on the geometric interpretation of the nonnegative rank the nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank one factors needed to reconstruct it exactly the problem of determining this rank and computing the corresponding nonnegative factors is difficult however it has many potential applications e g in data mining graph theory and computational geometry in particular it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program in this paper we introduce and study a related quantity called the restricted nonnegative rank we show that computing this quantity is equivalent to a problem in polyhedral combinatorics and fully characterize its computational complexity this in turn sheds new light on the nonnegative rank problem and in particular allows us to provide new improved lower bounds based on its geometric interpretation we apply these results to slack matrices and linear euclidean distance matrices and obtain counter examples to two conjectures of beasly and laffey namely we show that the nonnegative rank of linear euclidean distance matrices is not necessarily equal to their dimension and that the rank of a matrix is not always greater than the nonnegative rank of its square

parametric regression on the grassmannian we address the problem of fitting parametric curves on the grassmann manifold for the purpose of intrinsic parametric regression we start from the energy minimization formulation of linear least squares in euclidean space and generalize this concept to general nonflat riemannian manifolds following an optimal control point of view we then specialize this idea to the grassmann manifold and demonstrate that it yields a simple extensible and easy to implement solution to the parametric regression problem in fact it allows us to extend the basic geodesic model to 1 a time warped variant and 2 cubic splines we demonstrate the utility of the proposed solution on different vision problems such as shape regression as a function of age traffic speed estimation and crowd counting from surveillance video clips most notably these problems can be conveniently solved within the same framework without any specifically tailored steps along the processing pipeline

completely positive factorizations associated with euclidean distance matrices corresponding to an arithmetic progression abstract euclidean distance matrices corresponding to an arithmetic progression have rich spectral and structural properties we exploit those properties to develop completely positive factorizations of translations of those matrices we show that the minimal translation that makes such a matrix positive semidefinite results in a completely positive matrix we also discuss completely positive factorizations of such matrices over the integers methods developed in the paper can be used to find completely positive factorizations of other matrices with similar properties

a note on the computation of the cp rank abstract the purpose of this note is to address the computational question of determining whether or not a square nonnegative matrix over the reals is completely positive and finding its cp rank when it is we show that these questions can be resolved by finite algorithms and we provide non polynomial complexity bounds on the number of arithmetic boolean operations that these algorithms require we state several open questions including the existence of polynomial algorithms to resolve the above problems and availability of algorithms for addressing complete positivity over the rationals and over 0 1 matrices

on nonnegative factorization of matrices abstract it is shown that a sufficient condition for a nonnegative real symmetric matrix to be completely positive is that the matrix is diagonally dominant

a factorization method for completely positive matrices abstract a matrix a is called completely positive if there exists an entrywise nonnegative matrix b such that a b b t these matrices play a major role in combinatorial and quadratic optimization in this paper we study the problem of finding a nonnegative factorization b b t of a given completely positive matrix a we formulate this factorization problem as a nonconvex feasibility problem and develop a solution method based on alternating projections a local convergence result can be shown for this algorithm we also provide a heuristic extension which improves the numerical performance of the algorithm extensive numerical tests show that the factorization method is very fast in most of the test instances and performs better than existing algorithms

algorithmic copositivity detection by simplicial partition we present new criteria for copositivity of a matrix i e conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant these criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof we show that as the partition gets finer and finer the conditions eventually capture all strictly copositive matrices we propose an algorithmic implementation which considers several numerical aspects as an application we present results on the maximum clique problem we also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones

the fourier argand representation an optimal basis of steerable patterns computing the convolution between a 2d signal and a corresponding filter with variable orientations is a basic problem that arises in various tasks ranging from low level image processing e g ridge edge detection to high level computer vision e g pattern recognition through decades of research there still lacks an efficient method for solving this problem in this paper we investigate this problem from the perspective of approximation by considering the following problem what is the optimal basis for approximating all rotated versions of a given bivariate function surprisingly solely minimising the l 2 approximation error leads to a rotation covariant linear expansion which we name fourier argand representation this representation presents two major advantages 1 rotation covariance of the basis which implies a strong steerability rotating by an angle alpha corresponds to multiplying each basis function by a complex scalar e ik alpha 2 optimality of the fourier argand basis which ensures a few number of basis functions suffice to accurately approximate complicated patterns and highly direction selective filters we show the relation between the fourier argand representation and the radon transform leading to an efficient implementation of the decomposition for digital filters we also show how to retrieve accurate orientation of local structures patterns using a fast frequency estimation algorithm

rational factorizations of completely positive matrices abstract in this note it is proved that every rational matrix which lies in the interior of the cone of completely positive matrices also has a rational cp factorization

a simplex algorithm for rational cp factorization in this paper we provide an algorithm similar to the simplex algorithm which determines a rational cp factorization of a given matrix whenever the matrix allows such a factorization this algorithm can be used to show that every integral completely positive 2 times 2 matrix has an integral cp factorization

factorization and cutting planes for completely positive matrices by copositive projection we consider the problem of projecting a matrix onto the cones of copositive and completely positive matrices as this can not be done directly we use polyhedral approximations of the cones with the help of these projections we obtain a technique to compute factorizations of completely positive matrices we also describe a method to determine a cutting plane which cuts off an arbitrary matrix from the completely positive or copositive cone

higher order continuous solution of y f x y y and y f x y initial value problems abstract sarafyan s continuous method for approximate solution of initial value problem is applied to the case of systems of the form y f x y y and y f x y three formulas are presented here containing polynomials in c 0 1 such that y and y are continuously imbedded with x x0 ch as valid approximations of runge kutta type the first is fifth order four stage with continuous fourth order y and third order y the second is fifth order six stage with continuous fifth order y and fourth order y the third is sixth order seven stage with conntinuous fifth order y and fourth oder y at c 1 the order for the continuous y improves by 1 the formulas may even be used advantageously in the extended interval c 0 5 1 5 values of c greater than 1 can be used to extend the approximation for the current step and to see whether h can be increased for the next step

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

gaussian processes for ordinal regression we present a probabilistic kernel approach to ordinal regression based on gaussian processes a threshold model that generalizes the probit function is used as the likelihood function for ordinal variables two inference techniques based on the laplace approximation and the expectation propagation algorithm respectively are derived for hyperparameter learning and model selection we compare these two gaussian process approaches with a previous ordinal regression method based on support vector machines on some benchmark and real world data sets including applications of ordinal regression to collaborative filtering and gene expression analysis experimental results on these data sets verify the usefulness of our approach

stability of some continuously imbedded runge kutta methods of sarafyan abstract sarafyan and others have recently developed novel explicit runge kutta methods associated with each method is an imbedded polynomial which interpolates the runge kutta method and which is itself a runge kutta approximation at non meshpoints in this paper we show that the interpolation polynomials possess the desirable stability properties of the runge kutta methods

a sixth order imbedded runge kutta algorithm with continuously variable weights abstract sarafyan s continuous method for approximate solution of initial value problems is extended to the sixth order case the approximations obtained are continuous throughout an interval the main formula is a polynomial of fifth degree in c 0 c c a valid fifth order approximation of runge kutta type is obtained for the solution at x x 0 ch at c 1 the approximation improves to sixth order lower order continuous formulas are imbedded using polynomials of lesser degree which may be used for error estimation or step size selection these formulas are differentiated to give continuous approximations to the first and second derivatives of the solution the formulas are valid for systems as well as for a single differential equation

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

continuous approximate solution of ordinary differential equations and their systems abstract the solution of initial value problems in ordinary differential equations are continuously differentiable functions yet all effective methods presently used for the numerical solution of these problems either in scalar or vector form are discrete processes they yield a finite set of approximate values of the solution the process derived in this paper yields a continuously differentiable function a polynomial which gives fourth order runge kutta approximations at all interior points of each interval of application and a fifth order approximation at each end point like conventional fifth order runge kutta algorithms the process requires only six evaluations of the given directional function per application or step it is endowed with an imbedding property for step size control and provides valuable information about committed local truncation errors it may be called a continuous imbedded runge kutta process of maximum order five it is valid for both scalar and vector cases

real rank versus nonnegative rank abstract we consider the set of m n nonnegative real matrices and define the nonnegative rank of a matrix a to be the minimum k such that a bc where b is m k and c is k n given that the real rank of a is j for some j we give bounds on the nonnegative rank of a and a 2

on the nonnegative rank of euclidean distance matrices abstract the euclidean distance matrix for n distinct points in r r is generically of rank r 2 it is shown in this paper via a geometric argument that its nonnegative rank for the case r 1 is generically n

counting people with low level features and bayesian regression an approach to the problem of estimating the size of inhomogeneous crowds which are composed of pedestrians that travel in different directions without using explicit object segmentation or tracking is proposed instead the crowd is segmented into components of homogeneous motion using the mixture of dynamic texture motion model a set of holistic low level features is extracted from each segmented region and a function that maps features into estimates of the number of people per segment is learned with bayesian regression two bayesian regression models are examined the first is a combination of gaussian process regression with a compound kernel which accounts for both the global and local trends of the count mapping but is limited by the real valued outputs that do not match the discrete counts we address this limitation with a second model which is based on a bayesian treatment of poisson regression that introduces a prior distribution on the linear weights of the model since exact inference is analytically intractable a closed form approximation is derived that is computationally efficient and kernelizable enabling the representation of nonlinear functions an approximate marginal likelihood is also derived for kernel hyperparameter learning the two regression based crowd counting methods are evaluated on a large pedestrian data set containing very distinct camera views pedestrian traffic and outliers such as bikes or skateboarders experimental results show that regression based counts are accurate regardless of the crowd size outperforming the count estimates produced by state of the art pedestrian detectors results on 2 h of video demonstrate the efficiency and robustness of the regression based crowd size estimation over long periods of time