incremental support vector learning for ordinal regression support vector ordinal regression svor is a popular method to tackle ordinal regression problems however until now there were no effective algorithms proposed to address incremental svor learning due to the complicated formulations of svor recently an interesting accurate on line algorithm was proposed for training nu support vector classification nu svc which can handle a quadratic formulation with a pair of equality constraints in this paper we first present a modified svor formulation based on a sum of margins strategy the formulation has multiple constraints and each constraint includes a mixture of an equality and an inequality then we extend the accurate on line nu svc algorithm to the modified formulation and propose an effective incremental svor algorithm the algorithm can handle a quadratic formulation with multiple constraints where each constraint is constituted of an equality and an inequality more importantly it tackles the conflicts between the equality and inequality constraints we also provide the finite convergence analysis for the algorithm numerical experiments on the several benchmark and real world data sets show that the incremental algorithm can converge to the optimal solution in a finite number of steps and is faster than the existing batch and incremental svor algorithms meanwhile the modified formulation has better accuracy than the existing incremental svor algorithm and is as accurate as the sum of margins based formulation of shashua and levin

lipschitz algebras and derivations of von neumann algebras let m be an abelian von neumann algebra andean abelian operator m bimodule then the domain of any ultraweakly closed derivation from m intoeis a lipschitz algebra conversely every lipschitz algebra can be realized as such a domain we discuss the relevance of this result to the theory of noncommutative metric spaces

circular free spectrahedra abstract this paper considers matrix convex sets invariant under several types of rotations it is known that matrix convex sets that are free semialgebraic are solution sets of linear matrix inequalities lmis they are called free spectrahedra we classify all free spectrahedra that are circular that is closed under multiplication by e i t up to unitary equivalence the coefficients of a minimal lmi defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal a matrix convex set is called free circular if it is closed under left multiplication by unitary matrices as a consequence of a hahn banach separation theorem for free circular matrix convex sets we show the coefficients of a minimal lmi defining a free circular free spectrahedron have up to unitary equivalence a block decomposition as above with only two blocks this paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials

matricial quantum gromov hausdorff distance abstract we develop a matricial version of rieffel s gromov hausdorff distance for compact quantum metric spaces within the setting of operator systems and unital c algebras our approach yields a metric space of isometric unital complete order isomorphism classes of metrized operator systems which in many cases exhibits the same convergence properties as those in the quantum metric setting as for example in rieffel s approximation of the sphere by matrix algebras using berezin quantization within the metric subspace of metrized unital c algebras we establish the convergence of sequences which are cauchy with respect to a larger leibniz distance and we also prove an analogue of the precompactness theorems of gromov and rieffel

equivalence of quantum metrics with a common domain abstract we characterize lipschitz morphisms between quantum compact metric spaces as those morphisms which preserve the domain of certain noncommutative analogues of lipschitz seminorms namely lower semi continuous lip norms as a corollary lower semi continuous lip norms with a shared domain are in fact equivalent we then note that when a family of lower semi continuous lip norms are uniformly equivalent then they give rise to totally bounded classes of quantum compact metric spaces and we apply this observation to several examples of perturbations of quantum metric spaces we also construct the noncommutative generalization of the lipschitz distance between quantum compact metric spaces

an improved algorithm for the solution of the regularization path of support vector machine this paper describes an improved algorithm for the numerical solution to the support vector machine svm classification problem for all values of the regularization parameter c the algorithm is motivated by the work of hastie and follows the main idea of tracking the optimality conditions of the svm solution for ascending value of c it differs from hastie s approach in that the tracked path is not assumed to be 1 d instead a multidimensional feasible space for the optimality condition is used to solve the tracking problem such a treatment allows the algorithm to properly handle data sets which hastie s approach fails these data sets are characterized by the presence of linearly dependent points in the kernel space duplicate points or nearly duplicate points such data sets are quite common among many real world data especially those with nominal features other contributions of this paper include a unifying formulation of the tracking process in the form of a linear programming problem update formula for the linear programs considerations that guard against accumulation of errors resulting from the use of incremental updates and routines to speed up the algorithm the algorithm is implemented under the matlab environment and is available for download experiments with several data sets including data set having up to several thousand data points are reported

robust joint graph sparse coding for unsupervised spectral feature selection in this paper we propose a new unsupervised spectral feature selection model by embedding a graph regularizer into the framework of joint sparse regression for preserving the local structures of data to do this we first extract the bases of training data by previous dictionary learning methods and then map original data into the basis space to generate their new representations by proposing a novel joint graph sparse coding jgsc model in jgsc we first formulate its objective function by simultaneously taking subspace learning and joint sparse regression into account then design a new optimization solution to solve the resulting objective function and further prove the convergence of the proposed solution furthermore we extend jgsc to a robust jgsc rjgsc via replacing the least square loss function with a robust loss function for achieving the same goals and also avoiding the impact of outliers finally experimental results on real data sets showed that both jgsc and rjgsc outperformed the state of the art algorithms in terms of k nearest neighbor classification performance

dimension and dynamical entropy for metrized c algebras we introduce notions of dimension and dynamical entropy for unital c algebras metrized by means of which are complex scalar versions of the lip norms constitutive of rieffel s compact quantum metric spaces our examples involve the uhf algebras and noncommutative tori in particular we show that the entropy of a noncommutative toral automorphism with respect to the canonical coincides with the topological entropy of its commutative analogue

harmonic analysis approach to gromov hausdorff convergence for noncommutative tori we show that the rotation algebras are limits of matrix algebras in a very strong sense of convergence for algebras with additional lipschitz structure our results generalize to higher dimensional noncommutative tori and operator valued coefficients in contrast to previous results by rieffel li kerr and latremoliere we use lipschitz norms induced by the carre du champ of certain natural dynamical systems including the heat semigroup

sparse representation matching for person re identification the need for recognizing people across distributed surveillance cameras leads to the growth of recent research interest in person re identification person re identification aims at matching people in non overlapping cameras at different time and locations it is a difficult pattern matching task due to significant appearance variations in pose illumination or occlusion in different camera views to address this multi view matching problem we first learn a subspace using canonical correlation analysis cca in which the goal is to maximize the correlation between data from different cameras but corresponding to the same people given a probe from one camera view we represent it using a sparse representation from a jointly learned coupled dictionary in the cca subspace the 1 induced sparse representation are regularized by an 2 regularization term the introduction of 2 regularization allows learning a sparse representation while maintaining the stability of the sparse coefficients to compute the matching scores between probe and gallery their 2 regularized sparse representations are matched using a modified cosine similarity measure experimental results with extensive comparisons on challenging datasets demonstrate that the proposed method outperforms the state of the art methods and using 2 regularized sparse representation 1 2 is more accurate compared to use a single 1 or 2 regularization term

sparse regression with output correlation for cardiac ejection fraction estimation traditional regression methods minimize the sum of errors of samples with various regularization terms such as the 1 norm and 2 norm for the diagnosis of cardiovascular diseases the cardiac ejection fraction ef represents an essential measure however existing regularization terms do not consider the output correlation the correlation between ground truth volumes and estimated volumes w r t each subject which is beneficial in estimating the cardiac ef in this paper we first propose a sparse regression with two regularization terms of the 1 norm and output correlation sroc then we propose a one dimensional solution path algorithm for quickly finding two good regulation parameters in the formulation of sroc the solution path algorithm can effectively handle singularities and infinities in the key matrix finally we conduct experiments on a clinical cardiac image dataset with 100 subjects the experimental results show that our method produces competitive and strong results for estimating the cardiac ef based on quantitative and qualitative analyses

some remarks on c convexity abstract recently the study of completely positive maps has become important to the results of brown douglas and fillmore on ext a c algebra attempts to solve questions related to ext have often turned into questions about the matrix algebras mn in this paper we wish to discuss a notion of c convexity related to completely positive linear maps to state some facts about c convexity and to ask some questions about c convexity to a large degree the tone of this paper is expository

uniqueness universality and homogeneity of the noncommutative gurarij space we realize the noncommutative gurarij space ng defined by oikhberg as the fraisse limit of the class of finite dimensional 1 exact operator spaces as a consequence we deduce that the noncommutative gurarij space is unique up to completely isometric isomorphism homogeneous and universal among separable 1 exact operator spaces we also prove that ng is the unique separable nuclear operator space with the property that the canonical triple morphism from the universal tro to the triple envelope is an isomorphism we deduce from this fact that ng does not embed completely isometrically into an exact c algebra and it is not completely isometrically isomorphic to a c algebra or to a tro we also provide a canonical construction of ng which shows that the group of surjective complete isometries of ng is universal among polish groups analog results are proved in the commutative setting and more generally for m n spaces in particular we provide a new characterization and canonical construction of the gurarij banach space

spectral truncations in noncommutative geometry and operator systems in this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces i e imposing an ultraviolet cutoff in momentum space and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution in our new approach the traditional role played by c algebras is taken over by operator systems as part of the techniques we treat c envelopes dual operator systems and stable equivalence we define a propagation number for operator systems which we show to be an invariant under stable equivalence and use to compare approximations of the same space we illustrate our methods for concrete examples obtained by spectral truncations of the circle these are operator systems of finite dimensional toeplitz matrices and their dual operator systems which are given by functions in the group algebra on the integers with support in a fixed interval it turns out that the cones of positive elements and the pure state spaces for these operator systems possess a very rich structure which we analyze including for the algebraic geometry of the boundary of the positive cone and the metric aspect i e the distance on the state space associated to the dirac operator the main property of the spectral truncation is that it keeps the isometry group intact in contrast if one considers the other finite approximation provided by circulant matrices the isometry group becomes discrete even though in this case the operator system is a c algebra we analyze this in the context of the finite fourier transform on the cyclic group the extension of noncommutative geometry to operator systems allows one to deal with metric spaces up to finite resolution by considering the relation d x y varepsilon between two points or more generally a tolerance relation which naturally gives rise to an operator system

the kirchberg wassermann operator system is unique in 1998 kirchberg and wassermann constructed a separable nuclear operator system with the property that the canonical unital homomorphism from the maximal c algebra onto the c envelope is injective we show that such an operator system is unique and completely order isomorphic to the operator system associated with the noncommutative poulsen simplex

weighted sparse coding regularized nonconvex matrix regression for robust face recognition most existing regression based classification methods for robust face recognition usually characterize the representation error using l1 norm or frobenius norm for the pixel level noise or nuclear norm for the image level noise and code the coefficients vector by l1 norm or l2 norm to our best knowledge nuclear norm can be used to describe the low rank structural information but may lead to the suboptimal solution while l1 norm or l2 norm can promote the sparsity or cooperativity but may neglect the prior information e g the locality and similarity relationship among data to solve these drawbacks we propose two weighted sparse coding regularized nonconvex matrix regression models including weighted sparse coding regularized matrix norm based matrix regression wsmr for the structural noise and weighted sparse coding regularized matrix norm plus minimax concave plus mcp function based matrix regression wsm2r for the mixed noise e g structural noise plus sparse noise the mcp induced nonconvex function can overcome the imbalanced penalization of different singular values and entries of the error image matrix and the weighted sparse coding can consider the prior information by borrowing a novel distance metric the variants of inexact augmented lagrange multiplier ialm algorithm including nonconvex ialm ncialm and majorization minimization ialm mmialm are developed to solve the proposed models respectively the matrix norm based classifier is devised for classification finally experiments on four popular face image databases can validate the superiority of our methods compared with the state of the art regression methods

compressions of compact tuples abstract we study the matrix range of a tuple of compact operators on a hilbert space and examine the notions of minimal nonsingular and fully compressed tuples in this pursuit we refine previous results by characterizing nonsingular compact tuples in terms of matrix extreme points of the matrix range further we find that a compact tuple a is fully compressed if and only if it is multiplicity free and the shilov ideal is trivial which occurs if and only if a is minimal and nonsingular fully compressed compact tuples are therefore uniquely determined up to unitary equivalence by their matrix ranges we also produce a proof of this fact which does not depend on the concept of nonsingularity

metric aspects of noncommutative homogeneous spaces for a closed cocompact subgroup of a locally compact group g given a compact abelian subgroup k of g and a homomorphism k g satisfying certain conditions landstad and raeburn constructed equivariant noncommutative deformations c g of the homogeneous space g generalizing rieffel s construction of quantum heisenberg manifolds we show that when g is a lie group and g is connected given any norm on the lie algebra of g the seminorm on c g induced by the derivation map of the canonical g action defines a compact quantum metric furthermore it is shown that this compact quantum metric space depends on continuously with respect to quantum gromov hausdorff distances

dynamical approximation entropies and topological entropy in operator algebras dynamical entropy invariants based on a general approximation approach are introduced for c and w algebra automorphisms this includes a noncommutative extension of topological entropy

on continued fraction expansion of real roots of polynomial systems complexity and condition numbers we elaborate on a correspondence between the coefficients of a multivariate polynomial represented in the bernstein basis and in a tensor monomial basis which leads to homography representations of polynomial functions that use only integer arithmetic in contrast to the bernstein basis and are feasible over unbounded regions then we study an algorithm to split this representation and obtain a subdivision scheme for the domain of multivariate polynomial functions this implies a new algorithm for real root isolation mcf that generalizes the continued fraction cf algorithm of univariate polynomials a partial extension of vincent s theorem for multivariate polynomials is presented which allows us to prove the termination of the algorithm bounding functions projection and preconditioning are employed to speed up the scheme the resulting isolation boxes have optimized rational coordinates corresponding to the first terms of the continued fraction expansion of the real roots finally we present new complexity bounds for a simplified version of the algorithm in the bit complexity model and also bounds in the real ram model for a family of subdivision algorithms in terms of the real condition number of the system examples computed with our c implementation illustrate the practical aspects of our method

matrix convex sets without absolute extreme points abstract this article shows the existence of a class of closed bounded matrix convex sets which do not have absolute extreme points the sets we consider are noncommutative sets k x formed by taking matrix convex combinations of a single tuple x in the case that x is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces k x is a closed bounded matrix convex set with no absolute extreme points a central goal in the theory of matrix convexity is to find a natural notion of an extreme point in the dimension free setting which is minimal with respect to spanning matrix extreme points are the strongest type of extreme point known to span matrix convex sets however they are not necessarily the smallest set which does so absolute extreme points a more restricted type of extreme points that are closely related to arveson s boundary enjoy a strong notion of minimality should they span this result shows that matrix convex sets may fail to be spanned by their absolute extreme points

quantum duality unbounded operators and inductive limits in this paper we investigate the inductive limits of quantum normed or operator spaces this construction allows us to treat the space of all noncommutative continuous functions over a quantum domain as a quantum or local operator space of all matrix continuous linear operators equipped with s quantum topology in particular we classify all quantizations of the polynormed topologies compatible with the given duality proposing a noncommutative arens mackey theorem further the inductive limits of operator spaces are used to introduce locally compact and locally trace class unbounded operators on a quantum domain and prove the dual realization theorem for an abstract quantum space

robust nuclear norm regularized regression for face recognition with occlusion recently regression analysis based classification methods are popular for robust face recognition these methods use a pixel based error model which assumes that errors of pixels are independent this assumption does not hold in the case of contiguous occlusion where the errors are spatially correlated furthermore these methods ignore the whole structure of the error image nuclear norm as a matrix norm can describe the structural information well based on this point we propose a nuclear norm regularized regression model and use the alternating direction method of multipliers admm to solve it we thus introduce a novel robust nuclear norm regularized regression rnr method for face recognition with occlusion compared with the existing structured sparse error coding models which perform error detection and error support separately our method integrates error detection and error support into one regression model experiments on benchmark face databases demonstrate the effectiveness and robustness of our method which outperforms state of the art methods we propose a nuclear norm regularized regression model nr and use admm to solve it we also provide the complexity analysis and convergence analysis of nr the robust nr based classification rnr is introduced for face recognition experimental results demonstrate the advantages of our method

bounds for positive roots of polynomials a class of upper bounds for the positive roots of a polynomial is discussed and it is shown that the bound 15 is nearly optimal in this class

a gromov hausdorff distance between von neumann algebras and an application to free quantum fields abstract a distance between von neumann algebras is introduced depending on a further norm inducing the w topology on bounded sets such notion is related both with the gromov hausdorff distance for quantum metric spaces of rieffel 24 and with the effros marechal topology 10 19 on the von neumann algebras acting on a hilbert space this construction is tested on the local algebras of free quantum fields endowed with norms related with the buchholz wichmann nuclearity condition 3 showing the continuity of such algebras w r t the mass parameter

on the svmpath singularity this paper proposes a novel ridge adding based approach for handling singularities that are frequently encountered in the powerful svmpath algorithm unlike the existing method that performs linear programming as an additional step to track the optimality condition path in a multidimensional feasible space our new approach provides a simpler and computationally more efficient implementation which needs no extra time consuming procedures other than introducing a random ridge term to each data point contrary to the existing ridge adding method which fails to avoid singularities as the ridge terms tend to zero our novel approach for any small random ridge terms guarantees the existence of the inverse matrix by ensuring that only one index is added into or removed from the active set the performance of the proposed algorithm in terms of both computational complexity and the ability of singularity avoidance is manifested by rigorous mathematical analyses as well as experimental results

on the complexity of real root isolation using continued fractions we present algorithmic complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers one motivation is to explain the method s good performance in practice we derive an expected complexity bound of email protected b d 6 d email protected 2 where d is the polynomial degree and t bounds the coefficient bit size using a standard bound on the expected bit size of the integers in the continued fraction expansion thus matching the current worst case complexity bound for real root isolation by exact methods sturm descartes and bernstein subdivision moreover using a homothetic transformation we improve the expected complexity bound to email protected b d email protected we compute the multiplicities within the same complexity and extend the algorithm to non square free polynomials finally we present an open source c implementation in the algebraic library synaps and illustrate its completeness and efficiency as compared to some other available software for this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000

pure matrix states on operator systems abstract an operator system is a complex matricially ordered vector space that is completely order isomorphic to a unital selfadjoint subspace of a unital c algebra a matrix state on an operator system v is a unital completely positive linear map of v into a full matrix algebra pure matrix states are studied and a new and somewhat simplified proof of a krein milman type theorem of webster and winkler is given if v is 3 dimensional then the matrix state space of v is matrix affinely homeomorphic to the matricial range of some hilbert space operator with the aid of this representation pure matrix states on 3 dimensional operator systems are examined and in some cases completely determined

piecewise linear regularized solution paths we consider the generic regularized optimization problem argmin l y x j efron hastie johnstone and tibshirani ann statist 32 2004 407 499 have shown that for the lasso that is if l is squared error loss and j 1 is the l 1 norm of the optimal coefficient path is piecewise linear that is is piecewise constant we derive a general characterization of the properties of loss l penalty j pairs which give piecewise linear coefficient paths such pairs allow for efficient generation of the full regularized coefficient paths we investigate the nature of efficient path following algorithms which arise we use our results to suggest robust versions of the lasso for regression and classification and to develop new efficient algorithms for existing problems in the literature including mammen and van de geer s locally adaptive regression splines

metrics on group c algebras and a non commutative arzela ascoli theorem on a discrete group g a length function may implement a spectral triple on the reduced group c algebra following a connes the dirac operator of the triple then can induce a metric on the state space of the reduced group c algebra recent studies by m a rieffel raise several questions with respect to such a metric on the state space we propose a relaxation in the way a length function is used in the construction of a metric and we then show that for groups of rapid decay there are many metrics related to a length function which have all the expected properties at the end we show that this notion allows a non commutative version of the arzela ascoli theorem

quantized gromov hausdorff distance abstract a quantized metric space is a matrix order unit space equipped with an operator space version of rieffel s lip norm we develop for quantized metric spaces an operator space version of quantum gromov hausdorff distance we show that two quantized metric spaces are completely isometric if and only if their quantized gromov hausdorff distance is zero we establish a completeness theorem as applications we show that a quantized metric space with 1 exact underlying matrix order unit space is a limit of matrix algebras with respect to quantized gromov hausdorff distance and that matrix algebras converge naturally to the sphere for quantized gromov hausdorff distance

matrix convexity operator analogues of the bipolar and hahn banach theorems abstract several basic results of convexity theory are generalized to the quantized matrix convex sets of wittstock these include the bipolar theorem a gauge version of the hahn banach theorem and the existence theorem for support functionals

the quantum gromov hausdorff propinquity we introduce the quantum gromov hausdorff propinquity a new distance between quantum compact metric spaces which extends the gromovhausdorff distance to noncommutative geometry and strengthens rieffel s quantum gromov hausdorff distance and rieffel s proximity by making isomorphism a necessary condition for distance zero while being well adapted to leibniz seminorms this work offers a natural solution to the long standing problem of finding a framework for the development of a theory of leibniz lip norms over c algebras

c convexity of norm unit balls abstract we characterize the norms on b h whose unit balls are c convex we call such norms m norms and investigate their dual norms named l norms we show that the class of l norms greater than a given norm enjoys a minimum element and the class of m norms less than a given norm enjoys a maximum element these minimum and maximum elements will be determined in some cases finally we give a constructive result to obtain m norms and l norms on b h

fraisse limits in functional analysis we provide a unified approach to fraisse limits in functional analysis including the gurarij space the poulsen simplex and their noncommutative analogs we obtain in this general framework many known and new results about the gurarij space and the poulsen simplex and at the same time establish their noncommutative analogs particularly we construct noncommutative analogs of universal operators in the sense of rota

a note on operator probability theory involving numerical ranges abstract in this note the relationships between the expectation and variance in operator probability theory and numerical range of operators are considered

lipschitz isometries of compact quantum metric spaces abstract a compact quantum metric space is a complete order unit space endowed with a lip norm we investigate the category morphism between compact quantum metric spaces we show that a unital positive linear map between compact quantum metric spaces with lower semicontinuous lip norms is lipschitz if and only if it preserves the lipschitz elements we characterize when a unital positive linear map on a compact quantum metric space with lower semicontinuous lip norm is a lipschitz isometry

symmetric seminorms and the leibniz property abstract we show that certain symmetric seminorms on r n satisfy the leibniz inequality as an application we obtain that l p norms of centered bounded real functions defined on probability spaces have the same property even though this is well known for the standard deviation it seems that the complete result has never been established in addition we shall connect the results with the differential calculus introduced by cipriani and sauvageot and rieffel s non commutative riemann metric

compact quantum metric spaces and ergodic actions of compact quantum groups we show that for any co amenable compact quantum group a c g there exists a unique compact hausdorff topology on the set ea g of isomorphism classes of ergodic actions of g such that the following holds for any continuous field of ergodic actions of g over a locally compact hausdorff space t the map t ea g sending each t in t to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiplicity of in each fibre is continuous over t for every equivalence class of irreducible unitary representations of g generalizations for arbitrary compact quantum groups are also obtained in the case g is a compact group the restriction of this topology on the subset of isomorphism classes of ergodic actions of full multiplicity coincides with the topology coming from the work of landstad and wassermann podle spheres are shown to be continuous in the natural parameter as ergodic actions of the quantum su 2 group we also introduce a notion of regularity for quantum metrics on g and show how to construct a quantum metric from any ergodic action of g starting from a regular quantum metric on g furthermore we introduce a quantum gromov hausdorff distance between ergodic actions of g when g is separable and show that it induces the above topology

nuclear norm based matrix regression with applications to face recognition with occlusion and illumination changes recently regression analysis has become a popular tool for face recognition most existing regression methods use the one dimensional pixel based error model which characterizes the representation error individually pixel by pixel and thus neglects the two dimensional structure of the error image we observe that occlusion and illumination changes generally lead approximately to a low rank error image in order to make use of this low rank structural information this paper presents a two dimensional image matrix based error model namely nuclear norm based matrix regression nmr for face representation and classification nmr uses the minimal nuclear norm of representation error image as a criterion and the alternating direction method of multipliers admm to calculate the regression coefficients we further develop a fast admm algorithm to solve the approximate nmr model and show it has a quadratic rate of convergence we experiment using five popular face image databases the extended yale b ar eurecom multi pie and frgc experimental results demonstrate the performance advantage of nmr over the state of the art regression based methods for face recognition in the presence of occlusion and illumination variations

a robust regularization path algorithm for nu support vector classification the nu support vector classification has the advantage of using a regularization parameter nu to control the number of support vectors and margin errors recently a regularization path algorithm for nu support vector classification nu svcpath suffers exceptions and singularities in some special cases in this brief we first present a new equivalent dual formulation for nu svc and then propose a robust nu svcpath based on lower upper decomposition with partial pivoting theoretical analysis and experimental results verify that our proposed robust regularization path algorithm can avoid the exceptions completely handle the singularities in the key matrix and fit the entire solution path in a finite number of steps experimental results also show that our proposed algorithm fits the entire solution path with fewer steps and less running time than original one does

local operator spaces unbounded operators and multinormed c algebras abstract in this paper we propose a representation theorem for local operator spaces which extends ruan s representation theorem for operator spaces based upon this result we introduce local operator systems which are locally convex versions of the operator systems and prove stinespring theorem for local operator systems a local operator c algebra is an example of a local operator system finally we investigate the injectivity in both local operator space and local operator system senses and prove locally convex version of the known result by choi and effros that an injective local operator system possesses unique multinormed c algebra structure with respect to the original involution and matrix topology

complexity of real root isolation using continued fractions in this paper we provide polynomial bounds on the worst case bit complexity of two formulations of the continued fraction algorithm in particular for a square free integer polynomial of degree n with coefficients of bit length l we show that the bit complexity of akritas formulation is o n 8l 3 and the bit complexity of a formulation by akritas and strzebonski is o n 7l 2 here o indicates that we are omitting logarithmic factors the analyses use a bound by hong to compute the floor of the smallest positive root of a polynomial which is a crucial step in the continued fraction algorithm we also propose a modification of the latter formulation that achieves a bit complexity of o n 5l 2

regularization path for nu support vector classification the v support vector classification v svc proposed by scholkopf has the advantage of using a regularization parameter v for controlling the number of support vectors and margin errors however compared to c svc its formulation is more complicated and to date there are no effective methods for computing its regularization path in this paper we propose a new regularization path algorithm which is designed on the basis of a modified formulation of v svc and traces the solution path with respect to the parameter v through theoretical analysis and confirmatory experiments we show that our algorithm can avoid the infeasible updating path under several assumptions i e assumptions 1 and 2 and fit the entire solution path in a finite number of steps when the regularization path of v svc is available a novel approach proposed by yang and ong can be applied to obtain the global optimal solution of common validation functions for v svc and the computation for the whole process is minimal numerical experiments show that it is more efficient than various kinds of grid search methods for selecting the optimal regularization parameter v

efficient positivity test algorithms for parametric and nonparametric sequences of covariance estimates in statistical signal processing problems involving second order information of data such as covariance estimation spectral factorization and optimal filtering one often needs to test positivity of a real sequence obtained from the finite length of data as covariance estimates in this correspondence we present efficient time domain algorithms for testing nonnegativity of real finite nonparametric and linearly parametric sequences as valid covariance estimates for a parametric sequence the algorithm searches entire parameter space to find a unique set of parameters for which the sequence is positive definite examples show performance of the proposed algorithms versus direct use of dft fft

improved bounds for the cf algorithm we consider the problem of isolating the real roots of a square free polynomial with integer coefficients using the classic variant of the continued fraction algorithm cf introduced by akritas we compute a lower bound on the positive real roots of univariate polynomials using an exponential search this allows us to derive a worst case bound of email protected b d email protected 2 for isolating the real roots of a polynomial with integer coefficients using the classic variant of cf where d is the degree of the polynomial and t the maximum bitsize of its coefficients this improves the previous bound of sharma by a factor of d 3 and matches the bound derived by mehlhorn and ray for another variant of cf which is combined with subdivision it also matches the worst case bound of the classical subdivision based solvers sturm descartes and bernstein

arveson extreme points span free spectrahedra let sm n mathbb r g denote g tuples of n times n real symmetric matrices given tuples x x 1 ldots x g in sm n 1 mathbb r g and y y 1 ldots y g in sm n 2 mathbb r g a matrix convex combination of x and y is a sum of the form begin aligned v 1 xv 1 v 2 y v 2 qquad qquad v 1 v 1 v 2 v 2 i n end aligned where v 1 mathbb r n rightarrow mathbb r n 1 and v 2 mathbb r n rightarrow mathbb r n 2 are contractions matrix convex sets are sets which are closed under matrix convex combinations a key feature of matrix convex combinations is that the g tuples x y and v 1 xv 1 v 2 y v 2 do not need to have the same size as a result matrix convex sets are a dimension free analog of convex sets while in the classical setting there is only one notion of an extreme point there are three main notions of extreme points for matrix convex sets ordinary matrix and absolute extreme points absolute extreme points are closely related to the classical arveson boundary a central goal in the theory of matrix convex sets is to determine if one of these types of extreme points for a matrix convex set minimally recovers the set through matrix convex combinations this article shows that every real compact matrix convex set which is defined by a linear matrix inequality is the matrix convex hull of its absolute extreme points and that the absolute extreme points are the minimal set with this property furthermore we give an algorithm which expresses a tuple as a matrix convex combination of absolute extreme points with optimal bounds similar results hold when working over the field of complex numbers rather than the reals

subspaces of c algebras abstract basic properties of matricially normed spaces are considered and a simple matrix norm characterization of the subspaces of c algebras is given the latter result is used to study the haagerup tensor products and quotients of such subspaces

every convex free basic semi algebraic set has an lmi representation the matricial solution set of a linear matrix inequality lmi is a convex free basic open semi algebraic set the main theorem of this paper is a converse each such set arises from some lmi the result has implications for semi definite programming and systems engineering as well as for free semi algebraic geometry

the problem of completeness for gromov hausdorff metrics on c algebras it is proved that the family of equivalence classes of lip normed c algebras introduced by m rieffel up to isomorphisms preserving the lip seminorm is not complete w r t the matricial quantum gromov hausdorff distance introduced by d kerr this is shown by exhibiting a cauchy sequence whose limit which always exists as an operator system is not completely order isomorphic to any c algebra conditions ensuring the existence of a c structure on the limit are considered making use of the notion of ultraproduct more precisely a necessary and sufficient condition is given for the existence on the limiting operator system of a c product structure inherited from the approximating c algebra such condition can be considered as a generalisation of the f leibniz conditions introduced by kerr and li furthermore it is shown that our condition is not necessary for the existence of a c structure tout court namely there are cases in which the limit is a c algebra but the c structure is not inherited

θ deformations as compact quantum metric spaces let m be a compact spin manifold with a smooth action of the n torus connes and landi constructed deformations m of m parameterized by n n real skew symmetric matrices the m s together with the canonical dirac operator d open image in new window on m are an isospectral deformation of m the dirac operator d defines a lipschitz seminorm on c m which defines a metric on the state space of c m we show that when m is connected this metric induces the weak topology this means that m is a compact quantum metric space in the sense of rieffel

operator positivstellensatze for noncommutative polynomials positive on matrix convex sets abstract this article studies algebraic certificates of positivity for noncommutative nc operator valued polynomials on matrix convex sets such as the solution set d l called a free hilbert spectrahedron of the linear operator inequality loi l x a 0 i j 1 g a j x j 0 where a j are self adjoint linear operators on a separable hilbert space x j matrices and i is an identity matrix if a j are matrices then l x 0 is called a linear matrix inequality lmi and d l a free spectrahedron for monic lmis i e a 0 i and nc matrix valued polynomials the certificates of positivity were established by helton klep and mccullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry since the full strength of the theory of complete positivity is not restricted to finite dimensions but works well also in the infinite dimensional setting we use it to tackle our problems first we extend the characterization of the inclusion d l 1 d l 2 from monic lmis to monic lois l 1 and l 2 as a corollary one immediately obtains the description of a polar dual of a free hilbert spectrahedron d l and its projection called a free hilbert spectrahedrop further on using this characterization in a separation argument we obtain a certificate for multivariate matrix valued nc polynomials f positive semidefinite on a free hilbert spectrahedron defined by a monic loi replacing the separation argument by an operator fejer riesz theorem enables us to extend this certificate in the univariate case to operator valued polynomials f finally focusing on the algebraic description of the equality d l 1 d l 2 we remove the assumption of boundedness from the description in the lmis case by an extended analysis however the description does not extend to lois case by counterexamples