measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

generalizing discriminant analysis using the generalized singular value decomposition discriminant analysis has been used for decades to extract features that preserve class separability it is commonly defined as an optimization problem involving covariance matrices that represent the scatter within and between clusters the requirement that one of these matrices be nonsingular limits its application to data sets with certain relative dimensions we examine a number of optimization criteria and extend their applicability by using the generalized singular value decomposition to circumvent the nonsingularity requirement the result is a generalization of discriminant analysis that can be applied even when the sample size is smaller than the dimension of the sample data we use classification results from the reduced representation to compare the effectiveness of this approach with some alternatives and conclude with a discussion of their relative merits

random matrices in 2d laplacian growth and operator theory since it was first applied to the study of nuclear interactions by wigner and dyson almost 60 years ago random matrix theory rmt has developed into a field of its own within applied mathematics and is now essential to many parts of theoretical physics from condensed matter to high energy the fundamental results obtained so far rely mostly on the theory of random matrices in one dimension the dimensionality of the spectrum or equilibrium probability density in the last few years this theory has been extended to the case where the spectrum is two dimensional or even fractal with dimensions between 1 and 2 in this paper we review these recent developments and indicate some physical problems where the theory can be applied

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

incremental linear discriminant analysis a fast algorithm and comparisons it has always been a challenging task to develop a fast and an efficient incremental linear discriminant analysis ilda algorithm for this purpose we conduct a new study for linear discriminant analysis lda in this paper and develop a new ilda algorithm we propose a new batch lda algorithm called lda qr lda qr is a simple and fast lda algorithm which is obtained by computing the economic qr factorization of the data matrix followed by solving a lower triangular linear system the relationship between lda qr and uncorrelated lda ulda is also revealed based on lda qr we develop a new incremental lda algorithm called ilda qr the main features of our ilda qr include that 1 it can easily handle the update from one new sample or a chunk of new samples 2 it has efficient computational complexity and space complexity and 3 it is very fast and always achieves competitive classification accuracy compared with ulda algorithm and existing ilda algorithms numerical experiments based on some real world data sets demonstrate that our ilda qr is very efficient and competitive with the state of the art ilda algorithms in terms of classification accuracy computational complexity and space complexity

golden mean arithmetic in the fractal branching of diffusion limited aggregates we use the wavelet transform microscope to explore the intricate fractal geometry of witten and sander s diffusion limited aggregates we report on the discovery of fibonacci sequences in the internal hierarchical structure of these clusters we discuss the relevance of the golden mean arithmetic to the numerically well established statistical self similarity of these aggregates

a computationally efficient scheme for feature extraction with kernel discriminant analysis the kernel discriminant analysis kda an extension of the linear discriminant analysis lda and null space based lda into the kernel space generally provides good pattern recognition pr performance for both small sample size sss and non sss pr problems due to the eigen decomposition technique adopted however the original scheme for the feature extraction with the kda suffers from a high complexity burden in this paper we derive a transformation of the kda into a linear equation problem and propose a novel scheme for the feature extraction with the kda the proposed scheme is shown to provide us with a reduction of complexity without degradation of pr performance in addition to enhance the pr performance further we address the incorporation of regularization into the proposed scheme highlightswe propose a complexity reduced scheme for the kernel discriminant analysis the proposed scheme does not cause degradation of recognition accuracy performance the core is the transformation of the original problem into a linear equation problem we further address the incorporation of regularization into the proposed scheme

complexity reduced scheme for feature extraction with linear discriminant analysis owing to the singularity of the within class scatter linear discriminant analysis lda becomes ill posed for small sample size sss problems null space based lda nlda which is an extension of lda provides good discriminant performances for sss problems yet as the original scheme for the feature extractor fe of nlda suffers from a complexity burden a few modified schemes have since been proposed for complexity reduction in this brief by transforming the problem of finding the fe of nlda into a linear equation problem a novel scheme is derived offering a further reduction of the complexity

why can lda be performed in pca transformed space abstract pca plus lda is a popular framework for linear discriminant analysis lda in high dimensional and singular case in this paper we focus on building a theoretical foundation for this framework moreover we point out the weakness of the previous lda based methods and suggest a complete pca plus lda algorithm experimental results on orl face image database indicate that the proposed method is more effective than the previous ones

laplacian growth and whitham equations of soliton theory the laplacian growth the hele shaw problem of multiply connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of whitham equations known in soliton theory the whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations through this connection the laplacian growth is understood as a flow in the moduli space of riemann surfaces

radiating solitary waves of a model evolution equation in fluids of finite depth abstract the evolution equation proposed by kubota et al j hydronautics 12 1978 157 165 and joseph j phys a 10 1977 225 227 for long waves in stratified fluids of finite depth is considered under a small third order derivative dispersive perturbation solitary wave solutions to this model equation become weakly nonlocal they develop oscillatory tails of infinite extent and exponentially small amplitude here these tails are calculated asymptotically following a perturbation procedure in the wavenumber domain

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

convergent calculation of the asymptotic dimension of diffusion limited aggregates scaling and renormalization of small clusters diffusion limited aggregation dla is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes we present a convergent calculation of the fractal dimension d of dla based on a renormalization scheme for the first laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster the theory is applicable from very small 2 3 particles to asymptotically large n infinity clusters the computed dimension is d 1 713 0 003

noncreasy and uniformly noncreasy orlicz function spaces we show that in orlicz function spaces with orlicz luxemburg norm the criteria for being noncreasy and uniformly noncreasy are interesting combinations of conditions

computational and theoretical analysis of null space and orthogonal linear discriminant analysis dimensionality reduction is an important pre processing step in many applications linear discriminant analysis lda is a classical statistical approach for supervised dimensionality reduction it aims to maximize the ratio of the between class distance to the within class distance thus maximizing the class discrimination it has been used widely in many applications however the classical lda formulation requires the nonsingularity of the scatter matrices involved for undersampled problems where the data dimensionality is much larger than the sample size all scatter matrices are singular and classical lda fails many extensions including null space lda nlda and orthogonal lda olda have been proposed in the past to overcome this problem nlda aims to maximize the between class distance in the null space of the within class scatter matrix while olda computes a set of orthogonal discriminant vectors via the simultaneous diagonalization of the scatter matrices they have been applied successfully in various applications in this paper we present a computational and theoretical analysis of nlda and olda our main result shows that under a mild condition which holds in many applications involving high dimensional data nlda is equivalent to olda we have performed extensive experiments on various types of data and results are consistent with our theoretical analysis we further apply the regularization to olda the algorithm is called regularized olda or rolda for short an efficient algorithm is presented to estimate the regularization value in rolda a comparative study on classification shows that rolda is very competitive with olda this confirms the effectiveness of the regularization in rolda

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

an optimal transformation for discriminant and principal component analysis a general method is proposed to describe multivariate data sets using discriminant analysis and principal component analysis first the problem of finding k discriminant vectors in an l class data set is solved and compared to the solution proposed in the literature for two class problems and the classical solution for l class data sets it is shown that the method proposed is better than the classical method for l classes and is a generalization of the optimal set of discriminant vectors proposed for two class problems then the method is combined with a generalized principal component analysis to permit the user to define the properties of each successive computed vector all the methods were tested using measurements made on various kinds of flowers iris data

iterated conformal dynamics and laplacian growth the method of iterated conformal maps for the study of diffusion limited aggregates dla is generalized to the study of laplacian growth patterns and related processes we emphasize the fundamental difference between these processes dla is grown serially with constant size particles while laplacian patterns are grown by advancing each boundary point in parallel proportional to the gradient of the laplacian field we introduce a two parameter family of growth patterns that interpolates between dla and a discrete version of laplacian growth the ultraviolet putative finite time singularities are regularized here by a minimal tip size equivalently for all the models in this family with this we stress that the difference between dla and laplacian growth is not in the manner of ultraviolet regularization but rather in their deeply different growth rules the fractal dimensions of the asymptotic patterns depend continuously on the two parameters of the family giving rise to a phase diagram in which dla and discretized laplacian growth are at the extreme ends in particular we show that the fractal dimension of laplacian growth patterns is higher than the fractal dimension of dla with the possibility of dimension 2 for the former not excluded

multifractal structure of the harmonic measure of diffusion limited aggregates the method of iterated conformal maps allows one to study the harmonic measure of diffusion limited aggregates with unprecedented accuracy we employ this method to explore the multifractal properties of the measure including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing we resolve probabilities as small as 10 235 and present an accurate determination of the generalized dimensions and the spectrum of singularities we show that the generalized dimensions dq are infinite forq q where q is of the order of 20 2 in the language of f a this means that a max is finite the f a curve loses analyticity the phenomenon of phase transition at a max and a finite value of f a max we consider the geometric structure of the regions that support the lowest parts of the harmonic measure and thus offer an explanation for the phase transition rationalizing the value of q and f a max we thus offer a satisfactory physical picture of the scaling properties of this multifractal measure

face recognition based on the uncorrelated discriminant transformation abstract the extraction of discriminant features is the most fundamental and important problem in face recognition this paper presents a method to extract optimal discriminant features for face images by using the uncorrelated discriminant transformation and kl expansion experiments on the orl database and the nust603 database have been performed experimental results show that the uncorrelated discriminant transformation is superior to the foley sammon discriminant transformation and the new method to extract uncorrelated discriminant features for face images is very effective an error rate of 2 5 is obtained with the experiments on the orl database an average error rate of 1 2 is obtained with the experiments on the nust603 database experiments show that by extracting uncorrelated discriminant features face recognition could be performed with higher accuracy on lower than 16 16 resolution mosaic images it is suggested that for the uncorrelated discriminant transformation the optimal face image resolution can be regarded as the resolution m n which makes the dimensionality n mn of the original image vector space be larger and closer to the number of known face classes

fractal dimension and unscreened angles measured for radial viscous fingering we have examined fractal patterns formed by the injection of air into oil in a thin 0 127 phantom rule 0 3em 0ex mathrm mm layer contained between two cylindrical glass plates of 288 phantom rule 0 3em 0ex mathrm mm diameter a hele shaw cell for pressure differences in the range 0 25 ensuremath leqslant ensuremath delta p ensuremath leqslant 1 75 phantom rule 0 3em 0ex mathrm atm we find that an asymptotic structure is reached at large values of the ratio r b where r is the pattern radius and b the gap between the plates both the driving force and the size of the pattern which reaches r b 900 are far larger than in past experiments the fractal dimension d 0 of the pattern for large r b is 1 70 ifmmode pm else textpm fi 0 02 further the generalized dimensions d q of the pattern are independent of q d q ensuremath simeq 1 70 for the range examined ensuremath 11lql17 thus the pattern is self similar within the experimental uncertainty the results for d q agree well with recent calculations for diffusion limited aggregation dla clusters we have also measured the probability distribution of unscreened angles at late times the distribution approaches a universal i e forcing and size independent asymptotic form that has mean 145 ifmmode circ else textdegree fi and standard deviation 36 ifmmode circ else textdegree fi these results indicate that the distribution function for the unscreened angle is an invariant property of the growth process

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

selected topics on quadrature domains this is a selection of facts old and recent about quadrature domains the text written in the form of a survey is addressed to non experts and covers a variety of phenomena related to quadratur

incremental generalized discriminative common vectors for image classification subspace based methods have become popular due to their ability to appropriately represent complex data in such a way that both dimensionality is reduced and discriminativeness is enhanced several recent works have concentrated on the discriminative common vector dcv method and other closely related algorithms also based on the concept of null space in this paper we present a generalized incremental formulation of the dcv methods which allows the update of a given model by considering the addition of new examples even from unseen classes having efficient incremental formulations of well behaved batch algorithms allows us to conveniently adapt previously trained classifiers without the need of recomputing them from scratch the proposed generalized incremental method has been empirically validated in different case studies from different application domains faces objects and handwritten digits considering several different scenarios in which new data are continuously added at different rates starting from an initial model

diffusion limited aggregation and its response to anisotropy the stability of diffusion limited growths with n equivalent major fingers is investigated in two dimensions using a conformal mapping the results imply that square lattice but not hexagonal lattice bias are relevant in simple diffusion limited aggregation dla in general it is found that the maximum number of fingers stable with respect to finger loss by competition is given by nmax 2 2 d 1 where d is the apparent fractal dimension of the fingers at least when n is even computer simulations of dla with n fold symmetric growth rules are shown for n 5 8 12 finally it is argued that unbiased isotropic fractal dla should lie at the stability limit when combined with cone angle arguments independently relating d to the effective numbers of fingers this determines the value d 1 12 2 1 707 for dla in two dimensions this corresponds to a characteristic exterior half angle at the leading tips given by 2

noncreasy and uniformly noncreasy orlicz bochner function spaces abstract we point out that uniform noncreasiness is a super property in banach spaces moreover we prove that orlicz bochner function spaces are noncreasy if and only if they are rotund or smooth finally we obtain that orlicz bochner function spaces are uniformly noncreasy if and only if they are uniformly rotund or uniformly smooth

asymptotics beyond all orders asymptotology and borel summation asymptotics superasymptotics hyperasymptotics m berry the geometric model of crystal growth numerical analysis of the geometric model of crystal growth j m hammersley dendritic crystal growth an experimental assessment of continuum models of dendritic growth j p gollub directional solidification of solids directional growth of dilute mixtures and lamellar eutectics k kassner et al flow in a hele shaw cell viscous displacement in a hele shaw cell s tanveer the rapidly forced pendulum exponentially small estimate for separatrix splittings j scheurle et al ordinary differential equations singular perturbations of solitons y pomeau solitary water waves in the presence of small surface tension solitary waves with ripples beyond all orders j t beale problems in optics potpourri 23 additional articles index

extremal structure of the unit ball of direct sums of banach spaces abstract we characterize the extreme points and smooth points of the unit ball of certain direct sums of banach spaces we use these results to characterize noncreasiness and uniform noncreasiness of direct sums thereby extending results of the second author s saejung extreme points smooth points and noncreasiness of direct sum of banach spaces nonlinear anal 67 2007 1649 1653

regularized discriminant analysis and its application to face recognition the linear fisher discriminant analysis lda is a well known and popular statistical method in pattern recognition and classi5cation the basic idea is to optimize the discriminant criteria in which the ratio between the interand the intra class distance is maximized this approach is theoretically sound and a number of papers have shown its superior performance when applying in pattern recognition applications 1 5 however this approach su ers from the small size problem this problem occurs when the sample size is small compared with the size of feature vector which always appears in the face recognition applications this is because the typical image dimension for face image is greater than 32 32 and usually only 2 6 images are used for training thus the within class covariance estimation cw will be singular in this case the eigenvalue problem is ill posed the fisher index in t will be reached at the null space of cw and will always result in in5nity index in t a number of methods have been proposed in the last decade to overcome the limitation of lda on small sample size these methods in applying to face recognition can be roughly grouped into three categories the 5rst approach applied a dimension reduction method such as principal component analysis pca to extract important components for lda this approach is straightforward but the dimension reduction method may remove some useful information for recognition the second approach modi5es the fisher s optimization criteria theoretically this approach can solve the problem but the constraints 1 3 will limit the

feature extraction using a fast null space based linear discriminant analysis algorithm the small sample size problem is often encountered in pattern recognition several algorithms for null space based linear discriminant analysis nlda have been developed to solve the problem however these algorithms for nlda have high computational cost in this paper we simplify the recently proposed algorithm for nlda in chu and thye 2010 5 with the assumption that all the training data vectors are linearly independent and propose a new and fast algorithm for nlda our main observation is that two steps of economic qr decomposition with column pivoting can be replaced by one step of economic qr decomposition without column pivoting if the related matrix is of full column rank the main features of our algorithm for nlda include i our nlda algorithm is carried out by only one step of economic qr decomposition and does not compute any singular value decomposition svd when all the training data vectors are linearly independent ii the main cost of our method is the cost of an economic qr decomposition of an mx n 1 matrix here m is the dimension of the training data and n is the number of samples our method is a fast one experimental studies on orl feret and pie face databases demonstrate the effectiveness of our new algorithm for nlda

harmonic moment dynamics in laplacian growth harmonic moments are integrals of integer powers of z x iy over a domain here the domain is an exterior of a bubble of air growing in an oil layer between two horizontal closely spaced plates harmonic moments are a natural basis for such laplacian growth phenomena because unlike other representations these moments linearize the zero surface tension problem s richardson j fluid mech 56 609 1972 so that all moments except the lowest one the area of the bubble are conserved in time in our experiments we directly determine the harmonic moments and show that for nonzero surface tension all moments except the lowest one decay in time rather than exhibiting the divergences of other representations further we derive an expression that relates the derivative of the k th harmonic moment m k to measurable quantities surface tension viscosity the distance between the plates and a line integral over the contour encompassing the growing bubble the laboratory observations are in good accord with the expression we derive for d m k dt which is proportional to the surface tension thus in the zero surface tension limit the moments above k 0 are all conserved in accord with richardson s theory in addition from the measurements of the time evolution of the harmonic moments we obtain a value for the surface tension that is within 20 of the accepted value in conclusion our analysis and laboratory observations demonstrate that an interface dynamics description in terms of harmonic moments is physically realizable and robust

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

geometric mean for subspace selection subspace selection approaches are powerful tools in pattern classification and data visualization one of the most important subspace approaches is the linear dimensionality reduction step in the fisher s linear discriminant analysis flda which has been successfully employed in many fields such as biometrics bioinformatics and multimedia information management however the linear dimensionality reduction step in flda has a critical drawback for a classification task with c classes if the dimension of the projected subspace is strictly lower than c 1 the projection to a subspace tends to merge those classes which are close together in the original feature space if separate classes are sampled from gaussian distributions all with identical covariance matrices then the linear dimensionality reduction step in flda maximizes the mean value of the kullback leibler kl divergences between different classes based on this viewpoint the geometric mean for subspace selection is studied in this paper three criteria are analyzed 1 maximization of the geometric mean of the kl divergences 2 maximization of the geometric mean of the normalized kl divergences and 3 the combination of 1 and 2 preliminary experimental results based on synthetic data uci machine learning repository and handwriting digits show that the third criterion is a potential discriminative subspace selection method which significantly reduces the class separation problem in comparing with the linear dimensionality reduction step in flda and its several representative extensions

a theorem on the uncorrelated optimal discriminant vectors this paper proposes a theorem on the uncorrelated optimal discriminant vectors uodvs it is proved that the classical optimal discriminant vectors are equivalent to uodv which can be used to extract l 1 uncorrelated discriminant features for l class problems without losing any discriminant information in the meaning of fisher discriminant criterion function experiments on concordia university cenparmi handwritten numeral database indicate that uodvs are much more powerful than the foley sammon optimal discriminant vectors it is believed that when the number of training samples is large the conjugate orthogonal set of discriminant vectors can be much more powerful than the orthogonal set of discriminant vectors

an optimization criterion for generalized discriminant analysis on undersampled problems an optimization criterion is presented for discriminant analysis the criterion extends the optimization criteria of the classical linear discriminant analysis lda through the use of the pseudoinverse when the scatter matrices are singular it is applicable regardless of the relative sizes of the data dimension and sample size overcoming a limitation of classical lda the optimization problem can be solved analytically by applying the generalized singular value decomposition gsvd technique the pseudoinverse has been suggested and used for undersampled problems in the past where the data dimension exceeds the number of data points the criterion proposed in this paper provides a theoretical justification for this procedure an approximation algorithm for the gsvd based approach is also presented it reduces the computational complexity by finding subclusters of each cluster and uses their centroids to capture the structure of each cluster this reduced problem yields much smaller matrices to which the gsvd can be applied efficiently experiments on text data with up to 7 000 dimensions show that the approximation algorithm produces results that are close to those produced by the exact algorithm

a new lda based face recognition system which can solve the small sample size problem a new lda based face recognition system is presented in this paper linear discriminant analysis lda is one of the most popular linear projection techniques for feature extraction the major drawback of applying lda is that it may encounter the small sample size problem in this paper we propose a new lda based technique which can solve the small sample size problem we also prove that the most expressive vectors derived in the null space of the within class scatter matrix using principal component analysis pca are equal to the optimal discriminant vectors derived in the original space using lda the experimental results show that the new lda process improves the performance of a face recognition system signi cantly 2000 pattern recognition society published by elsevier science ltd all rights reserved

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

a stable numerical method for inverting shape from moments we derive a stable technique based upon matrix pencils for the reconstruction of or approximation by polygonal shapes from moments we point out that this problem can be considered the dual of 2 d numerical quadrature over polygonal domains an analysis of the sensitivity of the problem is presented along with some numerical examples illustrating the relevant points finally an application to the problem of gravimetry is explored where the shape of a gravitationally anomalous region is to be recovered from measurements of its exterior gravitational field

characterization of a family of algorithms for generalized discriminant analysis on undersampled problems a generalized discriminant analysis based on a new optimization criterion is presented the criterion extends the optimization criteria of the classical linear discriminant analysis lda when the scatter matrices are singular an efficient algorithm for the new optimization problem is presented the solutions to the proposed criterion form a family of algorithms for generalized lda which can be characterized in a closed form we study two specific algorithms namely uncorrelated lda ulda and orthogonal lda olda ulda was previously proposed for feature extraction and dimension reduction whereas olda is a novel algorithm proposed in this paper the features in the reduced space of ulda are uncorrelated while the discriminant vectors of olda are orthogonal to each other we have conducted a comparative study on a variety of real world data sets to evaluate ulda and olda in terms of classification accuracy

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

viscous fluid fingering on a negatively curved surface viscous fingering formation in flat hele shaw cells is a classical and widely studied fluid mechanical problem we examine the development of viscous fluid fingering on a two dimensional surface of constant negative gaussian curvature the hyperbolic plane h 2 a perturbative mode coupling formalism is applied to study the influence of the negative surface curvature on the two most important pattern formation mechanisms of the system fingertip splitting and finger competition we also report on a time dependent control strategy placed on the injection rate which is able to minimize viscous fingering growth on h 2

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

geometric representation of high dimension low sample size data summary high dimension low sample size data are emerging in various areas of science we find a common structure underlying many such data sets by using a non standard type of asymptotics the dimension tends to while the sample size is fixed our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex essentially all the randomness in the data appears only as a random rotation of this simplex this geometric representation is used to obtain several new statistical insights

local linear discriminant analysis framework using sample neighbors the linear discriminant analysis lda is a very popular linear feature extraction approach the algorithms of lda usually perform well under the following two assumptions the first assumption is that the global data structure is consistent with the local data structure the second assumption is that the input data classes are gaussian distributions however in real world applications these assumptions are not always satisfied in this paper we propose an improved lda framework the local lda llda which can perform well without needing to satisfy the above two assumptions our llda framework can effectively capture the local structure of samples according to different types of local data structure our llda framework incorporates several different forms of linear feature extraction approaches such as the classical lda and principal component analysis the proposed framework includes two llda algorithms a vector based llda algorithm and a matrix based llda mllda algorithm mllda is directly applicable to image recognition such as face recognition our algorithms need to train only a small portion of the whole training set before testing a sample they are suitable for learning large scale databases especially when the input data dimensions are very high and can achieve high classification accuracy extensive experiments show that the proposed algorithms can obtain good classification results

a finite polynomial solution of the two dimensional interface dynamics abstract the nonlocal nonlinear partial differential equation which describes the dynamics of the interface of the two dimensional stefan problem is solved explicitly with an ansatz of finite polynomials and reduced to a finite set of algebraic polynomial equations the constants of motion and recurrent formulas for the polynomial coefficients are presented the functional equation for the interfacial form is derived

multiscaling analysis of large scale off lattice dla abstract an improved algorithm has been used to simulate off lattice diffusion limited aggregation dla in two dimensions this enables us to grow clusters of up to 6 106 particles which are the largest such clusters made up to now the density profile in two dimensions is analyzed for the scaling form multiscaling g r r r d d x c x with x r r where r is the radius of gyration and r is the distance from the cluster origin in contrast with previous studies the dimension d x shows no symmetric tendency to drop at values of x but only fluctuates around the global fractal dimension d 1 712 0 003

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

interface dynamics in hele shaw flows with centrifugal forces preventing cusp singularities with rotation a class of exact solutions of hele shaw flows without surface tension in a rotating cell is reported we show that the interplay between injection and rotation modifies the scenario of formation of finite time cusp singularities for a subclass of solutions we show that for any given initial condition there exists a critical rotation rate above which cusp formation is suppressed we also find an exact sufficient condition to avoid cusps simultaneously for all initial conditions within the above subclass

multi cut solutions of laplacian growth a new class of solutions to laplacian growth lg with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases these solutions which are time dependent conformal maps with branch cuts inside the unit circle are governed by a nonlinear integral equation and describe oil fjords with non parallel walls in viscous fingering experiments in hele shaw cells integrals of motion for the multi cut lg solutions in terms of singularities of the schwarz function are found and the dynamics of densities jumps on the cuts are derived the subclass of these solutions with linear cauchy densities on the cuts of the schwarz function is of particular interest because in this case the integral equation for the conformal map becomes linear these solutions can also be of physical importance by representing oil air interfaces which form oil fjords with a constant opening angle in accordance with recent experiments in a hele shaw cell

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

laplacian growth and diffusion limited aggregation different universality classes it had been conjectured that diffusion limited aggregates and laplacian growth patterns with small surface tension are in the same universality class using iterated conformal maps we construct a one parameter family of fractal growth patterns with a continuously varying fractal dimension this family can be used to bound the dimension of laplacian growth patterns from below the bound value is higher than the dimension of diffusion limited aggregates showing that the two problems belong to two different universality classes

self similarity in laplacian growth abstract we consider laplacian growth of self similar domains in different geometries self similarity determines the analytic structure of the schwarz function of the moving boundary the knowledge of this analytic structure allows us to derive the integral equation for the conformal map it is shown that solutions to the integral equation obey also a second order differential equation which is the 1d schroedinger equation with the sinh 2 potential the solutions which are expressed through the gauss hypergeometric function characterize the geometry of self similar patterns in a wedge we also find the potential for the coulomb gas representation of the self similar laplacian growth in a wedge and calculate the corresponding free energy

structural five fold symmetry in the fractal morphology of diffusion limited aggregates abstract the statistical self similarity of the geometry of diffusion limited aggregates and the multifractal nature of the growth probability distribution on the surface of the growing clusters are investigated using the wavelet transform this study reveals the existence of a predominant structural five fold symmetry in the internal frozen region as well as in the active outer region of the interface this observation is corroborated by a statistical analysis of the screening effects that govern diffusion limited aggregation dla growth in linear and sector shaped cells the existence of this symmetry is likely to be a clue to a hierarchichal fractal ordering we report on the discovery of fibonacci sequences in the inner extinct region of large mass off lattice dla clusters with a branching ratio which converges asymptotically to the golden mean we suggest an interpretation of the dla morphology as a quasifractal counterpart of the well ordered snowflake fractal architecture

hamiltonian formulation towards minimization of viscous fluid fingering a variational approach has been recently employed to determine the ideal time dependent injection rate q t that minimizes fingering formation when a fluid is injected in a hele shaw cell filled with another fluid of much greater viscosity however such a calculation is approximate in nature since it has been performed by assuming a high capillary number regime in this work we go one step further and utilize a hamiltonian formulation to obtain an analytical exact solution for q t now valid for arbitrary values of the capillary number moreover this hamiltonian scheme is applied to calculate the corresponding injection rate that minimizes fingering formation in a uniform three dimensional porous media an analysis of the improvement offered by these exact injection rate expressions in comparison with previous approximate results is also provided

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed