an efficient threshold dynamics method for wetting on rough surfaces the threshold dynamics method developed by merriman bence and osher mbo is an efficient method for simulating the motion by mean curvature flow when the interface is away from the solid boundary direct generalization of mbo type methods to the wetting problem with interfaces intersecting the solid boundary is not easy because solving the heat equation in a general domain with a wetting boundary condition is not as efficient as it is with the original mbo method the dynamics of the contact point also follows a different law compared with the dynamics of the interface away from the boundary in this paper we develop an efficient volume preserving threshold dynamics method for simulating wetting on rough surfaces this method is based on minimization of the weighted surface area functional over an extended domain that includes the solid phase the method is simple stable with o n log n complexity per time step and is not sensitive to the inhomogeneity or roughness of the solid boundary

fitting and testing conditional multinormal partial credit models a multinormal partial credit model for factor analysis of polytomously scored items with ordered response categories is derived using an extension of the dutch identity holland in psychometrika 55 5 18 1990 in the model latent variables are assumed to have a multivariate normal distribution conditional on unweighted sums of item scores which are sufficient statistics attention is paid to maximum likelihood estimation of item parameters multivariate moments of latent variables and person parameters it is shown that the maximum likelihood estimates can be found without the use of numerical integration techniques more general models are discussed which can be used for testing the model and it is shown how models with different numbers of latent variables can be tested against each other in addition multi group extensions are proposed which can be used for testing both measurement invariance and latent population differences models and procedures discussed are demonstrated in an empirical data example

characterizing the manifest probabilities of latent trait models the problem of characterizing the manifest probabilities of a latent trait model is considered the item characteristic curve is transformed to the item passing odds curve and a corresponding transformation is made on the distribution of ability this results in a useful expression for the manifest probabilities of any latent trait model the result is then applied to give a characterization of the rasch model as a log linear model for a 2 j contingency table partial results are also obtained for other models the question of the identifiability of guessing parameters is also discussed

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

efficient algorithms for diffusion generated motion by mean curvature the problem of simulating the motion of evolving surfaces with junctions according to some curvature dependent speed arises in a number of applications by alternately diffusing and sharpening characteristic functions for each region a variety of motions have been obtained which allow for topological mergings and breakings and produce no overlapping regions or vacuums however the usual finite difference discretization of these methods is often excessively slow when accurate solutions are sought even in two dimensions we propose a new spectral discretization of these diffusion generated methods which obtains greatly improved efficiency over the usual finite difference approach these efficiency gains are obtained in part through the use of a quadrature based refinement technique by integrating fourier modes exactly and by neglecting the contributions of rapidly decaying solution transients indeed numerical studies demonstrate that the new algorithm is often more than 1000 times faster than the usual finite difference discretization our findings are demonstrated on several examples

threshold dynamics for the piecewise constant mumford shah functional we propose an efficient algorithm for minimizing the piecewise constant mumford shah functional of image segmentation it is based on the threshold dynamics of merriman bence and osher for evolving an interface by its mean curvature we show that a very fast minimization can be achieved by alternating the solution of a linear parabolic partial differential equation and simple thresholding

kernels with prescribed surface tension mobility for threshold dynamics schemes abstract we show how to construct a convolution kernel that has a desired anisotropic surface tension and desired anisotropic mobility to be used in threshold dynamics schemes for simulating weighted motion by mean curvature of interfaces including networks of them in both two and three dimensions moreover we discuss necessary and sufficient conditions for the positivity of the kernel which in the case of two phase flow ensures that the resulting scheme respects a comparison principle and implies convergence to the viscosity solution of the level set formulation of the flow in particular we show in a barrier type statement that the kernel cannot possibly be positive unless both the mobility and the surface tension satisfy necessary conditions in three dimensions and give a complete characterization among other results is a threshold dynamics scheme that is guaranteed to dissipate a non local approximation to the interfacial energy in the fully anisotropic multiphase setting using the new kernel construction

a rasch model for partial credit scoring a unidimensional latent trait model for responses scored in two or more ordered categories is developed this partial credit model is a member of the family of latent trait models which share the property of parameter separability and so permit specifically objective comparisons of persons and items the model can be viewed as an extension of andrich s rating scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item the difference between the parameters in this model and the category boundaries in samejima s graded response model is demonstrated an unconditional maximum likelihood procedure for estimating the model parameters is developed

mcmc estimation and some model fit analysis of multidimensional irt models a bayesian procedure to estimate the three parameter normal ogive model and a generalization of the procedure to a model with multidimensional ability parameters are presented the procedure is a generalization of a procedure by albert 1992 for estimating the two parameter normal ogive model the procedure supports analyzing data from multiple populations and incomplete designs it is shown that restrictions can be imposed on the factor matrix for testing specific hypotheses about the ability structure the technique is illustrated using simulated and real data

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

a mcmc method for models with continuous latent responses this paper introduces a new technique for estimating the parameters of models with continuous latent data using the rasch model as an example it is shown that existing bayesian techniques for parameter estimation such as the gibbs sampler are not always easy to implement then a new sampling based bayesian technique called the da t gibbs sampler is introduced the da t gibbs sampler relies on the particular latent data structure of latent response models to simplify the computations involved in parameter estimation

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

the spectrum of the billiard laplacian of a family of random billiards random billiards are billiard dynamical systems for which the reflection law giving the post collision direction of a billiard particle as a function of the pre collision direction is specified by a markov scattering operator p billiards with microstructure are random billiards whose markov operator is derived from a microscopic surface structure on the boundary of the billiard table the microstructure in turn is defined in terms of what we call a billiard cellq the shape of which completely determines the operator p this operator defined on an appropriate hilbert space is bounded self adjoint and for the examples considered here a hilbert schmidt operator a central problem in the statistical theory of such random billiards is to relate the geometric characteristics of q and the spectrum of p we show for a particular family of billiard cell shapes parametrized by a scale invariant curvature k fig 2 that the billiard laplacian p i is closely related to the ordinary spherical laplacian and indicate by partly analytical and partly numerical means how this provides asymptotic information about the spectrum of p for small values of k it is shown in particular that the second moment of scattering about the incidence angle closely approximates the spectral gap of p

the dutch identity a new tool for the study of item response models the dutch identity is a useful way to reexpress the basic equations of item response models that relate the manifest probabilities to the item response functions irfs and the latent trait distribution the identity may be exploited in several ways for example a to suggest how item response models behave for large numbers of items they are approximate submodels of second order loglinear models for 2 j tables b to suggest new ways to assess the dimensionality of the latent trait principle components analysis of matrices composed of second order interactions from loglinear models c to give insight into the structure of latent class models and d to illuminate the problem of identifying the irfs and the latent trait distribution from sample data

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

reverse auction and the solution of inequality constrained assignment problems in this paper we propose auction algorithms for solving several types of assignment problems with inequality constraints included are asymmetric problems with dierent numbers of persons and objects and multiassignment problems where persons may be assigned to several objects and reversely a central new idea in all these algorithms is to combine regular auction where persons bid for objects by raising their prices with reverse auction where objects compete for persons by essentially oering discounts reverse auction can also be used to accelerate substantially and

convolution kernels and stability of threshold dynamics methods threshold dynamics and its extensions have proven useful in computing interfacial motions in applications as diverse as materials science and machine learning certain desirable properties of the algorithm such as unconditional monotonicity in two phase flows and gradient stability more generally hinge on positivity properties of the convolution kernel and its fourier transform recent developments in the analysis of this class of algorithms indicate that sometimes as in the case of certain anisotropic curvature flows arising in materials science these properties of the convolution kernel cannot be expected other applications such as machine learning would benefit from as great a level of flexibility in choosing the convolution kernel as possible in this paper we establish certain desirable properties of threshold dynamics such as gamma convergence of its associated energy for a substantially wider class of kernels than has been hitherto possible we also present variants of the algorithm that

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

billiards in a general domain with random reflections we study stochastic billiards on general tables a particle moves according to its constant velocity inside some domain fancyscript d subset mathbb r d until it hits the boundary and bounces randomly inside according to some reflection law we assume that the boundary of the domain is locally lipschitz and almost everywhere continuously differentiable the angle of the outgoing velocity with the inner normal vector has a specified absolutely continuous density we construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location velocity we mainly focus on the case of bounded domains then we prove exponential ergodicity of these two markov processes we study their invariant distribution and their normal gaussian fluctuations of particular interest is the case of the cosine reflection law the stationary distributions for the two processes are uniform in this case the discrete time chain is reversible though the continuous time process is quasi reversible also in this case we give a natural construction of a chord picked at random in fancyscript d and we study the angle of intersection of the process with a d 1 dimensional manifold contained in fancyscript d

bayesian estimation of a multilevel irt model using gibbs sampling in this article a two level regression model is imposed on the ability parameters in an item response theory irt model the advantage of using latent rather than observed scores as dependent variables of a multilevel model is that it offers the possibility of separating the influence of item difficulty and ability level and modeling response variation and measurement error another advantage is that contrary to observed scores latent scores are test independent which offers the possibility of using results from different tests in one analysis where the parameters of the irt model and the multilevel model can be concurrently estimated the two parameter normal ogive model is used for the irt measurement model it will be shown that the parameters of the two parameter normal ogive model and the multilevel model can be estimated in a bayesian framework using gibbs sampling examples using simulated and real data are given

auction dynamics a volume constrained mbo scheme abstract we show how auction algorithms originally developed for the assignment problem can be utilized in merriman bence and osher s threshold dynamics scheme to simulate multi phase motion by mean curvature in the presence of equality and inequality volume constraints on the individual phases the resulting algorithms are highly efficient and robust and can be used in simulations ranging from minimal partition problems in euclidean space to semi supervised machine learning via clustering on graphs in the case of the latter application numerous experimental results on benchmark machine learning datasets show that our approach exceeds the performance of current state of the art methods while requiring a fraction of the computation time

spectral gap for a class of random billiards a random billiard is a random dynamical system similar to an ordinary billiard system except that the standard specular reflection law is replaced with a more general stochastic operator specifying the post collision distribution of velocities for any given pre collision velocity we consider such collision operators for certain random billiards that we call billiards with microstructure collisions modeled by these operators can still be thought of as elastic and time reversible the operators are canonically determined by a second deterministic billiard system that models microscopic roughness on the billiard table boundary our main purpose here is to develop some general tools for the analysis of the collision operator of such random billiards among the main results we give geometric conditions for these operators to be hilbert schmidt and relate their spectrum and speed of convergence to stationary markov chains with geometric features of the microscopic billiard structure the relationship between spectral gap and the shape of the microstructure is illustrated with several simple examples

capturing the behavior of bubbles and drops using the variational level set approach we reproduce the general behavior of complicated bubble and droplet motions using the variational level set formulation introduced by the authors earlier our approach here ignores inertial effects thus the motion is only correct as an approximation for very viscous problems however the steady states are true equilibrium solutions inertial forces will be added in future work the problems include soap bubbles colliding and merging drops falling or remaining attached to a generally irregular ceiling and liquid penetrating through a funnel in both two and three dimensions each phase is identified with a particular level set function the zero level set of this function is that particular phase boundary the level set functions all evolve in time through a constrained gradient descent procedure so as to minimize an energy functional the functions are coupled through physical constraints and through the requirements that different phases do not overlap and vacuum regions do not develop both boundary conditions and inequality constraints are cast in terms of either local or global equality constraints the gradient projection method leads to a system of perturbed by curvature if surface tension is involved hamilton jacobi equations coupled through a constraint the coupling is enforced using the lagrange multiplier associated with this constraint the numerical implementation requires much of the modern level set technology in particular we achieve a significant speed up by using the fast localization algorithm of h k zhao m kang b merriman d peng and s osher

threshold dynamics for networks with arbitrary surface tensions we present and study a new algorithm for simulating the n phase mean curvature motion for an arbitrary set of isotropic surface tensions the departure point is the threshold dynamics algorithm of merriman bence and osher for the two phase case a new energetic interpretation of this algorithm allows us to extend it in a natural way to the case of n phases for arbitrary surface tensions and arbitrary isotropic mobilities for a large class of surface tensions the algorithm is shown to be consistent in the sense that at every time step it decreases an energy functional that is an approximation in the sense of gamma convergence of the interfacial energy a broad range of numerical tests shows good convergence properties an important application is the motion of grain boundaries in polycrystalline materials it is also established that the above mentioned large class of surface tensions contains the read shockley surface tensions both in the two dimensional and three dimensional settings 2015 wiley periodicals inc

multiple scattering in random mechanical systems and diffusion approximation this paper is concerned with stochastic processes that model multiple or iterated scattering in classical mechanical systems of billiard type defined below from a given deterministic system of billiard type a random process with transition probabilities operator p is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions of particular interest are systems with weak scattering which are associated to parametric families of operators ph depending on a geometric or mechanical parameter h that approaches the identity as h goes to 0 it is shown that ph i h converges for small h to a second order elliptic differential operator mathcal l on compactly supported functions and that the markov chain process associated to ph converges to a diffusion with infinitesimal generator mathcal l both ph and mathcal l are self adjoint densely defined on the space l 2 mathbb h eta of square integrable functions over the lower half space mathbb h in mathbb r m where is a stationary measure this measure s density is either post collision maxwell boltzmann distribution or knudsen cosine law and the random processes with infinitesimal generator mathcal l respectively correspond to what we call mb diffusion and generalized legendre diffusion concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes

a gibbs sampler for the extended marginal rasch model in their seminal work on characterizing the manifest probabilities of latent trait models cressie and holland give a theoretically important characterization of the marginal rasch model because their representation of the marginal rasch model does not involve any latent trait nor any specific distribution of a latent trait it opens up the possibility for constructing a markov chain monte carlo method for bayesian inference for the marginal rasch model that does not rely on data augmentation such an approach would be highly efficient as its computational cost does not depend on the number of respondents which makes it suitable for large scale educational measurement in this paper such an approach will be developed and its operating characteristics illustrated with simulated data

a diffusion generated approach to multiphase motion in this article we present a diffusion generated approach for evolving multiple junctions this work generalizes an earlier method by merriman bence and osher which alternately diffuses and sharpens characteristic functions for each phase region to produce pure mean curvature flow specifically our new method produces a normal velocity which depends on a positive multiple of the curvature of the interface plus the difference in bulk energy densities for prescribed junction angles this simple method naturally treats topological mergings and breakings produces no overlapping regions or vacuums and can be made very fast numerical studies are provided which show that our method agrees with front tracking and a recent variational approach for a variety of examples asymptotic expansions are also carried out near junctions to justify our algorithms

multiclass data segmentation using diffuse interface methods on graphs we present two graph based algorithms for multiclass segmentation of high dimensional data on graphs the algorithms use a diffuse interface model based on the ginzburg landau functional related to total variation and graph cuts a multiclass extension is introduced using the gibbs simplex with the functional s double well potential modified to handle the multiclass case the first algorithm minimizes the functional using a convex splitting numerical scheme the second algorithm uses a graph adaptation of the classical numerical merriman bence osher mbo scheme which alternates between diffusion and thresholding we demonstrate the performance of both algorithms experimentally on synthetic data image labeling and several benchmark data sets such as mnist coil and webkb we also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph laplacian and take advantage of the sparsity of the matrix experiments indicate that the results are competitive with or better than the current state of the art in multiclass graph based segmentation algorithms for high dimensional data