computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

varieties of periodic attractor in cellular automata we apply three alternate definitions of attractor to cellular automata examples are given to show that using the different definitions can give different answers to the question does this cellular automaton have a periodic attractor the three definitions are the topological notion of attractor as used by c conley a more measure theoretic version given by j milnor and a variant of milnor s deenition that is based on the concept of the center of attraction of an orbit restrictions on the types of periodic orbits that can be periodic attractors for cellular automata are described with any of these definitions a cellular automaton has at most one periodic attractor additionally if conley s definition is used then a periodic attractor must be a fixed point using milnor s definition each point on a periodic attractor must be fixed by all shifts so the number of symbols used is an upper bound on the period whether the actual upper bound is 1 is unknown with the third definition this restriction is removed and examples are given of onedimensional cellular automata on three symbols that have finite attractors of arbitrarily large size with the third definition a finite attractor is not necessarily a single periodic orbit the purpose of this paper is to describe the types of periodic orbits that can be attractors of a cellular automaton we will consider three different definitions of attractor one that is based upon topological dynamics a second that is based upon a mixture of topological and measure theoretic dynamics and a third that is even more measure theoretic in nature examples will be given to show that these three definitions can lead to different answers to the question of the existence of a periodic attractor for cellular automata we begin with an informal discussion of some of the results details and precise definitions are given in later sections a cellular automaton is a type of endomorphism of a certain function space let z x zm s where zm is the integer lattice in rm and s is a finite set called the symbol set a shift on z is a map ct z z of the form vtx n x n t for some t e zm a cellular automaton is a continuous map f z z that commutes with all of these shifts cellular automata have received much attention in recent years one reason for this attention is the fact that especially for m 1 or 2 cellular automata are readily accessible received by the editors july 20 1989 1980 mathematics subject classi fication 1985 revision primary 58f12 58f21 research supported in part by nsf grant dms 880758 d 1991 american mathematical society 0002 9947 91 1 00 25 per page

some results about the chaotic behavior of cellular automata we study the behavior of cellular automata ca for short in the cantor besicovitch and weyl topologies we solve an open problem about the existence of transitive ca in the besicovitch topology the proof of this result has some interest of its own since it is obtained by using kolmogorov complexity to our knowledge it is the first result about discrete dynamical systems obtained using kolmogorov complexity we also prove that in the besicovitch topology every ca has either a unique periodic point thus a fixed point or an uncountable set of periodic points this result underlines the fact that ca have a great degree of stability it may be considered a further step towards the understanding of ca periodic behavior moreover we prove that in the besicovitch topology there is a special set of configurations the set of toeplitz configurations that plays a role similar to that of spatially periodic configurations in the cantor topology that is it is dense and has a central role in the study of surjectivity and injectivity finally it is shown that the set of spatially quasi periodic configurations is not dense in the weyl topology

discriminative least squares regression for multiclass classification and feature selection this paper presents a framework of discriminative least squares regression lsr for multiclass classification and feature selection the core idea is to enlarge the distance between different classes under the conceptual framework of lsr first a technique called e dragging is introduced to force the regression targets of different classes moving along opposite directions such that the distances between classes can be enlarged then the e draggings are integrated into the lsr model for multiclass classification our learning framework referred to as discriminative lsr has a compact model form where there is no need to train two class machines that are independent of each other with its compact form this model can be naturally extended for feature selection this goal is achieved in terms of l2 1 norm of matrix generating a sparse learning model for feature selection the model for multiclass classification and its extension for feature selection are finally solved elegantly and efficiently experimental evaluation over a range of benchmark datasets indicates the validity of our method

the attractor basin portrait of a cellular automaton local space time structures such as domains and the intervening dislocations dominate a wide class of cellular automaton ca behavior for such spatially extended dynamics regular domains vicinities and attractors are introduced as organizing principles to identify the discretized analogs of attractors basins and separatrices structures used in classifying dissipative continuous state dynamical systems we describe the attractor basin portrait of nonlinear elementary ca rule 18 whose global dynamics is largely determined by a single regular attracting domain the latter s basin is analyzed in terms of subbasin and portal structures associated with particle annihilation the conclusion is that the computational complexity of such ca is more apparent than real transducer machines are constructed that automatically identify domain and dislocation structures in space time count the number of dislocations in a spatial pattern and implement an isomorphism between rule 18 and rule 90 we use a transducer to trace dislocation trajectories and confirm that in rule 18 isolated dislocation trajectories as well as a dislocation gas agree extremely well with the classical model of annihilating diffusive particles the ca efficiently transforms randomness of an initial pattern ensemble into a random walk of dislocations in space time

limit sets of cellular automata associated to probability measures we introduce the concept of limit set associated to a cellular automaton ca and a shift invariant probability measure this is a subshift whose forbidden blocks are exactly those whose probabilities tend to zero as time tends to infinity we compare this probabilistic concept of limit set with the concepts of attractors both in topological and measure theoretic sense we also compare this notion with that of topological limit set in different dynamical situations

manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

conservation of some dynamical properties for operations on cellular automata we consider the family of all the cellular automata ca sharing the same local rule but having different memories this family contains also all ca with memory m 0 one sided ca which can act both on a z and on a n we study several set theoretical and topological properties for these classes in particular we investigate whether the properties of a given ca are preserved when considering the ca obtained by changing the memory of the original one shifting operation furthermore we focus our attention on the one sided ca acting on a z starting from the one sided ca acting on a n and having the same local rule lifting operation as a particular consequence of these investigations we prove that the long standing conjecture surjectivity dense periodic orbits dpo can be restated in several different but equivalent ways furthermore we give some results on properties conserved under the iteration of the ca global map

semi supervised bilinear subspace learning recent research has demonstrated the success of tensor based subspace learning in both unsupervised and supervised configurations e g 2 d pca 2 d lda and dater in this correspondence we present a new semi supervised subspace learning algorithm by integrating the tensor representation and the complementary information conveyed by unlabeled data conventional semi supervised algorithms mostly impose a regularization term based on the data representation in the original feature space instead we utilize graph laplacian regularization based on the low dimensional feature space an iterative algorithm referred to as adaptive regularization based semi supervised discriminant analysis with tensor representation arsda t is also developed to compute the solution in addition to handling tensor data a vector based variant arsda v is also presented in which the tensor data are converted into vectors before subspace learning comprehensive experiments on the cmu pie and yale b databases demonstrate that arsda t brings significant improvement in face recognition accuracy over both conventional supervised and semi supervised subspace learning algorithms

multiview vector valued manifold regularization for multilabel image classification in computer vision image datasets used for classification are naturally associated with multiple labels and comprised of multiple views because each image may contain several objects e g pedestrian bicycle and tree and is properly characterized by multiple visual features e g color texture and shape currently available tools ignore either the label relationship or the view complementarily motivated by the success of the vector valued function that constructs matrix valued kernels to explore the multilabel structure in the output space we introduce multiview vector valued manifold regularization rm mv 3 rm mr to integrate multiple features rm mv 3 rm mr exploits the complementary property of different features and discovers the intrinsic local geometry of the compact support shared by different features under the theme of manifold regularization we conduct extensive experiments on two challenging but popular datasets pascal voc 07 and mir flickr and validate the effectiveness of the proposed rm mv 3 rm mr for image classification

on the approach to statistical equilibrium in an infinite particle lattice dynamic model a deterministic reversible model dynamic system of infinite interacting particles on a lattice is studied the model is a modification by de haan of the kac ring model the time evolution of the statistical states can be described by a hierarchy for the reduced distribution functions it is shown that the system is isomorphic to a bernoulli shift the isomorphism is given in terms of the infinite process of observation of the states of one particle in a fixed site the approach to equilibrium is studied and time asymmetric initial states that approach equilibrium are constructed it is shown that these states contain nonvanishing correlations between infinitely distant particles the bernoulli property allows the construction of a markovian irreversible description of the model realized by a nonunitary transformation the action of this transformation on the statistical states the correlations and the rate of the approach to equilibrium is studied

semi supervised dimension reduction using trace ratio criterion in this brief we address the trace ratio tr problem for semi supervised dimension reduction we first reformulate the objective function of the recent work semi supervised discriminant analysis sda in a tr form we also observe that in sda the low dimensional data representation f is constrained to be in the linear subspace spanned by the training data matrix x i e f xt w in order to relax this hard constraint we introduce a flexible regularizer f xt w 2 which models the regression residual into the reformulated objective function with such relaxation our method referred to as tr based flexible sda tr fsda can better cope with data sampled from a certain type of nonlinear manifold that is somewhat close to a linear subspace in order to address the non trivial optimization problem in tr fsda we further develop an iterative algorithm to simultaneously solve for the low dimensional data representation f and the projection matrix w moreover we theoretically prove that our iterative algorithm converges to the optimum based on the newton raphson method the experiments on two face databases one shape image database and one webpage database demonstrate that tr fsda outperforms the existing semi supervised dimension reduction methods

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

aperiodicity in one dimensional cellular automata abstract cellular automata are a class of mathematical systems characterized by discreteness in space time and state values determinism and local interaction a certain class of one dimensional binary site valued nearest neighbor automata is shown to generate infinitely many aperiodic temporal sequences from arbitrary finite initial conditions on an infinite lattice the class of automaton rules that generate aperiodic temporal sequences are characterized by a particular form of injectivity in their interaction rules included are the nontrivial linear automaton rules that is rules for which the superposition principle holds certain nonlinear automata with injectivity properties similar to those of linear automata and a wider subset of nonlinear automata whose interaction rules satisfy a weaker form of injectivity together with certain symmetry conditions the evolution of this last subset of automata can be viewed as consisting of multiple domains within which behavior exactly mimics that of a linear automaton a linearization technique is used to establish that these domains coalesce in finite time to produce at most two domains and thereby to assert the aperiodicity of their temporal sequences

the theoretical analysis and algorithm on a class of optimal curve fitting problems in this paper the theory and algorithm on a class of optimal curve fitting problems which can be extensively applied to the engineering are established and completed with regard to the curve fitting problems the available theories and methods are largely concerned with the spline interpolation and linear or nonlinear least squares methods and there are lots of ripe results to my best knowledge however in view of the complexity the third method that is to resort to solve functional extremal problems remains completed both in theories and algorithms regarding the class of this problems that is here called as the optimal curve fitting problems under relaxation constraints the analytic expression of solution and solving algorithms were discussed in p cheng k c zhang on a class of constrained functional minimization problems and their numerical solution journal of optimization theory and applications 78 2 1993 p267 p287 p cheng kc zhang unified method for a kind of constrained functional minimization computing technology and automation 10 2 1991 13 25 k c zhang a numerical method for solving a class of variation problems with constraint and its application to engineering numerical mathematics a journal of chinese universities 11 2 1989 1 9 k c zhang the algorithm and analysis of numerical computation beijing science press 2003 pp 172 184 but the existence of optimal solution has not been solved so far in the paper with help of variational principle and convexity analysis we give the proof of existence in details and a different proof about the analytic expression of optimal solution moreover we prove that for every smoothing parameter m this class of optimal curve fitting problems is equivalent to solve a quadratic programming with a spherical constraint which usually appears in the trust region and sqp subproblems and its algorithms have been quite ripe therefore a novel algorithm different from zhang 2003 for this class problems is established finally we give an example of m 2 to validate our theory and method numerical test shows that the method is effective and practical

the dynamics of expansive invertible onesided cellular automata using textile systems we prove the conjecture of boyle and maass that the dynamical system defined by an expansive invertible onesided cellular automaton is topologically conjugate to a topological markov shift we also study expansive leftmost permutive onesided cellular automata and bipermutive endomorphisms of mixing topological markov shifts

combinatorial constructions associated to the dynamics of one sided cellular automata in this paper we study combinatorial constructions which lead to one sided invertible cellular automata with different dynamical behavior equicontinuity existence of equicontinuous points but not equicontinuity sensitivity and expansivity in particular we provide a simple characterization of the class of equicontinuous invertible one sided cellular automata and we construct families of expansive one sided cellular automata

discriminative semi supervised feature selection via manifold regularization feature selection has attracted a huge amount of interest in both research and application communities of data mining we consider the problem of semi supervised feature selection where we are given a small amount of labeled examples and a large amount of unlabeled examples since a small number of labeled samples are usually insufficient for identifying the relevant features the critical problem arising from semi supervised feature selection is how to take advantage of the information underneath the unlabeled data to address this problem we propose a novel discriminative semi supervised feature selection method based on the idea of manifold regularization the proposed approach selects features through maximizing the classification margin between different classes and simultaneously exploiting the geometry of the probability distribution that generates both labeled and unlabeled data in comparison with previous semi supervised feature selection algorithms our proposed semi supervised feature selection method is an embedded feature selection method and is able to find more discriminative features we formulate the proposed feature selection method into a convex concave optimization problem where the saddle point corresponds to the optimal solution to find the optimal solution the level method a fairly recent optimization method is employed we also present a theoretic proof of the convergence rate for the application of the level method to our problem empirical evaluation on several benchmark data sets demonstrates the effectiveness of the proposed semi supervised feature selection method

intriguing properties of global structure in some classes of finite cellular automata abstract the simplest one dimensional cellular automata introduced recently by wolfram are analyzed particularly the finite cellular automata with two special evolution rules are studied non trivial correspondence in global structure is found between the state transition diagrams under the two rules correspondence of cycle structure basin volume of each cycle and structure of transient trees etc origin of the correspondence is explored on the basis of local properties of the evolution rules it is shown that this correspondence becomes exact in the thermodynamic limit

a note on curvature variation minimizing cubic hermite interpolants in the paper 1 jaklic and agar studied curvature variation minimizing cubic hermite interpolants to match planar two point g 1 hermite data they obtained the optimal cubic curve by minimizing an approximate form of the curvature variation energy in this paper we present a simple method for this problem by minimizing the jerk energy which is also an approximate form of the curvature variation energy the unique solution can be easily obtained since the jerk energy is represented as a quadratic polynomial of two unknowns and is strictly convex finally we prove that our method is equivalent to their method

multiplicative cellular automata on nilpotent groups structure entropy and asymptotics if mathbb m mathbb z d and mathcal b is a finite nonabelian group then mathcal b mathbb m is a compact group a multiplicative cellular automaton mca is a continuous transformation mathfrak g b mathbb m to b mathbb m which commutes with all shift maps and where nearby coordinates are combined using the multiplication operation of mathcal b we characterize when mca are group endomorphisms of mathcal b mathbb m and show that mca on mathcal b mathbb m inherit a natural structure theory from the structure of mathcal b we apply this structure theory to compute the measurable entropy of mca and to study convergence of initial measures to haar measure

around probabilistic cellular automata we survey probabilistic cellular automata with approaches coming from combinatorics statistical physics and theoretical computer science each bringing a different viewpoint some of the questions studied are specific to a domain and some others are shared most notably the ergodicity problem

flexible manifold embedding a framework for semi supervised and unsupervised dimension reduction we propose a unified manifold learning framework for semi supervised and unsupervised dimension reduction by employing a simple but effective linear regression function to map the new data points for semi supervised dimension reduction we aim to find the optimal prediction labels f for all the training samples x the linear regression function h x and the regression residue f0 f h x simultaneously our new objective function integrates two terms related to label fitness and manifold smoothness as well as a flexible penalty term defined on the residue f0 our semi supervised learning framework referred to as flexible manifold embedding fme can effectively utilize label information from labeled data as well as a manifold structure from both labeled and unlabeled data by modeling the mismatch between h x and f we show that fme relaxes the hard linear constraint f h x in manifold regularization mr making it better cope with the data sampled from a nonlinear manifold in addition we propose a simplified version referred to as fme u for unsupervised dimension reduction we also show that our proposed framework provides a unified view to explain and understand many semi supervised supervised and unsupervised dimension reduction techniques comprehensive experiments on several benchmark databases demonstrate the significant improvement over existing dimension reduction algorithms

on cesaro limit distribution of a class of permutative cellular automata we study cesaro means time averages of the evolution measures of the class of permutative cellular automata over 0 1 defined by mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepekkfr0xfr x fr xb9adbaqaaegacigaaiaabeqaamaabaabaagcbagaaiikaiabew 9aqnaabaaaleaacawgcbaabeaakiaadihacaggpawaasbaasqaaiaa d6gaaeqaaogaeyypa0jaamieamaabaaaleaacawgubgaagjcvlaayi w7cqghrawkcaami8uaagjcvlaadkfaaeqaaogaey4kasiaeuioda1a a0baasqaaiaadqgacaami8uaagjcvlabg2da9iaayiw7caami8uaag imaaqaaiaadkfacaami8uaagjcvlabgkhitiaayiw7caami8uaagym aaaakiaayiw7caami8uaagjcvlaayiw7caami8uaagjcvlaacicaca aixagaey4kasiaamoyamaabaaaleaacawgqbaabeaakiabgucariaa dihadawgaawcbagaamobaiaayiw7caami8uaey4kasiaagjcvlaayi w7cawgqbaabeaakiaacmcaaaa 7530 varphi b x n x n kern 1pt kern 1pt kern 1pt kern 1pt r pi j kern 1pt kern 1pt kern 1pt kern 1pt 0 r kern 1pt kern 1pt kern 1pt kern 1pt 1 kern 1pt kern 1pt kern 1pt kern 1pt kern 1pt kern 1pt 1 b j x n kern 1pt kern 1pt kern 1pt kern 1pt j where b b 0 b r 1is an aperiodic block in 0 1 r and operations are taken mod 2 if the initial measure is bernoulli we prove that the limit of the cesaro mean of the first column distribution exists when r 1 and b 1 b is the mod 2 sum automaton for this automaton we show that the limit is the 1 2 1 2 bernoulli measure and if the initial measure is markov we show that the limit of cesaro mean of the one site distribution is equidistributed

fundamental study directional dynamics for cellular automata a sensitivity to initial condition approach a cellular automaton is a continuous function f defined on a full shift a z which commutes with the shift s often to study the dynamics of f one only considers implicitly s however it is possible to emphasize the spatio temporal structure produced by considering the dynamics of the zxn action induced by s f in this purpose we study the notion of directional dynamics in particular we are interested in directions of equicontinuity and expansivity which generalize the concepts introduced by gilman robert h gilman classes of linear automata ergodic theory dynam systems 7 1 1987 105 118 and p kurka petr kurka languages equicontinuity and attractors in cellular automata ergodic theory dynam systems 17 2 1997 417 433 we study the sets of directions which exhibit this special kind of dynamics showing that they induce a discrete geometry in space time diagrams

backtrackless walks on a graph the aim of this paper is to explore the use of backtrackless walks and prime cycles for characterizing both labeled and unlabeled graphs the reason for using backtrackless walks and prime cycles is that they avoid tottering and can increase the discriminative power of the resulting graph representation however the use of such methods is limited in practice because of their computational cost in this paper we present efficient methods for computing graph kernels which are based on backtrackless walks in a labeled graph and whose worst case running time is the same as that of kernels based on random walks for clustering unlabeled graphs we construct feature vectors using ihara coefficients since these coefficients are related to the frequencies of prime cycles in the graph to efficiently compute the low order coefficients we present an o v 3 algorithm which is better than the o v 6 worst case running time of previously known algorithms in the experimental evaluation we apply the proposed method to clustering both labeled and unlabeled graphs the results show that using backtrackless walks and prime cycles instead of random walks can increase the accuracy of recognition

prime discriminant simplicial complex the structure representation of data distribution plays an important role in understanding the underlying mechanism of generating data in this paper we propose the prime discriminant simplicial complex pdsc by utilizing persistent homology to capture such structures assuming that each class is represented with a prime simplicial complex we classify unlabeled samples based on the nearest projection distances from the samples to the simplicial complexes we also extend the extrapolation ability of these complexes with a projection constraint term experiments in simulated and practical datasets indicate that compared with several published algorithms the proposed pdsc approaches achieve promising performance without losing structure representation

clustered nystrom method for large scale manifold learning and dimension reduction kernel or similarity matrix plays a key role in many machine learning algorithms such as kernel methods manifold learning and dimension reduction however the cost of storing and manipulating the complete kernel matrix makes it infeasible for large problems the nystrom method is a popular sampling based low rank approximation scheme for reducing the computational burdens in handling large kernel matrices in this paper we analyze how the approximating quality of the nystrom method depends on the choice of landmark points and in particular the encoding powers of the landmark points in summarizing the data our non probabilistic error analysis justifies a clustered nystrom method that uses the k means clustering centers as landmark points our algorithm can be applied to scale up a wide variety of algorithms that depend on the eigenvalue decomposition of kernel matrix or its variant such as kernel principal component analysis laplacian eigenmap spectral clustering as well as those involving kernel matrix inverse such as least squares support vector machine and gaussian process regression extensive experiments demonstrate the competitive performance of our algorithm in both accuracy and efficiency

area preserving geometric hermite interpolation abstract in this paper we establish a framework for planar geometric interpolation with exact area preservation using cubic bezier polynomials we show there exists a family of such curves which are 5th order accurate one order higher than standard geometric cubic hermite interpolation we prove this result is valid when the curvature at the endpoints does not vanish and in the case of vanishing curvature the interpolation is 4th order accurate the method is computationally efficient and prescribes the parametrization speed at endpoints through an explicit formula based on the given data additional accuracy i e same order but lower error constant may be obtained through an iterative process to find optimal parametrization speeds which further reduces the error while still preserving the prescribed area exactly

applications of ergodic theory and sofic systems to cellular automata abstract we indicate a mathematical framework for analysing the evolution of cellular automata those automata obeying an additive rule are shown to be the same as endomorphisms of a compact abelian group and therefore their statistical and dynamical behavior can be told exactly by using fourier analysis and ergodic theory those obeying certain nonlinear rules are closely tied to a finitely described object called a sofic system but the underlying statistics appear to be more complicated and interesting we conclude by formulating several conjectures about one such system

planar cubic g1 interpolatory splines with small strain energy in this paper a classical problem of the construction of a cubic g 1 continuous interpolatory spline curve is considered the only data prescribed are interpolation points while tangent directions are unknown they are constructed automatically in such a way that a particular minimization of the strain energy of the spline curve is applied the resulting spline curve is constructed locally and is regular cusp loop and fold free numerical examples demonstrate that it is satisfactory as far as the shape of the curve is concerned

scaling up graph based semisupervised learning via prototype vector machines when the amount of labeled data are limited semisupervised learning can improve the learner s performance by also using the often easily available unlabeled data in particular a popular approach requires the learned function to be smooth on the underlying data manifold by approximating this manifold as a weighted graph such graph based techniques can often achieve state of the art performance however their high time and space complexities make them less attractive on large data sets in this paper we propose to scale up graph based semisupervised learning using a set of sparse prototypes derived from the data these prototypes serve as a small set of data representatives which can be used to approximate the graph based regularizer and to control model complexity consequently both training and testing become much more efficient moreover when the gaussian kernel is used to define the graph affinity a simple and principled method to select the prototypes can be obtained experiments on a number of real world data sets demonstrate encouraging performance and scaling properties of the proposed approach it also compares favorably with models learned via l1 regularization at the same level of model sparsity these results demonstrate the efficacy of the proposed approach in producing highly parsimonious and accurate models for semisupervised learning

c algebras associated with interval maps for each piecewise monotonic map r of 0 1 we associate a pair of c algebras f and or and calculate their k groups the algebra f is an ai algebra we characterize when f and or are simple in those cases f has a unique trace and or is purely infinite with a unique kms state in the case that is markov these algebras include the cuntz krieger algebras o a and the associated af algebras f a other examples for which the k groups are computed include tent maps quadratic maps multimodal maps interval exchange maps and transformations for the case of interval exchange maps and of transformations the c algebra or coincides with the algebras defined by putnam and katayama matsumoto watatani respectively

defect particle kinematics in one dimensional cellular automata let a z be the cantor space of bi infinite sequences in a finite alphabet a and let s be the shift map on a z a cellular automaton is a continuous s commuting self map f of a z and a f invariant subshift is a closed f s invariant subset s a z suppose a a z is s admissible everywhere except for some small region we call a defect it has been empirically observed that such defects persist under iteration of f and often propagate like particles we characterize the motion of these particles and show that it falls into several regimes ranging from simple deterministic motion to generalized random walks to complex motion emulating turing machines or pushdown automata one consequence is that some questions about defect behaviour are formally undecidable

laplacian embedded regression for scalable manifold regularization semi supervised learning ssl as a powerful tool to learn from a limited number of labeled data and a large number of unlabeled data has been attracting increasing attention in the machine learning community in particular the manifold regularization framework has laid solid theoretical foundations for a large family of ssl algorithms such as laplacian support vector machine lapsvm and laplacian regularized least squares laprls however most of these algorithms are limited to small scale problems due to the high computational cost of the matrix inversion operation involved in the optimization problem in this paper we propose a novel framework called laplacian embedded regression by introducing an intermediate decision variable into the manifold regularization framework by using insensitive loss we obtain the laplacian embedded support vector regression lapesvr algorithm which inherits the sparse solution from svr also we derive laplacian embedded rls laperls corresponding to rls under the proposed framework both lapesvr and laperls posses a simpler form of a transformed kernel which is the summation of the original kernel and a graph kernel that captures the manifold structure the benefits of the transformed kernel are two fold 1 we can deal with the original kernel matrix and the graph laplacian matrix in the graph kernel separately and 2 if the graph laplacian matrix is sparse we only need to perform the inverse operation for a sparse matrix which is much more efficient when compared with that for a dense one inspired by kernel principal component analysis we further propose to project the introduced decision variable into a subspace spanned by a few eigenvectors of the graph laplacian matrix in order to better reflect the data manifold as well as accelerate the calculation of the graph kernel allowing our methods to efficiently and effectively cope with large scale ssl problems extensive experiments on both toy and real world data sets show the effectiveness and scalability of the proposed framework

deformed graph laplacian for semisupervised learning graph laplacian has been widely exploited in traditional graph based semisupervised learning ssl algorithms to regulate the labels of examples that vary smoothly on the graph although it achieves a promising performance in both transductive and inductive learning it is not effective for handling ambiguous examples shown in fig 1 this paper introduces deformed graph laplacian dgl and presents label prediction via dgl lpdgl for ssl the local smoothness term used in lpdgl which regularizes examples and their neighbors locally is able to improve classification accuracy by properly dealing with ambiguous examples theoretical studies reveal that lpdgl obtains the globally optimal decision function and the free parameters are easy to tune the generalization bound is derived based on the robustness analysis experiments on a variety of real world data sets demonstrate that lpdgl achieves top level performance on both transductive and inductive settings by comparing it with popular ssl algorithms such as harmonic functions anchorgraph regularization linear neighborhood propagation laplacian regularized least square and laplacian support vector machine

limit behaviour of µ equicontinuous cellular automata the concept of equicontinuity was introduced in 12 to classify cellular automata we show that under some conditions the sequence of cesaro averages of a measure converge under the actions of a equicontinuous ca we address questions raised in 3 on whether the limit measure is either shift ergodic a uniform bernoulli measure or ergodic with respect to the ca many of our results hold for ca on multidimensional subshifts

fick s law assisted propagation for semisupervised learning how to propagate the label information from labeled examples to unlabeled examples is a critical problem for graph based semisupervised learning many label propagation algorithms have been developed in recent years and have obtained promising performance on various applications however the eigenvalues of iteration matrices in these algorithms are usually distributed irregularly which slow down the convergence rate and impair the learning performance this paper proposes a novel label propagation method called fick s law assisted propagation flap unlike the existing algorithms that are directly derived from statistical learning flap is deduced on the basis of the theory of fick s first law of diffusion which is widely known as the fundamental theory in fluid spreading we prove that flap will converge with linear rate and show that flap makes eigenvalues of the iteration matrix distributed regularly comprehensive experimental evaluations on synthetic and practical datasets reveal that flap obtains encouraging results in terms of both accuracy and efficiency

local structure theory for cellular automata abstract the mean field theory for cellular automata wolfram and schulman and seiden is generalized to the local structure theory the local structure theory is a sequence of finitely parameterized models of the statistical features of a cellular automaton s evolution the n th model in the sequence takes into account correlations in terms of the probability of blocks of n states a class of measures the n block measures is introduced the local structure operator of order n maps n block measures to n block measures in a manner which reflects the cellular automaton map on blocks of states the fixed points of the map on measures approximate the invariant measures of the cellular automaton the ability of the local structure theory to model evolution from uncorrelated initial distributions is studied the theory gives exact results in simple cases in more complex cases monte carlo numerical experiments suggest that an accurate statistical portrait of cellular automaton evolution is obtained the invariant measures of a cellular automaton and the stability of these measures may be obtained from the local structure theory the local structure theory appears to be a powerful method for characterization and classification of cellular automata nearest neighbor cellular automata with two states per cell are studied using this method

statistical mechanics of surjective cellular automata reversible cellular automata are seen as microscopic physical models and their states of macroscopic equilibrium are described using invariant probability measures we establish a connection between the invariance of gibbs measures and the conservation of additive quantities in surjective cellular automata namely we show that the simplex of shift invariant gibbs measures associated to a hamiltonian is invariant under a surjective cellular automaton if and only if the cellular automaton conserves the hamiltonian a special case is the well known invariance of the uniform bernoulli measure under surjective cellular automata which corresponds to the conservation of the trivial hamiltonian as an application we obtain results indicating the lack of non trivial gibbs or markov invariant measures for sufficiently chaotic cellular automata we discuss the relevance of the randomization property of algebraic cellular automata to the problem of approach to macroscopic equilibrium and pose several open questions as an aside a shift invariant pre image of a gibbs measure under a pre injective factor map between shifts of finite type turns out to be always a gibbs measure we provide a sufficient condition under which the image of a gibbs measure under a pre injective factor map is not a gibbs measure we point out a potential application of pre injective factor maps as a tool in the study of phase transitions in statistical mechanical models

on geometric interpolation by planar parametric polynomial curves in this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited the conjecture that a parametric polynomial curve of degree n can interpolate 2n given points in r 2 is confirmed for n 5 under certain natural restrictions this conclusion also implies the optimal asymptotic approximation order more generally the optimal order 2n can be achieved as soon as the interpolating curve exists

planar quintic g2 hermite interpolation with minimum strain energy in this paper we study planar quintic g 2 hermite interpolation with minimum strain energy to match arbitrary g 2 hermite data a quintic curve is expressed in terms of four free parameters that encode the local reparameterization at the endpoints and are available for further optimization we express the approximate strain energy as a quartic function in four parameters whose minimum can be found by solving an optimization problem of two parameters relating to the magnitudes of endpoint tangent vectors a feasible region is used while searching the optimal values of these two parameters such that the interpolating curve can preserve tangent directions and avoid singularities at the endpoints we then solve this constrained minimization problem via the proximal gradient method several comparative examples are provided to demonstrate the effectiveness of the proposed method and applications to shape design are also shown

hermite geometric interpolation by rational bezier spatial curves polynomial geometric interpolation by parametric curves has become one of the standard techniques for interpolation of geometric data an obvious generalization leads to rational geometric interpolation schemes which are a much less investigated research topic the aim of this paper is to present a general framework for hermite geometric interpolation by rational bezier spatial curves in particular cubic g 2 and quartic g 3 interpolations are analyzed in detail systems of nonlinear equations are derived in a simplified form and the existence of admissible solutions is studied for the cubic case geometric conditions implying solvability of the nonlinear system are also stated the asymptotic analysis is done in both cases and optimal approximation orders are proved numerical examples are given which confirm the theoretical results

generalized core vector machines kernel methods such as the support vector machine svm are often formulated as quadratic programming qp problems however given m training patterns a naive implementation of the qp solver takes o m 3 training time and at least o m2 space hence scaling up these qps is a major stumbling block in applying kernel methods on very large data sets and a replacement of the naive method for finding the qp solutions is highly desirable recently by using approximation algorithms for the minimum enclosing ball meb problem we proposed the core vector machine cvm algorithm that is much faster and can handle much larger data sets than existing svm implementations however the cvm can only be used with certain kernel functions and kernel methods for example the very popular support vector regression svr cannot be used with the cvm in this paper we introduce the center constrained meb problem and subsequently extend the cvm algorithm the generalized cvm algorithm can now be used with any linear nonlinear kernel and can also be applied to kernel methods such as svr and the ranking svm moreover like the original cvm its asymptotic time complexity is again linear in m and its space complexity is independent of m experiments show that the generalized cvm has comparable performance with state of the art svm and svr implementations but is faster and produces fewer support vectors on very large data sets

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

soliton like behavior in automata abstract we propose a new kind of automaton that uses newly computed site values as soon as they are available we call them filter automata fa they are analogous to infinite impulse response iir digital filters whereas the usual cellular automata ca correspond to finite impulse response fir digital filters it is shown that as a class the fa s are equivalent to ca s in the sense that the same array of space generation values can be produced they must be generated in a different order however a particular class of irreversible totalistic fa s are described that support a profusion of persistent structures that move at different speeds and these particle like patterns collide in nondestructive ways they often pass through one another with nothing more than a phase jump much like the solitons that arise in the solution of certain nonlinear differential equations histograms of speed displacement and period are given for neighborhood radii from 2 to 6 and particles with generators up to 16 bits wide we then present statistics for neighborhood radii 2 to 9 which show that collisions which preserve the identity of particles are very common

automated induction of heterogeneous proximity measures for supervised spectral embedding spectral embedding methods have played a very important role in dimensionality reduction and feature generation in machine learning supervised spectral embedding methods additionally improve the classification of labeled data using proximity information that considers both features and class labels however these calculate the proximity information by treating all intraclass similarities homogeneously for all classes and similarly for all interclass samples in this paper we propose a very novel and generic method which can treat all the intra and interclass sample similarities heterogeneously by potentially using a different proximity function for each class and each class pair to handle the complexity of selecting these functions we employ evolutionary programming as an automated powerful formula induction engine in addition for computational efficiency and expressive power we use a compact matrix tree representation equipped with a broad set of functions that can build most currently used similarity functions as well as new ones model selection is data driven because the entire model is symbolically instantiated using only problem training data and no user selected functions or parameters are required we perform thorough comparative experimentations with multiple classification datasets and many existing state of the art embedding methods which show that the proposed algorithm is very competitive in terms of classification accuracy and generalization ability

endomorphisms of expansive systems on compact metric spaces and the pseudo orbit tracing property we investigate the interrelationships between the dynamical properties of commuting continuous maps of a compact metric space let x be a compact metric space first we show the following if t x x is an expansive onto continuous map with the pseudo orbit tracing property potp and if there is a topologically mixing continuous map sp x x with r p 99t then t is topologically mixing if t x x and o x x are commuting expansive onto continuous maps with potp and if r is topologically transitive with period p then for some k dividing p x ui o bi where the b 0 x is a topologically transitive expansive homeomorphism having potp and o x x is a positively expansive onto continuous map with pw 9s then p has potp and is constant to one further we define essentially lr endomorphisms for systems of expansive onto continuous maps of compact metric spaces and prove that if r x x x is an expansive homeomorphism with canonical coordinates and s is an essentially lr automorphism of x r then po has canonical coordinates we add some discussions on basic properties of the essentially lr endomorphisms

curvature variation minimizing cubic hermite interpolants in this paper planar parametric hermite cubic interpolants with small curvature variation are studied by minimization of an appropriate approximate functional it is shown that a unique solution of the interpolation problem exists and has a nice geometric interpretation the best solution of such a problem is a quadratic geometric interpolant the optimal approximation order 4 of the solution is confirmed the approach is combined with strain energy minimization in order to obtain g 1 cubic interpolatory spline

modeling the manifolds of images of handwritten digits this paper describes two new methods for modeling the manifolds of digitized images of handwritten digits the models allow a priori information about the structure of the manifolds to be combined with empirical data accurate modeling of the manifolds allows digits to be discriminated using the relative probability densities under the alternative models one of the methods is grounded in principal components analysis the other in factor analysis both methods are based on locally linear low dimensional approximations to the underlying data manifold links with other methods that model the manifold are discussed

core vector machines fast svm training on very large data sets standard svm training has o m3 time and o m2 space complexities where m is the training set size it is thus computationally infeasible on very large data sets by observing that practical svm implementations only approximate the optimal solution by an iterative strategy we scale up kernel methods by exploiting such approximateness in this paper we first show that many kernel methods can be equivalently formulated as minimum enclosing ball meb problems in computational geometry then by adopting an efficient approximate meb algorithm we obtain provably approximately optimal solutions with the idea of core sets our proposed core vector machine cvm algorithm can be used with nonlinear kernels and has a time complexity that is linear in m and a space complexity that is independent of m experiments on large toy and real world data sets demonstrate that the cvm is as accurate as existing svm implementations but is much faster and can handle much larger data sets than existing scale up methods for example cvm with the gaussian kernel produces superior results on the kddcup 99 intrusion detection data which has about five million training patterns in only 1 4 seconds on a 3 2ghz pentium 4 pc