self scaling fast rotations for stiff and equality constrained linear least squares problems abstract we present algorithms which apply self scaling fast plane rotations to the qr decomposition for stiff least squares problems we show that both fast and standard givens rotation based algorithms produce accurate results regardless of row sorting and even with extremely large weights when equality constrained least squares problems are solved by the weighting method numerical test results show that the householder qr decomposition algorithm is sensitive to row sorting and produces less accurate results when the weights are large and that the modified gram schmidt algorithm is less sensitive to row sorting this makes the fast plane rotation a method of choice for the qr decomposition of stiff matrices since it is also competitive in computational complexity based on the above results an efficient algorithm is also derived for the application where the least squares solutions are required for various constrained matrices for each fixed data matrix

blind image deconvolution through bezoutians in this paper we introduce a fast algorithm for computing the univariate gcd of several polynomials not pairwise based on the generalized bezout matrix by using barnett s method this novel approach is devoted to presenting an algorithm that permits to solve the problem of blind image deconvolution by computing greatest common divisors gcd of several polynomials specifically we design a specialized algorithm for computing the gcd of bivariate polynomials of blurred images which correspond to z transforms to recover the original image all algorithms have been implemented in matlab and experimental results with synthetically blurred images are included to illustrate the effectiveness of our approach

large deviations for finite state markov jump processes with mean field interaction via the comparison principle for an associated hamilton jacobi equation we prove the large deviation principle ldp for the trajectory of a broad class of finite state mean field interacting markov jump processes via a general analytic approach based on viscosity solutions examples include generalized ehrenfest models as well as curie weiss spin flip dynamics with singular jump rates the main step in the proof of the ldp which is of independent interest is the proof of the comparison principle for an associated collection of hamilton jacobi equations additionally we show that the ldp provides a general method to identify a lyapunov function for the associated mckean vlasov equation

analogues of non gibbsianness in joint measures of disordered mean field models it is known that the joint measures on the product of spin space and disorder space are very often non gibbsian measures for lattice systems with quenched disorder at low temperature are there reflections of this non gibbsianness in the corresponding mean field models we study the continuity properties of the conditional probabilities in finite volume of the following mean field models a joint measures of random field ising b joint measures of dilute ising c decimation of ferromagnetic ising the conditional probabilities are functions of the empirical mean of the conditionings so we look at the large volume behavior of these functions to discover non trivial limiting objects for a we find 1 discontinuous dependence for almost any realization and 2 dependence of the conditional probabilities on the phase in contrast to that we see continuous behavior for b and c for almost any realization this is in complete analogy to the behavior of the corresponding lattice models in high dimensions it shows that non gibbsian behavior which seems a genuine lattice phenomenon can be partially understood already on the level of mean field models

blind image deconvolution using a robust gcd approach in this correspondence a new viewpoint is proposed for estimating an image from its distorted versions in presence of noise without the a priori knowledge of the distortion functions in z domain the desired image can be regarded as the greatest common polynomial divisor among the distorted versions with the assumption that the distortion filters are finite impulse response fir and relatively coprime in the absence of noise this becomes a problem of taking the greatest common divisor gcd of two or more two dimensional 2 d polynomials exact gcd is not desirable because even extremely small variations due to quantization error or additive noise can destroy the integrity of the polynomial system and lead to a trivial solution our approach to this blind deconvolution approximation problem introduces a new robust interpolative 2 d gcd method based on a one dimensional 1 d sylvester type gcd algorithm experimental results with both synthetically blurred images and real motion blurred pictures show that it is computationally efficient and moderately noise robust

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

a non linear structure preserving matrix method for the low rank approximation of the sylvester resultant matrix a non linear structure preserving matrix method for the computation of a structured low rank approximation s f g of the sylvester resultant matrix s f g of two inexact polynomials f f y and g g y is considered in this paper it is shown that considerably improved results are obtained when f y and g y are processed prior to the computation of s f g and that these preprocessing operations introduce two parameters these parameters can either be held constant during the computation of s f g which leads to a linear structure preserving matrix method or they can be incremented during the computation of s f g which leads to a non linear structure preserving matrix method it is shown that the non linear method yields a better structured low rank approximation of s f g and that the assignment of f y and g y is important because s f g may be a good structured low rank approximation of s f g but s g f may be a poor structured low rank approximation of s g f because its numerical rank is not defined examples that illustrate the differences between the linear and non linear structure preserving matrix methods and the importance of the assignment of f y and g y are shown

high order terms for condition estimation of univariate polynomials the established expressions for the condition numbers of a root of a polynomial assume that the perturbations in its coefficients are small such that only the lowest order terms in a taylor series need be considered this restriction to the lowest order terms is considered in this paper and it is shown that there exists a well defined class of polynomial for which it is not valid in particular if a root x 0 of a polynomial p x is close to but distinct from its neighboring roots x i i 1 dots r it is shown that it is necessary to consider higher order terms in the taylor series these additional terms imply that a closed form expression for the condition number of x 0 cannot be obtained and numerical methods must therefore be employed in this paper a high order taylor series is used to compute the region in the complex plane in which the perturbed root x 0 delta x 0 lies and this enables the variation of the condition number of x 0 with the distances to its neighboring roots to be established it is shown that this extra information which cannot be obtained if only the lowest order terms are used is significant

relative entropy and variational properties of generalized gibbsian measures we study the relative entropy density for generalized gibbs measures we first show its existence and obtain a familiar expression in terms of entropy and relative energy for a class of almost gibbsian measures almost sure continuity of conditional probabilities for quasilocal measures we obtain a full variational principle for the joint measures of the random field ising model we show that the weak gibbs property holds with an almost surely rapidly decaying translation invariant potential for these measures we show that the variational principle fails as soon as the measures lose the almost gibbs property these examples suggest that the class of weakly gibbsian measures is too broad from the perspective of a reasonable thermodynamic formalism

the computation of multiple roots of a bernstein basis polynomial this paper describes the algorithms of musser and gauss for the computation of multiple roots of a theoretically exact bernstein basis polynomial hat f y when the coefficients of its given for

on the mean field ising model in a random external field we use a method developed by van hemmen to obtain the free energy of the mean field ising model in a random external magnetic field some results of previous mean field calculations are confirmed and generalized the tricritical point in the global phase diagram is discussed in detail we also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point

gibbs non gibbs transitions via large deviations computable examples we give new and explicitly computable examples of gibbs non gibbs transitions of mean field type using the large deviation approach introduced in van enter et al in mosc math j 10 687 711 2010 these examples include brownian motion with small variance and related diffusion processes such as the ornstein uhlenbeck process as well as birth and death processes we show for a large class of initial measures and diffusive dynamics both short time conservation of gibbsianness and dynamical gibbs non gibbs transitions

variational description of gibbs non gibbs dynamical transitions for spin flip systems with a kac type interaction we continue our study of gibbs non gibbs dynamical transitions in the present paper we consider a system of ising spins on a large discrete torus with a kac type interaction subject to an independent spin flip dynamics infinite temperature glauber dynamics we show that in accordance with the program outlined in van enter et al moscow math j 10 687 711 2010 in the thermodynamic limit gibbs non gibbs dynamical transitions are equivalent to bifurcations in the set of global minima of the large deviation rate function for the trajectories of the empirical density conditional on their endpoint more precisely the time evolved measure is non gibbs if and only if this set is not a singleton for some value of the endpoint a partial description of the possible scenarios of bifurcation is given leading to a characterization of passages from gibbs to non gibbs and vice versa with sharp transition times our analysis provides a conceptual step up from our earlier work on gibbs non gibbs dynamical transitions for the curie weiss model where the mean field interaction allowed us to focus on trajectories of the empirical magnetization rather than the empirical density

class specific subspace kernel representations and adaptive margin slack minimization for large scale classification in kernel based classification models given limited computational power and storage capacity operations over the full kernel matrix becomes prohibitive in this paper we propose a new supervised learning framework using kernel models for sequential data processing the framework is based on two components that both aim at enhancing the classification capability with a subset selection scheme the first part is a subspace projection technique in the reproducing kernel hilbert space using a class specific subspace kernel representation for kernel approximation in the second part we propose a novel structural risk minimization algorithm called the adaptive margin slack minimization to iteratively improve the classification accuracy by an adaptive data selection we motivate each part separately and then integrate them into learning frameworks for large scale data we propose two such frameworks the memory efficient sequential processing for sequential data processing and the parallelized sequential processing for distributed computing with sequential data acquisition we test our methods on several benchmark data sets and compared with the state of the art techniques to verify the validity of the proposed techniques

error analysis and implementation aspects of deferred correction for equality constrained least squares problems the deferred correction procedure recently proposed by van loan on the method of weighting for equality constrained least squares problems siam j numer anal 22 1985 pp 851 864 solves the problem of minimizing f ex 2 subject to cx g by repeatedly solving the related least squares problem of minimizing left left begin array 20 c tau delta g delta f end array right left begin array 20 c tau c e end array right delta x right 2 for some large weight tau the convergence behavior of the procedure is shown to be greatly dependent upon error analysis considerations an analysis of its convergence with an approximation factorization of left begin array 20 c tau c e end array right is given an implementation improvement to the procedure that makes it more robust is also given van loan s conjecture that the method converges for all but ill conditioned problems is shown to be true

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

training a support vector machine in the primal most literature on support vector machines svms concentrates on the dual optimization problem in this letter we point out that the primal problem can also be solved efficiently for both linear and nonlinear svms and that there is no reason for ignoring this possibility on the contrary from the primal point of view new families of algorithms for large scale svm training can be investigated

low temperature dynamics of the curie weiss model periodic orbits multiple histories and loss of gibbsianness we consider the curie weiss model at initial temperature 0 1 in vanishing external field evolving under a glauber spin flip dynamics with temperature 0 1 we study the limiting conditional probabilities and their continuity properties and discuss their set of points of discontinuity bad points we provide a complete analysis of the transition between gibbsian and non gibbsian behavior as a function of time extending earlier work for the case of independent spin flip dynamics

qr factorization method for computing the greatest common divisor of polynomials with inexact coefficients this paper presents a novel means of computing the greatest common divisor gcd of two polynomials with real valued coefficients that have been perturbed by noise the method involves the qr factorization of a near to toeplitz matrix derived from the sylvester matrix of the two polynomials it turns out that the gcd to within a constant factor is contained in the last nonzero row of the upper triangular matrix r in the qr factorization of the near to toeplitz matrix the qr factorization is efficiently performed by an algorithm due to chun et al 1987 a condition number estimator due to bischof 1990 and an algorithm for rank estimation due to zarowski 1998 are employed to account for the effects of noise

method for finding multiple roots of polynomials conventional numerical methods for finding multiple roots of polynomials are inaccurate the accuracy is unsatisfactory because the derivatives of the polynomial in the intermediate steps of the associated root finding procedures are eliminated engineering applications require that this problem be solved this work presents an easy to implement method that theoretically completely resolves the multiple root issue the proposed method adopts the euclidean algorithm to obtain the greatest common divisor gcd of a polynomial and its fast derivative the gcd may be approximate because of computational inaccuracy the multiple roots are then deflated into simple ones and then determined by conventional root finding methods the multiplicities of the roots are accordingly calculated a detailed derivation and test examples are provided to demonstrate the efficiency of this method

short time gibbsianness for infinite dimensional diffusions with space time interaction we consider a class of infinite dimensional diffusions where the interaction between the components has a finite extent both in space and time we start the system from a gibbs measure with a finite range uniformly bounded interaction under suitable conditions on the drift we prove that there exists t0 0 such that the distribution at time t t0 is a gibbs measure with absolutely summable interaction the main tool is a cluster expansion of both the initial interaction and certain time reversed girsanov factors coming from the dynamics

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

fast givens rotations for orthogonal similarity transformations fast givens rotations with half as many multiplications are proposed for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily applications are proposed and numerical results are examined for the jacobi method the reduction to hessenberg form and the qr algorithm for hessenberg matrices it can be seen that in general fast givens rotations are competitive with householder reflexions and offer distinct advantages for sparse matrices

the computation of multiple roots of a polynomial this paper considers structured matrix methods for the calculation of the theoretically exact roots of a polynomial whose coefficients are corrupted by noise and whose exact form contains multiple roots the addition of noise to the exact coefficients causes the multiple roots of the exact form of the polynomial to break up into simple roots but the algorithms presented in this paper preserve the multiplicities of the roots in particular even though the given polynomial is corrupted by noise and all computations are performed on these inexact coefficients the algorithms sew together the simple roots that originate from the same multiple root thereby preserving the multiplicities of the roots of the theoretically exact form of the polynomial the algorithms described in this paper do not require that the noise level imposed on the coefficients be known and all parameters are calculated from the given inexact coefficients examples that demonstrate the theory are presented

attractor properties for irreversible and reversible interacting particle systems we consider translation invariant interacting particle systems on the lattice with finite local state space admitting at least one gibbs measure as a time stationary measure the dynamics can be irreversible but should satisfy some mild non degeneracy conditions we prove that weak limit points of any trajectory of translation invariant measures satisfying a non nullness condition are gibbs states for the same specification as the time stationary measure this is done under the additional assumption that zero entropy loss of the limiting measure w r t the time stationary measure implies that they are gibbs measures for the same specification we show how to prove the non nullness for a large number of cases and also give an alternate version of the last condition such that the non nullness requirement can be dropped as an application we obtain the attractor property if there is a reversible gibbs measure our method generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non ergodic ones

generalizations of laguerre s method higher order methods laguerre s method is an efficient and reliable method for finding zeros of polynomials and certain other functions a new derivation and motivation of laguerre s method is given which allows it to be included in a class of methods as general as methods of order three or more based on direct generalized hermite or hyperosculatory interpolation members of this new class share with laguerre s method the property of being globally convergent to zeros of polynomials with only real zeros and have the same order of convergence at simple zeros as the classic methods based on generalized hermite interpolation methods of order 4 and 3 303 are investigated and numerical results indicate that for large 100 times 100 eigenvalue problems the method of order 3 303 is as efficient and reliable as laguerre s method

on recursive edit distance kernels with application to time series classification this paper proposes some extensions to the work on kernels dedicated to string or time series global alignment based on the aggregation of scores obtained by local alignments the extensions that we propose allow us to construct from classical recursive definition of elastic distances recursive edit distance or time warp kernels that are positive definite if some sufficient conditions are satisfied the sufficient conditions we end up with are original and weaker than those proposed in earlier works although a recursive regularizing term is required to get proof of the positive definiteness as a direct consequence of the haussler s convolution theorem furthermore the positive definiteness is maintained when a symmetric corridor is used to reduce the search space and thus the algorithmic complexity which is quadratic in the worst case the classification experiment we conducted on three classical time warp distances two of which are metrics using support vector machine classifier leads to the conclusion that when the pairwise distance matrix obtained from the training data is far from definiteness the positive definite recursive elastic kernels outperform in general the distance substituting kernels for several classical elastic distances we have tested

direct blind deconvolution blind deconvolution seeks to deblur an image without knowing the cause of the blur iterative methods are commonly applied to that problem but the iterative process is slow uncertain and often ill behaved this paper considers a significant but limited class of blurs that can be expressed as convolutions of two dimensional symmetric levy stable probability density functions this class includes and generalizes gaussian and lorentzian distributions for such blurs methods are developed that can detect the point spread function from one dimensional fourier analysis of the blurred image a separate image deblurring technique uses this detected point spread function to deblur the image each of these two steps uses direct noniterative methods and requires interactive tuning of parameters as a result blind deblurring of 512 512 images can be accomplished in minutes of cpu time on current desktop workstations numerous blind experiments on synthetic data show that for a given blurred image several di

polynomial computations for blind image deconvolution this paper considers the problem of blind image deconvolution bid when the blur arises from a spatially invariant point spread function psf h which implies that a blurred image g is formed by the convolution of h and the exact form f of g since the multiplication of two bivariate polynomials is performed by convolving their coefficient matrices the equivalence of the formation of a blurred image and the product of two bivariate polynomials implies that bid can be performed by considering f g and h to be bivariate polynomials on which polynomial operations are performed these operations allow the psf to be computed which is then deconvolved from the blurred image g thereby obtaining a deblurred image that is a good approximation of the exact image f computational results show that the deblurred image obtained using polynomial computations is better than the deblurred image obtained using other methods for blind image deconvolution

an improved non linear method for the computation of a structured low rank approximation of the sylvester resultant matrix this paper reports on improvements to recent work on the computation of a structured low rank approximation of the sylvester resultant matrix s f g of two inexact polynomials f f y and g g y specifically it has been shown in previous work that these polynomials must be processed before a structured low rank approximation of s f g is computed the existing algorithm may still however yield a structured low rank approximation of s f g but not a structured low rank approximation of s g f which is unsatisfactory moreover a structured low rank approximation of s f g must be equal to apart from permutations of its columns a structured low rank approximation of s g f but the existing algorithm does not guarantee the satisfaction of this condition this paper addresses these issues by modifying the existing algorithm such that these deficiencies are overcome examples that illustrate these improvements are shown

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

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

the computation of the degree of the greatest common divisor of three bernstein basis polynomials abstract this paper considers the application of the sylvester resultant matrix to the computation of the degree of the greatest common divisor gcd of three bernstein basis polynomials f y g y and h y it is shown that the governing equations can be written in two forms which lead to different sylvester matrices the first form requires that the polynomials be considered in pairs but different pairs of polynomials may yield different computational answers for example the solution of the computations gcd f g and gcd g h may differ from the solution of the computations gcd f g and gcd f h depending on f y g y and h y this problem does not arise when the second form is considered which requires that the three polynomials be considered simultaneously complications arise in both forms because of the combinatorial terms in the bernstein basis functions which cause the entries of the matrices to span several orders of magnitude even if the coefficients of the polynomials are of the same order of magnitude it is shown that the adverse effects of this wide range of magnitudes can be mitigated by postmultiplying both forms of the sylvester matrix by a diagonal matrix of combinatorial terms and preprocessing f y g y and h y by three operations results of gcd computations from the two forms of the sylvester matrix when f y g y and h y are perturbed by noise and with the omission and inclusion of the preprocessing operations are shown

loss without recovery of gibbsianness during diffusion of continuous spins we consider a specific continuous spin gibbs distribution t 0 for a double well potential that allows for ferromagnetic ordering we study the time evolution of this initial measure under independent diffusions

linear and nonlinear image deblurring a documented study nonlinear image deblurring procedures based on probabilistic considerations are widely believed to outperform conventional linear methods this paper is exclusively concerned with nonsmooth images such as those that occur in biomedical imaging where reconstruction of high frequency detail is of prime interest and where avoidance of a priori smoothness constraints is a major concern the theoretical basis behind each of the following nonlinear procedures is examined the lucy richardson method the maximum likelihood e m algorithm the poisson maximum a posteriori method and the nunez llacer version of the maximum entropy method a linear iterative method vancittert s iteration is also studied it is shown that each of the first three methods as well as vancittert s method lack a necessary ingredient for successful solution of the ill posed deblurring problem while in the maximum entropy method the enforced smoothness may have adverse consequences in medical imaging a direct linear method the slow evolution from the continuation boundary secb method designed specifically for nonsmooth images is also considered that method is stabilized by constraining the blurring operator as well as the solution and does not require smoothness constraints it is shown that useful error estimates can be obtained in the secb method while this is impossible in tikhonov s method without a priori bounds on derivatives of the unknown solution reconstruction experiments on low noise synthetic mri data show that thousands of iterations are necessary to achieve sufficient resolution in the iterative procedures however the secb method provides higher resolution at considerable savings in computer time at high noise levels the iterative algorithms are shown to diverge at these same noise levels the secb method produces reconstructions comparable in quality to those that would be obtained in the iterative methods were one able to terminate the divergent algorithm at that iteration which best approximates the true solution in the l1 norm

multiclass semisupervised learning based upon kernel spectral clustering this paper proposes a multiclass semisupervised learning algorithm by using kernel spectral clustering ksc as a core model a regularized ksc is formulated to estimate the class memberships of data points in a semisupervised setting using the one versus all strategy while both labeled and unlabeled data points are present in the learning process the propagation of the labels to a large amount of unlabeled data points is achieved by adding the regularization terms to the cost function of the ksc formulation in other words imposing the regularization term enforces certain desired memberships the model is then obtained by solving a linear system in the dual furthermore the optimal embedding dimension is designed for semisupervised clustering this plays a key role when one deals with a large number of clusters

continuous spin mean field models limiting kernels and gibbs properties of local transforms we extend the notion of gibbsianness for mean field systems to the setup of general possibly continuous local state spaces we investigate the gibbs properties of systems arising from an initial mean field gibbs measure by application of given local transition kernels this generalizes previous case studies made for spins taking finitely many values to the first step in the direction to a general theory containing the following parts 1 a formula for the limiting conditional probability distributions of the transformed system it holds both in the gibbs and in the non gibbs regime and invokes a minimization problem for a constrained rate function 2 a criterion for gibbsianness of the transformed system for initial lipschitz hamiltonians involving concentration properties of the transition kernels and 3 a continuity estimate for the single site conditional distributions of the transformed system while 2 and 3 have provable lattice counterparts the characterization of 1 is stronger in m

variational description of gibbs non gibbs dynamical transitions for the curie weiss model we perform a detailed study of gibbs non gibbs transitions for the curie weiss model subject to independent spin flip dynamics infinite temperature dynamics we show that in this setup the program outlined in van enter fernandez den hollander and redig 3 can be fully completed namely that gibbs non gibbs transitions are equivalent to bifurcations in the set of global minima of the large deviation rate function for the trajectories of the magnetization conditioned on their endpoint as a consequence we show that the time evolved model is non gibbs if and only if this set is not a singleton for some value of the final magnetization a detailed description of the possible scenarios of bifurcation is given leading to a full characterization of passages from gibbs to non gibbs and vice versa with sharp transition times under the dynamics gibbsianness can be lost and can be recovered our analysis expands the work of ermolaev and kulske 7 who considered zero mag netic field and finite temperature spin flip dynamics we consider both zero and non zero magnetic field but restricted to infinite temperature spin flip dynamics our results re veal an interesting dependence on the interaction parameters including the presence of forbidden regions for the optimal trajectories and the possible occurrence of overshoots and undershoots in the optimal trajectories the numerical plots provided are obtained with the help of mathematica

on the method of weighting for equality constrained least squares problems the generalized singular value decomposition is used to analyze the problem of minimizing ax b 2 subject to the constraint bx d a byproduct of the analysis is a new iterative procedure that can be used to improve an approximate solution obtained via the method of weights all that is required to implement the procedure is a single qr factorization these developments turn out to be of interest when a and b are sparse and for the case when systolic architectures are used to carry out the computations

two connections between random systems and non gibbsian measures in this contribution we discuss the role disordered or random systems have played in the study of non gibbsian measures this role has two main aspects the distinction between which has not always been fully clear 1 from disordered systems disordered systems can be used as a tool analogies with as well as results and methods from the study of random systems can be employed to investigate non gibbsian properties of a variety of measures of physical and mathematical interest 2 of disordered systems non gibbsianness is a property of various joint measures describing quenched disordered systems we discuss and review this distinction and a number of results related to these issues moreover we discuss the mean field version of the non gibbsian property and present some ideas how a kac limit approach might connect the finite range and the mean field non gibbsian properties

hamiltonian and lagrangian for the trajectory of the empirical distribution and the empirical measure of markov processes we compute the hamiltonian and lagrangian associated to the large deviations of the trajectory of the empirical distribution for independent markov processes and of the empirical measure for translation invariant interacting markov processes we treat both the case of jump processes continuous time markov chains and interacting particle systems as well as diffusion processes for diffusion processes the lagrangian is a quadratic form of the deviation of the trajectory from the solution of the kolmogorov forward equation in all cases the lagrangian can be interpreted as a relative entropy or relative entropy density per unit time

efficient svm training using low rank kernel representations svm training is a convex optimization problem which scales with the training set size rather than the feature space dimension while this is usually considered to be a desired quality in large scale problems it may cause training to be impractical the common techniques to handle this difficulty basically build a solution by solving a sequence of small scale subproblems our current effort is concentrated on the rank of the kernel matrix as a source for further enhancement of the training procedure we first show that for a low rank kernel matrix it is possible to design a better interior point method ipm in terms of storage requirements as well as computational complexity we then suggest an efficient use of a known factorization technique to approximate a given kernel matrix by a low rank matrix which in turn will be used to feed the optimizer finally we derive an upper bound on the change in the objective function value based on the approximation error and the number of active constraints support vectors this bound is general in the sense that it holds regardless of the approximation method

asymptotic results for random walks in continuous time with alternating rates we investigate some large deviation problems for a random walk in continuous time n t t ge 0 with spatially inhomogeneous rates of alternating type we first deal with the large deviation principle for the convergence of n t t to a suitable constant then the case of moderate deviations is also discussed motivated by possible applications in chemical physics context we finally obtain an asymptotic lower bound for level crossing probabilities both in the case of finite and infinite horizon

computing multiple roots of inexact polynomials we present a combination of two algorithms that accurately calculate multiple roots of general polynomials algorithm i transforms the singular root finding into a regular nonlinear least squares problem on a pejorative manifold and it calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations to fulfill the input requirement of algorithm i we develop a numerical gcd finder containing a successive singular value updating and an iterative gcd refinement as the main engine of algorithm ii that calculates the multiplicity structure and the initial root approximation while limitations exist in certain situations the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic even if the coefficients are inexact this is perhaps the first blackbox type root finder with such capabilities to measure the sensitivity of the multiple roots a structure preserving condition number is proposed and error bounds are established according to our computational experiments and error analysis a polynomial being ill conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved and its multiple roots can be computed with high accuracy

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

qr factoring to compute the gcd of univariate approximate polynomials we present a stable and practical algorithm that uses qr factors of the sylvester matrix to compute the greatest common divisor gcd of univariate approximate polynomials over spl ropf x or spl copf x an approximate polynomial is a polynomial with coefficients that are not known with certainty the algorithm of this paper improves over previously published algorithms by handling the case when common roots are near to or outside the unit circle by splitting and reversal if necessary the algorithm has been tested on thousands of examples including pairs of polynomials of up to degree 1000 and is now distributed as the program qrgcd in the snap package of maple 9

structured matrix methods for the computation of multiple roots of a polynomial this paper considers the application of structured matrix methods for the computation of multiple roots of a polynomial in particular the given polynomial f y is formed by the addition of noise to the coefficients of its exact form email protected y and the noise causes multiple roots of email protected y to break up into simple roots it is shown that structured matrix methods enable the simple roots of f y that originate from the same multiple root of email protected y to be sewn together which therefore allows the multiple roots of email protected y to be computed the algorithm that achieves these results involves several greatest common divisor computations and polynomial deconvolutions and special care is required for the implementation of these operations because they are ill posed computational examples that demonstrate the theory are included and the results are compared with the results from multroot which is a suite of matlab programs for the computation of multiple roots of a polynomial

spin flip dynamics of the curie weiss model loss of gibbsianness with possibly broken symmetry we study the conditional probabilities of the curie weiss ising model in vanishing external field under a symmetric independent stochastic spin flip dynamics and discuss their set of points of discontinuity bad points we exhibit a complete analysis of the transition between gibbsian and non gibbsian behavior as a function of time extending the results for the corresponding lattice model where only partial answers can be obtained for initial temperature beta 1 geq 1 we prove that the time evolved measure is always gibbsian for frac 2 3 leq beta 1 1 the time evolved measure loses its gibbsian character at a sharp transition time for beta 1 frac 2 3 we observe the new phenomenon of symmetry breaking in the set of points of discontinuity bad points corresponding to non zero spin average appear at a sharp transition time and give rise to biased non gibbsianness of the time evolved measure these bad points become neutral at a later transition time while the measure stays non gibbs in our proof we give a detailed description of the phase diagram of a curie weiss random field ising model with possibly non symmetric random field distribution based on bifurcation analysis

the calculation of the degree of an approximate greatest common divisor of two polynomials the calculation of the degree of an approximate greatest common divisor agcd of two inexact polynomials f y and g y is a non trivial computation because it reduces to the estimation of the rank loss of a resultant matrix r f g this computation is usually performed by placing a threshold on the small singular values of r f g but this method suffers from disadvantages because the numerical rank of r f g may not be defined or the noise level imposed on the coefficients of f y and g y may not be known or it may only be known approximately this paper addresses this problem by considering two methods for estimating the degree of an agcd of f y and g y such that knowledge of the noise level is not required the first method involves the calculation of the smallest angle between two subspaces that are apparent from the structure of the sylvester resultant matrix s f g and the second method uses the theory of subresultant matrices which are derived from s f g by the deletion of some of its rows and columns the two methods are compared computationally on non trivial polynomials

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

a non linear structure preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two bernstein polynomials this paper describes a non linear structure preserving matrix method for the computation of the coefficients of an approximate greatest common divisor agcd of degree t of two bernstein polynomials f y and g y this method is applied to a modified form st f g qt of the tth subresultant matrix st f g of the sylvester resultant matrix s f g of f y and g y where qt is a diagonal matrix of combinatorial terms this modified subresultant matrix has significant computational advantages with respect to the standard subresultant matrix st f g and it yields better results for agcd computations it is shown that f y and g y must be processed by three operations before st f g qt is formed and the consequence of these operations is the introduction of two parameters and such that the entries of st f g qt are non linear functions of and the coefficients of f y and g y the values of and are optimised and it is shown that these optimal values allow an agcd that has a small error and a structured low rank approximation of s f g to be computed an approximate greatest common divisor of two bernstein polynomials is computed an approximate polynomial factorisation and the sylvester matrix are used the results from the two methods are very similar

soft margin multiple kernel learning multiple kernel learning mkl has been proposed for kernel methods by learning the optimal kernel from a set of predefined base kernels however the traditional l 1 rm mkl method often achieves worse results than the simplest method using the average of base kernels i e average kernel in some practical applications in order to improve the effectiveness of mkl this paper presents a novel soft margin perspective for mkl specifically we introduce an additional slack variable called kernel slack variable to each quadratic constraint of mkl which corresponds to one support vector machine model using a single base kernel we first show that l 1 rm mkl can be deemed as hard margin mkl and then we propose a novel soft margin framework for mkl three commonly used loss functions including the hinge loss the square hinge loss and the square loss can be readily incorporated into this framework leading to the new soft margin mkl objective functions many existing mkl methods can be shown as special cases under our soft margin framework for example the hinge loss soft margin mkl leads to a new box constraint for kernel combination coefficients using different hyper parameter values for this formulation we can inherently bridge the method using average kernel l 1 rm mkl and the hinge loss soft margin mkl the square hinge loss soft margin mkl unifies the family of elastic net constraint regularizer based approaches and the square loss soft margin mkl incorporates l 2 rm mkl naturally moreover we also develop efficient algorithms for solving both the hinge loss and square hinge loss soft margin mkl comprehensive experimental studies for various mkl algorithms on several benchmark data sets and two real world applications including video action recognition and event recognition demonstrate that our proposed algorithms can efficiently achieve an effective yet sparse solution for mkl

multiple kernel learning algorithms in recent years several methods have been proposed to combine multiple kernels instead of using a single one these different kernels may correspond to using different notions of similarity or may be using information coming from multiple sources different representations or different feature subsets in trying to organize and highlight the similarities and differences between them we give a taxonomy of and review several multiple kernel learning algorithms we perform experiments on real data sets for better illustration and comparison of existing algorithms we see that though there may not be large differences in terms of accuracy there is difference between them in complexity as given by the number of stored support vectors the sparsity of the solution as given by the number of used kernels and training time complexity we see that overall using multiple kernels instead of a single one is useful and believe that combining kernels in a nonlinear or data dependent way seems more promising than linear combination in fusing information provided by simple linear kernels whereas linear methods are more reasonable when combining complex gaussian kernels