feature interaction in subspace clustering using the choquet integral subspace clustering has recently emerged as a popular approach to removing irrelevant and redundant features during the clustering process however most subspace clustering methods do not consider the interaction between the features this unawareness limits the analysis performance in many pattern recognition problems in this paper we propose a novel subspace clustering technique by introducing the feature interaction using the concepts of fuzzy measures and the choquet integral this new framework of subspace clustering can provide optimal subsets of interacted features chosen for each cluster and hence can improve clustering based pattern recognition tasks various experimental results illustrate the effective performance of the proposed method

cross domain soft partition clustering with diversity measure and knowledge reference conventional soft partition clustering approaches such as fuzzy c means fcm maximum entropy clustering mec and fuzzy clustering by quadratic regularization fc qr are usually incompetent in those situations where the data are quite insufficient or much polluted by underlying noise or outliers in order to address this challenge the quadratic weights and gini simpson diversity based fuzzy clustering model qwgsd fc is first proposed as a basis of our work based on qwgsd fc and inspired by transfer learning two types of cross domain soft partition clustering frameworks and their corresponding algorithms referred to as type i type ii knowledge transfer oriented c means ti kt cm and tii kt cm are subsequently presented respectively the primary contributions of our work are four fold 1 the delicate qwgsd fc model inherits the most merits of fcm mec and fc qr with the weight factors in the form of quadratic memberships similar to fcm it can more effectively calculate the total intra cluster deviation than the linear form recruited in mec and fc qr meanwhile via gini simpson diversity index like shannon entropy in mec and equivalent to the quadratic regularization in fc qr qwgsd fc is prone to achieving the unbiased probability assignments 2 owing to the reference knowledge from the source domain both ti kt cm and tii kt cm demonstrate high clustering effectiveness as well as strong parameter robustness in the target domain 3 ti kt cm refers merely to the historical cluster centroids whereas tii kt cm simultaneously uses the historical cluster centroids and their associated fuzzy memberships as the reference this indicates that tii kt cm features more comprehensive knowledge learning capability than ti kt cm and tii kt cm consequently exhibits more perfect cross domain clustering performance and 4 neither the historical cluster centroids nor the historical cluster centroid based fuzzy memberships involved in ti kt cm or tii kt cm can be inversely mapped into the raw data this means that both ti kt cm and tii kt cm can work without disclosing the original data in the source domain i e they are of good privacy protection for the source domain in addition the convergence analyses regarding both ti kt cm and tii kt cm are conducted in our research the experimental studies thoroughly evaluated and demonstrated our contributions on both synthetic and real life data scenarios the novel qwgsd fc model is designed for soft partition clustering two cross domain clustering methods ti tii kt cm are subsequently proposed ti tii kt cm are superior to or competitive with other existing methods the convergences of both ti kt cm and tii kt cm are proved

fuzzy congruences on semigroups abstract in this note we discuss the fuzzy equivalence relation generated by a given fuzzy relation on a set x and the fuzzy congruence relation generated by a given fuzzy relation on a semigroup s we also study the lattice of fuzzy equivalence congruence relations on a semigroup and give certain lattice theoretic properties

fuzzy congruences and compatible fuzzy partitions the main purpose of this paper is to prove that fuzzy congruence relations and compatible fuzzy partitions on a fuzzy algebra f can be described with the help of crisp congruence relations and compatible partitions on the cuts of f

the ψs polar decomposition when the cosquare of s is normal abstract let s m n c be nonsingular such that s t s is normal that is the cosquare of s is normal set s a s 1 a t s for a m n c we determine conditions on a so that a has a s polar decomposition we also find the possible jordan canonical forms of a s orthogonal matrix and of a s skew symmetric matrix in the cases a 1 s t s and b 1 s t s

on t congruence l relations on groups and rings in this paper we further study t congruence l relations on groups and rings and give the formulas for calculating the t congruence l relations generated by l relations where t is an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l

a new canonical form for complex symmetric matrices it is well known that every real symmetric matrix and every complex hermitian matrix is diagonalizable i e orthogonally similar to a diagonal matrix however a complex symmetric matrix with repeated eigenvalues may fail to be diagonalizable we present a block diagonal canonical form in which each block is quasi diagonal to which every complex symmetric matrix is orthogonally similar as far as applications are concerned complex symmetric matrices as opposed to hermitian matrices play an important role in theories of wave propagation in continuous media e g elasticity thermoelasticity

fuzzy normal subgroups and fuzzy cosets abstract we introduce here the notion of 1 i a fuzzy normal subgroup and 2 ii a fuzzy coset we prove that for a group g a fuzzy subgroup is fuzzy normal if and only if it is constant on the conjugate classes of g further we show that the level subgroups of a fuzzy normal subgroup are all normal we obtain several results which are analogs of some basic theorems of group theory in particular a fuzzy version of lagrange s theorem is established

on the ϕj polar decomposition of matrices abstract we present new results on the j polar decomposition of matrices we show that every symplectic matrix may be written as a product of symplectic operation matrices we present a simple form attained under symplectic equivalence which makes it easy to determine if a matrix does not have a j polar decomposition we also determine the rank 4 matrices with j polar decomposition

convergence and consistency of fuzzy c means isodata algorithms the fuzzy c means isodata algorithm is usually described in terms of clustering a finite data set an equivalent point of view is that the algorithm clusters the support points of a finite support probability distribution motivated by recent work on the hard version of the algorithm this paper extends the definition to arbitrary distributions and considers asymptotic properties it is shown that fixed points of the algorithm are stationary points of the fuzzy objective functional and vice versa when the algorithm is iteratively applied to an initial prototype set the sequence of prototype sets produced approaches the set of fixed points if an unknown distribution is approximated by the empirical distribution of stationary ergodic observations then as the number of observations grows large fixed points of the algorithm based on the empirical distribution approach fixed points of the algorithm based on the true distribution furthermore with respect to minimizing the fuzzy objective functional the algorithm based on the empirical distribution is asymptotically at least as good as the algorithm based on the true distribution

tl subrings and tl ideals part 1 basic concepts abstract in 25 27 we put forward a rudimentary frame of the theory of tl subalgebras in 26 we established a theory of tl subgroups and tl subgroups in this paper we will continue our work in concern with basic ideas provided in 25 27 introduce the notions of tl subrings and tl ideals of a ring and study some of their properties where t is an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l

topological characterizations of generalized fuzzy rough sets in this paper topological characterizations of generalized fuzzy rough sets are investigated in the context of basic rough equalities various fuzzy topologies induced by different fuzzy relations are studied with respect to the lower fuzzy rough approximation operator determined by a fuzzy implicator which is left continuous in the first argument and right continuous in the second argument the relationships among fuzzy topologies induced by fuzzy relations are discussed the algebraic structures of t similarity set of fuzzy relations are investigated with respect to different fuzzy implicators where t is a left continuous t norm moreover the t transitive subset of t similarity set is proven to be a complete distributive lattice

a convergence theorem for the fuzzy subspace clustering fsc algorithm we establish the convergence of the fuzzy subspace clustering fsc algorithm by applying zangwill s convergence theorem we show that the iteration sequence produced by the fsc algorithm terminates at a point in the solution set s or there is a subsequence converging to a point in s in addition we present experimental results that illustrate the convergence properties of the fsc algorithm in various scenarios

contragredient equivalence a canonical form and some applications abstract let a and c be m by n complex matrices and let b and d be n by m complex matrices the pair a b is contragrediently equivalent to the pair c d if there are square nonsingular complex matrices x and y such that xay 1 c and ybx 1 d contragredient equivalence is a common generalization of four basic equivalence relations similarity consimilarity complex orthogonal equivalence and unitary equivalence we develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that a a t is contragrediently equivalent to c c t if and only if there are complex orthogonal matrices p and q such that c paq using this result we show that the following are equivalent for a given n by n complex matrix a 1 1 a qs for some complex orthogonal q and some complex symmetric s 2 2 a t a is similar to aa t 3 3 a a t is contragrediently equivalent to a t a 4 4 a q 1 a t q 2 for some complex orthogonal q 1 q 2 5 5 a pa t p for some complex orthogonal p we then consider a linear operator on n by n complex matrices that shares the following properties with transpose operators for every pair of n by n complex matrices a and b a preserves the spectrum of a b a a and c ab b a we show that a a is contragrediently similar to b b if and only if a x 1 bx 2 for some nonsingular x 1 x 2 that satisfy x 1 1 x 1 and x 1 2 x 2 we also consider a factorization of the form a xy where x 1 x and y y we use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of flanders concerning the relative sizes of the nilpotent jordan blocks of ab and ba we present a sufficient condition for the existence of square roots of ab and ba and close with a canonical form for complex orthogonal equivalence

fuzzy equivalence relations abstract in this note we define and discuss properties of fuzzy equivalence relations on a set in particular we show that there exists a correspondence between fuzzy equivalence relations and certain classes of fuzzy subsets we also study the cuts of fuzzy equivalence relations finally certain lattice theoretic properties of fuzzy equivalence relations are discussed

on fuzzy equivalence i fuzzy reflexive symmetric and transitive relations on fuzzy subsets are studied various types of reflexivities and corresponding equivalences are considered the possible fuzzy partitioning of a fuzzy subset is investigated

the jordan canonical forms of complex orthogonal and skew symmetric matrices abstract we study the jordan canonical forms of complex orthogonal and skew symmetric matrices and consider some related results

t type regular l relations on a complete brouwerian lattice abstract in this paper using a general infinitely distributive t norm t on a complete brouwerian lattice l we introduce the concept of t type regular l relations study some basic properties of them and give out some methods of calculating the maximal t type generalized inverse l relations of t type regular l relations

the sums of symplectic hamiltonian and skew hamiltonian matrices abstract a complex 2 n 2 n matrix a is called skew hamiltonian hamiltonian and symplectic if a j a a j a and a j a 1 respectively in which j 0 i n i n 0 and a j j 1 a t j we prove that each 2 n 2 n matrix is a sum of type symplectic hamiltonian a 2 n 2 n matrix a is a sum of type symplectic symplectic if and only if a a j is similar to a j a a 2 n 2 n matrix a is a sum of type symplectic skew hamiltonian if and only if the jordan blocks of a a j with eigenvalue 2i and size k 2 come in pairs of the form j k 2 i j k 2 i and j k 2 i j k 1 2 i

fuzzy weighted c ordered means clustering algorithm abstract in real life data sets some attributes may have lower importance or even may be completely noninformative the subspace clustering algorithms have been proposed to handle this the soft subspace algorithms are vulnerable to noise and outliers the paper presents a novel algorithm that handles both various importance of attributes and outliers the proposed fuzzy weighted c ordered mean fwcom clustering algorithm elaborates clusters in soft subspaces in each cluster each attribute is assigned a weight from interval 0 1 each attribute has its individual weight importance in each cluster the proposed algorithm applies the ordering technique to effectively reduce the influence of outliers and noise the paper is accompanied by numerical experiments

some properties of l fuzzy approximation spaces based on bounded integral residuated lattices in this paper we present a general framework for the study of l fuzzy rough sets based on bounded integral residuated lattices we introduce and discuss the notions of l fuzzy topological spaces left right lower and left right upper l fuzzy approximation operators based on a complete bounded integral residuated lattice investigate union intersection and composition operations of l fuzzy approximation spaces and study some properties of the l fuzzy topologies generated by these l fuzzy rough approximation operators and the l relations induced by an l fuzzy topology

fuzzy submonoids fuzzy preorders and quasi metrics let l be a residuated lattice then given a monoid m we define a galois connection from the lattice of the compatible l preorders in m and the lattice of l submonoids of m given a set s we define a galois connection between the lattice of the l preorders in s and the lattice of l submonoids of the monoid ss i a link with the notion of quasi metric is also established

unitary equivalence in an indefinite scalar product an analogue of singular value decomposition if h1 is an n n and h2 an m m invertible hermitian matrix and x and y are arbitrary complex m n matrices we call the last matrices equivalent if x u2yu1 for some h1 unitary matrix u1 and some h2 unitary matrix u2 it is well known that if h1 and h2 are positive definite then without loss of generality we can assume that they are identities and x and y are equivalent if and only if the diagonalizable matrices x x and y y have the same spectrum in the present paper we show that in general the jordan form of x x where x is the h1 h2 adjoint of x x h 11x h2 defines a finite number of nonequivalent classes of matrices and that each such class is defined by its integer matrix explicit formulas for all classes having the same jordan form of x x are presented the necessary and sufficient conditions for an operator to be presentable as uz with h unitary u and h self adjoint h consistent z h polar decomposition of the operator are found

tl filters of integral residuated l monoids in this paper we introduce and discuss the concept of tl filters of integral residuated l monoids first we study some basic properties of tl filters and give the formula for calculating the tl filters generated by l subsets then we discuss some properties of tl filters under homomorphisms and study the relationship between tl filters and t congruence l relations on integral commutative residuated l monoids

fuzzy prime ideals of rings abstract the notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers a procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established all fuzzy prime maximal ideals of a given ring are determined by establishing a one to one correspondence between fuzzy prime maximal ideals and the pairs p where p is a prime maximal ideal of the ring and is a prime element dual atom in the complete lattice

hamiltonian square roots of skew hamiltonian matrices revisited abstract recently h fassbender et al linear algebra appl 287 1999 125 proved the following theorem every real skew hamiltonian matrix w has a real hamiltonian square root h i e h 2 w we prove an analog of this theorem for complex matrices our approach may be of independent interest namely we use the polar decomposition of a nonsingular operator acting in a space with the symplectic inner product

fuzzy invariant subgroups and fuzzy ideals abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra as fuzzy invariant subgroups fuzzy ideals and to prove some fundamental properties in particular it will give a characteristic of a field by a fuzzy ideal

distance metric learning for soft subspace clustering in composite kernel space soft subspace clustering algorithms have been successfully used for high dimensional data in recent years however the existing algorithms often utilize only one distance function to evaluate the distance between data items on each feature which cannot deal with datasets with complex inner structures in this paper a composite kernel space cks is constructed based on a set of basis kernels and a novel framework of soft subspace clustering is proposed by integrating distance metric learning in the cks two soft subspace clustering algorithms i e entropy weighting fuzzy clustering in cks for kernel space cks ewfc k and feature space cks ewfc f are thus developed in both algorithms the prototype in the feature space is mapped into the cks by multiple simultaneous mappings one mapping for each cluster which is distinct from existing kernel based clustering algorithms by evaluating the distance on each feature in the cks both cks ewfc k and cks ewfc f learn the distance function adaptively during the clustering process experimental results have demonstrated that the proposed algorithms in general outperform classical clustering algorithms and are immune to ineffective kernels and irrelevant features in soft subspace the composite kernel space is constructed based on a set of basis kernels the general form of soft subspace clustering in cks is presented cks ewfc k and cks ewfc f are proposed under the framework of cks ssc the properties of cks ewfc k and cks ewfc f are investigated both cks ewfc k and cks ewfc f are immune to ineffective kernels

a criterion for determining the number of groups in a data set using sum of squares clustering marriott 1971 biometrics 27 501 514 used a heuristic argument to derive the criterion g2 l w i for determining the number of groups in a data set when the clustering objective function is the withingroup determinant i w 1 an analogous argument is employed to derive a criterion for use with the within group sum of squares objective function trace w the behaviour of both marriott s criterion and the new criterion is investigated by monte carlo methods for homogeneous data based on uniform and independent variables the performance of the new criterion is close to expectation while marriott s criterion shows much more extreme behaviour for grouped data the new criterion correctly identifies the number of groups in 85 of data sets under a wide range of conditions while marriott s criterion shows a success rate of less than 40 the new criterion is illustrated on the wellknown iris data and some cautionary comments are made about its use

a fuzzy k modes algorithm for clustering categorical data this correspondence describes extensions to the fuzzy k means algorithm for clustering categorical data by using a simple matching dissimilarity measure for categorical objects and modes instead of means for clusters a new approach is developed which allows the use of the k means paradigm to efficiently cluster large categorical data sets a fuzzy k modes algorithm is presented and the effectiveness of the algorithm is demonstrated with experimental results

divisible tl subgroups and pure tl subgroups abstract as a continuation of the study of divisible and pure fuzzy subgroups see sidky and mishref 1990 we introduce the concepts of divisible tl subgroups and pure tl subgroups of an additive group where t is an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l and show some of their properties

normality and congruence in fuzzy subgroups abstract it is proved that normal fuzzy subgroups and fuzzy congruence relations determine each other in a group theoretic situation this is analogous to the crisp set theoretic case of groups

fuzzy congruences and fuzzy normal subgroups abstract a complete description of the congruences on a group in terms of its normal subgroups can be seen in papers by a rosenfeld and w j howie in this paper we shall introduce the concept of fuzzy congruences on a groupoid and a group and characterize fuzzy congruences on a group in terms of fuzzy normal subgroups

polar decompositions in finite dimensional indefinite scalar product spaces general theory abstract several classes of polar decompositions of real and complex matrices with respect to a given indefinite scalar product are studied matrices that admit such polar decompositions are described in various ways in particular a full description of all polar decompositions of a given matrix up to the natural similarity between polar decompositions is given

tl subrings and tl ideals part 2 generated tl ideals abstract in this paper we introduce and discuss the tl left right two sided ideal generated by an l subset t being an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l

fuzzy congruence relations abstract if a x f is an universal algebra and if r x denotes the set of all fuzzy subsets of x x i e fuzzy relations on x then the algebraic operations fg f on x are extended to algebraic operations f on r x f is an algebra similar to a it is called relational algebra a fuzzy equivalence relation which is at the same time a fuzzy subalgebra of r x f is called a fuzzy congruence relation on x in this paper we study fuzzy congruence relations amongst other things we prove that the set of all fuzzy congruence relations on an algebra is a complete lattice and an algebraic closure system

convergence theorems for generalized alternating minimization procedures the em algorithm is widely used to develop iterative parameter estimation procedures for statistical models in cases where these procedures strictly follow the em formulation the convergence properties of the estimation procedures are well understood in some instances there are practical reasons to develop procedures that do not strictly fall within the em framework we study em variants in which the e step is not performed exactly either to obtain improved rates of convergence or due to approximations needed to compute statistics under a model family over which e steps cannot be realized since these variants are not em procedures the standard g em convergence results do not apply to them we present an information geometric framework for describing such algorithms and analyzing their convergence properties we apply this framework to analyze the convergence properties of incremental em and variational em for incremental em we discuss conditions under these algorithms converge in likelihood for variational em we show how the e step approximation prevents convergence to local maxima in likelihood

skew ϕ polar decompositions abstract we present necessary and sufficient conditions so that a square complex matrix a can be written as a q s with orthogonal q and skew symmetric s we also present necessary and sufficient conditions so that a 2n by 2n complex matrix b can be written as b w r with symplectic w and hamiltonian r

enhanced soft subspace clustering integrating within cluster and between cluster information while within cluster information is commonly utilized in most soft subspace clustering approaches in order to develop the algorithms other important information such as between cluster information is seldom considered for soft subspace clustering in this study a novel clustering technique called enhanced soft subspace clustering essc is proposed by employing both within cluster and between class information first a new optimization objective function is developed by integrating the within class compactness and the between cluster separation in the subspace based on this objective function the corresponding update rules for clustering are then derived followed by the development of the novel essc algorithm the properties of this algorithm are investigated and the performance is evaluated experimentally using real and synthetic datasets including synthetic high dimensional datasets uci benchmarking datasets high dimensional cancer gene expression datasets and texture image datasets the experimental studies demonstrate that the accuracy of the proposed essc algorithm outperforms most existing state of the art soft subspace clustering algorithms