formulation of elastostatic hybrid trefftz stress elements abstract the hybrid trefftz stress hts element is derived from the fundamental conditions of linear elastostatics the stress and boundary displacement fields are directly approximated consistent finite element variables are so defined as to obtain boundary integral expressions for the finite element equilibrium and compatibility conditions encoded by dual transformations independent of the constitutive relations reciprocity is preserved in the finite element elasticity relations which are also found to be described by boundary integral arrays when the trefftz constraint is enforced consequent upon duality and reciprocity the resulting boundary integral solving system is symmetric and sparse the energy statements associated with the hybrid trefftz stress element are established and the conditions for the existence uniqueness multiplicity and static and kinematic admissibility of the finite element solutions are presented the hts clement is compared with the displacement frame based hybrid trefftz ht d element of jirousek it is shown that the ht d element is in fact an hybrid stress element the numerical tests performed to assess the relative performances of the hts and ht d elements show that better accuracy levels are obtained using discontinuous boundary approximation bases

fast fourier transform method for partial differential equations case study the 2 d diffusion equation abstract the solution of time dependent partial differential equations using discrete i e finite difference versions of those equations traditionally has been done by marching forward in time one time unit per step the authors here extend earlier results to higher dimensions focusing on the solution of the discrete diffusion or heat equation in 2 d and show that the solution may be found using fast fourier transform fft techniques that treat the solution as the output of a linear filter this technique is analogous to an integral method for the differential equation and is related to the green s function of the difference equation it allows the solution to be determined with a single step computation that does not require the solution at intermediate time levels the number of numerical operations is essentially independent of the time level at which the solution is desired the results of numerical examples are given and computation times are shown to be much faster than those of the traditional method for all but small time levels

an efficient matrix factorization based low rank representation for subspace clustering in recent years robust subspace clustering is an important unsupervised clustering problem in machine learning and computer vision communities the recently proposed spectral clustering based approach called low rank representation lrr yields an optimal solution for the case of independent subspaces and partially corrupted data however it has to be solved iteratively and involves singular value decomposition svd at each iteration and then suffers from high computation cost of multiple svds in this paper we propose an efficient matrix tri factorization mtf approach with a positive semidefinite psd constraint to approximate the original nuclear norm minimization nnm problem and mitigate the computation cost of performing svds specially we introduce a matrix tri factorization idea into the original low rank representation framework and then convert it into a small scale matrix nuclear norm minimization problem finally we establish an alternating direction method adm based algorithm to efficiently solve the proposed problem experimental results on a variety of synthetic and real world data sets validate the efficiency robustness and effectiveness of the proposed mtf approach comparing with the state of the art algorithms

robust algebraic segmentation of mixed rigid body and planar motions from two views this paper studies segmentation of multiple rigid body motions in a 3 d dynamic scene under perspective camera projection we consider dynamic scenes that contain both 3 d rigid body structures and 2 d planar structures based on the well known epipolar and homography constraints between two views we propose a hybrid perspective constraint hpc to unify the representation of rigid body and planar motions given a mixture of k hybrid perspective constraints we propose an algebraic process to partition image correspondences to the individual 3 d motions called robust algebraic segmentation ras particularly we prove that the joint distribution of image correspondences is uniquely determined by a set of 2k th degree polynomials a global signature for the union of k motions of possibly mixed type the first and second derivatives of these polynomials provide a means to recover the association of the individual image samples to their respective motions finally using robust statistics we show that the polynomials can be robustly estimated in the presence of moderate image noise and outliers we conduct extensive simulations and real experiments to validate the performance of the new algorithm the results demonstrate that ras achieves notably higher accuracy than most existing robust motion segmentation methods including random sample consensus ransac and its variations the implementation of the algorithm is also two to three times faster than the existing methods the implementation of the algorithm and the benchmark scripts are available at http perception csl illinois edu ras

a multibody factorization method for independently moving objects the structure from motion problem has been extensively studied in the field of computer vision yet the bulk of the existing work assumes that the scene contains only a single moving object the more realistic case where an unknown number of objects move in the scene has received little attention especially for its theoretical treatment in this paper we present a new method for separating and recovering the motion and shape of multiple independently moving objects in a sequence of images the method does not require prior knowledge of the number of objects nor is dependent on any grouping of features into an object at the image level for this purpose we introduce a mathematical construct of object shapes called the shape interaction matrix which is invariant to both the object motions and the selection of coordinate systems this invariant structure is computable solely from the observed trajectories of image features without grouping them into individual objects once the matrix is computed it allows for segmenting features into objects by the process of transforming it into a canonical form as well as recovering the shape and motion of each object the theory works under a broad set of projection models scaled orthography paraperspective and affine but they must be linear so it excludes projective cameras

two view multibody structure from motion we present an algebraic geometric approach to 3 d motion estimation and segmentation of multiple rigid body motions from noise free point correspondences in two perspective views our approach exploits the algebraic and geometric properties of the so called multibody epipolar constraint and its associated multibody fundamental matrix which are natural generalizations of the epipolar constraint and of the fundamental matrix to multiple motions we derive a rank constraint on a polynomial embedding of the correspondences from which one can estimate the number of independent motions as well as linearly solve for the multibody fundamental matrix we then show how to compute the epipolar lines from the first order derivatives of the multibody epipolar constraint and the epipoles by solving a plane clustering problem using generalized pca gpca given the epipoles and epipolar lines the estimation of individual fundamental matrices becomes a linear problem the clustering of the feature points is then automatically obtained from either the epipoles and epipolar lines or from the individual fundamental matrices although our approach is mostly designed for noise free correspondences we also test its performance on synthetic and real data with moderate levels of noise

an index of topological preservation for feature extraction abstract this paper is about the ability of principal components analysis the sammon algorithm and an extension of the kohonen self organizing feature map to preserve spatial order during feature extraction on unlabeled data transformations to q space that preserve the order of all pairwise distances in any set of vectors in p space are defined as metric topology preserving mtp transformations we give a necessary and sufficient condition for this new property in terms of the spearman rank correlation coefficient unlike many other measures of extracted feature quality the mtp index is independent of the extraction method a modification of the kohonen self organizing feature map algorithm that extracts vectors in q space from data in p space is developed the extent to which principal components sammon s algorithm and our extension of the self organizing feature map sofm preserve the mtp property is discussed our mtp index shows that the first two methods preserve distance ranks on seven data sets much more effectively than extended sofm

quantifying the neighborhood preservation of self organizing feature maps it is shown that a topographic product p first introduced in nonlinear dynamics is an appropriate measure of the preservation or violation of neighborhood relations it is sensitive to large scale violations of the neighborhood ordering but does not account for neighborhood ordering distortions caused by varying areal magnification factors a vanishing value of the topographic product indicates a perfect neighborhood preservation negative positive values indicate a too small too large output space dimensionality in a simple example of maps from a 2d input space onto 1d 2d and 3d output spaces it is demonstrated how the topographic product picks the correct output space dimensionality in a second example 19d speech data are mapped onto various output spaces and it is found that a 3d output space instead of 2d seems to be optimally suited to the data this is an agreement with a recent speech recognition experiment on the same data set

completing low rank matrices with corrupted samples from few coefficients in general basis subspace recovery from the corrupted and missing data is crucial for various applications in signal processing and information theory to complete missing values and detect column corruptions the existing robust matrix completion mc methods mostly concentrate on recovering a low rank matrix from a few corrupted coefficients with respect to standard basis which however does not apply to more general basis e g fourier basis in this paper we prove that the range space of an m times n matrix with rank r can be exactly recovered from a few coefficients with respect to general basis though r and the number of corrupted samples are both as high as o min m n log 3 m n our model covers the previous ones as special cases and robust mc can recover the intrinsic matrix with a higher rank moreover we suggest a universal choice of the regularization parameter which is lambda 1 sqrt log n by our ell 2 1 filtering algorithm which has theoretical guarantees we can further reduce the computational cost of our model as an application we also find that the solutions to extended robust low rank representation and to our extended robust mc are mutually expressible so both our theory and algorithm can be applied to the subspace clustering problem with missing values under certain conditions the experiments verify our theories

highly articulated kinematic structure estimation combining motion and skeleton information in this paper we present a novel framework for unsupervised kinematic structure learning of complex articulated objects from a single view 2d image sequence in contrast to prior motion based methods which estimate relatively simple articulations our method can generate arbitrarily complex kinematic structures with skeletal topology via a successive iterative merging strategy the iterative merge process is guided by a density weighted skeleton map which is generated from a novel object boundary generation method from sparse 2d feature points our main contributions can be summarised as follows i an unsupervised complex articulated kinematic structure estimation method that combines motion segments with skeleton information ii an iterative fine to coarse merging strategy for adaptive motion segmentation and structural topology embedding iii a skeleton estimation method based on a novel silhouette boundary generation from sparse feature points using an adaptive model selection method iv a new highly articulated object dataset with ground truth annotation we have verified the effectiveness of our proposed method in terms of computational time and estimation accuracy through rigorous experiments with multiple datasets our experiments show that the proposed method outperforms state of the art methods both quantitatively and qualitatively

spectral clustering on multiple manifolds spectral clustering sc is a large family of grouping methods that partition data using eigenvectors of an affinity matrix derived from the data though sc methods have been successfully applied to a large number of challenging clustering scenarios it is noteworthy that they will fail when there are significant intersections among different clusters in this paper based on the analysis that sc methods are able to work well when the affinity values of the points belonging to different clusters are relatively low we propose a new method called spectral multi manifold clustering smmc which is able to handle intersections in our model the data are assumed to lie on or close to multiple smooth low dimensional manifolds where some data manifolds are separated but some are intersecting then local geometric information of the sampled data is incorporated to construct a suitable affinity matrix finally spectral method is applied to this affinity matrix to group the data extensive experiments on synthetic as well as real datasets demonstrate the promising performance of smmc

ordering of self organizing maps in multidimensional cases it has been proved that in one dimensional cases the weights of kohonen s self organizing maps som will become ordered with probability 1 once the weights are ordered they cannot become disordered in future training it is difficult to analyze kohonen s soms in multidimensional cases however it has been conjectured that similar results seem to be obtainable in multidimensional cases in this note we show that in multidimensional cases even though the weights are ordered at some time it is possible that they become disordered in the future

innovation pursuit a new approach to subspace clustering in subspace clustering a group of data points belonging to a union of subspaces are assigned membership to their respective subspaces this paper presents a new approach dubbed innovation pursuit ipursuit to the problem of subspace clustering using a new geometrical idea whereby subspaces are identified based on their relative novelties we present two frameworks in which the idea of innovation pursuit is used to distinguish the subspaces underlying the first framework is an iterative method that finds the subspaces consecutively by solving a series of simple linear optimization problems each searching for a direction of innovation in the span of the data potentially orthogonal to all subspaces except for the one to be identified in one step of the algorithm a detailed mathematical analysis is provided establishing sufficient conditions for ipursuit to correctly cluster the data the proposed approach can provably yield exact clustering even when the subspaces have significant intersections it is shown that the complexity of the iterative approach scales only linearly in the number of data points and subspaces and quadratically in the dimension of the subspaces the second framework integrates ipursuit with spectral clustering to yield a new variant of spectral clustering based algorithms the numerical simulations with both real and synthetic data demonstrate that ipursuit can often outperform the state of the art subspace clustering algorithms more so for subspaces with significant intersections and that it significantly improves the state of the art result for subspace segmentation based face clustering

constraint satisfaction neural networks for image segmentation abstract constraint satisfaction neural networks csnns are proposed for image recognition which is the process to identify objects of interest in an image the image recognition problem is formulated as a constraint satisfaction problem csp the regions resulting from a signal based image segmentation algorithm can be considered as objects and their entities as labels in the csp the proposed neural network is multi layered and the function of each layer is to identify one physical entity label from the input regions objects through cascading these layers each label can be assigned to the corresponding region s attention is focused on recognizing x ray ct computed tomography images of human heads e g anatomic structure in a medical image the proposed csnn is capable of identifying the components of a brain in a preliminarily segmented image the major components in a human head include bone ventricle and gyrus therefore three layers are needed for this task each layer is constructed based on the principles proposed in kohonen s self organizing feature maps and the optimal linear associative memories and their weights are trained by the features of each entity in the prestored model the experimental results show that the proposed approach is very promising

a multilayer neural network with piecewise linear structure and back propagation learning a multilayer neural network which is given a two layer piecewise linear structure for every cascaded section is proposed the neural networks have nonlinear elements that are neither sigmoidal nor of a signum type each nonlinear element is an absolute value operator it is almost everywhere differentiable which makes back propagation feasible in a digital setting both the feedforward signal propagation and the backward coefficient update rules belong to the class of regular iterative algorithms this form of neural network specializes in functional approximation and is anticipated to have applications in control communications and pattern recognition

parameterized principal component analysis abstract when modeling multivariate data one might have an extra parameter of contextual information that could be used to treat some observations as more similar to others for example images of faces can vary by age and one would expect the face of a 40 year old to be more similar to the face of a 30 year old than to a baby face we introduce a novel manifold approximation method parameterized principal component analysis ppca that models data with linear subspaces that change continuously according to the extra parameter of contextual information e g age instead of ad hoc atlases special care has been taken in the loss function and the optimization method to encourage smoothly changing subspaces across the parameter values the approach ensures that each observation s projection will share information with observations that have similar parameter values but not with observations that have large parameter differences we tested ppca on artificial data based on known smooth functions of an added parameter as well as on two real datasets with different types of parameters we compared ppca to pca sparse pca and to independent principal component analysis ipca an atlas based method that groups observations by their parameter values and projects each group using pca with no sharing of information between groups ppca recovers the known functions with less error and projects the datasets test set observations with consistently less reconstruction error than ipca does in some cases where the manifold is truly nonlinear pca outperforms all the other manifold approximation methods compared

relations among some low rank subspace recovery models recovering intrinsic low dimensional subspaces from data distributed on them is a key preprocessing step to many applications in recent years a lot of work has modeled subspace recovery as low rank minimization problems we find that some representative models such as robust principal component analysis r pca robust low rank representation r lrr and robust latent low rank representation r latlrr are actually deeply connected more specifically we discover that once a solution to one of the models is obtained we can obtain the solutions to other models in closed form formulations since r pca is the simplest our discovery makes it the center of low rank subspace recovery models our work has two important implications first r pca has a solid theoretical foundation under certain conditions we could find globally optimal solutions to these low rank models at an overwhelming probability although these models are nonconvex second we can obtain significantly faster algorithms for these models by solving r pca first the computation cost can be further cut by applying low complexity randomized algorithms for example our novel filtering algorithm to r pca although for the moment the formal proof of our filtering algorithm is not yet available experiments verify the advantages of our algorithm over other state of the art methods based on the alternating direction method

estimation of subspace arrangements with applications in modeling and segmenting mixed data recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high dimensional data that have very complicated structures from both geometric and statistical points of view such unsorted data are considered mixed as different parts of the data have significantly different structures which cannot be described by a single model in this paper we propose to use subspace arrangements a union of multiple subspaces for modeling mixed data each subspace in the arrangement is used to model just a homogeneous subset of the data thus multiple subspaces together can capture the heterogeneous structures within the data set in this paper we give a comprehensive introduction to a new approach for the estimation of subspace arrangements this is known as generalized principal component analysis gpca in particular we provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust even when the given data are corrupted by noise or contaminated with outliers this new method in many ways improves and generalizes extant methods for modeling or clustering mixed data there have been successful applications of this new method to many real world problems in computer vision image processing and system identification in this paper we will examine several of those representative applications this paper is intended to be expository in nature however in order that this may serve as a more complete reference for both theoreticians and practitioners we take the liberty of filling in several gaps between the theory and the practice in the existing literature

minimum error entropy based sparse representation for robust subspace clustering this paper addresses the problem of clustering data points that are approximately drawn from multiple subspaces recently a large family of spectral clustering based methods such as sparse subspace clustering ssc and low rank representation lrr have been proposed in this paper we present a general formulation to unify many of them within a common framework based on atomic representation since mean square error mse relies heavily on the gaussianity assumption the previous mse based subspace clustering methods have the limitation of being sensitive to non gaussian noise in this paper we develop a novel subspace clustering method termed meessc by specifying the minimum error entropy mee as the loss function and the sparsity inducing atomic set we show that meessc can well overcome the above limitation the experimental results on both synthetic and real data verify the effectiveness of the proposed method

operation of an associative memory abstract this paper describes the operation of an associative memory lyam governed by only ordinary differential equations useful for pattern clustering several computer simulations illustrate its operation as an unsupervised classifier vector quantizer and content addressable memory

a general framework for subspace detection in unordered multidimensional data the analysis of large volumes of unordered multidimensional data is a problem confronted by scientists and data analysts every day often it involves searching for data alignments that emerge as well defined structures or geometric patterns in datasets for example straight lines circles and ellipses represent meaningful structures in data collected from electron backscatter diffraction particle accelerators and clonogenic assays also customers with similar behavior describe linear correlations in e commerce databases we describe a general approach for detecting data alignments in large unordered noisy multidimensional datasets in contrast to classical techniques such as the hough transforms which are designed for detecting a specific type of alignment on a given type of input our approach is independent of the geometric properties of the alignments to be detected as well as independent of the type of input data thus it allows concurrent detection of multiple kinds of data alignments in datasets containing multiple types of data given its general nature optimizations developed for our technique immediately benefit all its applications regardless the type of input data

motion segmentation in the presence of outlying incomplete or corrupted trajectories in this paper we study the problem of segmenting tracked feature point trajectories of multiple moving objects in an image sequence using the affine camera model this problem can be cast as the problem of segmenting samples drawn from multiple linear subspaces in practice due to limitations of the tracker occlusions and the presence of nonrigid objects in the scene the obtained motion trajectories may contain grossly mistracked features missing entries or corrupted entries in this paper we develop a robust subspace separation scheme that deals with these practical issues in a unified mathematical framework our methods draw strong connections between lossy compression rank minimization and sparse representation we test our methods extensively on the hopkins155 motion segmentation database and other motion sequences with outliers and missing data we compare the performance of our methods to state of the art motion segmentation methods based on expectation maximization and spectral clustering for data without outliers or missing information the results of our methods are on par with the state of the art results and in many cases exceed them in addition our methods give surprisingly good performance in the presence of the three types of pathological trajectories mentioned above all code and results are publicly available at http perception csl uiuc edu coding motion

robust subspace clustering via thresholding the problem of clustering noisy and incompletely observed high dimensional data points into a union of low dimensional subspaces and a set of outliers is considered the number of subspaces their dimensions and their orientations are assumed unknown we propose a simple low complexity subspace clustering algorithm which applies spectral clustering to an adjacency matrix obtained by thresholding the correlations between data points in other words the adjacency matrix is constructed from the nearest neighbors of each data point in spherical distance a statistical performance analysis shows that the algorithm exhibits robustness to additive noise and succeeds even when the subspaces intersect specifically our results reveal an explicit tradeoff between the affinity of the subspaces and the tolerable noise level we furthermore prove that the algorithm succeeds even when the data points are incompletely observed with the number of missing entries allowed to be up to a log factor linear in the ambient dimension we also propose a simple scheme that provably detects outliers and we present numerical results on real and synthetic data

a class of competitive learning models which avoids neuron underutilization problem we study a qualitative property of a class of competitive learning cl models which is called the multiplicatively biased competitive learning mbcl model namely that it avoids neuron underutilization with probability one as time goes to infinity in the mbcl the competition among neurons is biased by a multiplicative term while only one weight vector is updated per learning step this is of practical interest since its instances have computational complexities among the lowest in existing cl models in addition in applications like classification vector quantizer design and probability density function estimation a necessary condition for optimal performance is to avoid neuron underutilization hence it is possible to define instances of mbcl to achieve optimal performance in these applications

toward a theory of statistical tree shape analysis to develop statistical methods for shapes with a tree structure we construct a shape space framework for tree shapes and study metrics on the shape space this shape space has singularities which correspond to topological transitions in the represented trees we study two closely related metrics on the shape space ted and qed qed is a quotient euclidean distance arising naturally from the shape space formulation while ted is the classical tree edit distance using gromov s metric geometry we gain new insight into the geometries defined by ted and qed we show that the new metric qed has nice geometric properties that are needed for statistical analysis geodesics always exist and are generically locally unique following this we can also show the existence and generic local uniqueness of average trees for qed ted while having some algorithmic advantages does not share these advantages along with the theoretical framework we provide experimental proof of concept results on synthetic data trees as well as small airway trees from pulmonary ct scans this way we illustrate that our framework has promising theoretical and qualitative properties necessary to build a theory of statistical tree shape analysis

a new approach to dimensionality reduction theory and algorithms this paper applies whitney s embedding theorem to the data reduction problem and introduces a new approach motivated in part by the constructive proof of the theorem the notion of a good projection is introduced which involves picking projections of the high dimensional system that are optimized such that they are easy to invert the basic theory of the approach is outlined and algorithms for finding the projections are presented and applied to several test cases a method for constructing the inverse projection is detailed and its properties including a new measure of complexity are discussed finally well known methods of data reduction are compared with our approach within the context of whitney s theorem

a geometric analysis of subspace clustering with outliers this paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes as is common in computer vision or unsupervised learning applications we do not know in advance how many subspaces there are nor do we have any information about their dimensions we develop a novel geometric analysis of an algorithm named sparse subspace clustering ssc 11 which signicantly broadens the range of problems where it is provably eective for instance we show that ssc can recover multiple subspaces each of dimension comparable to the ambient dimension we also prove that ssc can correctly cluster data points even when the subspaces of interest intersect further we develop an extension of ssc that succeeds when the data set is corrupted with possibly overwhelmingly many outliers underlying our analysis are clear geometric insights which may bear on other sparse recovery problems a numerical study complements our theoretical analysis and demonstrates the eectiveness of these methods

multiframe motion segmentation with missing data using powerfactorization and gpca we consider the problem of segmenting multiple rigid body motions from point correspondences in multiple affine views we cast this problem as a subspace clustering problem in which point trajectories associated with each motion live in a linear subspace of dimension two three or four our algorithm involves projecting all point trajectories onto a 5 dimensional subspace using the svd the powerfactorization method or ransac and fitting multiple linear subspaces representing different rigid body motions to the points in 5 using gpca unlike previous work our approach does not restrict the motion subspaces to be four dimensional and independent instead it deals gracefully with all the spectrum of possible affine motions from two dimensional and partially dependent to four dimensional and fully independent our algorithm can handle the case of missing data meaning that point tracks do not have to be visible in all images by using the powerfactorization method to project the data in addition our method can handle outlying trajectories by using ransac to perform the projection we compare our approach to other methods on a database of 167 motion sequences with full motions independent motions degenerate motions partially dependent motions missing data outliers etc on motion sequences with complete data our method achieves a misclassification error of less that 5 for two motions and 29 for three motions

segmentation of multivariate mixed data via lossy data coding and compression in this paper based on ideas from lossy data coding and compression we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of gaussian distributions which are allowed to be almost degenerate the goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data subject to a given distortion by analyzing the coding length rate of mixed data we formally establish some strong connections of data segmentation to many fundamental concepts in lossy data compression and rate distortion theory we show that a deterministic segmentation is approximately the asymptotically optimal solution for compressing mixed data we propose a very simple and effective algorithm that depends on a single parameter the allowable distortion at any given distortion the algorithm automatically determines the corresponding number and dimension of the groups and does not involve any parameter estimation simulation results reveal intriguing phase transition like behaviors of the number of segments when changing the level of distortion or the amount of outliers finally we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

exploiting unsupervised and supervised constraints for subspace clustering data in many image and video analysis tasks can be viewed as points drawn from multiple low dimensional subspaces with each subspace corresponding to one category or class one basic task for processing such kind of data is to separate the points according to the underlying subspace referred to as subspace clustering extensive studies have been made on this subject and nearly all of them use unconstrained subspace models meaning the points can be drawn from everywhere of a subspace to represent the data in this paper we attempt to do subspace clustering based on a constrained subspace assumption that the data is further restricted in the corresponding subspaces e g belonging to a submanifold or satisfying the spatial regularity constraint this assumption usually describes the real data better such as differently moving objects in a video scene and face images of different subjects under varying illumination a unified integer linear programming optimization framework is used to approach subspace clustering which can be efficiently solved by a branch and bound bb method we also show that various kinds of supervised information such as subspace number outlier ratio pairwise constraints size prior and etc can be conveniently incorporated into the proposed framework experiments on real data show that the proposed method outperforms the state of the art algorithms significantly in clustering accuracy the effectiveness of the proposed method in exploiting supervised information is also demonstrated

set membership identification of switched linear systems with known number of subsystems this paper addresses the problem of robust identification of a class of discrete time linear hybrid systems switched linear models in a set membership framework given a finite collection of input output data from a noisy process the objective is twofold i establish whether this data was generated by a system that switches amongst an a priori known number of subsystems and ii in that case identify a suitable set of linear models along with a switching sequence that can explain the available experimental information our main result shows that these problems are equivalent to minimizing the rank of a matrix whose entries are affine in the optimization variables subject to a convex constraint imposing that these variables are the moments of an unknown borel measure with finite support the use of well known tight convex relaxations of rank allows for further reducing the problem to a semidefinite optimization that can be efficiently solved in the second part of the paper we extend these results to handle sensor failures that result in corrupted input output measurements assuming that these failures are infrequent we show that the problem can be recast into an optimization form where the objective is to simultaneously minimize the rank of a matrix and the number of nonzero rows of a second one in both cases appealing to well known convex relaxations of rank and sparsity leads to overall semidefinite optimization problems that can be efficiently solved these results are illustrated with multiple examples showing substantially improved identification performance in the presence of noise and sensor faults

solving inverse problems with piecewise linear estimators from gaussian mixture models to structured sparsity a general framework for solving image inverse problems with piecewise linear estimations is introduced in this paper the approach is based on gaussian mixture models which are estimated via a maximum a posteriori expectation maximization algorithm a dual mathematical interpretation of the proposed framework with a structured sparse estimation is described which shows that the resulting piecewise linear estimate stabilizes the estimation when compared with traditional sparse inverse problem techniques we demonstrate that in a number of image inverse problems including interpolation zooming and deblurring of narrow kernels the same simple and computationally efficient algorithm yields results in the same ballpark as that of the state of the art

median radial basis function neural network radial basis functions rbfs consist of a two layer neural network where each hidden unit implements a kernel function each kernel is associated with an activation region from the input space and its output is fed to an output unit in order to find the parameters of a neural network which embeds this structure we take into consideration two different statistical approaches the first approach uses classical estimation in the learning stage and it is based on the learning vector quantization algorithm and its second order statistics extension after the presentation of this approach we introduce the median radial basis function mrbf algorithm based on robust estimation of the hidden unit parameters the proposed algorithm employs the marginal median for kernel location estimation and the median of the absolute deviations for the scale parameter estimation a histogram based fast implementation is provided for the mrbf algorithm the theoretical performance of the two training algorithms is comparatively evaluated when estimating the network weights the network is applied in pattern classification problems and in optical flow segmentation

the method of fundamental solutions for axisymmetric potential problems in this paper we investigate the application of the method of fundamental solutions mfs to two classes of axisymmetric potential problems in the first the boundary conditions as well as the domain of the problem are axisymmetric and in the second the boundary conditions are arbitrary in both cases the fundamental solutions of the governing equations and their normal derivatives which are required in the formulation of the mfs can be expressed in terms of complete elliptic integrals the method is tested on several axisymmetric problems from the literature and is also applied to an axisymmetric free boundary problem copyright 1999 john wiley sons ltd

transformation invariant subspace clustering subspace clustering has achieved great success in many computer vision applications however most subspace clustering algorithms require well aligned data samples which is often not straightforward to achieve this paper proposes a transformation invariant subspace clustering framework by jointly aligning data samples and learning subspace representation by alignment the transformed data samples become highly correlated and a better affinity matrix can be obtained the joint problem can be reduced to a sequence of least squares regression problems which can be efficiently solved we verify the effectiveness of the proposed method with extensive experiments on unaligned real data demonstrating its higher clustering accuracy than the state of the art subspace clustering and transformation invariant clustering algorithms highlightsa transformation invariant subspace clustering algorithm is proposed alignment and subspace clustering are mutually dependent state of the art clustering results on the unconstrained face dataset

a class of difference schemes with flexible local approximation solutions of many physical problems have salient local features that are qualitatively known a priori for example singularities at point sources edge and corners boundary layers derivative jumps at material interfaces strong dipole field components near polarized spherical particles cusps of electronic wavefunctions at the nuclei electrostatic double layers around colloidal particles etc the known methods capable of providing flexible local approximation of such features include the generalized finite element partition of unity method special variational difference schemes in broken sobolev spaces and a few other specialized techniques in the proposed new class of flexible local approximation methods flame a desirable set of local approximating functions such as cylindrical or spherical harmonics plane waves harmonic polynomials etc defines a finite difference scheme on a chosen grid stencil one motivation is to minimize the notorious staircase effect at curved and slanted interface boundaries however the new approach has much broader applications as illustrative examples the paper presents arbitrarily high order 3 point schemes for the 1d schrodinger equation and a 1d singular equation schemes for electrostatic interactions of colloidal particles electromagnetic wave propagation and scattering plasmon resonances moreover many classical finite difference schemes including the collatz mehrstellen schemes are direct particular cases of flame

three dimensional hybrid trefftz stress elements the stress model of the hybrid trefftz finite element formulation is applied to the linear elastostatic analysis of solids the stresses are approximated in the domain of the element and displacements on its boundary complete linearly independent hierarchical polynomial approximation functions are used in both domain and boundary approximations the displacement basis is defined independently on each inter element surface continuity at the edges and on the corners of the elements is not enforced a priori the stress basis is constrained to solve locally the beltrami governing differential equation it is derived from the associated papkovitch neuber elastic displacement solution generalized variables are used to ensure that the approximations are independent of the geometric description of the elements the solving system is derived directly from the fundamental relations of elastostatics the solving system is symmetric when the same property applies to the local elasticity condition sparse described by boundary integral arrays and well suited to p refinement and parallel processing the numerical implementation of these equations is discussed and numerical tests are presented to illustrate the performance of the finite element formulation copyright 2000 john wiley sons ltd

multiscale hybrid linear models for lossy image representation in this paper we introduce a simple and efficient representation for natural images we view an image in either the spatial domain or the wavelet domain as a collection of vectors in a high dimensional space we then fit a piece wise linear model i e a union of affine subspaces to the vectors at each downsampling scale we call this a multiscale hybrid linear model for the image the model can be effectively estimated via a new algebraic method known as generalized principal component analysis gpca the hybrid and hierarchical structure of this model allows us to effectively extract and exploit multimodal correlations among the imagery data at different scales it conceptually and computationally remedies limitations of many existing image representation methods that are based on either a fixed linear transformation e g dct wavelets or an adaptive uni modal linear transformation e g pca or a multimodal model that uses only cluster means e g vq we will justify both quantitatively and experimentally why and how such a simple multiscale hybrid model is able to reduce simultaneously the model complexity and computational cost despite a small overhead of the model our careful and extensive experimental results show that this new model gives more compact representations for a wide variety of natural images under a wide range of signal to noise ratios than many existing methods including wavelets we also briefly address how the same hybrid linear modeling paradigm can be extended to be potentially useful for other applications such as image segmentation

fuzzy hmc classifiers in this study we present a design strategy for hierarchical modular classifiers applicable to fuzzy and or overlapping categories the design includes techniques for decomposing a large scale problem into a hierarchy of subproblems a neural classification module is then designed for each of the subproblems the collection of classification modules and a rule book governing their use comprise the resultant design as an ancillary but necessary issue we demonstrate the uses of a specific neural network for generating membership functions we also explore the use of maximum membership as a classification criterion

generalized finite differences using fundamental solutions summary it is well known that solutions for linear partial differential equations may be given in terms of fundamental solutions the fundamental solutions solve the homogeneous equation exactly and are obtained from the solution of the inhomogeneous equation where the inhomogeneous term is described by a dirac delta distribution fundamental solutions are the building blocks of the boundary element method and of the method of fundamental solutions and are traditionally used to build boundary only global approximations in the domain of interest in this work the same characteristic of the fundamental solutions that of solving the homogeneous equation exactly is used but not to build a global approximation on the contrary local approximations are built in such a manner that it is possible to construct finite difference operators that are free from any form of structured grid copyright q 2009 john wiley sons ltd

trefftz method fitting boundary conditions the paper presents various ways of fitting the boundary conditions in the t complete functions method the authors point out the distinct advantages of the orthogonal collocation in comparison to the equidistant collocation and the integral fit the convergence of the collatz error measures and the conditioning of the solution matrices are investigated in detail

self organizing maps for the skeletonization of sparse shapes this paper presents a method for computing the skeleton of planar shapes and objects which exhibit sparseness lack of connectivity within their image regions such sparseness in images may occur due to poor lighting conditions incorrect thresholding or image sub sampling furthermore in document image analysis sparse shapes are characteristic of texts faded due to aging and or poor ink quality given the pixel distribution for a shape the proposed method involves an iterative evolution of a piecewise linear approximation of the shape skeleton by using a minimum spanning tree based self organizing map som by constraining the som to lie on the edges of the delaunay triangulation of the shape distribution the adjacency relationships between regions in the shape are detected and used in the evolution of the skeleton the som on convergence gives the final skeletal shape the skeletonization is invariant to euclidean transformations the potential of the method is demonstrated on a variety of sparse shapes from different application domains

generalized principal component analysis gpca this paper presents an algebro geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points we represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point when the number of subspaces is known we show that these polynomials can be estimated linearly from data hence subspace segmentation is reduced to classifying one point per subspace we select these points optimally from the data set by minimizing certain distance function thus dealing automatically with moderate noise in the data a basis for the complement of each subspace is then recovered by applying standard pca to the collection of derivatives normal vectors extensions of gpca that deal with data in a high dimensional space and with an unknown number of subspaces are also presented our experiments on low dimensional data show that gpca outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as k subspaces and expectation maximization we also present applications of gpca to computer vision problems such as face clustering temporal video segmentation and 3d motion segmentation from point correspondences in multiple affine views

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

trefftz difference schemes on irregular stencils the recently developed flexible local approximation method flame produces accurate difference schemes by replacing the usual taylor expansion with trefftz functions local solutions of the underlying differential equation this paper advances and casts in a general form a significant modification of flame proposed recently by pinheiro and webb a least squares fit instead of the exact match of the approximate solution at the stencil nodes as a consequence of that flame schemes can now be generated on irregular stencils with the number of nodes substantially greater than the number of approximating functions the accuracy of the method is preserved but its robustness is improved for demonstration the paper presents a number of numerical examples in 2d and 3d electrostatic magnetostatic particle interactions scattering of electromagnetic acoustic waves and wave propagation in a photonic crystal the examples explore the role of the grid and stencil size of the number of approximating functions and of the irregularity of the stencils

on the equivalence of the trefftz method and method of fundamental solutions for laplace and biharmonic equations in this paper it is proved that the two approaches known in the literature as the method of fundamental solutions mfs and the trefftz method are mathematically equivalent in spite of their essentially minor and apparent differences in formulation in deriving the equivalence of the trefftz method and the mfs for the laplace and biharmonic problems it is interesting to find that the complete set in the trefftz method for the laplace and biharmonic problems are embedded in the degenerate kernels of the mfs the degenerate scale appears using the mfs when the geometrical matrix is singular the occurring mechanism of the degenerate scale in the mfs is also studied by using circulant the comparison of accuracy and efficiency of the two methods was addressed

algebraic geometric comparison of probability distributions we propose a novel algebraic algorithmic framework for dealing with probability distributions represented by their cumulants such as the mean and covariance matrix as an example we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree instead of minimizing an objective function involving the estimated cumulants we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem at a lower computational cost and with higher accuracy moreover the algebraic viewpoint on probability distributions allows us to invoke the theory of algebraic geometry which we demonstrate in a compact proof for an identifiability criterion

a meshless method for kirchhoff plate bending problems in this work a meshless method for the analysis of bending of thin homogeneous plates is presented this meshless method is based on the use of radial basis functions to build an approximation of the general solution of the partial differential equations governing the kirchhoff plate bending problem in order to obtain a symmetric and non singular linear equation system the hermite collocation method is used to assess the formulation a series of plates with different boundary conditions are analysed comparisons are made with other results available in the literature copyright 2001 john wiley sons ltd

robust kernel low rank representation recently low rank representation lrr has shown promising performance in many real world applications such as face clustering however lrr may not achieve satisfactory results when dealing with the data from nonlinear subspaces since it is originally designed to handle the data from linear subspaces in the input space meanwhile the kernel based methods deal with the nonlinear data by mapping it from the original input space to a new feature space through a kernel induced mapping to effectively cope with the nonlinear data we first propose the kernelized version of lrr in the clean data case we also present a closed form solution for the resultant optimization problem moreover to handle corrupted data we propose the robust kernel lrr rklrr approach and develop an efficient optimization algorithm to solve it based on the alternating direction method in particular we show that both the subproblems in our optimization algorithm can be efficiently and exactly solved and it is guaranteed to obtain a globally optimal solution besides our proposed algorithm can also solve the original lrr problem which is a special case of our rklrr when using the linear kernel in addition based on our new optimization technique the kernelization of some variants of lrr can be similarly achieved comprehensive experiments on synthetic data sets and real world data sets clearly demonstrate the efficiency of our algorithm as well as the effectiveness of rklrr and the kernelization of two variants of lrr

spectral curvature clustering scc this paper presents novel techniques for improving the performance of a multi way spectral clustering framework govindu in proceedings of the 2005 ieee computer society conference on computer vision and pattern recognition cvpr 05 vol 1 pp 1150 1157 2005 chen and lerman 2007 preprint in the supplementary webpage for segmenting affine subspaces specifically it suggests an iterative sampling procedure to improve the uniform sampling strategy an automatic scheme of inferring the tuning parameter from the data a precise initialization procedure for k means as well as a simple strategy for isolating outliers the resulting algorithm spectral curvature clustering scc requires only linear storage and takes linear running time in the size of the data it is supported by theory which both justifies its successful performance and guides our practical choices we compare it with other existing methods on a few artificial instances of affine subspaces application of the algorithm to several real world problems is also discussed

green s function and stability of a linear partial difference scheme abstract this paper is concerned with a linear partial difference equation which arises from discretizing a heat and a first order wave equations a green s function for this equation is derived various representation theorems monotonicity and oscillatory properties of the green s functions are obtained then by means of the representation theorems we derive several stability criteria for arbitrary solutions of this equation

finite difference schemes for incompressible flow based on local pressure boundary conditions in this paper we discuss the derivation and use of local pressure boundary conditions for finite difference schemes for the unsteady incompressible navier stokes equations in the velocity pressure formulation their use is especially well suited for the computation of moderate to large reynolds number flows we explore the similarities between the implementation and use of local pressure boundary conditions and local vorticity boundary conditions in the design of numerical schemes for incompressible flow in 2d in their respective formulations when these local numerical boundary conditions are coupled with a fully explicit convectively stable time stepping procedure the resulting methods are simple to implement and highly efficient unlike the vorticity formulation the use of the local pressure boundary condition approach is readily applicable to 3d flows the simplicity of the local pressure boundary condition approach and its easy adaptation to more general flow settings make the resulting scheme an attractive alternative to the more popular methods for solving the navier stokes equations in the velocity pressure formulation we present numerical results of a second order finite difference scheme on a nonstaggered grid using local pressure boundary conditions stability and accuracy of the scheme applied to stokes flow is demonstrated using normal mode analysis also described is the extension of the method to variable density flows

decorrelated hebbian learning for clustering and function approximation this paper presents a new learning paradigm that consists of a hebbian and anti hebbian learning a layer of radial basis functions is adapted in an unsupervised fashion by minimizing a two element

bounds for solutions of a six point partial difference scheme abstract this paper is concerned with a class of partial difference equations which arise from discretising the one dimensional heat and the first order wave equations explicit bounds are found for their solutions and subexponential solutions are considered using the green s function we derive a stability criteria for arbitrary solutions of this equation

linear time subspace clustering via bipartite graph modeling we present a linear time subspace clustering approach that combines sparse representations and bipartite graph modeling the signals are modeled as drawn from a union of low dimensional subspaces and each signal is represented by a sparse combination of basis elements termed atoms which form the columns of a dictionary matrix the sparse representation coefficients are arranged in a sparse affinity matrix which defines a bipartite graph of two disjoint sets 1 atoms and 2 signals subspace clustering is obtained by applying low complexity spectral bipartite graph clustering that exploits the small number of atoms for complexity reduction the complexity of the proposed approach is linear in the number of signals thus it can rapidly cluster very large data collections performance evaluation of face clustering and temporal video segmentation demonstrates comparable clustering accuracies to state of the art at a significantly lower computational load

aggregating membership values by a choquet fuzzy integral based operator a new type of aggregator the choquet fuzzy integral based operator is introduced and applied in aggregation of membership values in this paper we generalized the definition of the fuzzy integral as a tool for membership aggregation the generalization involves replacing the max or min operator in information integration with a choquet fuzzy integral based aggregator resulting in increased flexibility the characteristics of the choquet fuzzy integral are studied and a two phase aggregating process is proposed also we demonstrate a good classification result using the proposed aggregator to integrate the memberships of several clusters in a complex unconstrained handwritten digit recognition domain

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

sce a manifold regularized set covering method for data partitioning cluster analysis plays a very important role in data analysis in these years cluster ensemble as a cluster analysis tool has drawn much attention for its robustness stability and accuracy many efforts have been done to combine different initial clustering results into a single clustering solution with better performance however they neglect the structure information of the raw data in performing the cluster ensemble in this paper we propose a structural cluster ensemble sce algorithm for data partitioning formulated as a set covering problem in particular we construct a laplacian regularized objective function to capture the structure information among clusters moreover considering the importance of the discriminative information underlying in the initial clustering results we add a discriminative constraint into our proposed objective function finally we verify the performance of the sce algorithm on both synthetic and real data sets the experimental results show the effectiveness of our proposed method sce algorithm

learning the nonlinear geometry of high dimensional data models and algorithms modern information processing relies on the axiom that high dimensional data lie near low dimensional geometric structures this paper revisits the problem of data driven learning of these geometric structures and puts forth two new nonlinear geometric models for data describing related objects phenomena the first one of these models straddles the two extremes of the subspace model and the union of subspaces model and is termed the metric constrained union of subspaces mc uos model the second one of these models suited for data drawn from a mixture of nonlinear manifolds generalizes the kernel subspace model and is termed the metric constrained kernel union of subspaces mc kuos model the main contributions of this paper in this regard include the following first it motivates and formalizes the problems of mc uos and mc kuos learning second it presents algorithms that efficiently learn an mc uos or an mc kuos underlying data of interest third it extends these algorithms to the case when parts of the data are missing last but not least it reports the outcomes of a series of numerical experiments involving both synthetic and real data that demonstrate the superiority of the proposed geometric models and learning algorithms over existing approaches in the literature these experiments also help clarify the connections between this work and the literature on subspace and kernel k means clustering

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

general solutions of a three level partial difference equation abstract this paper is concerned with a linear partial difference equation which includes the well known dufort frankel multilevel difference scheme for the heat equation by introducing green s functions for this equation we obtain via a novel formal approach an explicit formula for all its solutions given exponential initial conditions separable solutions are also found as applications we derive several stability criteria for the solutions of this equation

algebraic clustering of affine subspaces subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition prior work has studied this problem using algebraic iterative statistical low rank and sparse representation techniques while these methods have been applied to both linear and affine subspaces theoretical results have only been established in the case of linear subspaces for example algebraic subspace clustering asc is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal in this paper we study in a rigorous fashion the properties of asc in the case of affine subspaces using notions from algebraic geometry we prove that the homogenization trick which embeds points in a union of affine subspaces into points in a union of linear subspaces preserves the general position of the points and the transversality of the union of subspaces in the embedded space thus establishing the correctness of asc for affine subspaces

the effect of concave and convex weight adjustments on self organizing maps two nonlinear models of weight adjustments of self organizing maps are derived to obtain desirable densities of output units one that approaches the probability distribution p spl xi of the inputs and one that approaches a uniform distribution if a convex model is used to adjust weights the density of output units can be made to approach p spl xi instead of the p spl xi sup 2 3 which results from the linear weight adjustment of kohonen s self organizing maps if a concave model of weight adjustments is used the density approaches a uniform distribution and the winner frequency distribution of output units is proportional to p spl xi the former can provide more efficient data representations for vector quantization while the latter can provide more meaningful measures for cluster analysis numerical demonstrations validate the mathematical derivations the convergence of the concave model is faster than the linear and convex models while the convergence of the convex model is comparable to that of the linear model

dimension selective self organizing maps with time varying structure for subspace and projected clustering subspace clustering is the task of identifying clusters in subspaces of the input dimensions of a given dataset noisy data in certain attributes cause difficulties for traditional clustering algorithms because the high discrepancies within them can make objects appear too different to be grouped in the same cluster this requires methods specially designed for subspace clustering this paper presents our second approach to subspace and projected clustering based on self organizing maps soms which is a local adaptive receptive field dimension selective som by introducing a time variant topology our method is an improvement in terms of clustering quality computational cost and parameterization this enables the method to identify the correct number of clusters and their respective relevant dimensions and thus it presents nearly perfect results in synthetic datasets and surpasses our previous method in most of the real world datasets considered

meshfree explicit local radial basis function collocation method for diffusion problems this paper formulates a simple explicit local version of the classical meshless radial basis function collocation kansa method the formulation copes with the diffusion equation applicable in the solution of a broad spectrum of scientific and engineering problems the method is structured on multiquadrics radial basis functions instead of global the collocation is made locally over a set of overlapping domains of influence and the time stepping is performed in an explicit way only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node the computational effort thus grows roughly linearly with the number of the nodes the developed approach thus overcomes the principal large scale problem bottleneck of the original kansa method two test cases are elaborated the first is the boundary value problem nafems test associated with the steady temperature field with simultaneous involvement of the dirichlet neumann and robin boundary conditions on a rectangle the second is the initial value problem associated with the dirichlet jump problem on a square the accuracy of the method is assessed in terms of the average and maximum errors with respect to the density of nodes number of nodes in the domain of influence multiquadrics free parameter and timestep length on uniform and nonuniform node arrangements the developed meshless method outperforms the classical finite difference method in terms of accuracy in all situations except immediately after the dirichlet jump where the approximation properties appear similar

variants of self organizing maps self organizing maps have a bearing on traditional vector quantization a characteristic that makes them more closely resemble certain biological brain maps however is the spatial order of their responses which is formed in the learning process a discussion is presented of the basic algorithms and two innovations dynamic weighting of the input signals at each input of each cell which improves the ordering when very different input signals are used and definition of neighborhoods in the learning algorithm by the minimal spanning tree which provides a far better and faster approximation of prominently structured density functions it is cautioned that if the maps are used for pattern recognition and decision process it is necessary to fine tune the reference vectors so that they directly define the decision borders