branching index sum rules for simple lie algebras let l and l0 be a simple lie algebra and its sub lie algebra respectively then a given irreducible representation of l decomposes into a direct sum of irreducible components of l0 which is called the branching rule the general dynkin indices introduced earlier satisfy many sum rules for the branching rule these are found to be strong enough to uniquely determine the branching rule for many cases we have studied the sum rules are especially useful for cases of exceptional lie algebras

inverse branching rules a method is given for inverting branching rules which is based on a technique for computing kronecker products due to king 1981 this approach simplifies the calculation of plethysms and branching rules of representations of exceptional groups it may also be applied to basic spin plethysms

an extension of chebfun to two dimensions an object oriented matlab system is described that extends the capabilities of chebfun to smooth functions of two variables defined on rectangles functions are approximated to essentially machine precision by using iterative gaussian elimination with complete pivoting to form chebfun2 objects representing low rank approximations operations such as integration differentiation function evaluation and transforms are particularly efficient global optimization the singular value decomposition and rootfinding are also extended to chebfun2 objects numerical applications are presented

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

non gaussian random walks the authors present an explicit expression for the probability distribution for the position of a continuous time random walker in an arbitrary number of dimensions when the interjump density has a long time tail in contrast to earlier results which require numerical inversion of a fourier integral they replace this numerical procedure by one that relies on the method of steepest descents their results are applied to diffusion on a comb and on a percolation cluster generated on a cayley tree at criticality and are confirmed numerically

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

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

generating functions and elementary young tableaux when constructing polynomial bases for group representations one is led naturally to consider a stretched product of states elementary multiplets are states that cannot be expressed as the stretched product of any other two states and they can be identified with terms occurring in certain generating functions in this paper the authors define an analogous stretched product of young tableaux together with elementary tableaux these may be used in a manner similar to elementary multiplets to construct generating functions they consider several examples where this method can be applied and in particular give new large n rank free branching rule generating functions for the group subgroup pairs su n su m contained in implied by su n m first four labels non zero so n contained in implied by su n first three labels non zero and sp 2n contained in implied by su 2n first five labels non zero this method of finding these generating functions uses the theory of grobner bases which allows the authors to make precise the notion of compatibility relations or forbidden products of elementary tableaux

a topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon we show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium this result is directly applicable to mechanical systems having rotational degrees of freedom in particular the result applies to the attitude motion of a rigid body in light of this result we explain how attitude stabilizing controllers obtained using local coordinates lead to unwinding instead of global asymptotic stability

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

extended poincare supersymmetry rotation groups and branching rules the decomposition of the basic spin irreps of the special orthogonal group so2k into irreps of sod 2 k where d is the spacetime dimension of the extended d dimensional poincare supersymmetry and k is the appropriate automorphism group is determined general results for d 10 several new branching rules for subgroups of so2k are developed a number of specific results are tabulated

generalizing the column row matrix decomposition to multi way arrays abstract in this paper we provide two generalizations of the cur matrix decomposition y cur also known as pseudo skeleton approximation method 1 to the case of n way arrays tensors these generalizations which we called fiber sampling tensor decomposition types 1 and 2 fstd1 and fstd2 provide explicit formulas for the parameters of a rank r r r tucker representation the core tensor of size r r r and the matrix factors of sizes i n r n 1 2 n based only on some selected entries of the original tensor fstd1 uses p n 1 p r n mode fibers of the original tensor while fstd2 uses exactly r fibers in each mode as matrix factors as suggested by the existence theorem provided in oseledets et al 2008 2 with a core tensor defined in terms of the entries of a subtensor of size r r r for n 2 our results are reduced to the already known cur matrix decomposition where the core matrix is defined as the inverse of the intersection submatrix i e u w 1 additionally we provide an adaptive type algorithm for the selection of proper fibers in the fstd1 model which is useful for large scale applications several numerical results are presented showing the performance of our fstd1 adaptive algorithm compared to two recently proposed approximation methods for 3 way tensors

spectrum generating functions for strings and superstrings the spectrum generating functions for a variety of strings and superstrings are derived these are expressed in terms of characters of irreducible representations of the appropriate transverse spacetime symmetry group and in the case of heterotic strings the relevant gauge group e8 e8 or spin 32 z2 the expansions involve principal specialisations of characters of both gl n and gl m n in the limit as m and n tend to infinity by using schur function methods modification rules and various theta function identities a number of spectrum generating functions are derived and tabulated the equivalence of the gso projected neveu schwarz ramond superstring and the green schwarz superstring is proved and the connection between the lattice formulation and the gso projected neveu schwarz ramond formulation of heterotic strings is spelled out

fast directional multilevel algorithms for oscillatory kernels this paper introduces a new directional multilevel algorithm for solving n body or n point problems with highly oscillatory kernels these systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non uniformity of the particle distribution we address the problem by first proving that the interaction between a ball of radius r and a well separated region has an approximate low rank representation as long as the well separated region belongs to a cone with a spanning angle of o 1 r and is at a distance which is at least o r 2 away from from the ball we then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated low rank representation based on the resulting representations our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency the algorithm performs well on a large group of highly oscillatory kernels our algorithm is proved to have o n log n computational complexity for any given accuracy when the points are sampled from a two dimensional surface we also provide numerical results to demonstrate these properties

symmetrised powers of rotation group representations the results given permit the unambiguous evaluation of all possible kronecker products of the irreducible representations tensor and spinor of on and son for n 2 nu and n 2 nu 1 a complete resolution of the second and third powers of the basic spinor representations of so2 nu and so2 nu 1 is given together with a prescription for analysing the fourth power of these representations detailed application is made to the enumeration of properties of so10 relevant to grand unified theories and sufficient information given to resolve the fourth power of any representation of so10

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

branching rules and even dimensional rotation groups so2k unambiguous methods are developed for calculating branching rules for the classical subgroups of the even dimensional rotation group so2k complete results are given for the subgroups suk u1 so2k 2 u1 so2p so2q and so2p 1 so2q 1 a number of examples relevant to problems in supergravity and unification theories are given a complete resolution of the antisymmetric powers of the basic spinor irrep of so10 is given and the results extended to so11

technical communique retraction obstruction to time varying stabilization this paper addresses the problem of the global stabilization on a total space of a fiber bundle with a compact base space we prove that under mild assumptions existence of a continuous section and forward unicity of solutions no equilibrium of a continuous system defined on such a state space can be globally asymptotically uniformly stabilized using continuous time varying feedback

etale backstepping for control lyapunov function design on manifold abstract backstepping is one of the most popular nonlinear controller and differentiable control lyapunov function clf design techniques however for asymptotic stabilization of systems defined on noncontractible manifolds there exists no differentiable clf the semiconcave clf design problem based on the backstepping has not been discussed in this paper we propose a backstepping based controller design method for asymptotic stabilization of systems defined on noncontractible manifolds in the method we design a controller and a clf on an etale space then we obtain a semiconcave clf on the original space by the minimum projection method the effectiveness of the proposed method is confirmed by computer simulation

explicit design of time varying stabilizing control laws for a class of controllable systems without drift abstract this paper gives a systematic way to design time variant feedback control laws for a class of controllable nonlinear systems this class contains a lot of systems which cannot be stabilized via a time invariant feedback control law the interest of this work lies in the design method since a general existence results is already available the techniques employed here are basic they mainly involve classical lyapunov analysis

branching rules for e8 so16 branching rules for e8 down arrow so16 are derived from those of e8 down arrow su2 e7 down arrow su8 and so16 down arrow u1 su8 by noting that both group chains share a common u1 su8 subgroup additional branching rules are deduced from kronecker products of e8 and so16 irreps in each case the rules distinguish unambiguously between conjugate irreps of so16 an alternative labelling scheme for the irreps of so2k based on its maximal u1 suk subgroup is outlined

a fast algorithm for fourier continuation a new algorithm is presented which provides a fast method for the computation of recently developed fourier continuations a particular type of fourier extension method that yield superalgebraically convergent fourier series approximations of nonperiodic functions previously the coefficients of an approximating fourier series have been obtained by means of a regularized singular value decomposition svd based least squares solution to an overdetermined linear system of equations these svd methods are effective when the size of the system does not become too large but they quickly become unwieldy as the number of unknowns in the system grows we demonstrate a novel decoupling of the least squares problem which results in two systems of equations one of which may be solved quickly by means of fast fourier transforms ffts and another that is demonstrated to be well approximated by a low rank system utilizing randomized algorithms the low rank system is reduced to a significantly smaller system of equations this new system is then efficiently solved with drastically reduced computational cost and memory requirements while still benefiting from the advantages of using a regularized svd the computational cost of the new algorithm in on the order of the cost of a single fft multiplied by a slowly increasing factor that grows only logarithmically with the size of the system

low rank incremental methods for computing dominant singular subspaces computing the singular values and vectors of a matrix is a crucial kernel in numerous scientific and industrial applications as such numerous methods have been proposed to handle this problem in a computationally efficient way this paper considers a family of methods for incrementally computing the dominant svd of a large matrix a specifically we describe a unification of a number of previously independent methods for approximating the dominant svd after a single pass through a we connect the behavior of these methods to that of a class of optimization based iterative eigensolvers on ata an iterative procedure is proposed which allows the computation of an accurate dominant svd using multiple passes through a we present an analysis of the convergence of this iteration and provide empirical demonstration of the proposed method on both synthetic and benchmark data

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

natural labelling schemes for simple roots and irreducible representations of exceptional lie algebras the simple root system of each exceptional simple lie algebra is explicitly constructed in a variety of forms each construction is based on the natural embedding in the exceptional algebra of a classical semi simple algebra of the same rank the procedure adopted leads to the discovery of a number of new chains of subalgebras illustrated by means of supplemented dynkin diagrams the various sets of simple roots are then used to determine natural labels for the irreducible representations of each of the exceptional algebras for each such labelling scheme the modification rules for dealing with non standard representation labels are tabulated the connection between the natural labels and dynkin labels is given in detail and a comparison is made with the labels of b g wybourne and m j bowick 1977

positive semidefinite metric learning using boosting like algorithms the success of many machine learning and pattern recognition methods relies heavily upon the identification of an appropriate distance metric on the input data it is often beneficial to learn such a metric from the input training data instead of using a default one such as the euclidean distance in this work we propose a boosting based technique termed boostmetric for learning a quadratic mahalanobis distance metric learning a valid mahalanobis distance metric requires enforcing the constraint that the matrix parameter to the metric remains positive semidefinite semidefinite programming is often used to enforce this constraint but does not scale well and is not easy to implement boostmetric is instead based on the observation that any positive semidefinite matrix can be decomposed into a linear combination of trace one rank one matrices boostmetric thus uses rank one positive semidefinite matrices as weak learners within an efficient and scalable boosting based learning process the resulting methods are easy to implement efficient and can accommodate various types of constraints we extend traditional boosting algorithms in that its weak learner is a positive semidefinite matrix with trace and rank being one rather than a classifier or regressor experiments on various data sets demonstrate that the proposed algorithms compare favorably to those state of the art methods in terms of classification accuracy and running time

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

fast construction of hierarchical matrix representation from matrix vector multiplication we develop a hierarchical matrix construction algorithm using matrix vector multiplications based on the randomized singular value decomposition of low rank matrices the algorithm uses o logn applications of the matrix on structured random test vectors and o nlogn extra computational cost where n is the dimension of the unknown matrix numerical examples on constructing green s functions for elliptic operators in two dimensions show efficiency and accuracy of the proposed algorithm

the weyl groups and weight multiplicities of the exceptional lie groups the weyl group wg of each exceptional simple lie group g is described in detail its structure is defined in terms of its coset decomposition with respect to the weyl group wh of a classical semi simple lie group h embedded naturally in g the concepts of g dominance and g equivalence are defined and used to determine from the character formula of weyl the branching rule associated with the restriction of group elements from g to h the weyl group wg is used further to impose constraints on both the branching multiplicities for g to h and the weight multiplicities of g these constraints are used to evaluate the weight multiplicities of f4 e6 e7 and e8 together with the branching multiplicities for e8 to so 16

stabilization by artificial wiener processes in this technical note we restage the positive effects of gaussian white noises for continuous time driftless input affine nonholonomic systems compared with previous studies we found that the positive effects are hidden in the shadows of nonuniqueness of stochastic integrals over multi dimensional wiener processes furthermore we found that the positive effects are artificially restaged by numerical calculations with use of euler scheme for ordinary differential equations our results demonstrate that the positive effects act as hidden independent control inputs for nonholonomic systems this implies that our results provide a constructive procedure for designing control inputs such that the origins become p th moment exponentially or finite time stable

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

indexing hierarchical structures using graph spectra hierarchical image structures are abundant in computer vision and have been used to encode part structure scale spaces and a variety of multiresolution features in this paper we describe a framework for indexing such representations that embeds the topological structure of a directed acyclic graph dag into a low dimensional vector space based on a novel spectral characterization of a dag this topological signature allows us to efficiently retrieve a promising set of candidates from a database of models using a simple nearest neighbor search we establish the insensitivity of the signature to minor perturbation of graph structure due to noise occlusion or node split merge to accommodate large scale occlusion the dag rooted at each nonleaf node of the query votes for model objects that share that part effectively accumulating local evidence in a model dag s topological subspaces we demonstrate the approach with a series of indexing experiments in the domain of view based 3d object recognition using shock graphs

kronecker products for compact semisimple lie groups a review is given of the application of s function techniques to the evaluation of kronecker products of irreducible representations of compact semisimple lie groups explicit formulae are derived for all irreducible representations of all such groups recent developments involving composite young diagrams are brought to fruition and the vexed problem of so2k is dealt with completely new branching rules for the classical groups are given in an appendix these are exploited in the evaluation of kronecker products by means of a technique which is applied to both classical and exceptional groups a discussion is made of various modification rules which are needed to express the final results in standard form

low rank approximation and decomposition of large matrices using error correcting codes low rank approximation is an important tool used in many applications of signal processing and machine learning recently randomized sketching algorithms were proposed to effectively construct low rank approximations and obtain approximate singular value decompositions of large matrices similar ideas were used to solve least squares regression problems in this paper we show how matrices from error correcting codes can be used to find such low rank approximations and matrix decompositions and extend the framework to linear least squares regression problems the benefits of using these code matrices are the following first they are easy to generate and they reduce randomness significantly second code matrices with mild restrictions satisfy the subspace embedding property and have a better chance of preserving the geometry of an large subspace of vectors third for parallel and distributed applications code matrices have significant advantages over structured random matrices and gaussian random matrices fourth unlike fourier or hadamard transform matrices which require sampling o k log k columns for a rank k approximation the log factor is not necessary for certain types of code matrices in particular 1 epsilon optimal frobenius norm error can be achieved for a rank k approximation with o k epsilon samples fifth fast multiplication is possible with structured code matrices so fast approximations can be achieved for general dense input matrices sixth for least squares regression problem min ax b 2 where a in mathbb r n times d the 1 epsilon relative error approximation can be achieved with o d epsilon samples with high probability when certain code matrices are used

on the topological structure of attraction basins for differential inclusions abstract we show that when a compact set is globally asymptotically stable under the action of a differential inclusion satisfying certain regularity properties there exists a smooth differential equation rendering the same compact set globally asymptotically stable the regularity properties assumed in this work stem from the consideration of krasovskii filippov solutions to discontinuous differential equations and the robustness of asymptotic stability under perturbation in particular the results in this work show that when a compact set cannot be globally asymptotically stabilized by continuous feedback due to topological obstructions it cannot be robustly globally asymptotically stabilized by discontinuous feedback either the results follow from converse lyapunov theory and parallel what is known for the local stabilization problem

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

conley index condition for asymptotic stability in this paper we use conley index theory to develop necessary conditions for stability of equilibrium and periodic solutions of nonlinear continuous time systems the conley index is a topological generalization of the morse theory which has been developed to analyze dynamical systems using topological methods in particular the conley index of an invariant set with respect to a dynamical system is defined as the relative homology of an index pair for the invariant set the conley index can then be used to examine the structure of the system invariant set as well as the system dynamics within the invariant set including system stability properties efficient numerical algorithms using homology theory have been developed in the literature to compute the conley index and can be used to deduce the stability properties of nonlinear dynamical systems

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

quasioptimality of maximum volume cross interpolation of tensors we consider a cross interpolation of high dimensional arrays in the tensor train format we prove that the maximum volume choice of the interpolation sets provides the quasioptimal interpolation accuracy that differs from the best possible accuracy by the factor which does not grow exponentially with dimension for nested interpolation sets we prove the interpolation property and propose greedy cross interpolation algorithms we justify the theoretical results and measure speed and accuracy of the proposed algorithm with numerical experiments

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

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

four algorithms for the the efficient computation of truncated pivoted qr approximations to a sparse matrix in this paper we propose four algorithms to compute truncated pivoted qr approximations to a sparse matrix three are based on the gram schmidt algorithm and the other on householder triangularization all four algorithms leave the original matrix unchanged and the only additional storage requirements are arrays to contain the factorization itself thus the algorithms are particularly suited to determining low rank approximations to a sparse matrix

minimum projection method for nonsmooth control lyapunov function design on general manifolds abstract asymptotic stabilization on noncontractible manifolds is known as a difficult control problem on the other hand an important fact is every control system that is globally asymptotically stabilizable at a desired equilibrium must have nonsmooth control lyapunov functions this paper considers the problem of construction of nonsmooth control lyapunov functions on general manifolds and we propose a nonsmooth control lyapunov function design method called the minimum projection method the proposed method considers a simple structured smooth manifold associated with the original manifold by a surjective immersion and then a control lyapunov function defined on the simple structured manifold is projected to the original manifold a function on the original manifold is thus obtained in this paper we prove that the control system on another manifold associated with a surjective immersion is determined uniquely and the resulting function by the proposed method is a nonsmooth control lyapunov function on the original manifold the effectiveness of the proposed method is confirmed by examples

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

convergence analysis of markov chain monte carlo linear solvers using ulam von neumann algorithm the convergence of markov chain based monte carlo linear solvers using the ulam von neumann algorithm for a linear system of the form x hx b is investigated in this paper we analyze the convergence of the monte carlo solver based on the original ulam von neumann algorithm under the conditions that h 1 as well as rho h 1 where rho h is the spectral radius of h we find that although the monte carlo solver is based on sampling the neumann series the convergence of neumann series is not a sufficient condition for the convergence of the monte carlo solver actually properties of h are not the only factors determining the convergence of the monte carlo solver the underlying transition probability matrix plays an important role an improper selection of the transition matrix may result in divergence even though the condition h 1 holds however if the condition h 1 is satisfied we show that there always exist certain transition matrices that guarantee convergence of

an o nlogn hierarchical random compression method for kernel matrices by sampling partial matrix entries abstract in this paper we propose an o n log n hierarchical random compression method hrcm for kernel matrix compressing which only requires sampling o n log n entries of a matrix the hrcm combines the hierarchical framework of the h matrix and a randomized sampling technique of column and row spaces for far field interaction kernel matrix we show that a uniform column row sampling of a far field kernel matrix thus without the need and associated cost to pre compute a costly sampling distribution will give a low rank compression of such low rank matrix independent of the matrix size and only dependent on the separation of the source and target locations this far field random compression technique is then implemented at each level of the hierarchical decomposition for general kernel matrices resulting in an o n log n random compression method error and complexity analysis for the hrcm are included numerical results for electrostatic and low frequency helmholtz wave kernels have validated the efficiency and accuracy of the proposed method in comparison of direct o n 2 summations