unoriented simple tree we prove that the deficiency indices of any discrete schrodinger operator are either null or infinite we also prove that all deterministic discrete schrodinger operators which act on a random tree are almost surely self adjoint furthermore we provide several criteria of essential self adjointness we also address some importance to the case of the adjacency matrix and conjecture that given a locally finite unoriented simple graph its deficiency indices are either null or infinite besides that we consider some generalizations of trees and weighted graphs

fast computation of low rank matrix approximations given a matrix a it is often desirable to find a good approximation to a that has low rank we introduce a simple technique for accelerating the computation of such approximations when a has strong spectral features that is when the singular values of interest are significantly greater than those of a random matrix with size and entries similar to a our technique amounts to independently sampling and or quantizing the entries of a thus speeding up computation by reducing the number of nonzero entries and or the length of their representation our analysis is based on observing that the acts of sampling and quantization can be viewed as adding a random matrix n to a whose entries are independent random variables with zero mean and bounded variance since with high probability n has very weak spectral features we can prove that the effect of sampling and quantization nearly vanishes when a low rank approximation to a p n is computed we give high probability bounds on the quality of our approximation both in the frobenius and the 2 norm

forecasting using random subspace methods textabstractrandom subspace methods are a novel approach to obtain accurate forecasts in high dimensional regression settings we provide a theoretical justification of the use of random subspace methods and show their usefulness when forecasting monthly macroeconomic variables we focus on two approaches the first is random subset regression where random subsets of predictors are used to construct a forecast second we discuss random projection regression where artificial predictors are formed by randomly weighting the original predictors using recent results from random matrix theory we obtain a tight bound on the mean squared forecast error for both randomized methods we identify settings in which one randomized method results in more precise forecasts than the other and than alternative regularization strategies such as principal component regression partial least squares lasso and ridge regression the predictive accuracy on the high dimensional macroeconomic fred md data set increases substantially when using the randomized methods with random subset regression outperforming any one of the above mentioned competing methods for at least 66 of the series

a novel coupling of noise reduction algorithms for particle flow simulations proper orthogonal decomposition pod and its extension based on time windows have been shown to greatly improve the effectiveness of recovering smooth ensemble solutions from noisy particle data however to successfully de noise any molecular system a large number of measurements still need to be provided in order to achieve a better efficiency in processing time dependent fields we have combined pod with a well established signal processing technique wavelet based thresholding in this novel hybrid procedure the wavelet filtering is applied within the pod domain and referred to as wavinpod the algorithm exhibits promising results when applied to both synthetically generated signals and particle data in this work the simulations compare the performance of our new approach with standard pod or wavelet analysis in extracting smooth profiles from noisy velocity and density fields numerical examples include molecular dynamics and dissipative particle dynamics simulations of unsteady force and shear driven liquid flows as well as phase separation phenomenon simulation results confirm that wavinpod preserves the dimensionality reduction obtained using pod while improving its filtering properties through the sparse representation of data in wavelet basis this paper shows that wavinpod outperforms the other estimators for both synthetically generated signals and particle based measurements achieving a higher signal to noise ratio from a smaller number of samples the new filtering methodology offers significant computational savings particularly for multi scale applications seeking to couple continuum informations with atomistic models it is the first time that a rigorous analysis has compared de noising techniques for particle based fluid simulations

sensitivity analysis of an auto correlation function based damage index and its application in structural damage detection abstract structural damage detection using time domain vibration responses has advantages such as simplicity in calculation and no requirement of a finite element model which attracts more and more researchers in recent years in present paper a new approach to detect the damage based on the auto correlation function is proposed the maximum values of the auto correlation function of the vibration response signals from different measurement points are formulated as a vector called auto correlation function at maximum point value vector amv for short the relative change of the normalized amv before and after damage is used as the damage index to locate the damage sensitivity analysis of the normalized amv with respect to the local stiffness shows that the normalized amv has a sharp change around the local stiffness change location which means the normalized amv is a good indicator to detect the damage even when the damage is very small stiffness reduction detection of a 12 story frame structure is provided to illustrate the validity effectiveness and the anti noise ability of the proposed method comparison of the normalized amv and the other correlation function based damage detection method shows the normalized amv has a better detectability

generalized low rank approximations of matrices revisited compared to singular value decomposition svd generalized low rank approximations of matrices glram can consume less computation time obtain higher compression ratio and yield competitive classification performance glram has been successfully applied to applications such as image compression and retrieval and quite a few extensions have been successively proposed however in literature some basic properties and crucial problems with regard to glram have not been explored or solved yet for this sake we revisit glram in this paper first we reveal such a close relationship between glram and svd that glram s objective function is identical to svd s objective function except the imposed constraints second we derive a lower bound of glram s objective function and discuss when the lower bound can be touched moreover from the viewpoint of minimizing the lower bound we answer one open problem raised by ye machine learning 2005 i e a theoretical justification of the experimental phenomenon that under given number of reduced dimension the lowest reconstruction error is obtained when the left and right transformations have equal number of columns third we explore when and why glram can perform well in terms of compression which is a fundamental problem concerning the usability of glram

efficient low rank approximation of matrices based on randomized pivoted decomposition given a matrix bf a with numerical rank k the two sided orthogonal decomposition tsod computes a factorization bf a bf udv t where bf u and bf v are orthogonal and bf d is upper lower triangular tsod is rank revealing as the middle factor bf d reveals the rank of bf a the computation of tsod however is demanding in this paper we present an algorithm called randomized pivoted tsod rp tsod where the middle factor is lower triangular key in our work is the exploitation of randomization and rp tsod is primarily devised to efficiently construct an approximation to a low rank matrix we provide three different types of bounds for rp tsod i we furnish upper bounds on the error of the low rank approximation ii we bound the k approximate principal singular values and iii we derive bounds for the canonical angles between the approximate and the exact singular subspaces our bounds describe the characteristics and behavior of our proposed algorithm through numerical tests we show the effectiveness of the devised bounds as well as our proposed algorithm

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

approximating the minimum spanning tree weight in sublinear time we present a probabilistic algorithm that given a connected graph g represented by adjacency lists of average degree d with edge weights in the set 1 w and given a parameter 0 eps 1 2 estimates in time o dw varepsilon 2 log frac dw varepsilon the weight of the minimum spanning tree mst of g with a relative error of at most eps note that the running time does not depend on the number of vertices in g we also prove a nearly matching lower bound of omega dw varepsilon 2 on the probe and time complexity of any approximation algorithm for mst weight the essential component of our algorithm is a procedure for estimating in time o d eps 2 log frac d varepsilon the number of connected components of an unweighted graph to within an additive error of varepsilon n this becomes o eps 2 log frac 1 varepsilon for d o 1 the time bound is shown to be tight up to within the log frac d varepsilon factor our connected components algorithm picks o 1 varepsilon 2 vertices in the graph and then grows local spanning trees whose sizes are specified by a stochastic process from the local information collected in this way the algorithm is able to infer with high confidence an estimate of the number of connected components we then show how estimates on the number of components in various subgraphs of g can be used to estimate the weight of its mst

continuum versus discrete networks graph laplacians and reproducing kernel hilbert spaces abstract motivated by applications to machine learning we construct a reversible and irreducible markov chain whose state space is a certain collection of measurable sets of a chosen l c h space x we study the resulting network connected undirected graph including transience royden and riesz decompositions and kernel factorization we describe a construction for hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of l 2 functions by functions of finite energy the latter has applications to machine learning for markov random fields for example

completing any low rank matrix provably matrix completion i e the exact and provable recovery of a low rank matrix from a small subset of its elements is currently only known to be possible if the matrix satisfies a restrictive structural constraint known as incoherence on its row and column spaces in these cases the subset of elements is assumed to be sampled uniformly at random in this paper we show that any rank r n by n matrix can be exactly recovered from as few as o nr log2 n randomly chosen elements provided this random choice is made according to a specific biased distribution suitably dependent on the coherence structure of the matrix the probability of any element being sampled should be at least a constant times the sum of the leverage scores of the corresponding row and column moreover we prove that this specific form of sampling is nearly necessary in a natural precise sense this implies that many other perhaps more intuitive sampling schemes fail we further establish three ways to use the above result for the setting when leverage scores are not known a priori a we describe a provably correct sampling strategy for the case when only the column space is incoherent and no assumption or knowledge of the row space is required b we propose a two phase sampling procedure for general matrices that first samples to estimate leverage scores followed by sampling for exact recovery these two approaches assume control over the sampling procedure c by using our main theorem in a reverse direction we provide an analysis showing the advantages of the empirically successful weighted nuclear trace norm minimization approach over the vanilla un weighted formulation given non uniformly distributed observed elements this approach does not require controlled sampling or knowledge of the leverage scores

matrices vector spaces and information retrieval the evolution of digital libraries and the internet has dramatically transformed the processing storage and retrieval of information efforts to digitize text images video and audio now consume a substantial portion of both academic and industrial activity even when there is no shortage of textual materials on a particular topic procedures for indexing or extracting the knowledge or conceptual information contained in them can be lacking recently developed information retrieval technologies are based on the concept of a vector space data are modeled as a matrix and a user s query of the database is represented as a vector relevant documents in the database are then identified via simple vector operations orthogonal factorizations of the matrix provide mechanisms for handling uncertainty in the database itself the purpose of this paper is to show how such fundamental mathematical concepts from linear algebra can be used to manage and index large text collections

the round trip theory for reconstruction of green s functions at passive locations an expression for the green s function at an arbitrary set of passive locations no applied force is derived and validated by experiment three sets of points are involved the passive reconstruction points c which lie on a virtual boundary and two sets of auxiliary points denoted a and b located either side the reconstruction is achieved using green s functions forming a round trip from and to the reconstruction points via a and b a two stage measurement procedure is described involving excitation at b and a but with no excitation required at the reconstruction points a known round trip relationship is first introduced which is theoretically exact for points on a multi point interface between two linear time invariant subsystems experimental results for frequency response functions of a beam plate structure show that this relationship gives good results in practice it is then shown that the theory provides an nth order approximation for the green s function at arbitrary points where n is the number of points at b the expression is validated by reconstructing point and transfer frequency response functions at two passive points on an aluminum plate

iterative and active graph clustering using trace norm minimization without cluster size constraints this paper investigates graph clustering under the planted partition model in the presence of small clusters traditional results dictate that for an algorithm to provably correctly recover the underlying clusters all clusters must be sufficiently large in particular the cluster sizes need to be n where n is the number of nodes of the graph we show that this is not really a restriction by a refined analysis of a convex optimization based recovery approach we prove that small clusters under certain mild assumptions do not hinder recovery of large ones based on this result we further devise an iterative algorithm to provably recover almost all clusters via a peeling strategy we recover large clusters first leading to a reduced problem and repeat this procedure these results are extended to the partial observation setting in which only a chosen part of the graph is observed the peeling strategy gives rise to an active learning algorithm in which edges adjacent to smaller clusters are queried more often after large clusters are learned and removed we expect that the idea of iterative peeling that is sequentially identifying a subset of the clusters and reducing the problem to a smaller one is useful more broadly beyond the specific implementations based on convex optimization used in this paper

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

territorial design optimization for business sales plan abstract a well designed territory enhances customer coverage increases sales fosters fair performance and rewards systems and lower travel cost this paper considers a real life case study to design the sales territory for a business sales plan the business plan consist of assigning the optimal quantity of sellers to a territory including the scheduling and routing plans for each seller the problem is formulated as a combination of assignment scheduling and routing optimization models the solution approach considers a meta heuristic using stochastic iterative projection method for large systems several real life instances of different sizes were tested with stochastic data to represent raise fall in the customers demand as well as the appearance loss of customers

approximating the radii of point sets we consider the problem of computing the outer radii of point sets in this problem we are given integers n d and k where k le d and a set p of n points in re d the goal is to compute the outer k radius of p denoted by cal r k p which is the minimum over all d k dimensional flats f of max p in p d p f where d p f is the euclidean distance between the point p and flat f computing the radii of point sets is a fundamental problem in computational convexity with many significant applications the problem admits a polynomial time algorithm when the dimension d is constant u faigle w kern and m streng math program 73 1996 pp 1 5 here we are interested in the general case in which the dimension d is not fixed and can be as large as n where the problem becomes np hard even for k 1 it is known that r k p can be approximated in polynomial time by a factor of 1 varepsilon for any varepsilon 0 when d k is a fixed constant m babdoiu s har peled and p indyk in proceedings of the acm symposium on the theory of computing 2002 s har peled and k varadarajan in proceedings of the acm symposium on computing geometry 2002 a polynomial time algorithm that guarantees a factor of o sqrt log n approximation for r 1 p the width of the point set p is implied by the results of nemirovski roos and terlaky math program 86 1999 pp 463 473 and nesterov handbook of semidefinite programming theory algorithms kluwer academic publishers norwell ma 2000 in this paper we show that r k p can be approximated by a ratio of o sqrt log n for any 1 leq k leq d thus matching the previously best known ratio for approximating the special case r 1 p the width of point set p our algorithm is based on semidefinite programming relaxation with a new mixed deterministic and randomized rounding procedure we also prove an inapproximability result that gives evidence that our approximation algorithm is doing well for a large range of k we show that there exists a constant delta 0 such that the following holds for any 0 eps 1 there is no polynomial time algorithm that approximates r k p within log n delta for all k such that k leq d d varepsilon unless np subseteq dtime 2 log m o 1 our inapproximability result for r k p extends a previously known hardness result of brieden discrete comput geom 28 2002 pp 201 209 and is proved by modifying brieden s construction using basic ideas from probabilistically checkable proofs pcp theory

a sears type self adjointness result for discrete magnetic schrodinger operators abstract in the context of a weighted graph with vertex set v and bounded vertex degree we give a sufficient condition for the essential self adjointness of the operator w where is the magnetic laplacian and w v r is a function satisfying w x q x for all x v with q v 1 the condition is expressed in terms of completeness of a metric that depends on q and the weights of the graph the main result is a discrete analogue of the results of i oleinik and m a shubin in the setting of non compact riemannian manifolds

triggering expectation and residual excitation of distributed random decrement technique abstract a distributing process is proposed for random decrement technique rdt to accurate modal identification impulse response can be extracted from random response whose signal is overlap averaged under gaussian white noise excitation with better understanding of initial condition rdt can be improved with triggering expectation and residual excitation firstly the initial condition is defined as the expectation of random response on instants of triggering time series response of unit initial condition can be calculated from random decrement rd signals using matrix equations generated from multiple rd processes this response also keeps a unique expression for a particular system secondly the key of rdt error can be the residual excitation which refers to the residue of random excitation after rd process importantly the influence of residual excitation declines with increased overlap averaging times this modified rdt might be used to obtain impulse response from random response and its error can be estimated

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

noisy matrix completion understanding statistical guarantees for convex relaxation via nonconvex optimization this paper studies noisy low rank matrix completion given partial and noisy entries of a large low rank matrix the goal is to estimate the underlying matrix faithfully and efficiently arguably o

harmonic analysis for resistance forms abstract in this paper we define the green functions for a resistance form by using effective resistance and harmonic functions then the green functions and harmonic functions are shown to be uniformly lipschitz continuous with respect to the resistance metric making use of this fact we construct the green operator and the measure valued laplacian the domain of the laplacian is shown to be a subset of uniformly lipschitz continuous functions while the domain of the resistance form in general consists of uniformly 1 2 holder continuous functions

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

analyzing multiple probe microarray estimation and application of gene expression indexes gene expression index estimation is an essential step in analyzing multiple probe microarray data various modeling methods have been proposed in this area amidst all a popular method proposed in li and wong 2001 is based on a multiplicative model which is similar to the additive model discussed in irizarry et al 2003a at the logarithm scale along this line hu et al 2006 proposed data transformation to improve expression index estimation based on an ad hoc entropy criteria and naive grid search approach in this work we re examined this problem using a new profile likelihood based transformation estimation approach that is more statistically elegant and computationally efficient we demonstrate the applicability of the proposed method using a benchmark affymetrix u95a spiked in experiment moreover we introduced a new multivariate expression index and used the empirical study to shows its promise in terms of improving model fitting and power of detecting differential expression over the commonly used univariate expression index as the other important content of the work we discussed two generally encountered practical issues in application of gene expression index normalization and summary statistic used for detecting differential expression our empirical study shows somewhat different findings from the maqc project maqc 2006

improved graph clustering graph clustering involves the task of dividing nodes into clusters so that the edge density is higher within clusters as opposed to across clusters a natural classic and popular statistical setting for evaluating solutions to this problem is the stochastic block model also referred to as the planted partition model in this paper we present a new algorithm a convexified version of maximum likelihood for graph clustering we show that in the classic stochastic block model setting it outperforms existing methods by polynomial factors when the cluster size is allowed to have general scalings in fact it is within logarithmic factors of known lower bounds for spectral methods and there is evidence suggesting that no polynomial time algorithm would do significantly better we then show that this guarantee carries over to a more general extension of the stochastic block model our method can handle the settings of semirandom graphs heterogeneous degree distributions unequal cluster sizes unaffiliated nodes partially observed graphs planted clique coloring and so on in particular our results provide the best exact recovery guarantees to date for the planted partition planted k disjoint cliques and planted noisy coloring models with general cluster sizes in other settings we match the best existing results up to logarithmic factors

low rank approximation and regression in input sparsity time we design a new distribution over m n matrices s so that for any fixed n d matrix a of rank r with probability at least 9 10 psaxp2 e 1 e paxp2 simultaneously for all x rd here m is bounded by a polynomial in re 1 and the parameter e 0 1 such a matrix s is called a subspace embedding furthermore sa can be computed in o nnz a time where nnz a is the number of nonzero entries of a this improves over all previous subspace embeddings for which computing sa required at least ndlog d time we call these s sparse embedding matrices using our sparse embedding matrices we obtain the fastest known algorithms for overconstrained least squares regression low rank approximation approximating all leverage scores and ep regression more specifically let b be an n 1 vector e 0 a small enough value and integers k p g 1 our results include the following regression the regression problem is to find d 1 vector x for which pax bpp l 1 e min xpax bpp for the euclidean case p e 2 we obtain an algorithm running in o nnz a o d3e 2 time and another in o nnz a log 1 e o d3 log 1 e time here o f e f log o 1 f for p 1 more generally we obtain an algorithm running in o nnz a log n o r e 1 c time for a fixed c low rank approximation we give an algorithm to obtain a rank k matrix k such that pa kpf 1 e p a akpf where ak is the best rank k approximation to a that is ak is the output of principal components analysis produced by a truncated singular value decomposition useful for latent semantic indexing and many other statistical problems our algorithm runs in o nnz a o nk2e 4 k3e 5 time leverage scores we give an algorithm to estimate the leverage scores of a up to a constant factor in o nnz a log n o r3 time

a model selection approach for clustering a multinomial sequence with non negative factorization we consider a problem of clustering a sequence of multinomial observations by way of a model selection criterion we propose a form of a penalty term for the model selection procedure our approach subsumes both the conventional aic and bic criteria but also extends the conventional criteria in a way that it can be applicable also to a sequence of sparse multinomial observations where even within a same cluster the number of multinomial trials may be different for different observations in addition as a preliminary estimation step to maximum likelihood estimation and more generally to maximum l q estimation we propose to use reduced rank projection in combination with non negative factorization we motivate our approach by showing that our model selection criterion and preliminary estimation step yield consistent estimates under simplifying assumptions we also illustrate our approach through numerical experiments using real and simulated data

restricted strong convexity and weighted matrix completion optimal bounds with noise we consider the matrix completion problem under a form of row column weighted entrywise sampling including the case of uniformentrywise sampling as a special case we analyze the associated random observation operator and prove that with high probability it satisfies a form of restricted strong convexity with respect to weighted frobenius norm using this property we obtain as corollaries a number of error bounds on matrix completion in the weighted frobenius norm under noisy sampling and for both exact and near low rank matrices our results are based on measures of the spikiness and low rankness of matrices that are less restrictive than the incoherence conditions imposed in previous work our technique involves an m estimator that includes controls on both the rank and spikiness of the solution and we establish non asymptotic error bounds in weighted frobenius norm for recovering matrices lying with lq balls of bounded spikiness using information theoretic methods we show that no algorithm can achieve better estimates up to a logarithmic factor over these same sets showing that our conditions on matrices and associated rates are essentially optimal

a fast block low rank dense solver with applications to finite element matrices this article presents a fast solver for the dense frontal matrices that arise from the multifrontal sparse elimination process of 3d elliptic pdes the solver relies on the fact that these matrices can be efficiently represented as a hierarchically off diagonal low rank hodlr matrix to construct the low rank approximation of the off diagonal blocks we propose a new pseudo skeleton scheme the boundary distance low rank approximation that picks rows and columns based on the location of their corresponding vertices in the sparse matrix graph we compare this new low rank approximation method to the adaptive cross approximation aca algorithm and show that it achieves better speedup specially for unstructured meshes using the hodlr direct solver as a preconditioner with a low tolerance to the gmres iterative scheme we can reach machine accuracy much faster than a conventional lu solver numerical benchmarks are provided for frontal matrices arising from 3d finite element problems corresponding to a wide range of applications

a note on linear function approximation using random projections linear function approximations based on random projections are proposed and justified for a class of fixed point and minimization problems

a new method to extract modal parameters using output only responses this work proposes a new output only modal analysis method to extract mode shapes and natural frequencies of a structure the proposed method is based on an approach with a single degree of freedom in the time domain for a set of given mode isolated signals the un damped mode shapes are extracted utilizing the singular value decomposition of the output energy correlation matrix with respect to sensor locations the natural frequencies are extracted from a noise free signal that is projected on the estimated modal basis the proposed method is particularly efficient when a high resolution of mode shape is essential the accuracy of the method is numerically verified using a set of time histories that are simulated using a finite element method the feasibility and practicality of the method are verified using experimental data collected at the newly constructed king storm water bridge in california united states

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

a diffusion approach to network localization the localization of nodes on a network is a challenging research topic it arises in a variety of applications such as communications and sensor network analysis we propose a computational approach to recovering the positions of network nodes given partial and corrupted distance measurements and the positions of a small subset of anchor nodes first we show how to derive geometrically adaptive diffusion bases defined over the entire network given only partial distance measurements second we propose to utilize several diffusion bases simultaneously to derive multiscale diffusion frames last we utilize the diffusion frames to formulate a l1 regression based extension of the anchor points coordinates to the entire network we experimentally show that under a wide range of conditions our method compares favorably with state of the art approaches

improving cur matrix decomposition and the nystrom approximation via adaptive sampling the cur matrix decomposition and the nystrom approximation are two important low rank matrix approximation techniques the nystrom method approximates a symmetric positive semidefinite matrix in terms of a small number of its columns while cur approximates an arbitrary data matrix by a small number of its columns and rows thus cur decomposition can be regarded as an extension of the nystrom approximation in this paper we establish a more general error bound for the adaptive column row sampling algorithm based on which we propose more accurate cur and nystrom algorithms with expected relative error bounds the proposed cur and nystrom algorithms also have low time complexity and can avoid maintaining the whole data matrix in ram in addition we give theoretical analysis for the lower error bounds of the standard nystrom method and the ensemble nystrom method the main theoretical results established in this paper are novel and our analysis makes no special assumption on the data matrices

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

parametric estimation of the cross power spectral density a new cross spectral analysis procedure is proposed for the parametric estimation of the relationship between two time sequences in the frequency domain in this method the two observable outputs are modeled as a pair of autoregressive moving average and moving average armama models under the assumption that the two outputs are driven by a common input and independent ones simultaneously cross and auto power spectral densities psds of a pair of armama models can be derived as forms of rational polynomial functions the coefficients of these functions can be estimated from the cross correlation function or the auto correlation functions of the two observed sequences by using the method presented in this paper the main advantage of the present procedure is that the physical parameters of an unknown system can be easily estimated from the coefficients of the cross and auto psd functions to illustrate the effectiveness of the proposed procedure numerical and practical examples of a mechanical vibration problem are analyzed the results show that the proposed procedure gives accurate cross and auto psd estimates moreover the physical properties of the unknown system can be well estimated from the obtained cross and auto psds

efficient denoising algorithms for large experimental datasets and their applications in fourier transform ion cyclotron resonance mass spectrometry modern scientific research produces datasets of increasing size and complexity that require dedicated numerical methods to be processed in many cases the analysis of spectroscopic data involves the denoising of raw data before any further processing current efficient denoising algorithms require the singular value decomposition of a matrix with a size that scales up as the square of the data length preventing their use on very large datasets taking advantage of recent progress on random projection and probabilistic algorithms we developed a simple and efficient method for the denoising of very large datasets based on the qr decomposition of a matrix randomly sampled from the data this approach allows a gain of nearly three orders of magnitude in processing time compared with classical singular value decomposition denoising this procedure called urqrd uncoiled random qr denoising strongly reduces the computer memory footprint and allows the denoising algorithm to be applied to virtually unlimited data size the efficiency of these numerical tools is demonstrated on experimental data from high resolution broadband fourier transform ion cyclotron resonance mass spectrometry which has applications in proteomics and metabolomics we show that robust denoising is achieved in 2d spectra whose interpretation is severely impaired by scintillation noise these denoising procedures can be adapted to many other data analysis domains where the size and or the processing time are crucial

wave equation tomography the relation between ray trace and diffraction tomography is usually obscured by formulation of the two methods in different domains the former in space the latter in wavenumber here diffraction tomography is reformulated in the space domain under the title of wave equation tomography with this transformation wave equation tomography projects monochromatic scattered wavefields back over source receiver wavepaths just as ray trace tomography projects traveltime delays back over source receiver raypaths derived under the born approximation these wavepaths are wave theoretic back projection patterns for reflected energy derived under the rytov approximation they are wave theoretic back projection patterns for transmitted energy differences between ray trace and wave equation tomography are examined through comparison of wavepaths and raypaths followed by their application to a transmission geometry synthetic data set rytov wave equation tomography proves superior to ray trace tomography in de

non overlapped random decrement technique for parameter identification in operational modal analysis abstract the random decrement technique rdt is used to estimate free vibration response from output data generated by gaussian white noise the principle is to decay the excitation via averaging of segments in output data with rdt the triggering condition for determining the initial points of segments causes overlap during averaging the consequence is a residual excitation peaking at the first natural frequency this paper presents a modified rdt with non overlapped segments to eliminate this peak numerical comparison between non overlapped rdt nrdt and rdt shows the accuracy improvement of damping however time history data is sometimes not long enough in nrdt which results in an inevitable overlap in order to keep the accuracy of nrdt the first natural period is viewed as the critical length between adjacent initial points to distinguish the inevitable overlap from that in rdt

randomized lu decomposition using sparse projections abstract a fast algorithm that approximates a low rank lu decomposition is presented in order to achieve a low complexity the algorithm uses sparse random projections combined with fft based random projections the asymptotic approximation error of the algorithm is analyzed and a theoretical error bound is presented finally numerical examples illustrate that for a similar approximation error the sparse lu algorithm is faster than recent state of the art methods the algorithm is completely parallelizable and can fully run on a gpu the performance is tested on a gpu card showing a significant speed up improvement in the running time in comparison to a sequential execution

study on an auto correlation function based damage index sensitivity analysis and structural damage detection abstract the damage index based on the auto correlation function to detect the damage of the structure under white noise excitation is studied in detail in this paper the maximum values of the auto correlation function of the vibration response signals displacement velocity and acceleration from different measurement points of the structure are collected and formulated as a vector called auto correlation function at maximum point value vector amv which is expressed as a weighted combination of the hadamard product of two mode shapes amv is normalized by its root mean square value so that the influence of the excitation can be eliminated sensitivity analysis for the different parts of the normalized amv shows that the sensitivity of the normalized amv to the local stiffness is dependent most on the sensitivity of the hadamard product of the two lower order mode shapes to the local stiffness which has a sudden change of the value around the local stiffness change position the sensitivity of the normalized amv has the similar shape and same trend that shows it is a very good damage indicator even for the very small damage the relative change of the normalized amv before and after damage occurs in the structure is adopted as the damage index to show the damage location several examples of the stiffness reduction detection of a 12 story shear frame structure are utilized to validate the results in sensitivity analysis illustrate the effectiveness and anti noise ability of the amv based damage detection method and compare the effect of the response type on the detectability of the normalized amv

spectral reciprocity and matrix representations of unbounded operators abstract we study a family of unbounded hermitian operators in hilbert space which generalize the usual graph theoretic discrete laplacian for an infinite discrete set x we consider operators acting on hilbert spaces of functions on x and their representations as infinite matrices the focus is on l 2 x and the energy space h e in particular we prove that these operators are always essentially self adjoint on l 2 x but may fail to be essentially self adjoint on h e in the general case we examine the von neumann deficiency indices of these operators and explore their relevance in mathematical physics finally we study the spectra of the h e operators with the use of a new approximation scheme

damage localization from projections of free vibration signals abstract an approach for damage localization that operates with free vibration signals in the damaged state is presented the scheme does not directly point to the damage location but given a postulated distribution computes a metric that for ideal conditions approaches infinity when the distribution is correct the approach does not use identification operates without constraints between the number of sensors and the number of model degrees of freedom and differs from model updating in that only the damage distribution and not the severity enters the formulation analytical and experimental results are included

on selecting a maximum volume sub matrix of a matrix and related problems given a matrix email protected r m x n n vectors in m dimensions we consider the problem of selecting a subset of its columns such that its elements are as linearly independent as possible this notion turned out to be important in low rank approximations to matrices and rank revealing qr factorizations which have been investigated in the linear algebra community and can be quantified in a few different ways in this paper from a complexity theoretic point of view we propose four related problems in which we try to find a sub matrix email protected r m x k of a given matrix email protected r m x n such that i s m a x c the largest singular value of c is minimum ii s m i n c the smallest singular value of c is maximum iii k c email protected m a x c s m i n c the condition number of c is minimum and iv the volume of the parallelepiped defined by the column vectors of c is maximum we establish the np hardness of these problems and further show that they do not admit ptas we then study a natural greedy heuristic for the maximum volume problem and show that it has approximation ratio 2 o k l o g k our analysis of the greedy heuristic is tight to within a logarithmic factor in the exponent which we show by explicitly constructing an instance for which the greedy heuristic is 2 w k from optimal when a has unit norm columns a related problem is to select the maximum number of vectors with a given volume we show that if the optimal solution selects k columns then greedy will select w k logk columns providing a logk approximation

subspace iteration randomization and singular value problems a classical problem in matrix computations is the efficient and reliable approximation of a given matrix by a matrix of lower rank the truncated singular value decomposition svd is known to provide the best such approximation for any given fixed rank however the svd is also known to be very costly to compute among the different approaches in the literature for computing low rank approximations randomized algorithms have attracted researchers attention recently due to their surprising reliability and computational efficiency in different application areas typically such algorithms are shown to compute with very high probability low rank approximations that are within a constant factor from optimal and are known to perform even better in many practical situations in this paper we present a novel error analysis that considers randomized algorithms within the subspace iteration framework and show with very high probability that highly accurate low rank approximations as well as singular values ca

importance sampling for a monte carlo matrix multiplication algorithm with application to information retrieval we perform importance sampling for a randomized matrix multiplication algorithm by drineas kannan and mahoney and derive probabilities that minimize the expected value with regard to the distributions of the matrix elements of the variance we compare these optimized probabilities with uniform probabilities and derive conditions under which the actual variance of the optimized probabilities is lower numerical experiments with query matching in information retrieval applications illustrate that the optimized probabilities produce more accurate matchings than the uniform probabilities and that they can also be computed efficiently

structural change localization and monitoring through a perturbation based inverse problem structural change detection and characterization or structural health monitoring is generally based on modal analysis for detection localization and quantification of changes in structure classical methods combine both variations in frequencies and mode shapes which require accurate and spatially distributed measurements in this study the detection and localization of a local perturbation are assessed by analysis of frequency changes in the fundamental mode and overtones that are combined with a perturbation based linear inverse method and a deconvolution process this perturbation method is applied first to a bending beam with the change considered as a local perturbation of the young s modulus using a one dimensional finite element model for modal analysis localization is successful even for extended and multiple changes in a second step the method is numerically tested under ambient noise vibration from the beam support with local changes that are shifted step by step along the beam the frequency values are revealed using the random decrement technique that is applied to the time evolving vibrations recorded by one sensor at the free extremity of the beam finally the inversion method is experimentally demonstrated at the laboratory scale with data recorded at the free end of a plexiglas beam attached to a metallic support

a randomized algorithm for a tensor based generalization of the singular value decomposition an algorithm is presented and analyzed that when given as input a d mode tensor a computes an approximation a the approximation a is computed by performing the following for each of the d modes first form implicitly a matrix by unfolding the tensor along that mode then choose columns from the matrices thus generated and finally project the tensor along that mode onto the span of those columns an important issue affecting the quality of the approximation is the choice of the columns from the matrices formed by unfolding the tensor along each of its modes in order to address this issue two algorithms of independent interest are presented that given an input matrix a and a target rank k select columns that span a space close to the best rank k subspace of the matrix for example in one of the algorithms a number c that depends on k an error parameter and a failure probability of columns are chosen in c independent random trials according to a nonuniform probability distribution depending on the euclidean lengths of the columns when this algorithm for choosing columns is used in the tensor approximation then under appropriate assumptions bounds of the form

fast monte carlo algorithms for matrices i approximating matrix multiplication motivated by applications in which the data may be formulated as a matrix we consider algorithms for several common linear algebra problems these algorithms make more efficient use of computational resources such as the computation time random access memory ram and the number of passes over the data than do previously known algorithms for these problems in this paper we devise two algorithms for the matrix multiplication problem suppose a and b which are m times n and n times p respectively are the two input matrices in our main algorithm we perform c independent trials where in each trial we randomly sample an element of 1 2 ldots n with an appropriate probability distribution cal p on 1 2 ldots n we form an m times c matrix c consisting of the sampled columns of a each scaled appropriately and we form a c times n matrix r using the corresponding rows of b again scaled appropriately the choice of cal p and the column and row scaling are crucial features of the algorithm when these are chosen judiciously we show that cr is a good approximation to ab more precisely we show that left ab cr right f o left a right f left b right f sqrt c where cdot f denotes the frobenius norm i e a 2 f sum i j a ij 2 this algorithm can be implemented without storing the matrices a and b in ram provided it can make two passes over the matrices stored in external memory and use o c m n p additional ram to construct c and r we then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm in addition we present a model the pass efficient model in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets in this model the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm the input matrices may be presented in any order of the entries and not just row or column order as is the case in many applications where e g the data has been written in by multiple agents in addition the input matrices may be presented in a sparse representation where only the nonzero entries are written

randomized interpolative decomposition of separated representations we introduce an algorithm to compute tensor interpolative decomposition dubbed ctd id for the reduction of the separation rank of canonical tensor decompositions ctds tensor id selects for a user defined accuracy a near optimal subset of terms of a ctd to represent the remaining terms via a linear combination of the selected terms ctd id can be used as an alternative to or in combination with the alternating least squares als algorithm we present examples of its use within a convergent iteration to compute inverse operators in high dimensions we also briefly discuss the spectral norm as a computational alternative to the frobenius norm in estimating approximation errors of tensor id we reduce the problem of finding tensor ids to that of constructing interpolative decompositions of certain matrices these matrices are generated via randomized projection of the terms of the given tensor we provide cost estimates and several examples of the new approach to the reduction of separation rank

information recovery from pairwise measurements this paper is concerned with jointly recovering n node variables left x i right 1 leq i leq n from a collection of pairwise difference measurements imagine we acquire a few observations taking the form of x i x j the observation pattern is represented by a measurement graph mathcal g with an edge set mathcal e such that x i x j is observed if and only if i j in mathcal e to account for noisy measurements in a general manner we model the data acquisition process by a set of channels with given input output transition measures employing information theoretic tools applied to channel decoding problems we develop a unified framework to characterize the fundamental recovery criterion which accommodates general graph structures alphabet sizes and channel transition measures in particular our results isolate a family of minimum channel divergence measures to characterize the degree of measurement corruption which together with the size of the minimum cut of mathcal g dictates the feasibility of exact information recovery for various homogeneous graphs the recovery condition depends almost only on the edge sparsity of the measurement graph irrespective of other graphical metrics alternatively the minimum sample complexity required for these graphs scales like n log n mathsf hel 1 2 min for certain information metric mathsf hel 1 2 min defined in the main text as long as the alphabet size is not super polynomial in n we apply our general theory to three concrete applications including the stochastic block model the random corruption model and the haplotype assembly problem our theory leads to orderwise tight recovery conditions for all these scenarios

statistical theory of the vector random decrement technique the vector random decrement technique has previously been introduced as an efficient method to transform ambient responses of linear structures into vector random decrement functions which are equivalent to free decays of the current structure the modal parameters can be extracted from the free decays due to the speed and or accuracy of the vector random decrement technique it was introduced as an attractive alternative to the random decrement technique in this paper the theory of the vector random decrement technique is extended by applying a statistical description of the stochastic processes describing the ambient measurements the vector random decrement functions are linked to the correlation functions of the stochastic processes provided they are stationary and gaussian distributed furthermore a new approach for quality assessment of the vector random decrement functions is given on the basis of the derived results the work presented in this paper makes the theory of the vector random decrement technique equivalent to the theory of the random decrement technique the theoretical derivations are illustrated by the analysis of the response of a 3dof system loaded by white noise

a fast tri factorization method for low rank matrix recovery and completion in recent years matrix rank minimization problems have received a significant amount of attention in machine learning data mining and computer vision communities and these problems can be solved by a convex relaxation of the rank minimization problem which minimizes the nuclear norm instead of the rank of the matrix and has to be solved iteratively and involves singular value decomposition svd at each iteration therefore those algorithms for nuclear norm minimization problems suffer from high computation cost of multiple svds in this paper we propose a fast tri factorization ftf method to approximate the nuclear norm minimization problem and mitigate the computation cost of performing svds the proposed ftf method can be used to reliably solve a wide range of low rank matrix recovery and completion problems such as robust principal component analysis rpca low rank representation lrr and low rank matrix completion mc we also present three specific models for rpca lrr and mc problems respectively moreover we develop two alternating direction method adm based iterative algorithms for solving the above three problems experimental results on a variety of synthetic and real world data sets validate the efficiency robustness and effectiveness of our ftf method comparing with the state of the art nuclear norm minimization algorithms

on the nystrom method for approximating a gram matrix for improved kernel based learning a problem for many kernel based methods is that the amount of computation required to find the solution scales as o n3 where n is the number of training examples we develop and analyze an algorithm to compute an easily interpretable low rank approximation to an n n gram matrix g such that computations of interest may be performed more rapidly the approximation is of the form gk cwk ct where c is a matrix consisting of a small number c of columns of g and wk is the best rank k approximation to w the matrix formed by the intersection between those c columns of g and the corresponding c rows of g an important aspect of the algorithm is the probability distribution used to randomly sample the columns we will use a judiciously chosen and data dependent nonuniform probability distribution let 2 and f denote the spectral norm and the frobenius norm respectively of a matrix and let gk be the best rank k approximation to g we prove that by choosing o k e4 columns g cwk ct g gk e i 1n gii2 both in expectation and with high probability for both 2 f and for all k 0 k rank w this approximation can be computed using o n additional space and time after making two passes over the data from external storage the relationships between this algorithm other related matrix decompositions and the nystrom method from integral equation theory are discussed

system identification from ambient vibration measurements on a bridge abstract the cross correlation function between two response measurements made on an ambiently excited structure is shown to have the same form as the system s impulse response function therefore standard time domain curve fitting procedures which are typically applied to impulse response functions can now be applied to the cross correlation functions to estimate the resonant frequencies and modal damping of the structure this derivation is based on the assumption that the ambient vibration source is a white noise random process curve fitting cross correlation functions to obtain modal properties offers advantages over standard procedures that identify resonant frequencies from peaks in the power spectrum and damping from the width of the power spectrum the primary advantage is the ability to identify closely spaced modes and their associated damping the resonant frequencies of a highway bridge that were identified by curve fitting the cross correlation functions using traffic excitation as the ambient vibration source are compared to modal properties identified by standard forced vibration testing methods results of this comparison showed a maximum discrepancy of 3 63 percent similar comparisons for the average modal damping values identified by the two methods showed a 9 82 percent difference this experimental verification implies that the proposed method of analyzing ambient vibration data can be used to accurately assess the dynamic properties of structures in a non intrusive manner

subspace orbit randomized decomposition for low rank matrix approximations an efficient accurate and reliable approximation of a matrix by one of lower rank is a fundamental task in numerical linear algebra and signal processing applications in this paper we introduce a new matrix decomposition approach termed subspace orbit randomized singular value decomposition sor svd which makes use of random sampling techniques to give a low rank approximation to an input matrix given a large and dense data matrix of size m times n the algorithm requires a few passes through data to compute a rank k approximation in o mnk floating point operations moreover the sor svd algorithm can utilize advanced computer architectures and as a result it can be optimized for maximum efficiency the sor svd algorithm is simple accurate and provably correct and outperforms previously reported techniques in terms of accuracy and efficiency our numerical experiments support these claims

analysis of the physical laplacian and the heat flow on a locally finite graph we study the physical laplacian and the corresponding heat flow on an infinite locally finite graph with possibly unbounded valence

fast monte carlo algorithms for matrices ii computing a low rank approximation to a matrix in many applications the data consist of or may be naturally formulated as an m times n matrix a it is often of interest to find a low rank approximation to a i e an approximation d to the matrix a of rank not greater than a specified rank k where k is much smaller than m and n methods such as the singular value decomposition svd may be used to find an approximation to a which is the best in a well defined sense these methods require memory and time which are superlinear in m and n for many applications in which the data sets are very large this is prohibitive two simple and intuitive algorithms are presented which when given an m times n matrix a compute a description of a low rank approximation d to a and which are qualitatively faster than the svd both algorithms have provable bounds for the error matrix a d for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively in the first algorithm c columns of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling then it is shown that from c tc approximations to the top singular values and corresponding singular vectors may be computed from the computed singular vectors a description d of the matrix a may be computed such that mathrm rank d le k and such that left a d right xi 2 le min d mathrm rank d le k left a d right xi 2 poly k 1 c left a right 2 f holds with high probability for both xi 2 f this algorithm may be implemented without storing the matrix a in random access memory ram provided it can make two passes over the matrix stored in external memory and use o cm c 2 additional ram the second algorithm is similar except that it further approximates the matrix c by randomly sampling r rows of c to form a r times c matrix w thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a 2 f both algorithms take time which is polynomial in k 1 epsilon and log 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n our bounds improve previously published results with respect to the rank parameter k for both the frobenius and spectral norms in addition the proofs for the error bounds use a novel method that makes important use of matrix perturbation theory the probability distribution over columns of a and the rescaling are crucial features of the algorithms which must be chosen judiciously

hardy inequality and asymptotic eigenvalue distribution for discrete laplacians abstract in this paper we study in detail some spectral properties of the magnetic discrete laplacian we identify its form domain characterize the absence of essential spectrum and provide the asymptotic eigenvalue distribution

fast direct methods for gaussian processes a number of problems in probability and statistics can be addressed using the multivariate normal gaussian distribution in the one dimensional case computing the probability for a given mean and variance simply requires the evaluation of the corresponding gaussian density in the n dimensional setting however it requires the inversion of an n times n covariance matrix c as well as the evaluation of its determinant det c in many cases such as regression using gaussian processes the covariance matrix is of the form c sigma 2 i k where k is computed using a specified covariance kernel which depends on the data and additional parameters hyperparameters the matrix c is typically dense causing standard direct methods for inversion and determinant evaluation to require mathcal o n 3 work this cost is prohibitive for large scale modeling here we show that for the most commonly used covariance functions the matrix c can be hierarchically factored into a product of block low rank updates of the identity matrix yielding an mathcal o n log 2 n algorithm for inversion more importantly we show that this factorization enables the evaluation of the determinant det c permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining k our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single cpu core the combination of nearly optimal scaling in terms of problem size with high performance computing resources will permit the modeling of previously intractable problems we illustrate the performance of the scheme on standard covariance kernels