bounds for effective parameters of multicomponent media by analytic continuation recently d bergman introduced a method for obtaining bounds for the effective dielectric constant or conductivity of a two component medium this method does not rely on a variational principle but instead exploits the properties of the effective parameter as an analytic function of the ratio of the component parameters we extend the method to multicomponent media using techniques of several complex variables

the cavity of the optimal shape under the shear stresses the problem of optimal shape of a single cavity in an infinite 2 d elastic domain is analyzed an elastic plane is subjected to a uniform load at infinity the cavity of the fixed area is said to be optimal if it provides the minimal energy change between the homogeneous plane and the plane with the cavity we show that for the case of shear loading the contour of the optimal cavity is not smooth but is shaped as a curved quadrilateral the shape is specified in terms of conformal mapping coefficients and explicit analytical representations for components of the dipole tensor associated with the cavity are employed we also find the exact values of angles at the corners of the optimal contour the applications include the problems of optimal design for dilute composites

structured multicategory support vector machines with analysis of variance decomposition the support vector machine has been a popular choice of classification method for many applications in machine learning while it often outperforms other methods in terms of classification accuracy the implicit nature of its solution renders the support vector machine less attractive in providing insights into the relationship between covariates and classes use of structured kernels can remedy the drawback borrowing the flexible model building idea of functional analysis of variance decomposition we consider multicategory support vector machines with analysis of variance kernels in this paper an additional penalty is imposed on the sum of weights of functional subspaces which encourages a sparse representation of the solution incorporation of the additional penalty enhances the interpretability of a resulting classifier with often improved accuracy the proposed method is demonstrated through simulation studies and an application to real data

statistical analysis of some multi category large margin classification methods the purpose of this paper is to investigate statistical properties of risk minimization based multi category classification methods these methods can be considered as natural extensions of binary large margin classification we establish conditions that guarantee the consistency of classifiers obtained in the risk minimization framework with respect to the classification error examples are provided for four specific forms of the general formulation which extend a number of known methods using these examples we show that some risk minimization formulations can also be used to obtain conditional probability estimates for the underlying problem such conditional probability information can be useful for statistical inferencing tasks beyond classification

optimal bounds on the effective bulk modulus of polycrystals the classical problem of bounding the effective bulk modulus of isotropic polycrystalline aggregates built from a single anisotropic phase is studied for three dimensional composites we find that the voigt and reuss bounds are each attained by multi layered laminate geometries and hence are optimal for two dimensional composites the reuss lower bound is proved to be optimal while a new optimal upper bound is derived

the entire regularization path for the support vector machine the support vector machine svm is a widely used tool for classification many efficient implementations exist for fitting a two class svm model the user has to supply values for the tuning parameters the regularization cost parameter and the kernel parameters it seems a common practice is to use a default value for the cost parameter often leading to the least restrictive model in this paper we argue that the choice of the cost parameter can be critical we then derive an algorithm that can fit the entire path of svm solutions for every value of the cost parameter with essentially the same computational cost as fitting one svm model we illustrate our algorithm on some examples and use our representation to give further insight into the range of svm solutions

multicategory support vector machines two category support vector machines svm have been very popular in the machine learning community for classification problems solving multicategory problems by a series of binary classifiers is quite common in the svm paradigm however this approach may fail under various circumstances we propose the multicategory support vector machine msvm which extends the binary svm to the multicategory case and has good theoretical properties the proposed method provides a unifying framework when there are either equal or unequal misclassification costs as a tuning criterion for the msvm an approximate leave one out cross validation function called generalized approximate cross validation is derived analogous to the binary case the effectiveness of the msvm is demonstrated through the applications to cancer classification using microarray data and cloud classification with satellite radiance profiles

from microstructure independent formulas for composite materials to rank one convex non quasiconvex functions examples of non quasiconvex functions that are rank one convex are rare in this paper we construct a family of such functions by means of the algebraic methods of the theory of exact relations for polycrystalline composite materials developed to identify g closed sets of positive codimensions the algebraic methods are used to construct a set of materials of positive codimension that is closed under lamination but is not g closed the well known link between g closed sets and quasiconvex functions and sets closed under lamination and rank one convex functions is then used to construct a family of rotationally invariant nonnegative and 2 homogeneous rank one convex functions that are not quasiconvex

multicomponent composites electrical networks and new types of continued fraction i the development of bounds on the complex effective conductivity tensor that relates the average current to the average electric field in a multicomponent composite has been hindered by lack of a suitable continued fraction representation for here a new field equation recursion method is developed which gives an expression for as a continued fraction of a novel form incorporating as coefficients the component conductivities and a set of fundamental geometric parameters reflecting the composite geometry a hierarchy of field equations is set up such that the solutions of the j 1 th order equation generate the solutions of thejth order equation consequently the effective tensor j associated with thejth order field equation is expressible as a fractional linear matrix transformation of j 1 these transformations combine to form the continued fraction expansion for 0 which is exploited in the following paper part ii to obtain bounds crude bounds on j forj 1 give narrow bounds on the continued fraction is a generalization to multivariate functions of the continued fraction expansion of single variable stieltjes functions that proved important in the development of the theory of pade approximants asymptotic analysis and the theory of orthogonal polynomials in the last century the results extend to other transport problems including conduction in polycrystalline media the viscoelasticity of composites and the response of multicomponent multiterminal linear electrical networks

multicategory ψ learning in binary classification margin based techniques usually deliver high performance as a result a multicategory problem is often treated as a sequence of binary classifications in the absence of a dominating class this treatment may be suboptimal and may yield poor performance such as for support vector machines svms we propose a novel multicategory generalization of learning that treats all classes simultaneously the new generalization eliminates this potential problem while at the same time retaining the desirable properties of its binary counterpart we develop a statistical learning theory for the proposed methodology and obtain fast convergence rates for both linear and nonlinear learning examples we demonstrate the operational characteristics of this method through a simulation our results indicate that the proposed methodology can deliver accurate class prediction and is more robust against extreme observations than its svm counterpart

on reject and refine options in multicategory classification abstractin many real applications of statistical learning a decision made from misclassification can be too costly to afford in this case a reject option which defers the decision until further investigation is conducted is often preferred in recent years there has been much development for binary classification with a reject option yet little progress has been made for the multicategory case in this article we propose margin based multicategory classification methods with a reject option in addition and more importantly we introduce a new and unique refine option for the multicategory problem where the class of an observation is predicted to be from a set of class labels whose cardinality is not necessarily one the main advantage of both options lies in their capacity of identifying error prone observations moreover the refine option can provide more constructive information for classification by effectively ruling out implausible classes efficient implementations have been developed

a scalar inequality which bounds the effective conductivity of composites new bounds on the eigenvalues of the effective conductivity tensor in spatial dimension d 3 are proved using the compensated compactness method the inequalities are obtained using the nine quadratic null lagrangians given by the 2 times 2 minors of a jacobian matrix partial phi i partial x j it is proved that in general the bounds depend at most on four parameters

entropic formulation of statistical mechanics we present an alternative formulation of equilibrium statistical mechanics which follows the method based on the maximum statistical entropy principle in information theory combined with the use of massieu planck functions the different statistical ensembles are obtained by a suitable restriction of the whole set of available microstates the main advantage is that all of the equations that relate the average values with derivatives of the partition function are formally identical in the different ensembles moreover einstein s fluctuation formula is also derived within the same framework this provides a suitable starting point for the calculation of fluctuations of extensive and intensive variables in any statistical ensemble

a new and fast implementation for null space based linear discriminant analysis in this paper we present a new implementation for the null space based linear discriminant analysis the main features of our implementation include i the optimal transformation matrix is obtained easily by only orthogonal transformations without computing any eigendecomposition and singular value decomposition svd consequently our new implementation is eigendecomposition free and svd free ii its main computational complexity is from a economic qr factorization of the data matrix and a economic qr factorization of a nxn matrix with column pivoting here n is the sample size thus our new implementation is a fast one the effectiveness of our new implementation is demonstrated by some real world data sets

variational principles for complex conductivity viscoelasticity and similar problems in media with complex moduli linear processes in media with dissipation arising in conductivity optics viscoelasticity etc are considered time periodic fields in such media are described by linear differential equations for complex valued potentials the properties of the media are characterized by complex valued tensors for example by complex conductivity or complex elasticity tensors variational formulations are suggested for such problems the functionals whose euler equations coincide with the original ones are constructed four equivalent variational principles are obtained two minimax and two minimal ones the functionals of the obtained minimal variational principles are proportional to the energy dissipation averaged over the period of oscillation the last principles can be used in the homogenization theory to obtain the bounds on the effective properties of composite materials with complex valued properties tensors

convergence rate of sieve estimates in this paper we develop a general theory for the convergence rate of sieve estimates maximum likelihood estimates mle s and related estimates obtained by optimizing certain empirical criteria in general parameter spaces in many cases especially when the parameter space is infinite dimensional maximization over the whole parameter space is undesirable in such cases one has to perform maximization over an approximating space sieve of the original parameter space and allow the size of the approximating space to grow as the sample size increases this method is called the method of sieves in the case of the maximum likelihood estimation an mle based on a sieve is called a sieve mle we found that the convergence rate of a sieve estimate is governed by a the local expected values variances and l 2 entropy of the criterion differences and b the approximation error of the sieve a robust nonparametric regression problem a mixture problem and a nonparametric regression problem are discussed as illustrations of the theory we also found that when the underlying space is too large the estimate based on optimizing over the whole parameter space may not achieve the best possible rates of convergence whereas the sieve estimate typically does not suffer from this difficulty

convexity classification and risk bounds many of the classification algorithms developed in the machine learning literature including the support vector machine and boosting can be viewed as minimum contrast methods that minimize a convex surrogate of the 0 1 loss function the convexity makes these algorithms computationally efficient the use of a surrogate however has statistical consequences that must be balanced against the computational virtues of convexity to study these issues we provide a general quantitative relationship between the risk as assessed using the 0 1 loss and the risk as assessed using any nonnegative surrogate loss function we show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function that it satisfies a pointwise form of fisher consistency for classification the relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications we also present a refined version of this result in the

force distribution in a semiflexible loop loops undergoing thermal fluctuations are prevalent in nature ringlike or cross linked polymers cyclic macromolecules and protein mediated dna loops all belong to this category stability of these molecules are generally described in terms of free energy an average quantity but it may also be impacted by local fluctuating forces acting within these systems the full distribution of these forces can thus give us insights into mechanochemistry beyond the predictive capability of thermodynamics in this paper we study the force exerted by an inextensible semiflexible polymer constrained in a looped state by using a simulation method termed phase space sampling we generate the equilibrium distribution of chain conformations in both position and momentum space we compute the constraint forces between the two ends of the loop in this chain ensemble using lagrangian mechanics and show that the mean of these forces is equal to the thermodynamic force by analyzing kinetic and potential contributions to the forces we find that the mean force acts in the direction of increasing extension not because of bending stress but in spite of it furthermore we obtain a distribution of constraint forces as a function of chain length extension and stiffness notably increasing contour length decreases the average force but the additional freedom allows fluctuations in the constraint force to increase the force distribution is asymmetric and falls off less sharply than a gaussian distribution our work exemplifies a system where large amplitude fluctuations occur in a way unforeseen by a purely thermodynamic framework and offers computational tools useful for efficient unbiased simulation of a constrained system

on the effective elasticity of a two dimensional homogenised incompressible elastic composite the set of effective elasticity tensors for all two dimensional mixtures of two isotropic incompressible elastic materials taken in prescribed proportion is described in two dimensions the effective tensors are completely characterised by bounds on their eigenvalues

multiphase interchange inequalities phase interchange inequalities have been studied since the early work of keller j math phys 5 548 1964 they constrain the effective conductivity of composite materials which are obtained from each other for fixed configuration by interchanging the position of the phases optimal results in this direction for the case of a two phase composite are due to keller in spatial dimension d 2 and to avellaneda et al j appl phys 63 4989 1988 in dimension d 3 in this paper new inequalities in spatial dimension d 2 and d 3 which are valid when an arbitrary number of phases is present are proven when specialized to two phase composites they agree with those of keller in d 2 and of avellaneda et al in d 3 respectively

a moment bound for multi hinge classifiers the success of support vector machines in binary classification relies on the fact that hinge loss employed in the risk minimization targets the bayes rule recent research explores some extensions of this large margin based method to the multicategory case we show a moment bound for the socalled multi hinge loss minimizers based on two kinds of complexity constraints entropy with bracketing and empirical entropy obtaining such a result based on the latter is harder than finding one based on the former we obtain fast rates of convergence that adapt to the unknown margin

end to end distribution for a wormlike chain in arbitrary dimensions we construct an efficient methodology for calculating wormlike chain statistics in arbitrary d dimensions over all chain rigidities from fully rigid to completely flexible the structure of our exact analytical solution for the end to end distribution function for a wormlike chain in arbitrary d dimensions in fourier laplace space i e fourier transformed end position and laplace transformed chain length adopts the form of an infinite continued fraction which is advantageous for its compact structure and stability for numerical implementation we then proceed to present a step by step methodology for performing the fourier laplace inversion in order to make full use of our results in general applications asymptotic methods for evaluating the laplace inversion power law expansion and rayleigh schrodinger perturbation theory are employed in order to improve the accuracy of the numerical inversions of the end to end distribution function in real space we adapt our results to the evaluation of the single chain structure factor rendering simple closed form expressions that facilitate comparison with scattering experiments using our techniques the accuracy of the end to end distribution function is enhanced up to the limit of the machine precision we demonstrate the utility of our methodology with realizations of the chain statistics giving a general methodology that can be applied to a wide range of biophysical problems

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

smoothly clipped absolute deviation on high dimensions the smoothly clipped absolute deviation scad estimator proposed by fan and li has many desirable properties including continuity sparsity and unbiasedness the scad estimator also has the asymptotically oracle property when the dimension of covariates is fixed or diverges more slowly than the sample size in this article we study the scad estimator in high dimensional settings where the dimension of covariates can be much larger than the sample size first we develop an efficient optimization algorithm that is fast and always converges to a local minimum second we prove that the scad estimator still has the oracle property on high dimensional problems we perform numerical studies to compare the scad estimator with the lasso and sis scad estimators in terms of prediction accuracy and variable selectivity when the true model is sparse through the simulation we show that the variance estimator of fan and li still works well for some limited high dimensional cases where the true nonzero coeffic

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

on characterizing the set of possible effective tensors of composites the variational method and the translation method a general algebraic framework is developed for characterizing the set of possible effective tensors of composites a transformation to the polarization problem simplifies the derivation of the hashin shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials a general connection is established between two methods for bounding effective tensors of composites the first method is based on the variational principles of hashin and shtrikman the second method due to tartar murat lurie and cherkaev uses translation operators or equivalently quadratic quasiconvex functions a correspondence is established between these translation operators and bounding operators on the relevant non local projection operator t1 an important class of bounds namely trace bounds on the effective tensors of two component media are given a geometrical interpretation after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors a wide class of translation operators that generate these bounds is found the extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors these extremal translations generate attainable trace bounds even when the tensors of the two components are not well ordered in particular they generate the bounds of walpole on the effective bulk modulus the variational principles of gibiansky and cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by golden and papanicolaou an isomorphism is shown to underlie their variational principles this isomorphism is used to obtain dirichlet type variational principles and bounds for the effective tensors of general non selfadjoint problems it is anticipated that these variational principles which stem from the work of gibiansky and cherkaev will have applications in many fields of science

regularized orthogonal linear discriminant analysis in this paper the regularized orthogonal linear discriminant analysis rolda is studied the major issue of the regularized linear discriminant analysis is to choose an appropriate regularization parameter in existing regularized linear discriminant analysis methods they all select the best regularization parameter from a given parameter candidate set by using cross validation for classification an obvious limitation of such regularized linear discriminant analysis methods is that it is not clear how to choose an appropriate candidate set therefore up to now there is no concrete mathematical theory available in selecting an appropriate regularization parameter in practical applications of the regularized linear discriminant analysis the present work is to fill this gap here we derive the mathematical relationship between orthogonal linear discriminant analysis and the regularized orthogonal linear discriminant analysis first and then by means of this relationship we find a mathematical criterion for selecting the regularization parameter in rolda and consequently we develop a new regularized orthogonal linear discriminant analysis method in which no candidate set of regularization parameter is needed the effectiveness of our proposed regularized orthogonal linear discriminant analysis is illustrated by some real world data sets

optimal bounds and microgeometries for elastic two phase composites consider two isotropic linearly elastic materials with elasticity tensors c 1 c 2 and assume c 1 c 2 we prove that the class of effective tensors of composites built from these materials with given volume fractions that are invariant under a given symmetry group is such that each of its elements is bounded above and below by tensors in the same class corresponding to finite rank laminates this implies that for any imposed uniform strain or stress field optimal bounds on the effective strain or stress energy per unit volume are attained by finite rank laminates explicit bounds on the strain energy with no symmetry assumptions are given in dimensions 2 and 3 these bounds are more stringent than the classical voigt reuss bounds finally explicit bounds and microgeometries are given for the effective moduli of composites with cubic symmetry

on the equivalence of thermodynamics ensembles for flexible polymer chains abstract although the problem of the ensembles equivalence for flexible polymers has aroused considerable interest there is not an overall consensus on this topic in this work we present a theoretical investigation on the asymptotic equivalence of two ensembles for single flexible polymer chains without confinement effects i e fluctuating in the entire space the first is the gibbs or isotensional ensemble with one end terminal of the chain tethered to a given point and the other subjected to an applied force the other ensemble is the helmholtz or isometric one characterized by both terminals tethered to fixed points the equivalence property is rigorously proved for a class of potentials characterized by a continuous pairing interaction between neighboring monomers to approach the problem we adopted an original analytical formalism based on the stationary phase technique and on the exact determination of the eigenvalues sign of the hessian matrix of the phase function to give some examples of application the general result is successively applied to freely jointed chains to flexible polymers with extensible bonds and to chains with domains that exhibit conformational transitions between two stable states

optimal bounds for conduction in two dimensional multiphase polycrystalline media a complete geometrical characterization is given of the set of all possible effective conductivity tensors for two dimensional composites made of an arbitrary number of given anisotropic phases it is rigorously established that any polycrystalline composite formed from an arbitrary possibly infinite number of phases can be replaced by a composite formed from only two of the given phases without altering the effective conductivity tensor

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

multicategory angle based large margin classification large margin classifiers are popular methods for classification among existing simultaneous multicategory large margin classifiers a common approach is to learn k different functions for a k class problem with a sum to zero constraint such a formulation can be inefficient we propose a new multicategory angle based large margin classification framework the proposed angle based classifiers consider a simplex based prediction rule without the sum to zero constraint and enjoy more efficient computation many binary large margin classifiers can be naturally generalized for multicategory problems through the angle based framework theoretical and numerical studies demonstrate the usefulness of the angle based methods

on l1 norm multiclass support vector machines binary support vector machines svms have been proven to deliver high performance in multiclass classification however issues remain with respect to variable selection one challenging issue is classification and variable selection in the presence of variables in the magnitude of thousands greatly exceeding the size of training sample this often occurs in genomics classification to meet the challenge this article proposes a novel multiclass support vector machine which performs classification and variable selection simultaneously through an l1 norm penalized sparse representation the proposed methodology together with the developed regularization solution path permits variable selection in such a situation for the proposed methodology a statistical learning theory is developed to quantify the generalization error in an attempt to gain insight into the basic structure of sparse learning permitting the number of variables to greatly exceed the sample size the operating characteristics of the

on ψ learning the concept of large margins have been recognized as an important principle in analyzing learning methodologies including boosting neural networks and support vector machines svms however this concept alone is not adequate for learning in nonseparable cases we propose a learning methodology called learning that is derived from a direct consideration of generalization errors we provide a theory for learning and show that it essentially attains the optimal rates of convergence in two learning examples finally results from simulation studies and from breast cancer classification confirm the ability of learning to outperform svm in generalization

a regularization framework for multiclass classification a deterministic annealing approach in this paper we propose a general regularization framework for multiclass classification based on discriminant functions since the objective function in the primal optimization problem of this framework is always not differentiable the optimal solution cannot be obtained directly with the aid of the deterministic annealing approach a differentiable objective function is derived subject to a constraint on the randomness of the solution the problem can be approximated by solving a sequence of differentiable optimization problems and such approximation converges to the original problem asymptotically based on this approach class conditional posterior probabilities can be calculated directly without assuming the underlying probabilistic model we also notice that there is a connection between our approach and some existing statistical models such as fisher discriminant analysis and logistic regression

on quantile regression in reproducing kernel hilbert spaces with the data sparsity constraint for spline regressions it is well known that the choice of knots is crucial for the performance of the estimator as a general learning framework covering the smoothing splines learning in a reproducing kernel hilbert space rkhs has a similar issue however the selection of training data points for kernel functions in the rkhs representation has not been carefully studied in the literature in this paper we study quantile regression as an example of learning in a rkhs in this case the regular squared norm penalty does not perform training data selection we propose a data sparsity constraint that imposes thresholding on the kernel function coefficients to achieve a sparse kernel function representation we demonstrate that the proposed data sparsity method can have competitive prediction performance for certain situations and have comparable performance in other cases compared to that of the traditional squared norm penalty therefore the data sparsity method can serve as a competitive alternative to the squared norm penalty method some theoretical properties of our proposed method using the data sparsity constraint are obtained both simulated and real data sets are used to demonstrate the usefulness of our data sparsity constraint