pattern recognition by graph matching using the potts mft neural networks this paper is concerned with programming of the potts mean field theory neural networks for pattern recognition by homomorphic mapping of the attributed relational graphs arg in order to generate the homomorphic mapping from the scene relational graph to the model graph we make use of the recently introduced suganthan technical report nanyang technical university 1994 compatibility functions in relation to the hopfield network an efficient pose clustering algorithm is used to separate and localize different occurrences of any particular object model in the scene the pose clustering algorithm also eliminates spurious hypotheses generated by the network and resolves ambiguities in the final interpretation the performance of the proposed approach to pattern recognition by homomorphism is demonstrated using a number of line patterns silhouette images and circle patterns

rank equalities related to the generalized inverses at s 2 bt 1 s1 2 of two matrices a and b for given matrices a and b by applying the group inverse expressions of generalized inverses a t s 2 and b t 1 s 1 2 we present some rank equalities related to a t s 2 and b t 1 s 1 2 as applications we obtain a variety of rank equalities related to the moore penrose inverses the drazin inverses the group inverses and the weighted moore penrose inverses of a and b

representation of tripotents and representations via tripotents abstract let a be an algebra an element a a is called tripotent if a 3 a we study the questions if both a and b are tripotents then under what conditions are a b and ab tripotent under what conditions do a and b commute we extend the partial order from the hilbert space idempotents to the set of all tripotents and show that every normal tripotent is self adjoint for a m n c we describe the set of all finite sums of tripotents the convex hull of tripotents and the set of all tripotents averages we also give the new proof of rational trace matrix representations by choi and wu 2

systems of generalized nonlinear variational inequalities and its projection methods abstract based on relaxed cocoercive monotonicity a new generalized class of nonlinear variational inequality problems is presented our results improve and extend the recent ones announced by h y huang m a noor an explicit projection method for a system of nonlinear variational inequalities with different r cocoercive mappings appl math comput 190 2007 356 361 s s chang h w joseph lee c k chan generalized system for relaxed cocoercive variational inequalities in hilbert spaces appl math lett 20 2007 329 334 r u verma generalized system for relaxed cocoercive variational inequalities and its projection methods j optim theory appl 121 2004 203 210 m a noor general variational inequalities appl math lett 1 1988 119 121 and many others

on angles and distances between subspaces abstract the angles and distances between two given subspaces of c n 1 are investigated on the basis of a joint decomposition of the corresponding orthogonal projectors several new results are established with the particular attention paid to the notions of inclinedness and minimal angle to demonstrate the usefulness of the approach utilized some results known to be valid in hilbert space are reestablished in c n 1 either in generalized form or with considerably shorter proofs than in the original sources

a graduated assignment algorithm for graph matching a graduated assignment algorithm for graph matching is presented which is fast and accurate even in the presence of high noise by combining graduated nonconvexity two way assignment constraints and sparsity large improvements in accuracy and speed are achieved its low order computational complexity o lm where l and m are the number of links in the two graphs and robustness in the presence of noise offer advantages over traditional combinatorial approaches the algorithm not restricted to any special class of graph is applied to subgraph isomorphism weighted graph matching and attributed relational graph matching to illustrate the performance of the algorithm attributed relational graphs derived from objects are matched then results from twenty five thousand experiments conducted on 100 mode random graphs of varying types graphs with only zero one links weighted graphs and graphs with node attributes and multiple link types are reported no comparable results have been reported by any other graph matching algorithm before in the research literature twenty five hundred control experiments are conducted using a relaxation labeling algorithm and large improvements in accuracy are demonstrated

on the product of orthogonal projectors abstract in this paper we introduce a canonical form to represent the product of two orthogonal projectors this form is useful for deriving many known results which are scattered in the literature we summarize many of those results and present new ones

applications of cs decomposition in linear combinations of two orthogonal projectors abstract we study the spectrum and the rank of a linear combination of two orthogonal projectors we characterize when this linear combination is ep diagonalizable idempotent tripotent involutive nilpotent generalized projector and hypergeneralized projector also we derive the moore penrose inverse of a linear combination of two orthogonal projectors in a particular case the main tool used here is the cs decomposition

characterizations and linear combinations of k generalized projectors abstract we extend generalized projectors introduced by gros and trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 to k generalized projectors and we characterize them obtaining results in the aforesaid paper as a consequence moreover we list all situations when a linear combination of commuting k generalized projectors is a k generalized projector the method for obtaining this result permits to give a revisited version of the main result by baksalary and baksalary on linear combinations of generalized projectors linear algebra appl 388 2004 17 24 in addition the case of orthogonal projectors is also analyzed

nonnegative idempotent matrices and the minus partial order abstract we describe the structure of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus order

functions of orthogonal projectors involving the moore penrose inverse several results scattered in the literature express an oblique projector having given onto and along spaces in terms of a pair of orthogonal projectors the results were established in various settings including finite and infinite dimensional vector spaces over either real or complex numbers but their common feature is that they are valid merely under the assumption of the nonsingularity of certain functions of the involved projectors in the present paper these results are unified and reestablished in a generalized form in a complex euclidean vector space with the generalization obtained by relaxing the nonsingularity assumption and use of the moore penrose inverse instead of the ordinary inverse additionally several new formulae of the type are provided

subsets and subspaces several rules known to be valid within the set theory are revisited and considered from the point of view of the question whether they remain satisfied when sets are replaced with subspaces from c n 1 it is shown that some of the set relationships hold true upon such a replacement however there is also an extensive class of relationships whose validity is no longer guaranteed in this class the commutativity of the orthogonal projectors onto the involved subspaces proves to play an important role some related results inspired by probability theory are established as well

structural matching by discrete relaxation this paper describes a bayesian framework for performing relational graph matching by discrete relaxation our basic aim is to draw on this framework to provide a comparative evaluation of a number of contrasting approaches to relational matching broadly speaking there are two main aspects to this study firstly we focus on the issue of how relational inexactness may be quantified we illustrate that several popular relational distance measures can be recovered as specific limiting cases of the bayesian consistency measure the second aspect of our comparison concerns the way in which structural inexactness is controlled we investigate three different realizations of the matching process which draw on contrasting control models the main conclusion of our study is that the active process of graph editing outperforms the alternatives in terms of its ability to effectively control a large population of contaminating clutter

general implicit variational inclusion problems based on a maximal m relaxed monotonicity amrm frameworks a general framework for an algorithmic procedure based on the variational convergence of operator sequences involving a maximal m relaxed monotone amrm mappings in a hilbert space setting is developed and then it is applied to approximating the solution of a general class of nonlinear implicit inclusion problems involving a maximal m relaxed monotone mappings furthermore some specializations of interest on existence theorems and corresponding approximation solvability theorems on h maximal monotone mappings are included that may include several other results for general variational inclusion problems on general maximal monotonicity in the literature

graph matching with a dual step em algorithm this paper describes a new approach to matching geometric structure in 2d point sets the novel feature is to unify the tasks of estimating transformation geometry and identifying point correspondence matches unification is realized by constructing a mixture model over the bipartite graph representing the correspondence match and by affecting optimization using the em algorithm according to our em framework the probabilities of structural correspondence gate contributions to the expected likelihood function used to estimate maximum likelihood transformation parameters these gating probabilities measure the consistency of the matched neighborhoods in the graphs the recovery of transformational geometry and hard correspondence matches are interleaved and are realized by applying coupled update operations to the expected log likelihood function we evaluate the technique on two real world problems

characterizations of ep matrices and weighted ep matrices abstract a square complex matrix a is said to be ep if a and its conjugate transpose a have the same range in this paper we first collect a group of known characterizations of ep matrix and give some new characterizations of ep matrices then we define weighted ep matrix and present a wealth of characterizations for weighted ep matrix through various rank formulas for matrices and their generalized inverses

symbolic graph matching with the em algorithm abstract this paper describes how relational graph matching can be effected using the expectation and maximisation algorithm according to this viewpoint matching is realised as a two step iterative em like process firstly updated symbolic matches are located so as to minimise the divergence between the model and data graphs secondly with the updated matches to hand probabilities describing the affinity between nodes in the model and data graphs may be computed the probability distributions underpinning this study are computed using a simple model of uniform matching errors as a result the expected likelihood function is defined over a family of exponential distributions of hamming distance we evaluate our matching method and offer comparison with both mean field annealing and quadratic assignment

on the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings abstract the purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in banach spaces the results presented in this paper extend and improve the corresponding results of h bauschke the approximation of fixed points of compositions of nonexpansive mappings in hilbert space j math anal appl 202 1996 150 159 b halpern fixed points of nonexpansive maps bull amer math soc 73 1967 957 961 p l lions approximation de points fixes de contractions c r acad sci paris ser a 284 1977 1357 1359 s reich strong convergence theorems for resolvents of accretive operators in banach spaces j math anal appl 75 1980 287 292 z h sun strong convergence of an implicit iteration process for a finite family of asymptotically quasi nonexpansive mappings j math anal appl 286 2003 351 358 r wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491 h k xu m g ori an implicit iterative process for nonexpansive mappings numer funct anal optimiz 22 2001 767 773 y y zhou s s chang convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in banach spaces numer funct anal optimiz 23 2002 911 921

some convergence theorems of non implicit iteration process with errors for a finite families of i asymptotically nonexpansive mappings the purpose of this paper is to study the weak and strong convergence of non implicit iteration process with errors to a common fixed point for a finite family of i asymptotically quasi nonexpansive mappings in banach spaces the results presented in this paper extend and improve the corresponding results of several authors 1 2 7 14 17 19 22 29

a new method for solving a system of generalized nonlinear variational inequalities in banach spaces abstract the purpose of this paper is by using the generalized projection approach to introduce an iterative scheme for finding a solution to a system of generalized nonlinear variational inequality problem under suitable conditions some existence and strong convergence theorems are established in uniformly smooth and strictly convex banach spaces the results presented in the paper improve and extend some recent results

an image understanding system using attributed symbolic representation and inexact graph matching this paper presents a powerful image understanding system that utilizes a semantic syntactic or attributed synibolic representation scheme in the form of attributed relational graphs arg s for comprehending the global information contents of images nodes in the arg represent the global image features while the relations between those features are represented by attributed branches between their corresponding nodes the extraction of arg representation from images is achieved by a multilayer graph transducer scheme this scheme is basically a rule based system that uses a combination of model driven and data driven concepts in performing a hierarchical symbolic mapping of the image information content from the spatial domain representation into a global representation further analysis and inter pretation of the imagery data is performed on the extracted arg representation a distance measure between images is defined in terms of the distance between their respective arg representations the distance between two arg s and the inexact matching of their respective components are calculated by an efficient dynamic programming technique the system handles noise distortion and ambiguity in real world images by two means namely through modeling and embedding them into the transducer s mapping rules as well as through the appropriate cost of error transformation for the inexact matching of the arg image representation two illustrative experiments are presented to demonstrate some capabilities of the proposed system experiment i deals with locating objects in multiobject scenes while experiment ii is concerned with target detection in sar images

rank equalities and inequalities for kronecker products of matrices with applications some rank equalities and inequalities are established for kronecker products of matrices as applications upper and lower bounds are presented for the dimensions of the ranges of the two well known linear transformations t 1 x x axb and t 2 x ax xb

on hybrid projection methods for asymptotically quasi φ nonexpansive mappings the purpose of this paper is to consider the problem of approximating a common fixed point of two asymptotically quasi f nonexpansive mappings based on hybrid projection methods strong convergence theorems are established in a real banach space

nonsingularity of the difference and the sum of two idempotent matrices abstract gro s and trenkler 1 pointed out that if a difference of idempotent matrices p and q is nonsingular then so is their sum and koliha et al 2 expressed explicitly a condition which combined with the nonsingularity of p q ensures the nonsingularity of p q in the present paper these results are strengthened by showing that the nonsingularity of p q is in fact equivalent to the nonsingularity of any combination ap bq cpq where a 0 b 0 a b c and the nonsingularity of p q is equivalent to the nonsingularity of any combination ap bq cpq where a 0 b 0 a b c

convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications in this paper we consider the problem of convergence of an iterative algorithm for a system of generalized variational inequalities and a nonexpansive mapping strong convergence theorems are established in the framework of real banach spaces

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

graph matching by relaxation of fuzzy assignments graphs are very powerful and widely used representational tools in computer applications we present a relaxation approach to sub graph matching based on a fuzzy assignment matrix the algorithm has a computational complexity of o n sup 2 m sup 2 where n and m are the number of nodes in the two graphs being matched and can perform both exact and inexact matching to illustrate the performance of the algorithm we summarize the results obtained for more than 12 000 pairs of graphs of varying types weighted graphs attributed graphs and noisy graphs we also compare our results with those obtained using the graduated assignment algorithm

generalized system for relaxed cocoercive mixed variational inequalities in hilbert spaces the approximate solvability of a generalized system for relaxed cocoercive mixed variational inequality is studied by using the resolvent operator technique the results presented in this paper are more general and include many previously known results as special cases

on the existence of solutions of variational inequalities in banach spaces abstract in this paper we study the existence of the solution of the variational inequality tx y x 0 by applying the generalized projection operator k b b where b is a banach space with dual space b and by using the well known fankkm theorem

on a subspace metric based on matrix rank abstract the metric between subspaces m n c n 1 defined by m n rk p m p n where rk denotes rank of a matrix argument and p m and p n are the orthogonal projectors onto the subspaces m and n respectively is investigated such a metric takes integer values only and is not induced by any vector norm by exploiting partitioned representations of the projectors several features of the metric m n are identified it turns out that the metric enjoys several properties possessed also by other measures used to characterize subspaces such as distance also called gap frobenius distance direct distance angle or minimal angle

convergence theorem for i asymptotically quasi nonexpansive mapping in hilbert space abstract let h be a hilbert space with inner product and norm and let k be weakly compact a subset of h let t k k be nonlinear mapping and i k k be a nonlinear bounded mapping in this paper we define the i asymptotically quasi nonexpansive mapping in hilbert space if t is an i asymptotically quasi nonexpansive mapping then we prove that 1 n i 0 n 1 t i u for u k as n is weakly almost convergent to its asymptotic center

column space equalities for orthogonal projectors in their paper y tian g p h styan rank equalities for idempotent and involutory matrices linear algebra appl 335 2001 101 117 tian and styan established several rank equalities involving a pair of idempotent matrices p and q subsequently these results are reinvestigated from the point of view of the following question provided that idempotent p q are hermitian which relationships given in the aforementioned paper remain valid when ranks are replaced with column spaces simultaneously some related results are established which shed additional light on the links between subspaces attributed to various functions of a pair of orthogonal projectors

on chidume s open questions and approximate solutions of multivalued strongly accretive mapping equations in banach spaces abstract some new convergence theorems of the ishikawa iteration process for multi valued pseudo contractive mappings and multi valued accretive mappings are obtained which affirmatively answer the open questions put forth by 4

rank equalities for idempotent matrices with applications we show by elementary block matrix operations a variety of rank equalities related to sums of two idempotent matrices and give various consequences and applications

rank equalities for idempotent and involutary matrices abstract we establish several rank equalities for idempotent and involutary matrices in particular we obtain new formulas for the rank of the difference the sum the product and the commutator of idempotent or involutary matrices extensions to scalar potent matrices are also included our matrices are complex and are not necessarily hermitian

revisitation of the product of two orthogonal projectors abstract several results involving a product of two orthogonal projectors i e hermitian idempotent matrices are established by exploring a representation of the product as a partitioned matrix these results concern for instance rank trace range null space generalized inverses and spectral properties of the product and its various functions particular attention is paid to the conditions equivalent to the requirement that the product of two orthogonal projectors is an orthogonal projector itself and these characterizations refer to such known classes of matrices as hermitian involutory normal star dagger unitary as well as partial isometries and semi orthogonal projectors moreover some results dealing with the notions of parallel sum and spectral norm are obtained the variety of problems considered shows that the approach utilized in the paper provides a powerful tool of wide applicability

using rank formulas to characterize equalities for moore penrose inverses of matrix products a group of rank formulas related to the moore penrose inverses ab and b a are established various interesting consequences of these rank formulas are obtained the extensions of these results to the weighted moore penrose inverses ab m n and b p na m p are considered

properties of the combinations of commutative idempotents this paper deals with idempotent operators i e p2 p and n potent operators i e pn p we characterize the invertibility the group invertibility and idempotency of the linear combinations of idempotent operators or n potent operators under certain commutativity property imposed on them

eigenvalues of functions of orthogonal projectors by representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices several characterizations concerning eigenvalues of various functions of the pair are obtained these results substantially extend the ones already available in the literature additionally some related results dealing with the functions of a pair of orthogonal projectors are provided with the emphasis laying on the problem of invertibility

generalized projection method for a variational inequality system with different mapping in banach spaces the approximate solution of a system for variational inequality with different mapping in hilbert spaces has been studied based on the convergence of projection methods however little research has been done in banach space the primary reason is that projection mapping lacks preferable properties in banach space in this paper we introduce generalized projection methods by using these methods the results presented in this paper have extended the main results of chang et al s and huang and noor s recent works s s chang h w joseph lee c k chan generalized system for relaxed cocoercive variational inequalities in hilbert spaces appl math lett 20 2007 329 334 zhenyu huang muhammad aslam noor an explicit projection method for a system of nonlinear variational inequalities with different c r cocoercive mappings appl math comput 190 2007 356 361 from hilbert spaces to banach space

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

expansion formulas for the inertias of hermitian matrix polynomials and matrix pencils of orthogonal projectors abstract this paper gives a group of expansion formulas for the inertias of hermitian matrix polynomials a a 2 i a 2 and a a 3 through some congruence transformations for block matrices where a is a hermitian matrix then the paper derives various expansion formulas for the ranks and inertias of some matrix pencils generated from two or three orthogonal projectors and hermitian unitary matrices as applications the paper establishes necessary and sufficient conditions for many matrix equalities to hold as well as many inequalities in the lowner partial ordering to hold

improving consistency and reducing ambiguity in stochastic labeling an optimization approach we approach the problem of labeling a set of objects from a quantitative standpoint we define a world model in terms of transition probabilities and propose a definition of a class of global criteria that combine both ambiguity and consistency a projected gradient algorithm is developed to minimize the criterion we show that the minimization procedure can be implemented in a highly parallel manner results are shown on several examples and comparisons are made with relaxation labeling techniques

on oblique projectors abstract oblique projectors are useful in many contexts especially in the instrumental variable estimation in regression models in econometrics where the disturbance term tends to be correlated with predictor variables this paper addresses two important questions regarding oblique projectors 1 products of two oblique projectors and 2 decompositions of oblique projectors in 1 we investigate what determines the onto and the along spaces of an oblique projector defined as a product of two oblique projectors under a variety of conditions in 2 we examine various decompositions of oblique projectors when both predictor variables and instrumental variables consist of two distinct sets of variables these decompositions are analogous to those of orthogonal projectors

the difference and sum of projectors abstract the aim of this paper is to present new results on the invertibility of the sum of projectors and new relations between the nonsingularity of the difference and the sum of projectors we also give simple proofs without referring to rank theory of some results on the nonsingularity of the difference of projectors obtained by gros and trenkler siam j matrix anal appl 21 1999 390

a mann type iterative scheme for variational inequalities in noncompact subsets of banach spaces abstract in this paper we study the existence of solutions and approximation of the solutions by using a mann type iterative scheme for variational inequalities in noncompact subsets of banach spaces the results presented in this paper generalize the corresponding results of j li j li on the existence of solutions of variational inequalities in banach spaces j math anal appl 295 2004 115 126

idempotency of linear combinations of two idempotent matrices abstract a complete solution is established to the problem of characterizing all situations where a linear combination of two different idempotent matrices p 1 and p 2 is also an idempotent matrix including naturally three such situations known in the literature viz if the combination is either the sum p 1 p 2 or one of the differences p 1 p 2 p 2 p 1 and appropriate additional conditions are fulfilled the solution asserts that in the particular case where p 1 and p 2 are complex matrices such that p 1 p 2 is hermitian these three situations exhaust the list of all possibilities and that this list extends when the above assumption on p 1 and p 2 is violated a statistical interpretation of the idempotency problem considered in this note is also pointed out

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

an energy function and continuous edit process for graph matching the contributions of this article are twofold first we develop a new nonquadratic energy function for graph matching the starting point is a recently reported mixture model that gauges relational consistency using a series of exponential functions of the hamming distances between graph neighborhoods we compute the effective neighborhood potentials associated with the mixture model by identifying the single probability function of zero kullback divergence this new energy function is simply a weighted sum of graph hamming distances the second contribution is to locate matches by graduated assignment rather than solving the meanfield saddle point equations which are intractable for our nonquadratic energy function we apply the soft assign ansatz to the derivatives of our energy function here we introduce a novel departure from the standard graduated assignment formulation of graph matching by allowing the connection strengths of the data graph to update themselves the aim is to provide a means by

twosetsofnewcharacterizationsfornormaland ep matrices two sets of new characterizations for normal matrices and ep matrices are presented which are derived through ranks and generalized inverses of matrices

on the convergence theorems of implicit iteration process for a finite family of i asymptotically nonexpansive mappings in this paper we consider the weak and strong convergence of implicit iteration process to a common fixed point of i asymptotically nonexpansive mappings the main results extend to a finite family of i asymptotically nonexpansive mappings in a banach space

the approximation of fixed points of compositions of nonexpansive mappings in hilbert space determining fixed points of nonexpansive mappings is a frequent problem in mathematics and physical sciences an algorithm for finding common fixed points of nonexpansive mappings in hilbert space essentially due to halpern is analyzed the main theorem extends wittmann s recent work and partially generalizes a result by lions algorithms of this kind have been applied to the convex feasibility problem