learning eigenfunctions links spectral embedding and kernel pca in this letter we show a direct relation between spectral embedding methods and kernel principal components analysis and how both are special cases of a more general learning problem learning the principal eigenfunctions of an operator defined from a kernel and the unknown data generating density whereas spectral embedding methods provided only coordinates for the training points the analysis justifies a simple extension to out of sample examples the nystrom formula for multidimensional scaling mds spectral clustering laplacian eigenmaps locally linear embedding lle and isomap the analysis provides for all such spectral embedding methods the definition of a loss function whose empirical average is minimized by the traditional algorithms the asymptotic expected value of that loss defines a generalization performance and clarifies what these algorithms are trying to learn experiments with lle isomap spectral clustering and mds show that this out of sample embedding formula generalizes well with a level of error comparable to the effect of small perturbations of the training set on the embedding

on the spectra of some weighted rooted trees and applications abstract let t be a weighted rooted tree of k levels such that 1 the vertices in level j have a degree equal to d k j 1 for j 1 2 k and 2 the edges joining the vertices in level j with the vertices in level j 1 have a weight equal to w k j for j 1 2 k 1 we give a complete characterization of the eigenvalues of the laplacian matrix and adjacency matrix of t they are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k moreover we give some results concerning their multiplicities by application of the above mentioned results we derive upper bounds on the largest eigenvalue of any weighted tree and the spectra of some weighted bethe trees

spectral variation bounds in hyperbolic geometry abstract we derive new estimates for distances between optimal matchings of eigenvalues of non normal matrices in terms of the norm of their difference we introduce and estimate a hyperbolic metric analogue of the classical spectral variation distance the result yields a qualitatively new and simple characterization of the localization of eigenvalues our bound improves on the best classical spectral variation bounds due to krause if the distance of matrices is sufficiently small and is sharp for asymptotically large matrices our approach is based on the theory of model operators which provides us with strong resolvent estimates the latter naturally lead to a chebyshev type interpolation problem with finite blaschke products which can be solved explicitly and gives stronger bounds than the classical chebyshev interpolation with polynomials as compared to the classical approach our method does not rely on hadamard s inequality and immediately generalizes to algebraic operators on hilbert space

ten methods to bound multiple roots of polynomials given a univariate polynomial p with a k fold multiple root or a k fold root cluster near some z we discuss nine different methods to compute a disc near z which either contains exactly or contains at least k roots of p many of the presented methods are known some are new we are especially interested in the behaviour of methods when implemented in a rigorous way that is when taking into account all possible effects of rounding errors in other words every result shall be mathematically correct we display extensive test sets comparing the methods under different circumstances based on the results we present a tenth a hybrid method combining five of the previous methods which for given z i detects the number k of roots near z and ii computes an including disc with in most cases a radius of the order of the numerical sensitivity of the root cluster therefore the resulting discs are numerically nearly optimal

twist based acquisition and tracking of animal and human kinematics this paper demonstrates a new visual motion estimation technique that is able to recover high degree of freedom articulated human body configurations in complex video sequences we introduce the use and integration of a mathematical technique the product of exponential maps and twist motions into a differential motion estimation this results in solving simple linear systems and enables us to recover robustly the kinematic degrees of freedom in noise and complex self occluded configurations a new factorization technique lets us also recover the kinematic chain model itself we are able to track several human walk cycles several wallaby hop cycles and two walk cycels of the famous movements of eadweard muybridge s motion studies from the last century to the best of our knowledge this is the first computer vision based system that is able to process such challenging footage

unsupervised learning of human motion an unsupervised learning algorithm that can obtain a probabilistic model of an object composed of a collection of parts a moving human body in our examples automatically from unlabeled training data is presented the training data include both useful foreground features as well as features that arise from irrelevant background clutter the correspondence between parts and detected features is unknown the joint probability density function of the parts is represented by a mixture of decomposable triangulated graphs which allow for fast detection to learn the model structure as well as model parameters an em like algorithm is developed where the labeling of the data part assignments is treated as hidden variables the unsupervised learning technique is not limited to decomposable triangulated graphs the efficiency and effectiveness of our algorithm is demonstrated by applying it to generate models of human motion automatically from unlabeled image sequences and testing the learned models on a variety of sequences

infinite dimensional features of matrices and pseudospectra abstract given a hilbert space operator t the level sets of function t z t z 1 1 determine the so called pseudospectra of t we set t to be zero on the spectrum of t after giving some elementary properties of t which as it seems were not noticed before we apply them to the study of the approximation we prove that for any operator t there is a sequence t n of finite matrices such that t n z tends to t z uniformly on c in this proof quasitriangular operators play a special role this is merely an existence result we do not give a concrete construction of this sequence of matrices one of our main points is to show how to use infinite dimensional operator models in order to produce examples and counterexamples in the set of finite matrices of large size in particular we get a result which means in a sense that the pseudospectrum of a nilpotent matrix can be anything one can imagine we also study the norms of the multipliers in the context of cowen douglas class operators we use these results to show that to the opposite to the function s the function s z for certain finite matrices s may oscillate arbitrarily fast even far away from the spectrum

real time visual tracking of complex structures presents a framework for three dimensional model based tracking graphical rendering technology is combined with constrained active contour tracking to create a robust wire frame tracking system it operates in real time at video frame rate 25 hz on standard hardware it is based on an internal cad model of the object to be tracked which is rendered using a binary space partition tree to perform hidden line removal a lie group formalism is used to cast the motion computation problem into simple geometric terms so that tracking becomes a simple optimization problem solved by means of iterative reweighted least squares a visual servoing system constructed using this framework is presented together with results showing the accuracy of the tracker the paper then describes how this tracking system has been extended to provide a general framework for tracking in complex configurations the adjoint representation of the group is used to transform measurements into common coordinate frames the constraints are then imposed by means of lagrange multipliers results from a number of experiments performed using this framework are presented and discussed

a note on the variation of permanents it is shown that for any two n by n complex matrices a b the inequality per a per b n a b max a b n 1 holds if is either the row sum or the column sum norm it is conjectured that this result holds for any operator norm

inclusion of the roots of a polynomial based on gerschgorin s theorem in this note a new companion matrix is presented which can be interpreted as a product of werner s companion matrices 13 gerschgorin s theorem yields an inclusion of the roots of a polynomial which is best in the sense of 4 and generalizes a result of l elsner 5 this inclusion is better than the one due to w borsch supan in 1

bounds for the variation of the roots of a polynomial and the eigenvalues of a matrix we derive some new bounds for the distance between the roots of two polynomials in terms of their coefficients and for the distance between the eigenvalues of two matrices in terms of the norm of their difference

resolvent bounds and spectral variation abstract a new spectral variation bound is given for linear operators acting on normed linear spaces this bound is derived using the cayley hamilton theorem and chebychev polynomials

on condition numbers of polynomial eigenvalue problems in this paper we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients generalizing classical results of matrix perturbation theory we provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate we obtain that if a simple eigenvalue of a matrix polynomial is ill conditioned in some respects then it is close to be multiple and we construct an upper bound for this distance measured in the euclidean norm we also derive a new expression for the condition number of a simple eigenvalue which does not involve eigenvectors moreover an elsner like perturbation bound for matrix polynomials is presented

topology adaptive mesh deformation for surface evolution morphing and multiview reconstruction triangulated meshes have become ubiquitous discrete surface representations in this paper we address the problem of how to maintain the manifold properties of a surface while it undergoes strong deformations that may cause topological changes we introduce a new self intersection removal algorithm transformesh and propose a mesh evolution framework based on this algorithm numerous shape modeling applications use surface evolution in order to improve shape properties such as appearance or accuracy both explicit and implicit representations can be considered for that purpose however explicit mesh representations while allowing for accurate surface modeling suffer from the inherent difficulty of reliably dealing with self intersections and topological changes such as merges and splits as a consequence a majority of methods rely on implicit representations of surfaces e g level sets that naturally overcome these issues nevertheless these methods are based on volumetric discretizations which introduce an unwanted precision complexity trade off the method that we propose handles topological changes in a robust manner and removes self intersections thus overcoming the traditional limitations of mesh based approaches to illustrate the effectiveness of transformesh we describe several challenging applications surface morphing and 3d reconstruction

multi view stereo reconstruction and scene flow estimation with a global image based matching score we present a new variational method for multi view stereovision and non rigid three dimensional motion estimation from multiple video sequences our method minimizes the prediction error of the shape and motion estimates both problems then translate into a generic image registration task the latter is entrusted to a global measure of image similarity chosen depending on imaging conditions and scene properties rather than integrating a matching measure computed independently at each surface point our approach computes a global image based matching score between the input images and the predicted images the matching process fully handles projective distortion and partial occlusions neighborhood as well as global intensity information can be exploited to improve the robustness to appearance changes due to non lambertian materials and illumination changes without any approximation of shape motion or visibility moreover our approach results in a simpler more flexible and more efficient implementation than in existing methods the computation time on large datasets does not exceed thirty minutes on a standard workstation finally our method is compliant with a hardware implementation with graphics processor units our stereovision algorithm yields very good results on a variety of datasets including specularities and translucency we have successfully tested our motion estimation algorithm on a very challenging multi view video sequence of a non rigid scene

universal fir mmse filtering we consider the problem of causal estimation i e filtering of a real valued signal corrupted by zero mean time independent real valued additive noise under the mean squared error mse criterion we build a universal filter whose per symbol squared error for every bounded underlying signal is essentially as small as that of the best finite duration impulse response fir filter of a given order we do not assume a stochastic mechanism generating the underlying signal and assume only that the variance of the noise is known to the filter the regret of the expected mse of our scheme is shown to decay as o logn n where n is the length of the signal moreover we present a stronger concentration result which guarantees the performance of our scheme not only in expectation but also with high probability our result implies a conventional stochastic setting result i e when the underlying signal is a stationary process our filter achieves the performance of the optimal fir filter we back our theoretical findings with several experiments showcasing the potential merits of our universal filter in practice our analysis combines tools from the problems of universal filtering and competitive online regression

robust stabilization of sampled data systems to structured lti perturbations sampled data systems with stable structured lti linear time invariant perturbations in the half plane algebra applied to the continuous time plant are considered necessary and sufficient conditions are derived for robust l sub 2 stability of such systems an example is provided to illustrate the results which shows that the small gain theorem can be an extremely conservative robustness test in this sampled data context

shape from silhouette across time part i theory and algorithms shape from silhouette sfs is a shape reconstruction method which constructs a 3d shape estimate of an object using silhouette images of the object the output of a sfs algorithm is known as the visual hull vh traditionally sfs is either performed on static objects or separately at each time instant in the case of videos of moving objects in this paper we develop a theory of performing sfs across time estimating the shape of a dynamic object with unknown motion by combining all of the silhouette images of the object over time we first introduce a one dimensional element called a bounding edge to represent the visual hull we then show that aligning two visual hulls using just their silhouettes is in general ambiguous and derive the geometric constraints in terms of bounding edges that govern the alignment to break the alignment ambiguity we combine stereo information with silhouette information and derive a temporal sfs algorithm which consists of two steps 1 estimate the motion of the objects over time visual hull alignment and 2 combine the silhouette information using the estimated motion visual hull refinement the algorithm is first developed for rigid objects and then extended to articulated objects in the part ii of this paper we apply our temporal sfs algorithm to two human related applications 1 the acquisition of detailed human kinematic models and 2 marker less motion tracking

higher order derivatives and perturbation bounds for determinants the first derivative of the determinant function is given by the well known jacobi s formula we obtain three different expressions for all higher order derivatives norms of these derivatives are then evaluated exactly

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

human motion tracking with a kinematic parameterization of extremal contours this paper addresses the problem of human motion tracking from multiple image sequences the human body is described by five articulated mechanical chains and human body parts are described by volumetric primitives with curved surfaces if such a surface is observed with a camera an extremal contour appears in the image whenever the surface turns smoothly away from the viewer we describe a method that recovers human motion through a kinematic parameterization of these extremal contours the method exploits the fact that the observed image motion of these contours is a function of both the rigid displacement of the surface and of the relative position and orientation between the viewer and the curved surface first we describe a parameterization of an extremal contour point velocity for the case of developable surfaces second we use the zero reference kinematic representation and we derive an explicit formula that links extremal contour velocities to the angular velocities associated with the kinematic model third we show how the chamfer distance may be used to measure the discrepancy between predicted extremal contours and observed image contours moreover we show how the chamfer distance can be used as a differentiable multi valued function and how the tracker based on this distance can be cast into a continuous non linear optimization framework fourth we describe implementation issues associated with a practical human body tracker that may use an arbitrary number of cameras one great methodological and practical advantage of our method is that it relies neither on model to image nor on image to image point matches in practice we model people with 5 kinematic chains 19 volumetric primitives and 54 degrees of freedom we observe silhouettes in images gathered with several synchronized and calibrated cameras the tracker has been successfully applied to several complex motions gathered at 30 frames second

on the variation of the spectra of matrix pencils several distances between the spectra of two regular matrix pencils are discussed and compared the relations obtained enable us to estimate upper bounds on one distance via known upper bounds on others we also give some perturbation bounds and then show how to apply the relations obtained in this note

permanental polynomials of graphs abstract the first section surveys recent results on the permanental polynomial of a square matrix a i e per xi a the second section concerns the permanental polynomial of the adjacency matrix of a graph the final section is an introduction to the permanental polynomial of the laplacian matrix of a graph an appendix lists some of these latter polynomials

robust temporally coherent laplacian protrusion segmentation of 3d articulated bodies in motion analysis and understanding it is important to be able to fit a suitable model or structure to the temporal series of observed data in order to describe motion patterns in a compact way and to discriminate between them in an unsupervised context i e no prior model of the moving object s is available such a structure has to be learned from the data in a bottom up fashion in recent times volumetric approaches in which the motion is captured from a number of cameras and a voxel set representation of the body is built from the camera views have gained ground due to attractive features such as inherent view invariance and robustness to occlusions automatic unsupervised segmentation of moving bodies along entire sequences in a temporally coherent and robust way has the potential to provide a means of constructing a bottom up model of the moving body and track motion cues that may be later exploited for motion classification spectral methods such as locally linear embedding can be useful in this context as they preserve protrusions i e high curvature regions of the 3d volume of articulated shapes while improving their separation in a lower dimensional space making them in this way easier to cluster in this paper we therefore propose a spectral approach to unsupervised and temporally coherent body protrusion segmentation along time sequences volumetric shapes are clustered in an embedding space clusters are propagated in time to ensure coherence and merged or split to accommodate changes in the body s topology experiments on both synthetic and real sequences of dense voxel set data are shown this supports the ability of the proposed method to cluster body parts consistently over time in a totally unsupervised fashion its robustness to sampling density and shape quality and its potential for bottom up model construction

an analysis of the rayleigh ritz method for approximating eigenspaces this paper concerns the rayleigh ritz method for computing an approximation to an eigenspace x of a general matrix a from a subspace w that contains an approximation to x the method produces a pair n x that purports to approximate a pair l x where x is a basis for x and ax xl in this paper we consider the convergence of n x as the sine e of the angle between x and w approaches zero it is shown that under a natural hypothesis called the uniform separation condition the ritz pairs n x converge to the eigenpair l x when one is concerned with eigenvalues and eigenvectors one can compute certain refined ritz vectors whose convergence is guaranteed even when the uniform separation condition is not satisfied an attractive feature of the analysis is that it does not assume that a has distinct eigenvalues or is diagonalizable

a measure for the decentralized assignability of eigenvalues abstract the results of 7 are generalized to produce a continuous measure of how close a mode is to being a decentralized fixed mode 13 implications of this measure on controller design are also explored

estimates for the spectrum near algebraic elements abstract we extend a result of s friedland linear algebra appl 12 1982 81 98 on the variation of eigenvalues of matrices to show that if a b are elements of a banach algebra both algebraic of degree at most n then the hausdorff distance between their spectra satisfies a b n c n 2m n 1 a b where m max a b and c n 23n 13 the same technique also re proves a local form of this result obtained earlier by b aupetit and j zemanek linear algebra appl 52 53 1983 39 44 but with improved bounds on the constants we further investigate the sharpness of these bounds

a constructive version of the boyle handelman theorem on the spectra of nonnegative matrices abstract a constructive version of the celebrated boyle handelman theorem on the non zero spectra of nonnegative matrices is presented

unitary invariance and spectral variation abstract we call a norm on operators or matrices weakly unitarily invariant if its value at operator a is not changed by replacing a by u au provided only that u is unitary this class includes such norms as the numerical radius we extend to all such norms an inequality that bounds the spectral variation when a normal operator a is replaced by another normal b in terms of the arclength of any normal path from a to b computed using the norm in question related results treat the local metric geometry of the manifold of normal operators we introduce a representation for weakly unitarily invariant matrix norms in terms of function norms over the unit ball and identify this correspondence explicitly in certain cases

a spectral approach to the randic index abstract the higher randic index rt of a simple graph is defined as r t v i 1 v i 2 v i t 1 1 i 1 i 2 i t 1 where i denotes the degree of the vertex vi and v i 1 v i 2 v i t 1 runs over all paths of length t in in this paper we introduce a suitable version of the laplacian of a graph and we formulated r1 in terms of its spectrum moreover bounds on r2 from the eigenvalues either the adjacency matrix or the laplacian matrix of the graph are obtained in the paper

reduced stability of parameter dependent matrices abstract the stability properties of parameter dependent linear systems z a e z with a 0 block diag 0 f f a stable matrix and 0 of order n n are analyzed near e 0 by a reduction principle which amounts to considering the schur complement g e of a e some sufficient conditions are given for a e and g e to have the same stability properties the so called principle of reduced stability in terms of the asymptotic expansion of g e explicit necessary and sufficient conditions are given in the case f i n 2 in terms of the location of the spectrum of g e allowing for geometrical interpretation

computation of eigenpair partial derivatives by rayleigh ritz procedure based on rayleigh ritz procedure a new method is proposed for a few eigenpair partial derivatives of large matrices this method simultaneously computes the approximate eigenpairs and their partial derivatives the linear systems of equations that are solved for eigenvector partial derivatives are greatly reduced from the original matrix size and the left eigenvectors are not required moreover errors of the computed eigenpairs and their partial derivatives are investigated hausdorff distance and containment gap are used to measure the accuracy of approximate eigenpair partial derivatives error bounds on the computed eigenpairs and their partial derivatives are derived finally numerical experiments are reported to show the efficiency of the proposed method

upper bound on randic energy of some graphs abstract applying the cauchy schwarz inequality we obtain a sharp upper bound on the randic energy of a bipartite graph and of graphs whose adjacency matrix is partitioned into blocks with constant row sum

perturbation bounds for polynomials using two different elementary approaches we derive a global and a local perturbation theorem on polynomial zeros that significantly improve the results of ostrowski acta math 72 99 257 1940 elsner et al linear algebra appl 142 195 209 1990 a comparison of different perturbation bounds shows that our results are better in many cases than the similar local result of beauzamy can math bull 42 1 3 12 1999 using the matrix theoretical approach we also improve the backward stability result of edelman and murakami proceedings of the fifth siam conference on applied linear algebra siam philapdelphia 1994 math comput 64 210 763 1995

explicit upper bounds for the spectral distance of two trace class operators abstract given two trace class operators a and b on a separable hilbert space we provide an upper bound for the hausdorff distance of their spectra involving only the distance of a and b in operator norm and the singular values of a and b by specifying particular asymptotics of the singular values our bound reproduces or improves existing bounds for the spectral distance the proof is based on lower and upper bounds for determinants of trace class operators of independent interest

lipschitz properties of structure preserving matrix perturbations abstract it is proved that under small perturbations that preserve the jordan structure canonical transformation matrices have the lipschitz property analogous result is proved for matrices under additional structure such as hamiltonian skew hamiltonian and symplectic both real and complex cases are considered

analysis of spectral variation and some inequalities a geometric method based on a decomposition of the space of complex matrices is employed to study the variation of the spectrum of a matrix when adapted to special cases this leads to some classical inequalities as well as some new ones as an example of the latter we show that if u v are unitary matrices and k is a skew hermitian matrix such that uv exp k then for every unitary invariant norm the distance between the eigenvalues of v and those of vis bounded by k this generalises two earlier results which used particular unitary invariant norms

variation of induced linear operators abstract let v be an n dimensional inner product space let be an irreducible character of the symmetric group s m and let v be the symmetry class of tensors associated with it let a be a linear operator on v and let k a be the operator it induces on v we obtain an explicit expression for the norm of the derivative of the map a k a in terms of the singular values of a two special cases of this problem antisymmetric and symmetric tensor products have been studied earlier and our results reduce to the earlier ones in these cases

variation of grassman powers and spectra the operator norm of the derivative of the map which takes a finite dimensional linear operator to its kth grassman power the kth compound is evaluated this leads to a bound for the distance between the grassman powers of two operators as an important application a bound for the distance between the eigenvalues of two operators is obtained

optimal bounds for the spectral variation of two regular matrix pairs for the generalized eigenvalue problem we establish upper bounds for the spectral variation of two regular matrix pairs some of which are optimal we describe the set of regular matrix pairs for which the bounds are attained

the spectra of the adjacency matrix and laplacian matrix for some balanced trees abstract let t be an unweighted rooted tree of k levels such that in each level the vertices have equal degree let dk j 1 denotes the degree of the vertices in the level j we find the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k the codiagonal entries for both matrices are d j 1 2 j k 1 and d k while the diagonal entries are zeros in the case of the adjacency matrix and dj 1 j k in the case of the laplacian matrix moreover we give some results concerning to the multiplicity of the above mentioned eigenvalues

on the variation of the spectra of matrices abstract several distances between the spectra of two matrices are discussed and compared optimal bounds are given which enable us to reduce certain bounds on the eigenvalue variation of matrices by a factor of about two the results of bhatia and mukherjea and of bhatia and friedland on the eigenvalue variation are derived in an elementary way using results of henrici on the spectral variation

on the normalized laplacian energy and general randic index r 1 of graphs in this paper we consider the energy of a simple graph with respect to its normalized laplacian eigenvalues which we call the l energy over graphs of order n that contain no isolated vertices we characterize the graphs with minimal l energy of 2 and maximal l energy of 2 n 2 we provide upper and lower bounds for l energy based on its general randic index r 1 g we highlight known results for r 1 g most of which assume g is a tree we extend an upper bound of r 1 g known for trees to connected graphs we provide bounds on the l energy in terms of other parameters one of which is the energy with respect to the adjacency matrix finally we discuss the maximum change of l energy and r 1 g upon edge deletion

efficient polyhedral modeling from silhouettes modeling from silhouettes is a popular and useful topic in computer vision many methods exist to compute the surface of the visual hull from silhouettes but few address the problem of ensuring good topological properties of the surface such as manifoldness this article provides an efficient algorithm to compute such a surface in the form of a polyhedral mesh it relies on a small number of geometric operations to compute a visual hull polyhedron in a single pass such simplicity enables the algorithm to combine the advantages of being fast producing pixel exact surfaces and repeatably yield manifold and watertight polyhedra in general experimental conditions with real data as verified with all datasets tested the algorithm is fully described its complexity analyzed and modeling results given

topology invariant similarity of nonrigid shapes this paper explores the problem of similarity criteria between nonrigid shapes broadly speaking such criteria are divided into intrinsic and extrinsic the first referring to the metric structure of the object and the latter to how it is laid out in the euclidean space both criteria have their advantages and disadvantages extrinsic similarity is sensitive to nonrigid deformations while intrinsic similarity is sensitive to topological noise in this paper we approach the problem from the perspective of metric geometry we show that by unifying the extrinsic and intrinsic similarity criteria it is possible to obtain a stronger topology invariant similarity suitable for comparing deformed shapes with different topology we construct this new joint criterion as a tradeoff between the extrinsic and intrinsic similarity and use it as a set valued distance numerical results demonstrate the efficiency of our approach in cases where using either extrinsic or intrinsic criteria alone would fail

improving spectral variation bounds with chebyshev polynomials a new spectral variation bound is established for linear operators a and b on finite dimensional vector spaces this bound is v a b c a b 1n a b n 1 n with c 8 this result affirmatively answers a conjecture of friedland

bounds for the variation of matrix eigenvalues and polynomial roots abstract for two given complex matrices a b upper bounds are derived for the optimal matching distance between the spectra a and b in terms of a b2 where 2 is the spectral norm the case of arbitrary matrix norms is treated a similar result estimates the optimal matching distance between the roots of two polynomials these bounds replace a factor of 4 in earlier results by the value 16 3 3 3 08

probabilistic tracking with exemplars in a metric space a new exemplar based probabilistic paradigm for visual tracking is presented probabilistic mechanisms are attractive because they handle fusion of information especially temporal fusion in a principled manner exemplars are selected representatives of raw training data used here to represent probabilistic mixture distributions of object configurations their use avoids tedious hand construction of object models and problems with changes of topology using exemplars in place of a parameterized model poses several challenges addressed here with what we call the metric mixture m2 approach which has a number of attractions principally it provides alternatives to standard learning algorithms by allowing the use of metrics that are not embedded in a vector space secondly it uses a noise model that is learned from training data lastly it eliminates any need for an assumption of probabilistic pixelwise independence experiments demonstrate the effectiveness of the m2 model in two domains tracking walking people using chamfer distances on binary edge images and tracking mouth movements by means of a shuffle distance

counterexample to a conjecture of elsner on the spectral variation of matrices abstract in 1985 elsner proved that the hausdorff distance between the spectra of two n n matrices a and b satisfies a b n a b n 1 a b where denotes the operator norm with respect to the euclidean norm on c n he further conjectured that the same inequality holds for all operator norms we disprove this conjecture and also the weaker conjecture where a b is replaced by 2max a b

optimal periodic feedback design for continuous time lti systems with constrained control structure this paper aims to design a high performance controller with any predefined structure for continuous time lti systems the control law employed is the generalized sampled data hold function gshf which can have any special form e g polynomial exponential piecewise constant etc the gshf is first written as a linear combination of a set of basis functions obtained in accordance with its desired form and structure the objective is to find the coefficients of this linear combination such that a prespecified linear quadratic performance index is minimized a necessary and sufficient condition for the existence of such gshf is first obtained in the form of matrix inequality which can be solved by using the existing methods to obtain a set of stabilizing initial values for the coefficients or to conclude the non existence of such structurally constrained gshf an efficient algorithm is then presented to compute the optimal coefficients from their initial values so that the performance index is minimi

linear construction of companion matrices this note is concerned with the following problem for a given matrix a e c ix and a vector a e c does there exist a mapping 2 assigning to each manic polynomial f of degree n a vector x f e c such that the matrix b a a x f is a companion matrix off i e the characteristic polynomial of b is t l ftf the classes of suitable matrices a and vectors a are characterized and some properties of b are described the corresponding unique mapping is determined by a system of linear equations the cases of a triangular bidiagonal or diagonal matrix a are discussed explicitly and many known companion matrices are obtained as particular cases then gershgorin s theorem is applied yielding error estimates for polynomial roots finally the extension to block companion matrices and an example of nonlinear construction are discussed

an elsner like perturbation theorem for generalized eigenvalues in 1985 elsner established a general bound on the distance between an eigenvalue of a matrix and the closest eigenvalue of a perturbation of that matrix in this note we show that a similar result holds for the generalized eigenvalue problem

performance analysis of adaptive eigenanalysis algorithms we present a rigorous analysis of several popular forms of short memory adaptive eigenanalysis algorithms using a stochastic averaging method a first order analysis shows that the algorithms do not have any equilibrium points despite published claims to the contrary through averaging analysis we show that they hover around an appropriate eigenvector a second order analysis is also given without the gaussian noise assumption and our results greatly outperform an earlier approximation in the literature the second order analysis has been of much interest in the offline study but in the dynamic adaptive case is uncommon

ludwig elsner and his contributions to core applied and numerical linear algebra ludwig elsner was bona on 17 january 1939 in grog strehlitz upper silesia in germany he studied mathematics physics and astronomy at the universities of hamburg and freiburg and received his diploma in mathematics in 1963 at the university of hamburg as a member of the lively and inspiring group of lothar collatz in hamburg he worked on his ph d while holding a position at the computing center of the university of hamburg he received his ph d in 1965 and became an assistant professor at collatz institute of applied mathematics at the university of hamburg for six years during this period he spent one year 1968 1969 as senior officer at the csir in pretoria south africa in 1971 ludwig eisner became associate professor at the university of erlangen where he worked at the institute for applied mathematics until 1976 since 1976 he has been a full professor at the university of bielefeld where he has built up the numerical analysis group among many other activities in bielefeld he has twice been chairman of the department of mathematics and has planned and formed the numerical analysis group of the sonderforschungsbereich diskrete strukturen in der mathematik which since 1989 has attracted many specialists in linear algebra and numerical linear algebra to work with eisner in bielefeld numerous publications prove the success of these collaborative efforts

an optimal bound for the spectral variation of two matrices abstract we establish a bound for the spectral variation of two complex n n matrices a b in terms of a b and a b here denotes the spectral norm it is always better than a bound previously given by bhatia and friedland and it is optimal we describe the set of pairs a b for which the bound is attained

building roadmaps of minima and transitions in visual models becoming trapped in suboptimal local minima is a perennial problem when optimizing visual models particularly in applications like monocular human body tracking where complicated parametric models are repeatedly fitted to ambiguous image measurements we show that trapping can be significantly reduced by building roadmaps of nearby minima linked by transition pathways paths leading over low mountain passes in the cost surface found by locating the transition state codimension 1 saddle point at the top of the pass and then sliding downhill to the next minimum we present two families of transition state finding algorithms based on local optimization in eigenvector tracking unconstrained newton minimization is modified to climb uphill towards a transition state while in hypersurface sweeping a moving hypersurface is swept through the space and moving local minima within it are tracked using a constrained newton method these widely applicable numerical methods which appear not to be known in vision and optimization generalize methods from computational chemistry where finding transition states is critical for predicting reaction parameters experiments on the challenging problem of estimating 3d human pose from monocular images show that our algorithms find nearby transition states and minima very efficiently but also underline the disturbingly large numbers of minima that can exist in this and similar model based vision problems

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

perturbations of invariant subspaces of unreduced hessenberg matrices non structured perturbation of invariant subspaces of unreduced i e nonderogatory hessenberg matrices is considered some perturbation results for the generalized eigenvectors and the characteristic polynomial of unreduced upper hessenberg matrices are given two theorems are on the perturbation of invariant subspaces which are somewhat similar to the sin q theorems of davis and kahan apart from the residual which we do not have here dense perturbations of unreduced hessenberg matrices are also considered finally we prove an invariant subspace perturbation theorem for nonderogatory matrices

variation of symmetric tensor powers and permanents the norm of the derivative of the symmetric tensor power of an operator is evaluated exactly a bound for the distance between the permanents of two matrices is obtained as a consequence

approximation of haar distributed matrices and limiting distributions of eigenvalues of jacobi ensembles we develop a tool to approximate the entries of a large dimensional complex jacobi ensemble with independent complex gaussian random variables based on this and the author s earlier work in this direction we obtain the tracy widom law of the largest singular values of the jacobi emsemble moreover the circular law the marchenko pastur law the central limit theorem and the laws of large numbers for the spectral norms are also obtained

classification and recognition of dynamical models the role of phase independent components kernels and optimal transport we address the problem of performing decision tasks and in particular classification and recognition in the space of dynamical models in order to compare time series of data motivated by the application of recognition of human motion in image sequences we consider a class of models that include linear dynamics both stable and marginally stable periodic both minimum and nonminimum phases driven by non gaussian processes this requires extending existing learning and system identification algorithms to handle periodic modes and nonminimum phase behavior while taking into account higher order statistics of the data once a model is identified we define a kernel based cord distance between models which includes their dynamics their initial conditions and input distribution this is made possible by a novel kernel defined between two arbitrary non gaussian distributions which is computed by efficiently solving an optimal transport problem we validate our choice of models inference algorithm and distance on the tasks of human motion synthesis sample paths of the learned models and recognition nearest neighbor classification in the computed distance however our work can be applied more broadly where one needs to compare historical data while taking into account periodic trends nonminimum phase behavior and non gaussian input distributions

diffusion maps and coarse graining a unified framework for dimensionality reduction graph partitioning and data set parameterization we provide evidence that nonlinear dimensionality reduction clustering and data set parameterization can be solved within one and the same framework the main idea is to define a system of coordinates with an explicit metric that reflects the connectivity of a given data set and that is robust to noise our construction which is based on a markov random walk on the data offers a general scheme of simultaneously reorganizing and subsampling graphs and arbitrarily shaped data sets in high dimensions using intrinsic geometry we show that clustering in embedding spaces is equivalent to compressing operators the objective of data partitioning and clustering is to coarse grain the random walk on the data while at the same time preserving a diffusion operator for the intrinsic geometry or connectivity of the data set up to some accuracy we show that the quantization distortion in diffusion space bounds the error of compression of the operator thus giving a rigorous justification for k means clustering in diffusion space and a precise measure of the performance of general clustering algorithms

recovering 3d human pose from monocular images we describe a learning based method for recovering 3d human body pose from single images and monocular image sequences our approach requires neither an explicit body model nor prior labeling of body parts in the image instead it recovers pose by direct nonlinear regression against shape descriptor vectors extracted automatically from image silhouettes for robustness against local silhouette segmentation errors silhouette shape is encoded by histogram of shape contexts descriptors we evaluate several different regression methods ridge regression relevance vector machine rvm regression and support vector machine svm regression over both linear and kernel bases the rvms provide much sparser regressors without compromising performance and kernel bases give a small but worthwhile improvement in performance the loss of depth and limb labeling information often makes the recovery of 3d pose from single silhouettes ambiguous to handle this the method is embedded in a novel regressive tracking framework using dynamics from the previous state estimate together with a learned regression value to disambiguate the pose we show that the resulting system tracks long sequences stably for realism and good generalization over a wide range of viewpoints we train the regressors on images resynthesized from real human motion capture data the method is demonstrated for several representations of full body pose both quantitatively on independent but similar test data and qualitatively on real image sequences mean angular errors of 4 6 spl deg are obtained for a variety of walking motions

derivatives for antisymmetric tensor powers and perturbation bounds abstract in an earlier paper r bhatia t jain higher order derivatives and perturbation bounds for determinants linear algebra appl 431 2009 2102 2108 we gave formulas for derivatives of all orders for the map that takes a matrix to its determinant in this paper we continue that work and find expressions for the derivatives of all orders for the antisymmetric tensor powers and for the coefficients of the characteristic polynomial we then evaluate norms of these derivatives and use them to obtain perturbation bounds

the energy of graphs and matrices abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman let 2 m n a be an m n nonnegative matrix with maximum entry and a 1 n extending previous results of koolen and moulton for graphs we prove that e a a 1 m n m 1 a 2 2 a 1 2 m n n m m 2 furthermore if a is any nonconstant matrix then e a 1 a a 2 2 1 2 a 2 a finally we note that wigner s semicircle law implies that e g 4 3 o 1 n 3 2 for almost all graphs g

maximizing numerical radii of weighted shifts under weight permutations abstract let w i c 1 i n and l s n the symmetric group of all permutations of 1 2 n suppose a l is the weighted cyclic matrix a l 0 w l 1 0 w l n 1 w l n 0 and w a l denotes its numerical radius we characterize those s n which satisfy w a max l s n w a l the characterizations for unilateral and bilateral weighted backward shifts are also obtained

on the sum of k largest singular values of graphs and matrices in the recent years the trace norm of graphs has been extensively studied under the name of graph energy the trace norm is just one of the ky fan k norms given by the sum of the k largest singular values which are studied more generally in the present paper several relations to chromatic number spectral radius spread and to other fundamental parameters are outlined some results are extended to more general matrices

subsystem interconnection and near optimum control of discrete balanced systems the relationship between subsystem interconnection of discrete balanced systems and the hankel singular values is explored upper bounds on the spectral norm of the submatrices of the discrete balanced representation are derived by approximating the matrix riccati equation by its power series expansion with respect to two scalar parameters related to the hankel singular values for diagonalizable symmetric discrete time systems the relation between the eigenvalues of the subsystems and the eigenvalues of the overall system is explored it is shown that an upper bound on the distance between corresponding eigenvalues in these two sets is related to the ratio of the hankel singular values

interpreting line drawings as three dimensional surfaces understanding how line drawings convey tri dimensionality is of fundamental importance in explaining surface perception when photometry is either uninformative or too complex to model analytically we put forward here a computational model for interpreting line drawings as three dimensional surfaces based on constraints on local surface orientation along extremal and discontinuity boundaries specific techniques are described for two key processes recovering the three dimensional conformation of a space curve e g a surface boundary from its two dimensional projection in an image and interpolating smooth surfaces from orientation constraints along extremal boundaries the relevance of the model to a general theory of low level vision is discussed

canonical correlation analysis of video volume tensors for action categorization and detection this paper addresses a spatiotemporal pattern recognition problem the main purpose of this study is to find a right representation and matching of action video volumes for categorization a novel method is proposed to measure video to video volume similarity by extending canonical correlation analysis cca a principled tool to inspect linear relations between two sets of vectors to that of two multiway data arrays or tensors the proposed method analyzes video volumes as inputs avoiding the difficult problem of explicit motion estimation required in traditional methods and provides a way of spatiotemporal pattern matching that is robust to intraclass variations of actions the proposed matching is demonstrated for action classification by a simple nearest neighbor classifier we moreover propose an automatic action detection method which performs 3d window search over an input video with action exemplars the search is speeded up by dynamic learning of subspaces in the proposed cca experiments on a public action data set kth and a self recorded hand gesture data showed that the proposed method is significantly better than various state of the art methods with respect to accuracy our method has low time complexity and does not require any major tuning parameters

asymptotically ordinary linear volterra difference equations with infinite delay abstract a class of linear volterra difference equations with infinite delay is considered it is shown that if the coefficient matrices are sufficiently small then the volterra difference equation is asymptotically equivalent to a linear ordinary difference equation at infinity the coefficient matrix of the ordinary difference equation is a solution of an associated matrix equation which can be obtained by successive approximations the eigenvalues of the approximating matrices converge exponentially to the characteristic roots of the volterra difference equation as a corollary an efficient new method is obtained for the computation of the characteristic roots with an explicit error estimate