regression methods for stochastic control problems and their convergence analysis in this paper we develop several regression algorithms for solving general stochastic optimal control problems via monte carlo this type of algorithm is particularly useful for problems with a high dimensional state space and complex dependence structure of the underlying markov process with respect to some control the main idea behind the algorithms is to simulate a set of trajectories under some reference measure and to use the bellman principle combined with fast methods for approximating conditional expectations and functional optimization theoretical properties of the presented algorithms are investigated and the convergence to the optimal solution is proved under some assumptions finally the presented methods are applied in a numerical example of a high dimensional controlled bermudan basket option in a financial market with a large investor

self calibration under the cayley framework the cayley framework here is meant to tackle the vision problems under the infinite cayley transformation ict its main advantage lies in its numerical stability in this work the stratified self calibration under the cayley framework is investigated it is well known that the main difficulty of the stratified self calibration in multiple view geometry is to upgrade a projective reconstruction to an affine one in other words to estimate the unknown 3 vector of the plane at infinity called the normal vector to our knowledge without any prior knowledge about the scene or the camera motion the only available constraint on a moving camera with constant intrinsic parameters is the well known modulus constraint in the literature do other kinds of constraints exist if yes what they are how could they be used in this work such questions will be systematically investigated under the cayley framework our key contributions include 1 the original projective expression of the ict is simplified and a new projective expression is derived to make the upgrade easier from a projective reconstruction to a metric reconstruction 2 the constraints on the normal vector are systematically investigated for two views two constraints on the normal vector are derived one of them is the well known modulus constraint while the other is a new inequality constraint there are only these two constraints for two views for three views besides the constraints for two views two groups of new constraints are derived and each of them contains three constraints in other words there are 12 constraints in total for three views 3 based on our projective expression and these constraints a stratified cayley algorithm and a total cayley algorithm are proposed for the metric reconstruction from images it is experimentally shown that they both improve significantly the numerical stability of the classical algorithms compared with the global optimal algorithm under the infinite homography framework the cayley algorithms have comparable calibration accuracy but substantially reduce the computational load

a nonlinear programming algorithm for solving semidefinite programs via low rank factorization in this paper we present a nonlinear programming algorithm for solving semidefinite programs sdps in standard form the algorithm s distinguishing feature is a change of variables that replaces the symmetric positive semidefinite variable x of the sdp with a rectangular variable r according to the factorization x rr t the rank of the factorization i e the number of columns of r is chosen minimally so as to enhance computational speed while maintaining equivalence with the sdp fundamental results concerning the convergence of the algorithm are derived and encouraging computational results on some large scale test problems are also presented

a central limit theorem and hypotheses testing for risk averse stochastic programs we study statistical properties of the optimal value and optimal solutions of the sample average approximation of risk averse stochastic problems central limit theorem type results are derived for the optimal value when the stochastic program is expressed in terms of a law invariant coherent risk measure having a discrete kusuoka representation the obtained results are applied to hypotheses testing problems aiming at comparing the optimal values of several risk averse convex stochastic programs on the basis of samples of the underlying random vectors we also consider nonasymptotic tests based on confidence intervals on the optimal values of the stochastic programs obtained using the stochastic mirror descent algorithm numerical simulations show how to use our developments to choose among different distributions and on the considered class of risk averse stochastic programs the asymptotic tests show better results

critical loci for projective reconstruction from multiple views in higher dimension a comprehensive theoretical approach abstract a class of determinantal varieties arising as critical loci for a natural generalization of a classical problem in computer vision is introduced their ideals are investigated these varieties are the critical loci for projective reconstruction from multiple views in higher dimension and under suitable genericity assumptions turn out to be hypersurfaces of degree r in p 2 r 1 or varieties of codimension 2 and degree r 2 r 1 2 in p 2 r

probabilistic analysis of a generalized bin packing problem and applications we give a unified probabilistic analysis for a general class of bin packing problems by directly analyzing corresponding mathematical programs in this general class of packing problems objects are described by a given number of attribute values some attributes may be discrete others may be continuous bins are sets of objects and the collection of feasible bins is merely required to satisfy some general consistency properties we characterize the asymptotic optimal value as the value of an easily specified linear program whose size is independent of the number of objects to be packed or as the limit of a sequence of such linear program values we also provide bounds for the rate of convergence of the average cost to its asymptotic value the analysis suggests an a s asymptotically optimal heuristic that runs in linear time the heuristics can be designed to be asymptotically optimal while still running in polynomial time we also show that in several important cases the algorithm has both po

sample boundedness of stochastic processes under increment conditions let t d be a compact metric space of diameter d and be an orlicz norm when is it true that all separable processes x t t t that satisfy the increment condition x t x s d t s for all s t in t are sample bounded we give optimal necessary conditions and optimal sufficient conditions in terms of the existence of a probability measure on t that satisfies an integral condition when t is a compact group and d is translation invariant we are able to compute the necessary and sufficient condition in several cases

sur l integrabilite des vecteurs gaussiens lett be any set and let x t t t be a separable gaussian process with mean zero ont assume thatx is almost surely bounded and let mathop n sup limits t in x text x t text sigma mathop sup limits t in t ex t 2 1 2 and let tau sup lambda geqq 0 p n lambda 0 let 2 we prove that e left exp left frac 1 2 sigma 2 n 2 beta n right right infty examples are given to show that this result cannot be readily improved when 0 we also show that the distribution ofn has a continuous density with respect to lebesgue measure

probability inequalities for empirical processes and a law of the iterated logarithm given a class f of functions on x we can view vn as a stochastic process indexed by f and consider limit theorems for this process to prove such theorems it is often helpful to have bounds on the tail of the r v sup w i n f i see for example dudley 1978 kuelbs and dudley 1980 dudley and phillip 1983 or gine and zinn 1983 our main question of interest is for what classes f can best possible bounds be obtained so first we must ask what are these best possible bounds of course

caustics of the harmonic double cusp near an e_6 point the harmonic double cusp is the real singularity x 4 6x 2 y 2 y 4 its unfolding is eight dimensional and contains a stratum of codimension five corresponding to the e 6 singularity x 3 y 4 a complete description of the bifurcation set in a neighbourhood of the e 6 stratum is given by means of a tableau of two dimensional sections over the three dimensional base the tableau also gives information about the bifurcation set near other strata of high codimension notably d 6 x 2 y y 5 and a 7 x 2 y 8 and it shows how these strata fit together in optics the harmonic double cusp organizes a large family of caustics produced by generic thin lenses some predictions made by the tableau have already been confirmed by observations of caustics from liquid drops points in the tableau base correspond to experimental set ups and the fibre corresponds to the viewing screen

stochastic integer programming limit theorems and confidence intervals we consider empirical approximations sample average approximations of two stage stochastic mixed integer linear programs and derive central limit theorems for the objectives and optimal values the limit theorems are based on empirical process theory and the functional delta method we also show how these limit theorems can be used to derive confidence intervals for optimal values via resampling methods bootstrap subsampling

quasi perspective projection model theory and application to structure and motion factorization from uncalibrated image sequences this paper addresses the problem of factorization based 3d reconstruction from uncalibrated image sequences previous studies on structure and motion factorization are either based on simplified affine assumption or general perspective projection the affine approximation is widely adopted due to its simplicity whereas the extension to perspective model suffers from recovering projective depths to fill the gap between simplicity of affine and accuracy of perspective model we propose a quasi perspective projection model for structure and motion recovery of rigid and nonrigid objects based on factorization framework the novelty and contribution of this paper are as follows firstly under the assumption that the camera is far away from the object with small lateral rotations we prove that the imaging process can be modeled by quasi perspective projection which is more accurate than affine model from both geometrical error analysis and experimental studies secondly we apply the model to establish a framework of rigid and nonrigid factorization under quasi perspective assumption finally we propose an extended cholesky decomposition to recover the rotation part of the euclidean upgrading matrix we also prove that the last column of the upgrading matrix corresponds to a global scale and translation of the camera thus may be set freely the proposed method is validated and evaluated extensively on synthetic and real image sequences and improved results over existing schemes are observed

characterization of lil behavior in banach space in a recent paper by the authors a general result characterizing two sided lil behavior for real valued random variables has been established in this paper we look at the corresponding problem in the banach space setting we show that there are analogous results in this more general setting in particularly we provide a necessary and sufficient condition for lil behavior with respect to sequences of the form p nh n where h is from a suitable subclass of the positive nondecreasing slowly varying functions to prove these results we have to use a different method one of our main tools is an improved fuk nagaev type inequality in banach space which should be of independent interest

on the path density of a gradient field we consider the problem of reliably finding filaments in point clouds realistic data sets often have numerous filaments of various sizes and shapes statistical techniques exist for finding one or a few filaments but these methods do not handle noisy data sets with many filaments other methods can be found in the astronomy literature but they do not have rigorous statistical guarantees we propose the following method starting at each data point we construct the steepest ascent path along a kernel density estimator we locate filaments by finding regions where these paths are highly concentrated formally we define the density of these paths and we construct a consistent estimator of this path density

a central limit theorem under metric entropy with l_2 bracketing let mathbf s rho be a metric space mathbf v mathscr v mu be a probability space and f mathbf s times mathbf v rightarrow mathbb r be a real valued function on mathbf s times mathbf v which has mean zero and is lipschitz in l 2 mu with respect to rho let v be a random variable defined on mathbf v mathscr v mu and let v i i geq 1 be a sequence of independent copies of v the limiting behavior of the process s n s n 1 2 sum n i 1 f s v i is studied under an integrability condition on the metric entropy with bracketing in l 2 mu this metric entropy condition is analogous to one which implies the continuity of the limiting gaussian process a tightness result is derived which in conjunction with the results of andersen and dobric 1987 shows that a central limit theorem holds for the s n process this result generalizes those of dudley 1978 dudley 1981 and jain and marcus 1975

multiple pass streaming algorithms for learning mixtures of distributions in rd we present a multiple pass streaming algorithm for learning the density function of a mixture of k uniform distributions over rectangles in r d for any d 0 our learning model is samples drawn according to the mixture are placed in arbitrary order in a data stream that may only be accessed sequentially by an algorithm with a very limited random access memory space our algorithm makes 2 2 passes for any 0 and requires memory at most o e 2 k 2d 4 4k d where e is the tolerable error of the algorithm this exhibits a strong memory pass tradeoff in terms of e a few more passes significantly lower its memory requirements thus trading one of the two most important resources in streaming computation for the other chang and kannan first considered this problem for d 1 2 our learning algorithm is especially appropriate for situations where massive data sets of samples are available but computation with such large inputs requires very restricted models of computation

on the computational complexity of mcmc based estimators in large samples in this paper we examine the implications of the statistical large sample theory for the computational complexity of bayesian and quasi bayesian estimation carried out using metropolis random walks our anal ysis is motivated by the laplace bernstein von mises central limit theo rem which states that in large samples the posterior or quasi posterior approaches a normal density using this observation we establish polyno mial bounds on the computational complexity of general metropolis ran dom walks methods in large samples our analysis covers cases where the underlying log likelihood or extremum criterion function is possibly non concave discontinuous and of increasing dimension however the central limit theorem restricts the deviations from continuity and log concavity of the log likelihood or extremum criterion function in a very specific manner under minimal assumptions for the central limit theorem framework to hold we show that the metropolis algorithm is theoretically efficient even for the canonical gaussian walk which is studied in detail specifically we show that the running time of the algorithm in large samples is bounded in probability by a polynomial in the parameter dimension d and in partic ular is of stochastic order d2 in the leading cases after the burn in period we then give an application to exponential and curved exponential families of increasing dimension

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

projective reconstruction and invariants from multiple images this correspondence investigates projective reconstruction of geometric configurations seen in two or more perspective views and the computation of projective invariants of these configurations from their images a basic tool in this investigation is the fundamental matrix that describes the epipolar correspondence between image pairs it is proven that once the epipolar geometry is known the configurations of many geometric structures for instance sets of points or lines are determined up to a collineation of projective 3 space spl pscrsup 3 by their projection in two independent images this theorem is the key to a method for the computation of invariants of the geometry invariants of six points in spl pscrsup 3 and of four lines in spl pscrsup 3 are defined and discussed an example with real images shows that they are effective in distinguishing different geometrical configurations since the fundamental matrix is a basic tool in the computation of these invariants new methods of computing the fundamental matrix from seven point correspondences in two images or six point correspondences in three images are given

a simple prior free method for non rigid structure from motion factorization this paper proposes a simple prior free method for solving the non rigid structure from motion nrsfm factorization problem other than using the fundamental low order linear combination model assumption our method does not assume any extra prior knowledge either about the non rigid structure or about the camera motions yet it works effectively and reliably producing optimal results and not suffering from the inherent basis ambiguity issue which plagued most conventional nrsfm factorization methods our method is very simple to implement which involves solving a very small sdp semi definite programming of fixed size and a nuclear norm minimization problem we also present theoretical analysis on the uniqueness and the relaxation gap of our solutions extensive experiments on both synthetic and real motion capture data assuming following the low order linear combination model are conducted which demonstrate that our method indeed outperforms most of the existing non rigid factorization methods this work offers not only new theoretical insight but also a practical everyday solution to nrsfm

laws of large numbers and tail inequalities for random tries and patricia trees we consider random tries and random patricia trees constructed from n independent strings of symbols drawn from any distribution on any discrete space if hn is the height of this tree we show that hn e hn tends to one in probability additional tail inequalities are given for the height depth size internal path length and profile of these trees and ordinary tries that apply without any conditions on the string distributions they need not even be identically distributed

multistep stochastic mirror descent for risk averse convex stochastic programs based on extended polyhedral risk measures we consider risk averse convex stochastic programs expressed in terms of extended polyhedral risk measures we derive computable confidence intervals on the optimal value of such stochastic programs using the robust stochastic approximation and the stochastic mirror descent smd algorithms when the objective functions are uniformly convex we also propose a multistep extension of the stochastic mirror descent algorithm and obtain confidence intervals on both the optimal values and optimal solutions numerical simulations show that our confidence intervals are much less conservative and are quicker to compute than previously obtained confidence intervals for smd and that the multistep stochastic mirror descent algorithm can obtain a good approximate solution much quicker than its nonmultistep counterpart

asymptotics for panel quantile regression models with individual effects this paper studies panel quantile regression models with individual fixed effects we formally establish sufficient conditions for consistency and asymptotic normality of the quantile regression estimator when the number of individuals n and the number of time periods t jointly go to infinity the estimator is shown to be consistent under similar conditions to those found in the nonlinear panel data literature nevertheless due to the non smoothness of the objective function we had to impose a more restrictive condition on t to prove asymptotic normality than that usually found in the literature the finite sample performance of the estimator is evaluated by monte carlo simulations

learning without concentration we obtain sharp bounds on the estimation error of the empirical risk minimization procedure performed in a convex class and with respect to the squared loss without assuming that class members and the target are bounded functions or have rapidly decaying tails rather than resorting to a concentration based argument the method used here relies on a small ball assumption and thus holds for classes consisting of heavy tailed functions and for heavy tailed targets the resulting estimates scale correctly with the noise level of the problem and when applied to the classical bounded scenario always improve the known bounds

on aggregation for heavy tailed classes we introduce an alternative to the notion of fast rate in learning theory which coincides with the optimal error rate when the given class happens to be convex and regular in some sense while it is well known that such a rate cannot always be attained by a learning procedure i e a procedure that selects a function in the given class we introduce an aggregation procedure that attains that rate under rather minimal assumptions for example that the l q and l 2 norms are equivalent on the linear span of the class for some q 2 and the target random variable is square integrable the key components in the proof include a two sided isomorphic estimator on distances between class members which is based on the median of means and an almost isometric lower bound of the form n 1 sum i 1 n f 2 x i ge 1 zeta mathbb e f 2 which holds uniformly in the class both results only require that the l q and l 2 norms are equivalent on the linear span of the class for some q 2

change point estimation under adaptive sampling we consider the problem of locating a jump discontinuity change point in a smooth parametric regression model with a bounded covariate it is assumed that one can sample the covariate at different values and measure the corresponding responses budget constraints dictate that a total of n such measurements can be obtained a multistage adaptive procedure is proposed where at each stage an estimate of the change point is obtained and new points are sampled from its appropriately chosen neighborhood it is shown that such procedures accelerate the rate of convergence of the least squares estimate of the change point further the asymptotic distribution of the estimate is derived using empirical processes techniques the latter result provides guidelines on how to choose the tuning parameters of the multistage procedure in practice the improved efficiency of the procedure is demonstrated using real and synthetic data this problem is primarily motivated by applications in engineering systems

smoothed quantile regression for panel data abstract this paper studies fixed effects estimation of quantile regression models for panel data under an asymptotic framework where both the numbers of individuals and time periods grow at the same rate we show that the fixed effects estimator for the smoothed objective function has a limiting normal distribution with a bias in the mean and provide the analytic form of the asymptotic bias we propose a one step bias correction estimator based on the analytic bias formula obtained from the asymptotic analysis importantly our results cover the case that observations are dependent over time we illustrate the effects of the bias correction through simulations

concentration for multidimensional diffusions and their boundary local times we prove that probability laws of certain multidimensional semimartingales which includes time inhomogenous diffusions under suitable assumptions satisfy quadratic transportation cost inequality under the uniform metric from this we derive concentration properties of lipschitz functions of process paths that depend on the entire history in particular we estimate concentration of boundary local time of reflected brownian motions on a polyhedral domain we work out explicit applications of consequences of measure concentration for the case of brownian motion with rank based drifts

on the complexity of learning from drifting distributions abstract we consider two models of on line learning of binary valued functions from drifting distributions due to bartlett we show that if each example is drawn from a joint distribution which changes in total variation distance by at mosto e3 d log 1 e between trials then an algorithm can achieve a probability of a mistake at mosteworse than the best function in a class of vc dimensiond we prove a corresponding necessary condition ofo e3 d finally in the case that a fixed function is to be learned from noise free examples we show that if the distributions on the domain generating the examples change by at mosto e2 d log 1 e then any consistent algorithm learns to within accuracye

a stochastic immersed boundary method for fluid structure dynamics at microscopic length scales in modeling many biological systems it is important to take into account flexible structures which interact with a fluid at the length scale of cells and cell organelles thermal fluctuations of the aqueous environment become significant in this work it is shown how the immersed boundary method of c s peskin the immersed boundary method acta num 11 2002 1 39 for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations a stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps an important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved partially resolved or fully resolved while retaining a good level of accuracy error estimates in each of these regimes are given for the method a number of theoretical and numerical checks are furthermore performed to assess its physical fidelity for a conservative force the method is found to simulate particles with the correct boltzmann equilibrium statistics it is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters the method is also shown to reproduce a well known hydrodynamic effect of a brownian particle in which the velocity autocorrelation function exhibits an algebraic 3 2 decay for long times b j alder t e wainwright decay of the velocity autocorrelation function phys rev a 1 1 1970 18 21 a few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method specifically we present simulations of osmotic effects of molecular dimers worm like chain polymer knots and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force the theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales

universal prediction of individual binary sequences in the presence of noise the problem of predicting the next outcome of an individual binary sequence based on noisy observations of the past is considered the goal of the predictor is to perform for each individual sequence almost as well as the best in a set of experts where performance is evaluated using a general loss function a comprehensive approach to prediction in this noisy setting is presented and proven generally efficient under appropriate conditions as an illustration of the applicability of the approach suggested for concrete situations two important special cases are explicitly treated the first is the case where the data corrupting noise process is binary valued where the observed bit is the bitwise xor of the clean bit and the noise bit the second case is that of real valued additive noise it is shown that even in this more challenging situation where the information available to the predictor regarding the past sequence is incomplete a predictor can be guaranteed to successfully compete with a whole set of experts in considerably strong senses

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

single camera structure and motion a reduced order nonlinear observer is proposed for the problem of structure and motion sam estimation of a stationary object observed by a moving calibrated camera in comparison to existing work which requires some knowledge of the euclidean geometry of an observed object or full knowledge of the camera motion the developed reduced order observer only requires one camera linear velocity and corresponding acceleration to asymptotically identify the euclidean coordinates of the feature points attached to an object with proper scale reconstruction and the remaining camera velocities the unknown linear velocities are assumed to be generated using a model with unknown parameters the unknown angular velocities are determined from a robust estimator which uses a standard homography decomposition algorithm applied to tracked feature points a lyapunov analysis is provided to prove the observer asymptotically estimates the unknown states under a persistency of excitation condition

u processes and preference learning preference learning has caused great attention in machining learning in this letter we propose a learning framework for pairwise loss based on empirical risk minimization of u processes via rademacher complexity we first establish a uniform version of bernstein inequality of u processes of degree 2 via the entropy methods then we estimate the bound of the excess risk by using the bernstein inequality and peeling skills finally we apply the excess risk bound to the pairwise preference and derive the convergence rates of pairwise preference learning algorithms with squared loss and indicator loss by using the empirical risk minimization with respect to u processes

on the shape from moments problem and recovering edges from noisy radon data we consider the problem of reconstructing a planar convex set from noisy observations of its moments an estimation method based on pointwise recovering of the support function of the set is developed we study intrinsic accuracy limitations in the shape from moments estimation problem by establishing a lower bound on the rate of convergence of the mean squared error it is shown that the proposed estimator is near optimal in the sense of the order an application to tomographic reconstruction is discussed and it is indicated how the proposed estimation method can be used for recovering edges from noisy radon data

the quasi perspective model geometric properties and 3d reconstruction the paper investigates geometric properties of quasi perspective projection model in one and two view geometry the main results are as follows i quasi perspective projection matrix has nine degrees of freedom dof and the parallelism along x and y directions in world system are preserved in images ii quasi fundamental matrix can be simplified to a special form with only six dofs the fundamental matrix is invariant to any non singular projective transformation iii plane induced homography under quasi perspective model can be simplified to a special form defined by six dofs the quasi homography may be recovered from two pairs of corresponding points with known fundamental matrix iv any two reconstructions in quasi perspective space are defined up to a non singular quasi perspective transformation the results are validated experimentally on both synthetic and real images

recovering low rank and sparse components of matrices from incomplete and noisy observations many problems can be characterized by the task of recovering the low rank and sparse components of a given matrix recently it was discovered that this nondeterministic polynomial time hard np hard task can be well accomplished both theoretically and numerically via heuristically solving a convex relaxation problem where the widely acknowledged nuclear norm and l 1 norm are utilized to induce low rank and sparsity this paper studies the recovery task in the general settings that only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and gaussian noise we show that the resulting model falls into the applicable scope of the classical augmented lagrangian method moreover the separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented lagrangian function hence some splitting numerical algorithms are developed for solving the new recovery model some preliminary numerical experiments verify that these augmented lagrangian based splitting algorithms are easily implementable and surprisingly efficient for tackling the new recovery model

on local u statistic processes and the estimation of densities of functions of several sample variables a notion of local u statistic process is introduced and central limit theorems in various norms are obtained for it this involves the development of several inequalities for u processes that may be useful in other contexts this local u statistic process is based on an estimator of the density of a function of several sample variables proposed by frees j amer statist assoc 89 1994 517 525 and as a consequence uniform in bandwidth central limit theorems in the sup and in the lp norms are obtained for these estimators

rank classification of linear line structures from images by trifocal tensor determinability the problem we address is given line correspondences over three views what is the condition of the line correspondences for the spatial relation of the three associated camera positions to be uniquely recoverable the observed set of lines in space is called critical if there are multiple projectively nonequivalent configurations of the camera positions that can picture the same image triplet of the lines we tackle the problem from the perspective of trifocal tensor a quantity that captures the relative pose of the cameras in relation to the captured views we show that the rank of a matrix that leads to the estimation of the tensor is reduced to 7 11 15 if the observed lines come from a line pencil a line bundle and a line field respectively which are line families belonging to linear line space and 12 19 23 if the lines come from a general linear ruled surface a general linear line congruence and a general linear line complex which are subclasses of linear line structures we show that the above line structures with the exception of linear line congruence and linear line complex ought to be critical line structures all of these structures are quite typical in reality and thus the findings are important to the validity and stability of practically all algorithms related to structure from motion and projective reconstruction using line correspondences

a note on density model size testing let f sub k sub k spl ges 1 be a nested family of parametric classes of densities with finite vapnik chervonenkis dimension let f be a probability density belonging to f sub k sup where k sup is the unknown smallest integer such that f spl isin f sub k given a random sample x sub 1 x sub n drawn from f an integer k sub 0 spl ges 1 and a real number spl alpha spl isin 0 1 we introduce a new simple explicit spl alpha level consistent testing procedure of the null hypothesis h sub 0 k sup k sub 0 versus the alternative h sub 1 k sup spl ne k sub 0 our method is inspired by the combinatorial tools developed in devroye and lugosi and it includes a wide range of density models such as mixture models neural networks or exponential families

adaptive confidence interval for pointwise curve estimation we present a procedure associated with nonlinear wavelet methods that provides adaptive confidence intervals around f x 0 in either a white noise model or a regression setting a suitable modification in the truncation rule for wavelets allows construction of confidence intervals that achieve optimal coverage accuracy up to a logarithmic factor the procedure does not require knowledge of the regularity of the unknown function f it is also efficient for functions with a low degree of regularity

in defense of the eight point algorithm the fundamental matrix is a basic tool in the analysis of scenes taken with two uncalibrated cameras and the eight point algorithm is a frequently cited method for computing the fundamental matrix from a set of eight or more point matches it has the advantage of simplicity of implementation the prevailing view is however that it is extremely susceptible to noise and hence virtually useless for most purposes this paper challenges that view by showing that by preceding the algorithm with a very simple normalization translation and scaling of the coordinates of the matched points results are obtained comparable with the best iterative algorithms this improved performance is justified by theory and verified by extensive experiments on real images

direct methods for recovering motion we have developed direct methods for recovering the motion of an observer in a static environment in the case of pure rotation pure translation and arbitrary motion when the rotation is known some of these methods are based on the minimization of the difference between the observed time derivative of brightness and that predicted from the spatial brightness gradient given the estimated motion we minimize the square of the integral of this difference taken over the image region of interest other methods presented here exploit the fact that surfaces have to be in front of the observer in order to be seen

pass efficient algorithms for learning mixtures of uniform distributions we present multiple pass streaming algorithms for a basic statistical clustering problem for massive data sets if our algorithm is allotted 2 ell passes it will produce an approximation with error at most epsilon using tilde o k 3 epsilon 2 ell bits of memory the most critical resource for streaming computation we demonstrate that this tradeoff between passes and memory allotted is intrinsic to the problem and model of computation by proving lower bounds on the memory requirements of any ell pass randomized algorithm that are nearly matched by our upper bounds in this problem we are given a set of n points drawn randomly according to a mixture of k uniform distributions and wish to approximate the density function of the mixture the points are placed in a data stream possibly in adversarial order which may only be read in sequential passes by the algorithm the algorithm is quite general and can be adapted to solve the problems of learning a mixture of linear distributions in mathbb r and a mixture of uniform distributions in mathbb r 2

discussion local rademacher complexities and oracle inequalities in risk minimization in this magnificent paper professor koltchinskii offers general and powerful performance bounds for empirical risk minimization a fundamental principle of statistical learning theory since the elegant pioneering work of vapnik and chervonenkis in the early 1970s various such bounds have been known that relate the performance of empirical risk minimizers to combinatorial and geometrical features of the class over which the minimization is performed this area of research has been a rich source of motivation and a major field of applications of empirical process theory the appearance of advanced concentration inequalities in the 1990s primarily thanks to talagrand s influential work provoked major advances in both empirical process theory and statistical learning theory and led to a much deeper understanding of some of the basic phenomena in the discussed paper professor koltchinskii develops a powerful new methodology iterative localization which with the help of concentration inequalities is able to explain most of the recent results and go significantly beyond them in many cases the main motivation behind professor koltchinskii s paper is based on classical problems of statistical learning theory such as binary classification and regression in which given a sample xi y i i 1 n of independent and identically distributed pairs of random variables where the xi take their values in some feature space x and the yi are say real valued the goal is to find a function f x r whose risk defined in terms of the expected value of an appropriately chosen loss function is as small as possible in the remaining part of this discussion we point out how the performance bounds of professor koltchinskii s paper can be used to study a seemingly different model motivated by nonparametric ranking problems which has received increasing attention both in the statistical and machine learning literature indeed in several applications such as the search engine problem or credit risk screening the goal is to learn how to rank or to score observations rather than just classify them in this case performance measures involve pairs of observations as can be seen for instance with the auc area under an roc curve criterion in this

uniform bounds for norms of sums of independent random functions in this paper we develop a general machinery for finding explicit uniform probability and moment bounds on sub additive positive functionals of random processes using the developed general technique we derive uniform bounds on the ls norms of empirical and regression type processes usefulness of the obtained results is illustrated by application to the processes appearing in kernel density estimation and in nonparametric estimation of regression functions

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

cayley transformation and numerical stability of calibration equation the application of cayley transformation to enhance the numerical stability of camera calibration is investigated first a new calibration equation called the standard calibration equation is introduced using the cayley transformation and its analytical solution is obtained the standard calibration equation is equivalent to the classical calibration equation but it exhibits remarkable better numerical stability second a one parameter calibration family called the cayley calibration family which is equivalent to the standard calibration equation is obtained using also the cayley transformation and it is found that this family is composed of those infinite homographies whose rotation has the same axis with the rotation between the two given views the condition number of equations in the cayley calibration family varies with the parameter value and an algorithm to determine the best parameter value is provided third the generalized cayley calibration families equivalent to the standard calibration equation are also introduced via generalized cayley transformations an example of the generalized cayley transformations is illustrated called the s cayley calibration family as in the cayley calibration family the numerical stability of equations in a generalized cayley calibration family also depends on the parameter value in addition a more generic calibration family is also proposed and it is proved that the standard calibration equation the cayley calibration family and the s cayley calibration family are all some special cases of this generic calibration family

generating experimental data for the generalized assignment problem the generalized assignment problem gap is the problem of finding the minimal cost assignment of jobs to machines such that each job is assigned to exactly one machine subject to capacity restrictions on the machines we propose a new stochastic model for the gap a tight condition on this stochastic model under which the gap is feasible with probability one when the number of jobs goes to infinity is derived this new stochastic model enables us to analyze the adequacy of most of the random generators given for the gap in the literature we demonstrate that the random generators commonly used to test solution procedures for the gap tend to create easier problem instances when the number of machines m increases we describe a greedy heuristic for the gap and use it to illustrate the results from the paper

imperfection sensitivity of elastically supported beams and its relation to the double cusp instability model this paper deals with the two mode initial postbuckling analysis of geometrically imperfect beams on elastic foundation under compression by adjusting the beam length to the characteristic length ratio it is found that the resulting two mode stability problem may be classified as the double cusp model in catastrophe theory the appropriate postbuckling coefficients are computed with the use of koiter s theory of elastic stability the equilibrium paths and the critical sets are plotted and analysed eight independent control parameters are suggested the paper is the first in the open literature to apply the double cusp model to a two mode stability problem which displays imperfection sensitivity

robust factorization methods using a gaussian uniform mixture model in this paper we address the problem of building a class of robust factorization algorithms that solve for the shape and motion parameters with both affine weak perspective and perspective camera models we introduce a gaussian uniform mixture model and its associated em algorithm this allows us to address parameter estimation within a data clustering approach we propose a robust technique that works with any affine factorization method and makes it resilient to outliers in addition we show how such a framework can be further embedded into an iterative perspective factorization scheme we carry out a large number of experiments to validate our algorithms and to compare them with existing ones we also compare our approach with factorization methods that use m estimators

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

on ranking and generalization bounds the problem of ranking is to predict or to guess the ordering between objects on the basis of their observed features in this paper we consider ranking estimators that minimize the empirical convex risk we prove generalization bounds for the excess risk of such estimators with rates that are faster than 1 n we apply our results to commonly used ranking algorithms for instance boosting or support vector machines moreover we study the performance of considered estimators on real data sets

the shattering dimension of sets of linear functionals we evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets the proofs here relay mostly on methods from the local theory of normed spaces and include volume estimates factorization techniques and tail estimates of norms viewed as random variables on euclidean spheres the estimates of shattering dimensions can be applied to obtain error bounds for certain classes of functions a fact which was the original motivation of this study although this can probably be done in a more traditional manner we also use the approach presented here to determine whether several classes of linear functionals satisfy the uniform law of large numbers and the uniform central limit theorem

general nested and constrained wiberg minimization wiberg matrix factorization breaks a matrix y into low rank factors u and v by solving for v in closed form given u linearizing v u about u and iteratively minimizing y uv u 2 with respect to u only this approach factors the matrix while effectively removing v from the minimization recently eriksson and van den hengel extended this approach to l 1 minimizing y uv u 1 we generalize their approach beyond factorization to minimize y f u v 1 for more general functions f u v that are nonlinear in each of two sets of variables we demonstrate the idea with a practical wiberg algorithm for l 1 bundle adjustment one wiberg minimization can be nested inside another effectively removing two of three sets of variables from a minimization we demonstrate this idea with a nested wiberg algorithm for l 1 projective bundle adjustment solving for camera matrices points and projective depths wiberg minimization also generalizes to handle nonlinear constraints and we demonstrate this idea with constrained wiberg minimization for multiple instance learning cwm mil which removes one set of variables from the constrained optimization our experiments emphasize isolating the effect of wiberg by comparing against the algorithm it modifies successive linear programming

uniform in bandwidth consistency of kernel type function estimators we introduce a general method to prove uniform in bandwidth consistency of kernel type function estimators examples include the kernel density estimator the nadaraya watson regression estimator and the conditional empirical process our results may be useful to establish uniform consistency of data driven bandwidth kernel type function estimators

a generalized projective reconstruction theorem and depth constraints for projective factorization this paper presents a generalized version of the classic projective reconstruction theorem which helps to choose or assess depth constraints for projective depth estimation algorithms the theorem shows that projective reconstruction is possible under a much weaker constraint than requiring all estimated projective depths to be nonzero this result enables us to present classes of depth constraints under which any reconstruction of cameras and points projecting into given image points is projectively equivalent to the true camera point configuration it also completely specifies the possible wrong configurations allowed by other constraints we demonstrate the application of the theorem by analysing several constraints used in the literature as well as presenting new constraints with desirable properties we mention some of the implications of our results on iterative depth estimation algorithms and projective reconstruction via rank minimization our theory is verified by running experiments on both synthetic and real data

critical configurations for projective reconstruction from multiple views this paper investigates a classical problem in computer vision given corresponding points in multiple images when is there a unique projective reconstruction of the 3d geometry of the scene points and the camera positions a set of points and cameras is said to be critical when there is more than one way of realizing the resulting image points for two views it has been known for almost a century that the critical configurations consist of points and camera lying on a ruled quadric surface we give a classification of all possible critical configurations for any number of points in three images and show that in most cases the ambiguity extends to any number of cameras the underlying framework for deriving the critical sets is projective geometry using a generalization of pascal s theorem we prove that any number of cameras and scene points on an elliptic quartic form a critical set another important class of critical configurations consists of cameras and points on rational quartics the theoretical results are accompanied by many examples and illustrations

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

bilinear modeling via augmented lagrange multipliers balm this paper presents a unified approach to solve different bilinear factorization problems in computer vision in the presence of missing data in the measurements the problem is formulated as a constrained optimization where one of the factors must lie on a specific manifold to achieve this we introduce an equivalent reformulation of the bilinear factorization problem that decouples the core bilinear aspect from the manifold specificity we then tackle the resulting constrained optimization problem via augmented lagrange multipliers the strength and the novelty of our approach is that this framework can seamlessly handle different computer vision problems the algorithm is such that only a projector onto the manifold constraint is needed we present experiments and results for some popular factorization problems in computer vision such as rigid non rigid and articulated structure from motion photometric stereo and 2d 3d non rigid registration

on weakly bounded empirical processes let f be a class of functions on a probability space and let x 1 x k be independent random variables distributed according to we establish an upper bound that holds with high probability on rm sup f in f i f x i geq t for every t 0 and that depends on a natural geometric parameter associated with f we use this result to analyze the supremum of empirical processes of the form z f left k 1 sum i 1 k f p x i mathbb e f p right for p 1 using the geometry of f we also present some geometric applications of this approach based on properties of the random operator gamma k 1 2 sum i 1 k x i e i where x i i 1 k are sampled according to an isotropic log concave measure on mathbb r n

how to use expert advice we analyze algorithms that predict a binary value by combining the predictions of several prediction strategies called experts our analysis is for worst case situations i e we make no assumptions about the way the sequence of bits to be predicted is generated we measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence where the expectation is taken with respect to the randomization in the predictins we show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert and we give efficient algorithms that achieve this our upper and lower bounds have matching leading constants in most cases we then show how this leads to certain kinds of pattern recognition learning algorithms with performance bounds that improve on the best results currently know in this context we also compare our analysis to the case in which log loss is used instead of the expected number of mistakes

stable distributions pseudorandom generators embeddings and data stream computation in this article we show several results obtained by combining the use of stable distributions with pseudorandom generators for bounded space in particular we show that for any p 0 2 one can maintain using only o log n e2 words of storage a sketch c q of a point q lnp under dynamic updates of its coordinates the sketch has the property that given c q and c s one can estimate vq svp up to a factor of 1 p e with large probability this solves the main open problem of feigenbaum et al 1999 we show that the aforementioned sketching approach directly translates into an approximate algorithm that for a fixed linear mapping a and given x n and y m estimates vax yvp in o n p m time for any p 0 2 this generalizes an earlier algorithm of wasserman and blum 1997 which worked for the case p e 2 we obtain another sketch function c which probabilistically embeds ln1 into a normed space lm1 the embedding guarantees that if we set m e log 1 o 1 e then for any pair of points q s ln1 the distance between q and s does not increase by more than 1 p e with constant probability and it does not decrease by more than 1 e with probability 1 this is the only known dimensionality reduction theorem for the l1 norm in fact stronger theorems of this type i e that guarantee very low probability of expansion as well as of contraction cannot exist brinkman and charikar 2003 we give an explicit embedding of ln2 into lno log n 1 with distortion 1 p 1 n 1

lower bounds in pattern recognition and learning lower bounds are derived for the performance of any pattern recognition algorithm which using training data selects a discrimination rule from a certain class of rules the bounds involve the vapnik chervonenkis dimension of the class and l the minimal error probability within the class we provide lower bounds when l 0 the usual assumption in valiant s theory of learning and l 0

fast and accurate algorithms for projective multi image structure from motion we describe algorithms for computing projective structure and motion from a multi image sequence of tracked points the algorithms are essentially linear work for any motion of moderate size and give accuracies similar to those of a maximum likelihood estimate they give better results than the factorization approach of sturm and triggs 1996 and are equally fast and they are much faster than bundle adjustment our experiments show that the iterated sturm triggs approach often fails for linear camera motions in addition we study experimentally the common situation where the calibration is fixed and approximately known comparing the projective versions of our algorithms to mixed projective euclidean strategies we clarify the nature of dominant plane compensation showing that it can be considered a small translation approximation rather than an approximation that the scene is planar we show that projective algorithms accurately recover the projected inverse depths and homographies despite the possibility of transforming the structure and motion by a projective transformation

a flexible new technique for camera calibration we propose a flexible technique to easily calibrate a camera it only requires the camera to observe a planar pattern shown at a few at least two different orientations either the camera or the planar pattern can be freely moved the motion need not be known radial lens distortion is modeled the proposed procedure consists of a closed form solution followed by a nonlinear refinement based on the maximum likelihood criterion both computer simulation and real data have been used to test the proposed technique and very good results have been obtained compared with classical techniques which use expensive equipment such as two or three orthogonal planes the proposed technique is easy to use and flexible it advances 3d computer vision one more step from laboratory environments to real world use

sample compression schemes for vc classes sample compression schemes were defined by littlestone and warmuth 1986 as an abstraction of the structure underlying many learning algorithms roughly speaking a sample compression scheme of size k means that given an arbitrary list of labeled examples one can retain only k of them in a way that allows us to recover the labels of all other examples in the list they showed that compression implies probably approximately correct learnability for binary labeled classes and asked whether the other direction holds we answer their question and show that every concept class c with vc dimension d has a sample compression scheme of size exponential in d

planar structure from motion with affine camera models closed form solutions ambiguities and degeneracy analysis planar structure from motion sfm is the problem of reconstructing a planar object or surface from a set of 2d images using motion information the problem is well understood with the perspective camera model and can be solved with homography decomposition hd however when the structure is small and or viewed far from the camera the perspective effects diminish and in the limit the projections become affine in these situations hd fails because the problem itself becomes ill posed we propose a stable alternative using affine camera models these have been used extensively to reconstruct non planar structures however a general accurate and closed form method for planar structures has been missing the problem is fundamentally different with planar structures because the types of affine camera models one can use are more restricted and it is inherently more ambiguous and non linear we provide a closed form method for the orthographic camera model that solves the general problem three or more views with three or more correspondences and missing correspondences and returns all metric structure solutions and corresponding camera poses the method does not require initialisation and optimises an objective function that is very similar to the reprojection error in fact there is no clear benefit in refining its solutions with bundle adjustment which is a remarkable result we also present a new theoretical analysis that deepens our understanding of the problem the main result is the necessary and sufficient geometric conditions for the problem to be degenerate with the orthographic camera we also show there can exist up to two solutions for metric structure with four or more views previously it was assumed to be unique and we give the necessary and sufficient geometric conditions for disambiguation other theoretical results include showing that in the case of three images the optimal reconstruction with respect to reprojection error can usually be found in closed form and additional prior knowledge needed to solve with non orthographic affine cameras

the lil for canonical u statistics of order 2 let x xi i 2 n be independent identically distributed random variables and let h x y h y x be a measurable function of two variables it is shown that the bounded law of the iterated logarithm lim supn n log log n 1 p 1i jn h xi xj 1 a s holds if and only if the following three conditions are satised h is canonical for the law of x that is eh x y 0 for almost all y and there exists c 1 such that both e h 2 x1 x2 u c log log u for all large u and supfeh x1 x2 f x1 g x2 kf x k21 kg x k21 kfk1 1 kgk1 1gc