discriminant analysis by gaussian mixtures fisher rao linear discriminant analysis lda is a valuable tool for multigroup classification lda is equivalent to maximum likelihood classification assuming gaussian distributions for each class in this paper we fit gaussian mixtures to each class to facilitate effective classification in non normal settings especially when the classes are clustered low dimensional views are an important by product of lda our new techniques inherit this feature we can control the within class spread of the subclass centres relative to the between class spread our technique for fitting these models permits a natural blend with nonparametric versions of lda

an iterative scheme for a class of quasi variational inequalities an iterative scheme is given to obtain the approximate solution of a class of quasi variational inequalities it is shown that the approximate solution obtained by the iterative scheme converges strongly in the hilbert space to the exact solution as a special case we obtain the corresponding iterative scheme for variational inequalities

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

discriminant subspace analysis a fukunaga koontz approach the fisher linear discriminant fld is commonly used in pattern recognition it finds a linear subspace that maximally separates class patterns according to the fisher criterion several methods of computing the fld have been proposed in the literature most of which require the calculation of the so called scatter matrices in this paper we bring a fresh perspective to fld via the fukunaga koontz transform fkt we do this by decomposing the whole data space into four subspaces with different discriminabilities as measured by eigenvalue ratios by connecting the eigenvalue ratio with the generalized eigenvalue we show where the fisher criterion is maximally satisfied we prove the relationship between fld and fkt analytically and propose a unified framework to understanding some existing work furthermore we extend our theory to the multiple discriminant analysis mda this is done by transforming the data into intraclass and extraclass spaces followed by maximizing the bhattacharyya distance based on our fkt analysis we identify the discriminant subspaces of mda fkt and propose an efficient algorithm which works even when the scatter matrices are singular or too large to be formed our method is general and may be applied to different pattern recognition problems we validate our method by experimenting on synthetic and real data

learning a mahalanobis metric from equivalence constraints many learning algorithms use a metric defined over the input space as a principal tool and their performance critically depends on the quality of this metric we address the problem of learning metrics using side information in the form of equivalence constraints unlike labels we demonstrate that this type of side information can sometimes be automatically obtained without the need of human intervention we show how such side information can be used to modify the representation of the data leading to improved clustering and classification specifically we present the relevant component analysis rca algorithm which is a simple and efficient algorithm for learning a mahalanobis metric we show that rca is the solution of an interesting optimization problem founded on an information theoretic basis if dimensionality reduction is allowed within rca we show that it is optimally accomplished by a version of fisher s linear discriminant that uses constraints moreover under certain gaussian assumptions rca can be viewed as a maximum likelihood estimation of the within class covariance matrix we conclude with extensive empirical evaluations of rca showing its advantage over alternative methods

regularized coplanar discriminant analysis for dimensionality reduction abstract the dimensionality reduction methods based on linear embedding such as neighborhood preserving embedding npe sparsity preserving projections spp and collaborative representation based projections crp try to preserve a certain kind of linear representation for each sample after projection however in the transformed low dimensional space the linear relationship between the samples may be changed which cannot make the linear representation based classifiers such as sparse representation based classifier src to achieve higher recognition accuracy in this paper we propose a new linear dimensionality reduction algorithm called regularized coplanar discriminant analysis rcda to address this problem it simultaneously seeks a linear projection matrix and some linear representation coefficients that make the samples from the same class coplanar and the samples from different classes not coplanar the proposed regularization term balances the bias from the optimal linear representation and that from the class mean to avoid overfitting the training data and overcomes the matrix singularity in solving the linear representation coefficients an alternative optimization approach is proposed to solve the rcda model experiments are done on several benchmark face databases and hyperspectral image databases and results show that rcda can obtain better performance than other dimensionality reduction methods

on an iterative method for variational inequalities a number of numerical solutions are presented as examples of a new iterative method for variational inequalities the iterative method is based on the reduction of variational inequalities to the wiener hopf equations for obstacle problems the convergence is guaranteed inw 1 p spaces forp 2 the examples presented are one and two dimensional obstacle problems in cases when the greens function is known but the method is very general

reducing nfas by invariant equivalences we give new general methods for constructing small non deterministic finite automata nfa from arbitrary ones given an nfa we compute the largest right invariant equivalence on the set of states and then merge the equivalent states to obtain a smaller automaton when applying this method to position automata we get a way to convert regular expressions into nfas which are always smaller than or equal to the position partial derivative and follow automata it can be arbitrarily smaller the construction can be dually made for left invariant equivalences and then the two can be combined for even better results

iterative methods for linear complementarity problems with upperbounds on primary variables this paper is concerned with iterative methods for the linear complementarity problem lcp of findingx and y in rn such thatc dx y 0 b x 0 andxt c dx y yt b x 0 whenb 0 this is the lcp m q withm l 0d t which is in turn equivalent to a linear variational inequality over a rectangle this type of problem arises for example from quadratic programming qp problems with upper and lower bounds ifd is symmetric and from multicommodity market equilibrium problems with institutional price controls imposed ifd is not necessarily symmetric iterative methods for the lcp m q with symmetricm are well developed and studied using qp applications for nonsymmetric cases withm being either an h matrix with positive diagonals or a z matrix there exists a robust iterative method with guaranteed convergence this paper extends this algorithm so that the lcp m q withm l 0d t which is neither symmetric nor an h matrix with positive diagonals nor a z matrix can be processed when onlyd notm satisfies such properties the case whered is nonsymmetric is explicitly discussed

solving third order obstacle problems with quartic b splines in this paper we develop a new numerical method for solving a system of third order boundary value problems associated with third order obstacle problems using the quartic b splines the convergence order of the method is considered and numerical results are also given

generalized nonlinear variational inequalities abstract in this paper we investigate the existence results of solutions for a class of generalized nonlinear variational inequalities gnvi with nonmonotone set valued operators defined on noncompact sets in locally convex hausdorff topological vector spaces results that generalize the corresponding recent results of chang and zhang are derived

pruning noisy bases in discriminant analysis the success of linear discriminant analysis lda is due in part to the simplicity of its formulation which reduces to a simultaneous diagonalization of two symmetric matrices a and b however a fundamental drawback of this approach is that it cannot be efficiently applied wherever the matrix a is singular or when some of the smallest variances in are due to noise in this paper we present a factorization a 1b of and a correlation based criterion that can be readily employed to solve these problems we provide detailed derivations for the linear and nonlinear classification problems the usefulness of the proposed approach is demonstrated thoroughly using a large variety of databases

a new class of completely generalized quasi variational inclusions in banach spaces in this paper a new notion of j proximal mapping for a nonconvex lower semicontinuous subdifferentiable proper functional on banach space is introduced the existence and lipschitz continuity of j proximal mapping of a lower semicontinuous subdifferentiable proper functional are proved by applying the concept we introduce and study a new class of completely generalized quasi variational inclusions in reflexive banach spaces a novel and innovative iterative algorithm for finding the approximate solutions is suggested and analyzed the convergence criteria of the iterative sequences generated by the new iterative algorithm is also given these algorithm and existence result generalize many known results under hilbert space setting in recent literature to reflexive banach spaces

iterative methods for a class of complementarity problems abstract in this paper we propose and study an algorithm for a new class of complementarity problems of finding u r n such that u 0 tu a u 0 u tu a u 0 where t is a continuous mapping and a is a nonlinear transformation from r n into itself it is proved that the approximate solution obtained from the iterative scheme converges to the exact solution several special cases are also discussed

graph embedding and extensions a general framework for dimensionality reduction a large family of algorithms supervised or unsupervised stemming from statistics or geometry theory has been designed to provide different solutions to the problem of dimensionality reduction despite the different motivations of these algorithms we present in this paper a general formulation known as graph embedding to unify them within a common framework in graph embedding each algorithm can be considered as the direct graph embedding or its linear kernel tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided furthermore the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms by utilizing this framework as a tool we propose a new supervised dimensionality reduction algorithm called marginal fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class while the penalty graph connects the marginal points and characterizes the interclass separability we show that mfa effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions real face recognition experiments show the superiority of our proposed mfa in comparison to lda also for corresponding kernel and tensor extensions

an iterative algorithm for variational inequalities abstract in this paper we introduce and study a new class of nonlinear variational inequalities this new class enables us to apply variational techniques to the solution of differential equations of both odd and even orders a projection method is used to suggest an iterative algorithm for finding the approximate solution of this class we also discuss the convergence criteria of the proposed iterative algorithm several special cases are discussed which can be obtained from the general result

generalized nash games and quasi variational inequalities abstract a generalized nash game is an n person noncooperative game with nondisjoint strategy sets other names for this game form include social equilibria and pseudo nash games this paper explores both the qualitative and quantitative properties of such games through the use of quasi variational inequality theory several interesting relationships between the variational and quasi variational inequality forms of this class of games are described and the practical implementation of generalized nash games are explored at length

sensitivity analysis in variational inequalities in this paper we derive conditions for the local uniqueness continuity and differentiability of a solution of parametric variational inequalities our approach is new and has a strong geometric flavor as an illustration we discuss the sensitivity of the classical complementarity problem

perturbed ishikawa type iterative algorithm for generalized quasivariational inclusions in this paper a perturbed ishikawa type iterative algorithm of solving a new class of generalized nonlinear implicit quasivariational inclusions are suggested and analysed first an existence theorem of solutions for generalized nonlinear implicit quasivariational inclusion is proved under suitable assumptions then the convergence of the perturbed ishikawa type iterative sequences generated by the new algorithm is proved as special cases some known results are also discussed

existence and algorithm for solving some generalized mixed variational inequalities abstract in this paper by applying the auxiliary variational principle technique an existence theorem of solutions for a new class of generalized mixed variational inequalities is proved in hilbert spaces a novel and innovative iterative algorithm to compute approximate solutions is suggested and analyzed the convergence criteria and error estimates are also given these results of existence algorithm and convergence are new and generalize some corresponding results involving single valued mappings in recent literatures

pareto models for discriminative multiclass linear dimensionality reduction we address the class masking problem in multiclass linear discriminant analysis lda in the multiclass setting lda does not maximize each pairwise distance between classes but rather maximizes the sum of all pairwise distances this results in serious overlaps between classes that are close to each other in the input space and degrades classification performance our research proposes pareto discriminant analysis parda an approach for multiclass discriminative analysis that builds over multiobjective optimizing models parda decomposes the multiclass problem to a set of objective functions each representing the distance between every pair of classes unlike existing lda extensions that maximize the sum of all distances parda maximizes each pairwise distance to maximally separate all class means while minimizing the class overlap in the lower dimensional space experimental results on various data sets show consistent and promising performance of parda when compared with well known multiclass lda extensions highlightswe address the class masking problem cmp in multiclass lda mlda we model mlda as a set of objective functions for pairwise distances between classes we optimize the compound objective function using multiobjective optimization our models overcome the cmp and are extended to multimodal non gaussian data empirical analysis shows consistent and promising performance in favor of our models

max min distance analysis by using sequential sdp relaxation for dimension reduction we propose a new criterion for discriminative dimension reduction max min distance analysis mmda given a data set with c classes represented by homoscedastic gaussians mmda maximizes the minimum pairwise distance of these c classes in the selected low dimensional subspace thus unlike fisher s linear discriminant analysis flda and other popular discriminative dimension reduction criteria mmda duly considers the separation of all class pairs to deal with general case of data distribution we also extend mmda to kernel mmda kmmda dimension reduction via mmda kmmda leads to a nonsmooth max min optimization problem with orthonormal constraints we develop a sequential convex relaxation algorithm to solve it approximately to evaluate the effectiveness of the proposed criterion and the corresponding algorithm we conduct classification and data visualization experiments on both synthetic data and real data sets experimental results demonstrate the effectiveness of mmda kmmda associated with the proposed optimization algorithm

quasi variational inequalities and applications to the traffic equilibrium problem discussion of a paradox abstract this paper attempts to explore the possibility of solving the traffic assignment problem with elastic demands by way of a quasi variational inequality it is shown that the solving procedure proposed by maugeri 1987 with respect to the fixed demands model is also available when the equilibrium problem with elastic demands is expressed in terms of a quasi variational inequality a numerical example is presented to illustrate the computational procedure and to discuss a paradox moreover a theorem that ensures the existence of a solution of the quasi variational inequality is exhibited

from regular expressions to dfa s using compressed nfa s we show how to turn a regular expression r of length r into an o s space representation of mcnaughton and yamada s nfa where s is the number of occurrences of alphabet symbols in r and s 1 is the number of nfa states the standard adjacency list representation of mcnaughton and yamada s nfa takes up 1 2s s sp2 space in the worst case the adjacency list representation of the nfa produced by thompson takes up between 2r and 6r space where r can be arbitrarily larger than s given any subset v of states in mcnaughton and yamada s nfa our representation can be used to compute the set u of states one transition away from the states in v in optimal time o vert v vert vert u vert mcnaughton and yamada s nfa requires theta vert v vert times vert u vert time in the worst case using thompson s nfa the equivalent calculation requires theta r time in the worst case an implementation of our nfa representation confirms that it takes up an order of magnitude less space than mcnaughton and yamada s machine an implementation to produce a dfa from our nfa representation by subset construction shows linear and quadratic speedups over subset construction starting from both thompson s and mcnaughton and yamada s nfa s it also shows that the dfa produced from our nfa is as much as one order of magnitude smaller than dfa s constructed from the two other nfa s an unix egrep compatible software called cgrep based on our nfa representation is implemented a benchmark shows that cgrep is dramatically faster than both unix egrep and gnu e grep throughout this thesis the importance of syntax is stressed in the design of our algorithms in particular we exploit a method of program improvement in which costly repeated calculations can be avoided by establishing and maintaining program invariants this method of symbolic finite differencing has been used previously by douglas smith to derive efficient functional programs

from regular expressions to deterministic automata abstract the main theorem allows an elegant algorithm to be refined into an efficient one the elegant algorithm for constructing a finite automaton from a regular expression is based on derivatives of regular expressions the efficient algorithm is based on marking of regular expressions derivatives of regular expressions correspond to state transitions in finite automata when a finite automaton makes a transition under input symbol a a leading a is stripped from the remaining input correspondingly if the input string is generated by a regular expression e then the derivative of e by a generates the remaining input after a leading a is stripped brzozowski 1964 used derivatives to construct finite automata the state for expression e has a transition under a to the state for the derivative of e by a this approach extends to regular expressions with new operators including intersection and complement however explicit computation of derivatives can be expensive marking of regular expressions yields an expression with distinct input symbols following mcnaughton and yamada 1960 we attach subscripts to each input symbol in an expression ab b ba becomes a 1 b 2 b 3 b 4 a 5 conceptually the efficient algorithm constructs an automaton for the marked expression the marks on the transitions are then erased resulting in a nondeterministic automaton for the original unmarked expression this approach works for the usual operations of union concatenation and iteration however intersection and complement cannot be handled because marking and unmarking do not preserve the languages generated by regular expressions with these operators

general nonlinear variational inequalities abstract the fixed point technique is used to prove the existence of a unique solution of a new unified and general class of nonlinear variational inequalities several special cases which can be obtained from our main result are also discussed

exchange price equilibria and variational inequalities the aim of this paper is to show the relevance of the concept and the theory of variational inequalities in the study of economic equilibria

solution of minimax problems using equivalent differentiable functions abstract a method is proposed for the solution of minimax optimization problems in which the individual functions involved are convex the method consists of solving a problem with an objective function which is the sum of high powers or strong exponentials of the separate components of the original objective function the resulting objective function which is equivalent at the limit to the minimax one is shown to be smooth as well as convex any efficient nonlinear programming method can be utilized for solving the equivalent problem a number of examples are discussed

sensitivity analysis framework for variational inequalities in this paper a sensitivity analysis framework is developed for variational inequalities the perturbed solution to a parametric variational inequality problem is shown to be continuous and directionally differentiable under appropriate second order and regularity assumptions moreover this solution is once continuously differentiable if strict complementarity also holds

bayes optimality in linear discriminant analysis we present an algorithm that provides the one dimensional subspace where the bayes error is minimized for the c class problem with homoscedastic gaussian distributions our main result shows that the set of possible one dimensional spaces v for which the order of the projected class means is identical defines a convex region with associated convex bayes error function g v this allows for the minimization of the error function using standard convex optimization algorithms our algorithm is then extended to the minimization of the bayes error in the more general case of heteroscedastic distributions this is done by means of an appropriate kernel mapping function this result is further extended to obtain the d dimensional solution for any given d by iteratively applying our algorithm to the null space of the d l dimensional solution we also show how this result can be used to improve upon the outcomes provided by existing algorithms and derive a low computational cost linear approximation extensive experimental validations are provided to demonstrate the use of these algorithms in classification data analysis and visualization

variational inequalities for fuzzy mappings ii in this paper we establish the equivalence between the variational inequalities for fuzzy mappings and the wiener hopf equations for fuzzy mappings we use this equivalence to suggest a number of new iterative algorithms for solving the variational inequalities for fuzzy mappings we also study the convergence criteria of these iterative algorithms the auxiliary principle technique is used to study the existence of a solution of the variational inequality for fuzzy mappings and to suggest a novel and innovative iterative algorithm for computing the approximate solution the results proved in this paper represent an improvement of previously known results

subclass discriminant analysis over the years many discriminant analysis da algorithms have been proposed for the study of high dimensional data in a large variety of problems each of these algorithms is tuned to a specific type of data distribution that which best models the problem at hand unfortunately in most problems the form of each class pdf is a priori unknown and the selection of the da algorithm that best fits our data is done over trial and error ideally one would like to have a single formulation which can be used for most distribution types this can be achieved by approximating the underlying distribution of each class with a mixture of gaussians in this approach the major problem to be addressed is that of determining the optimal number of gaussians per class i e the number of subclasses in this paper two criteria able to find the most convenient division of each class into a set of subclasses are derived extensive experimental results are shown using five databases comparisons are given against linear discriminant analysis lda direct lda dlda heteroscedastic lda hlda nonparametric da nda and kernel based lda k lda we show that our method is always the best or comparable to the best

some resolvent iterative methods for variational inclusions and nonexpansive mappings in this paper we suggest and analyze three step iterations for finding the common element of the set of fixed points of a nonexpansive mappings and the set of the solutions of the variational inclusions using the resolvent operator technique we also study the convergence criteria of three step iterative method under some mild conditions our results include the previous results of noor m aslam noor general variational inequalities and nonexpansive mappings j math anal appl 331 2007 810 822 and noor and huang m aslam noor zhenyu huang three step methods for nonexpansive mappings and variational inequalities appl math comput in press doi 10 1016 j amc 2006 08 088 as special cases and may be considered as an improvement and refinement of the previously known results

generalized multivalued quasi variational inequalities in this paper we introduce and study a new class of variational inequalities known as generalized multivalued quasi variational inequalities this class is the most general and unifying one using essentially the projection technique we establish the equivalence between this class and the implicit wiener hopf equations this equivalence is used to suggest a number of iterative algorithms for solving the generalized multivalued quasi variational inequalities the convergence analysis of these algorithms is also studied the results proved in this paper represent a significant improvement and refinement of the previously known results in this area

multiclass linear dimension reduction by weighted pairwise fisher criteria we derive a class of computationally inexpensive linear dimension reduction criteria by introducing a weighted variant of the well known k class fisher criterion associated with linear discriminant analysis lda it can be seen that lda weights contributions of individual class pairs according to the euclidean distance of the respective class means we generalize upon lda by introducing a different weighting function

parametric completely generalized mixed implicit quasi variational inclusions involving h maximal monotone mappings a new class of parametric completely generalized mixed implicit quasi variational inclusions involving h maximal monotone mappings is introduced by applying resolvent operator technique of h maximal monotone mapping and the property of fixed point set of set valued contractive mappings the behavior and sensitivity analysis of the solution set of the parametric completely generalized mixed implicit quasi variational inclusions involving h maximal monotone mappings are studied the continuity and lipschitz continuity of the solution set with respect to the parameter are proved under suitable assumptions our approach and results are new and improve unify and extend previous many known results in this field

a new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings in this paper we introduce and study a new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings an existence theorem of solutions is proved without compactness assumptions a new iterative algorithm is suggested and analysed the convergence of iterative sequences generated by the new algorithm are also given as special cases some known results are also discussed

a decision cognizant kullback leibler divergence in decision making systems involving multiple classifiers there is the need to assess classifier in congruence that is to gauge the degree of agreement between their outputs a commonly used measure for this purpose is the kullback leibler kl divergence we propose a variant of the kl divergence named decision cognizant kullback leibler divergence dc kl to reduce the contribution of the minority classes which obscure the true degree of classifier incongruence we investigate the properties of the novel divergence measure analytically and by simulation studies the proposed measure is demonstrated to be more robust to minority class clutter its sensitivity to estimation noise is also shown to be considerably lower than that of the classical kl divergence these properties render the dc kl divergence a much better statistic for discriminating between classifier congruence and incongruence in pattern recognition systems

some aspects of variational inequalities abstract in this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence uniqueness computational properties various generalizations sensitivity analysis and their applications we also propose some open problems with sufficient information and references so that someone may attempt solution s in his her area of special interest we also include some new results which we have recently obtained

generalized discriminant analysis using a kernel approach we present a new method that we call generalized discriminant analysis gda to deal with nonlinear discriminant analysis using kernel function operator the underlying theory is close to the support vector machines svm insofar as the gda method provides a mapping of the input vectors into high dimensional feature space in the transformed space linear properties make it easy to extend and generalize the classical linear discriminant analysis lda to nonlinear discriminant analysis the formulation is expressed as an eigenvalue problem resolution using a different kernel one can cover a wide class of nonlinearities for both simulated data and alternate kernels we give classification results as well as the shape of the decision function the results are confirmed using real data to perform seed classification

on the generalized nonlinear variational inequalities and quasi variational inequalities the purpose of this paper is to investigate the existence of solutions for a class of generalized nonlinear inequalities gnvi and generalized nonlinear quasi variational inequalities gnqvi with non monotone set valued operators defined on non compact sets in locally convex hausdorff topological vector spaces results that generalize and unify corresponding recent results of ding and tarafdar are derived

face recognition using lda based algorithms low dimensional feature representation with enhanced discriminatory power is of paramount importance to face recognition fr systems most of traditional linear discriminant analysis lda based methods suffer from the disadvantage that their optimality criteria are not directly related to the classification ability of the obtained feature representation moreover their classification accuracy is affected by the small sample size sss problem which is often encountered in fr tasks in this paper we propose a new algorithm that deals with both of the shortcomings in an efficient and cost effective manner the proposed method is compared in terms of classification accuracy to other commonly used fr methods on two face databases results indicate that the performance of the proposed method is overall superior to those of traditional fr approaches such as the eigenfaces fisherfaces and d lda methods

on a class of variational inequalities in this paper we introduce and study a new unified and general class of variational inequalities we derive the general error estimates for the finite element solutions of variational inequalities it has been shown that a class of contact problems with friction terms arising in elastostatics can be studied in the framework of variational inequalities several special cases which can be obtained from the general results are also discussed

fractional step dimensionality reduction linear projections for dimensionality reduction computed using linear discriminant analysis lda are commonly based on optimization of certain separability criteria in the output space the resulting optimization problem is linear but these separability criteria are not directly related to the classification accuracy in the output space consequently a trial and error procedure has to be invoked experimenting with different separability criteria that differ in the weighting function used and selecting the one that performed best on the training set often even the best weighting function among the trial choices results in poor classification of data in the subspace in this short paper we introduce the concept of fractional dimensionality and develop an incremental procedure called the fractional step lda f lda to reduce the dimensionality in fractional steps the f lda algorithm is more robust to the selection of weighting function and for any given weighting function it finds a subspace in which the classification accuracy is higher than that obtained using lda

browder hartman stampacchia variational inequalities for multi valued monotone operators abstract an existence theorem of browder hartman stampacchia variational inequalities is extended to multi valued monotone operators surjectivity for multi valued monotone operators and properties of solution sets are discussed

linear dimensionality reduction via a heteroscedastic extension of lda the chernoff criterion we propose an eigenvector based heteroscedastic linear dimension reduction ldr technique for multiclass data the technique is based on a heteroscedastic two class technique which utilizes the so called chernoff criterion and successfully extends the well known linear discriminant analysis lda the latter which is based on the fisher criterion is incapable of dealing with heteroscedastic data in a proper way for the two class case the between class scatter is generalized so to capture differences in co variances it is shown that the classical notion of between class scatter can be associated with euclidean distances between class means from this viewpoint the between class scatter is generalized by employing the chernoff distance measure leading to our proposed heteroscedastic measure finally using the results from the two class case a multiclass extension of the chernoff criterion is proposed this criterion combines separation information present in the class mean as well as the class covariance matrices extensive experiments and a comparison with similar dimension reduction techniques are presented

from regular expressions to smaller nfas several methods have been developed to construct l free automata that represent a regular expression among the most widely known are the position automaton glushkov the partial derivatives automaton antimirov and the follow automaton ilie and yu all these automata can be obtained with quadratic time complexity thus the comparison criterion is usually the size of the resulting automaton the methods that obtain the smallest automata although for general expressions they are not comparable are the follow and the partial derivatives methods in this paper we propose another method to obtain a l free automaton from a regular expression the number of states of the automata we obtain is bounded above by the size of both the partial derivatives automaton and of the follow automaton our algorithm also runs with the same time complexity of these methods

enhanced fisher discriminant criterion for image recognition many previous studies have shown that image recognition can be significantly improved by fisher linear discriminant analysis flda technique however flda ignores the variation of data points from the same class which characterizes the most important modes of variability of patterns and helps to improve the generalization capability of flda thus the performance of flda on testing data is not good enough in this paper we propose an enhanced fisher discriminant criterion efdc efdc explicitly considers the intra class variation and incorporates the intra class variation into the fisher discriminant criterion to build a robust and efficient dimensionality reduction function efdc obtains a subspace which best detects the discriminant structure and simultaneously preserves the modes of variability of patterns which will result in stable intraclass representation experimental results on four image database show a clear improvement over the results of flda based methods

linear dimensionality reduction using relevance weighted lda the linear discriminant analysis lda is one of the most traditional linear dimensionality reduction methods this paper incorporates the inter class relationships as relevance weights into the estimation of the overall within class scatter matrix in order to improve the performance of the basic lda method and some of its improved variants we demonstrate that in some specific situations the standard multi class lda almost totally fails to find a discriminative subspace if the proposed relevance weights are not incorporated in order to estimate the relevance weights of individual within class scatter matrices we propose several methods of which one employs the evolution strategies