quantum superimposing multiple cloning cum complementing machine we prove that the nonorthogonal states randomly selected from a set can evolve into a linear superposition of multiple original and orthogonal complementing states with failure branch if and only if the input states are linearly independent the results for a single input state are also generalized into the case of several copies of an input state

perceptron winnow and pac learning we analyze the performance of the widely studied perceptron and winnow algorithms for learning linear threshold functions under valiant s probably approximately correct pac model of concept learning we show that under the uniform distribution on boolean examples the perceptron algorithm can efficiently pac learn nested functions a class of linear threshold functions known to be hard for perceptron under arbitrary distributions but cannot efficiently pac learn arbitrary linear threshold functions we also prove that littlestone s winnow algorithm is not an efficient pac learning algorithm for the class of positive linear threshold functions thus answering an open question posed by schmitt neural comput 10 1998 pp 235 250 based on our results we conjecture that no local algorithm can learn linear threshold functions efficiently

l_ 1 norm low rank matrix decomposition by neural networks and mollifiers the l 1 norm cost function of the low rank approximation of the matrix with missing entries is not smooth and also cannot be transformed into a standard linear or quadratic programming problem and thus the optimization of this cost function is still not well solved to tackle this problem first a mollifier is used to smooth the cost function high closeness of the smoothed function to the original one can be obtained by tuning the parameters contained in the mollifier next a recurrent neural network is proposed to optimize the mollified function which will converge to a local minimum in addition to boost the speed of the system the mollifying process is implemented by a filtering procedure the influence of two mollifier parameters is theoretically analyzed and experimentally confirmed showing that one of the parameters is critical to computational efficiency and accuracy while the other not a large number of experiments on synthetic data show that the proposed method is competitive to the state of the art methods in particular the experiments on large matrices and a real application in the structure from motion indicate that the memory requirement of the proposed algorithm is mild making it suitable for real applications that often involve large scale matrix decomposition

a paraperspective factorization method for shape and motion recovery the factorization method first developed by tomasi and kanade 1992 recovers both the shape of an object and its motion from a sequence of images using many images and tracking many feature points to obtain highly redundant feature position information the method robustly processes the feature trajectory information using singular value decomposition svd taking advantage of the linear algebraic properties of orthographic projection however an orthographic formulation limits the range of motions the method can accommodate paraperspective projection first introduced by ohta et al 1981 is a projection model that closely approximates perspective projection by modeling several effects not modeled under orthographic projection while retaining linear algebraic properties our paraperspective factorization method can be applied to a much wider range of motion scenarios including image sequences containing motion toward the camera and aerial image sequences of terrain taken from a low altitude airplane

learnability of quantified formulas we consider the following classes of quantified formulas fix a set of basic relations called a basis take conjunctions of these basic relations applied to variables and constants in arbitrary ways finally quantify existentially or universally some of the variables we introduce some conditions on the basis that guarantee efficient learnability furthermore we show that with certain restrictions on the basis the classification is complete we introduce as an intermediate tool a link between this class of quantified formulas and some well studied structures in universal algebra called clones more precisely we prove that the computational complexity of the learnability of these formulas is completely determined by a simple algebraic property of the basis of relations their clone of polymorphisms finally we use this technique to give a simpler proof of the already known dichotomy theorem over boolean domains

the inverse satisfiability problem we study the complexity of telling whether a set of bit vectors represents the set of all satisfying truth assignments of a boolean expression of a certain type we show that the problem is conp complete when the expression is required to be in conjunctive normal form with three literals per clause 3cnf we also prove a dichotomy theorem analogous to the classical one by schaefer stating that unless p np the problem can be solved in polynomial time if and only if the clauses allowed are all horn or all anti horn or all 2cnf or all equivalent to equations modulo two

the complexity of minimal satisfiability problems a dichotomy theorem for a class of decision problems is a result asserting that certain problems in the class are solvable in polynomial time while the rest are np complete the first remarkable such dichotomy theorem was proved by schaefer in 1978 it concerns the class of generalized satisfiability problems sat s whose input is a cnf s formula i e a formula constructed from elements of a fixed set s of generalized connectives using conjunctions and substitutions by variables here we investigate the complexity of minimal satisfiability problems min sat s where s is a fixed set of generalized connectives the input to such a problem is a cnf s formula and a satisfying truth assignment the question is to decide whether there is another satisfying truth assignment that is strictly smaller than the given truth assignment with respect to the coordinate wise partial order on truth assignments minimal satisfiability problems were first studied by researchers in artificial intelligence while investigating the computational complexity of propositional circumscription the question of whether dichotomy theorems can be proved for these problems was raised at that time but was left open we settle this question affirmatively by establishing a dichotomy theorem for the class of all min sat s problems where s is a finite set of generalized connectives we also prove a dichotomy theorem for a variant of min sat s in which the minimization is restricted to a subset of the variables whereas the remaining variables may vary arbitrarily this variant is related to extensions of propositional circumscription and was first studied by cadoli moreover we show that similar dichotomy theorems hold also when some of the variables are assigned constant values finally we give simple criteria that tell apart the polynomial time solvable cases of these minimal satisfiability problems from the np complete ones

projective multiview structure and motion from element wise factorization the sturm triggs type iteration is a classic approach for solving the projective structure from motion sfm factorization problem which iteratively solves the projective depths scene structure and camera motions in an alternated fashion like many other iterative algorithms the sturm triggs iteration suffers from common drawbacks such as requiring a good initialization the iteration may not converge or may only converge to a local minimum and so on in this paper we formulate the projective sfm problem as a novel and original element wise factorization i e hadamard factorization problem as opposed to the conventional matrix factorization thanks to this formulation we are able to solve the projective depths structure and camera motions simultaneously by convex optimization to address the scalability issue we adopt a continuation based algorithm our method is a global method in the sense that it is guaranteed to obtain a globally optimal solution up to relaxation gap another advantage is that our method can handle challenging real world situations such as missing data and outliers quite easily and all in a natural and unified manner extensive experiments on both synthetic and real images show comparable results compared with the state of the art methods

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

rigid structure from motion from a blind source separation perspective we present an information theoretic approach to define the problem of structure from motion sfm as a blind source separation one given that for almost all practical joint densities of shape points the marginal densities are non gaussian we show how higher order statistics can be used to provide improvements in shape estimates over the methods of factorization via singular value decomposition svd bundle adjustment and bayesian approaches previous techniques have either explicitly or implicitly used only second order statistics in models of shape or noise a further advantage of viewing sfm as a blind source problem is that it easily allows for the inclusion of noise and shape models resulting in maximum likelihood ml or maximum a posteriori map shape and motion estimates a key result is that the blind source separation approach has the ability to recover the motion and shape matrices without the need to explicitly know the motion or shape pdf we demonstrate that it suffices to know whether the pdf is sub or super gaussian i e semi parametric estimation and derive a simple formulation to determine this from the data we provide extensive experimental results on synthetic and real tracked points in order to quantify the improvement obtained from this technique

novel cloning machine with supplementary information probabilistic cloning was first proposed by duan and guo then pati established a novel cloning machine ncm for copying superposition of multiple clones simultaneously in this paper we deal with the novel cloning machine with supplementary information ncmsi for the case of cloning two states we demonstrate that the optimal efficiency of the ncmsi in which the original party and the supplementary party can perform quantum communication equals that achieved by a two step cloning protocol wherein classical communication is only allowed between the original and the supplementary parties from this equivalence it follows that ncmsi may increase the success probabilities for copying also an upper bound on the unambiguous discrimination of two nonorthogonal pure product states is derived our investigation generalizes and completes the results in the literature

multi frame correspondence estimation using subspace constraints when a rigid scene is imaged by a moving camera the set of all displacements of all points across multiple frames often resides in a low dimensional linear subspace linear subspace constraints have been used successfully in the past for recovering 3d structure and 3d motion information from multiple frames e g by using the factorization method of tomasi and kanade 1992 international journal of computer vision 9 137 154 these methods assume that the 2d correspondences have been precomputed however correspondence estimation is a fundamental problem in motion analysis in this paper we show how the multi frame subspace constraints can be used for constraining the 2d correspondence estimation process itself we show that the multi frame subspace constraints are valid not only for affine cameras but also for a variety of imaging models scene models and motion models the multi frame subspace constraints are first translated from constraints on correspondences to constraints directly on image measurements e g image brightness quantities these brightness based subspace constraints are then used for estimating the correspondences by requiring that all corresponding points across all video frames reside in the appropriate low dimensional linear subspace the multi frame subspace constraints are geometrically meaningful and are not violated at depth discontinuities nor when the camera motion changes abruptly these constraints can therefore replace heuristic constraints commonly used in optical flow estimation such as spatial or temporal smoothness

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

recovering the missing components in a large noisy low rank matrix application to sfm in computer vision it is common to require operations on matrices with missing data for example because of occlusion or tracking failures in the structure from motion sfm problem such a problem can be tackled allowing the recovery of the missing values if the matrix should be of low rank when noise free the filling in of missing values is known as imputation imputation can also be applied in the various subspace techniques for face and shape classification online recommender systems and a wide variety of other applications however iterative imputation can lead to the recovery of data that is seriously in error in this paper we provide a method to recover the most reliable imputation in terms of deciding when the inclusion of extra rows or columns containing significant numbers of missing entries is likely to lead to poor recovery of the missing parts although the proposed approach can be equally applied to a wide range of imputation methods this paper addresses only the sfm problem the performance of the proposed method is compared with jacobs and shum s methods for sfm

principal component analysis with missing data and its application to polyhedral object modeling observation based object modeling often requires integration of shape descriptions from different views to overcome the problems of errors and their accumulation we have developed a weighted least squares wls approach which simultaneously recovers object shape and transformation among different views without recovering interframe motion we show that object modeling from a range image sequence is a problem of principal component analysis with missing data pcamd which can be generalized as a wls minimization problem an efficient algorithm is devised after we have segmented planar surface regions in each view and tracked them over the image sequence we construct a normal measurement matrix of surface normals and a distance measurement matrix of normal distances to the origin for all visible regions over the whole sequence of views respectively these two matrices which have many missing elements due to noise occlusion and mismatching enable us to formulate multiple view merging as a combination of two wls problems a two step algorithm is presented after surface equations are extracted spatial connectivity among the surfaces is established to enable the polyhedral object model to be constructed experiments using synthetic data and real range images show that our approach is robust against noise and mismatching and generates accurate polyhedral object models

probabilistic cloning and orthogonal complementing of an unknown epr state abstract in this paper a novel scheme is proposed where with the help of victor a state preparer an unknown epr pair is copied in a certain probability so that three parties of communication can obtain the epr state respectively the first stage of this scheme is to teleport an unknown epr state to carla one receiver the second stage of this scheme alice a sender and bob the other receiver get the same epr pair or an orthogonal complement copy of the epr pair with the help of victor

trace distance from the viewpoint of quantum operation techniques in the present paper the trace distance is exposed within the quantum operations formalism the definition of the trace distance in terms of a maximum over all quantum operations is given it is shown that for any pair of different states there are an uncountably infinite number of maximizing quantum operations conversely for any operation of the described type there are an uncountably infinite number of those pairs of states that the maximum is reached by the operation the behaviour of the trace distance under considered operations is studied relations and distinctions between the trace distance and the sine distance are discussed

projective reconstruction from multiple views with minimization of 2d reprojection error the problem of projective reconstruction by minimization of the 2d reprojection error in multiple images is considered although bundle adjustment techniques can be used to minimize the 2d reprojection error these methods being based on nonlinear optimization algorithms require a good starting point quasi linear algorithms with better global convergence properties can be used to generate an initial solution before submitting it to bundle adjustment for refinement in this paper we propose a factorization based method to integrate the initial search as well as the bundle adjustment into a single algorithm consisting of a sequence of weighted least squares problems in which a control parameter is initially set to a relaxed state to allow the search of a good initial solution and subsequently tightened up to force the final solution to approach a minimum point of the 2d reprojection error the proposed algorithm is guaranteed to converge our method readily handles images with missing points

on learning perceptrons with binary weights we present an algorithm that pac learns any perceptron with binary weights and arbitrary threshold under the family of product distributions the sample complexity of this algorithm is of o n e 4 ln n and its running time increases only linearly with the number of training examples the algorithm does not try to find an hypothesis that agrees with all of the training examples rather it constructs a binary perceptron based on various probabilistic estimates obtained from the training examples we show that under the restricted case of the uniform distribution and zero threshold the algorithm reduces to the well known clipped hebb rule we calculate exactly the average generalization rate i e the learning curve of the algorithm under the uniform distribution in the limit of an infinite number of dimensions we find that the error rate decreases exponentially as a function of the number of training examples hence the average case analysis gives a sample complexity of o n ln 1 e a large improvement over the pac learning analysis the analytical expression of the learning curve is in excellent agreement with the extensive numerical simulations in addition the algorithm is very robust with respect to classification noise

computing smooth time trajectories for camera and deformable shape in structure from motion with occlusion we address the classical computer vision problems of rigid and nonrigid structure from motion sfm with occlusion we assume that the columns of the input observation matrix w describe smooth 2d point trajectories over time we then derive a family of efficient methods that estimate the column space of w using compact parameterizations in the discrete cosine transform dct domain our methods tolerate high percentages of missing data and incorporate new models for the smooth time trajectories of 2d points affine and weak perspective cameras and 3d deformable shape we solve a rigid sfm problem by estimating the smooth time trajectory of a single camera moving around the structure of interest by considering a weak perspective camera model from the outset we directly compute euclidean 3d shape reconstructions without requiring postprocessing steps such as euclidean upgrade and bundle adjustment our results on real sfm data sets with high percentages of missing data compared positively to those in the literature in nonrigid sfm we propose a novel 3d shape trajectory approach that solves for the deformable structure as the smooth time trajectory of a single point in a linear shape space a key result shows that compared to state of the art algorithms our nonrigid sfm method can better model complex articulated deformation with higher frequency dct components while still maintaining the low rank factorization constraint finally we also offer an approach for nonrigid sfm when w is presented with missing data

iterative extensions of the sturm triggs algorithm convergence and nonconvergence we give the first complete theoretical convergence analysis for the iterative extensions of the sturm triggs algorithm we show that the simplest extension siesta converges to nonsense results another proposed extension has similar problems and experiments with balanced iterations show that they can fail to converge or become unstable we present ciesta an algorithm that avoids these problems it is identical to siesta except for one simple extra computation under weak assumptions we prove that ciesta iteratively decreases an error and approaches fixed points with one more assumption we prove it converges uniquely our results imply that ciesta gives a reliable way of initializing other algorithms such as bundle adjustment a descent method such as gauss newton can be used to minimize the ciesta error combining quadratic convergence with the advantage of minimizing in the projective depths experiments show that ciesta performs better than other iterations

a simple polynomial time rescaling algorithm for solving linear programs the perceptron algorithm developed mainly in the machine learning literature is a simple greedy method for finding a feasible solution to a linear program alternatively for learning a threshold function in spite of its exponential worst case complexity it is often quite useful in part due to its noise tolerance and also its overall simplicity in this paper we show that a randomized version of the perceptron algorithm along with periodic rescaling runs in polynomial time the resulting algorithm for linear programming has an elementary description and analysis

low rank matrix fitting based on subspace perturbation analysis with applications to structure from motion the task of finding a low rank r matrix that best fits an original data matrix of higher rank is a recurring problem in science and engineering the problem becomes especially difficult when the original data matrix has some missing entries and contains an unknown additive noise term in the remaining elements the former problem can be solved by concatenating a set of r column matrices that share a common single r dimensional solution space unfortunately the number of possible submatrices is generally very large and hence the results obtained with one set of r column matrices will generally be different from that captured by a different set ideally we would like to find that solution that is least affected by noise this requires that we determine which of the r column matrices i e which of the original feature points are less influenced by the unknown noise term this paper presents a criterion to successfully carry out such a selection our key result is to formally prove that the more distinct the r vectors of the r column matrices are the less they are swayed by noise this key result is then combined with the use of a noise model to derive an upper bound for the effect that noise and occlusions have on each of the r column matrices it is shown how this criterion can be effectively used to recover the noise free matrix of rank r finally we derive the affine and projective structure from motion sfm algorithms using the proposed criterion extensive validation on synthetic and real data sets shows the superiority of the proposed approach over the state of the art

projection pursuit approach to robust dispersion matrices and principal components primary theory and monte carlo abstract this article proposes and discusses a type of new robust estimators for covariance correlation matrices and principal components via projection pursuit techniques the most attractive advantage of the new procedures is that they are of both rotational equivariance and high breakdown point besides they are qualitatively robust and consistent at elliptic underlying distributions the monte carlo study shows that the best of the new estimators compare favorably with other robust methods they provide as good a performance as m estimators and somewhat better empirical breakdown properties

shape and motion from image streams under orthography a factorization method inferring scene geometry and camera motion from a stream of images is possible in principle but is an ill conditioned problem when the objects are distant with respect to their size we have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step an image stream can be represented by the 2fxp measurement matrix of the image coordinates of p points tracked through f frames we show that under orthographic projection this matrix is of rank 3 based on this observation the factorization method uses the singular value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively two of the three translation components are computed in a preprocessing stage the method can also handle and obtain a full solution from a partially filled in measurement matrix that may result from occlusions or tracking failures the method gives accurate results and does not introduce smoothing in either shape or motion we demonstrate this with a series of experiments on laboratory and outdoor image streams with and without occlusions

structure identification in relational data this paper presents several investigations into the prospects for identifying meaningful structures in empirical data namely structures permitting effective organization of the data to meet requirements of future queries we propose a general framework whereby the notion of identifiability is given a precise formal definition similar to that of learnability using this framework we then explore if a tractable procedure exists for deciding whether a given relation is decomposable into a constraint network or a cnf theory with desirable topology and if the answer is positive identifying the desired decomposition finally we address the problem of expressing a given relation as a horn theory and if this is impossible finding the best k horn approximation to the given relation we show that both problems can be solved in time polynomial in the length of the data

some dichotomy theorems for neural learning problems the computational complexity of learning from binary examples is investigated for linear threshold neurons we introduce combinatorial measures that create classes of infinitely many learning problems with sample restrictions we analyze how the complexity of these problems depends on the values for the measures the results are established as dichotomy theorems showing that each problem is either np complete or solvable in polynomial time in particular we consider consistency and maximum consistency problems for neurons with binary weights and maximum consistency problems for neurons with arbitrary weights we determine for each problem class the dividing line between the np complete and polynomial time solvable problems moreover all efficiently solvable problems are shown to have constructive algorithms that require no more than linear time on a random access machine model similar dichotomies are exhibited for neurons with bounded threshold the results demonstrate on the one hand that the consideration of sample constraints can lead to the discovery of new efficient algorithms for non trivial learning problems on the other hand hard learning problems may remain intractable even for severely restricted samples

training a single sigmoidal neuron is hard we first present a brief survey of hardness results for training feedforward neural networks these results are then completed by the proof that the simplest architecture containing only a single neuron that applies a sigmoidal activation function r satisfying certain natural axioms e g the standard logistic sigmoid or saturated linear function to the weighted sum of n inputs is hard to train in particular the problem of finding the weights of such a unit that minimize the quadratic training error within 2 or its average over a training set within 5 2 12n of its infimum proves to be np hard hence the well known backpropagation learning algorithm appears not to be efficient even for one neuron which has negative consequences in constructive learning

quantum state separation unambiguous discrimination and exact cloning unambiguous discrimination and exact cloning reduce the square overlap between quantum states exemplifying the more general type of procedure we term state separation we obtain the maximum probability with which two equiprobable quantum states can be separated by an arbitrary degree and find that the established bounds on the success probabilities for discrimination and cloning are special cases of this general bound the latter also gives the maximum probability of successfully producing n exact copies of a quantum system whose state is chosen secretly from a known pair given m initial realizations of the state where n m we also discuss the relationship between this bound and that on unambiguous state discrimination

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

applications of circumscription to formalizing common sense knowledge we present a new and more symmetric version of the circumscription method of nonmonotonic reasoning first described in mccarthy 1980 and some applications to formalizing common sense knowledge the applications in this paper are mostly based on minimizing the abnormality of different aspects of various entities included are nonmonotonic treatments of is a hierarchies the unique names hypothesis and the frame problem the new circumscription may be called formula circumscription to distinguish it from the previously defined domain circumscription and predicate circumscription a still more general formalism called prioritized circumscription is briefly explored

a fast tri factorization method for low rank matrix recovery and completion in recent years matrix rank minimization problems have received a significant amount of attention in machine learning data mining and computer vision communities and these problems can be solved by a convex relaxation of the rank minimization problem which minimizes the nuclear norm instead of the rank of the matrix and has to be solved iteratively and involves singular value decomposition svd at each iteration therefore those algorithms for nuclear norm minimization problems suffer from high computation cost of multiple svds in this paper we propose a fast tri factorization ftf method to approximate the nuclear norm minimization problem and mitigate the computation cost of performing svds the proposed ftf method can be used to reliably solve a wide range of low rank matrix recovery and completion problems such as robust principal component analysis rpca low rank representation lrr and low rank matrix completion mc we also present three specific models for rpca lrr and mc problems respectively moreover we develop two alternating direction method adm based iterative algorithms for solving the above three problems experimental results on a variety of synthetic and real world data sets validate the efficiency robustness and effectiveness of our ftf method comparing with the state of the art nuclear norm minimization algorithms

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given

probabilistic broadcasting of mixed states it is well known that the non broadcasting theorem proved by barnum et al is a fundamental principle of quantum communication as we are aware optimal broadcasting ob is the only method to broadcast noncommuting mixed states approximately in this paper motivated by the probabilistic cloning of quantum states proposed by duan and guo we propose a new way for broadcasting noncommuting mixed states probabilistic broadcasting pb and we present a sufficient condition for pb of mixed states to a certain extent we generalize the probabilistic cloning theorem from pure states to mixed states and in particular we generalize the non broadcasting theorem since the case that commuting mixed states can be exactly broadcast can be thought of as a special instance of pb where the success ratio is 1 moreover we discuss probabilistic local broadcasting plb of separable bipartite states

statistics of equally weighted random paths on a class of self similar structures we study the statistics of equally weighted random walk paths on a family of sierpinski gasket lattices whose members are labelled by an integer b 2 b 2 mean path end to end distance grows more slowly than any power of its length n we provide arguments for the emergence of usual power law critical behaviour in the limit b when fractal lattices become almost compact