object representation and recognition in shape spaces in this paper we describe a shape space based approach for invariant object representation and recognition in this approach an object and all its similarity transformed versions are identified with a single point in a high dimensional manifold called the shape space object recognition is achieved by measuring the geodesic distance between an observed object and a model in the shape space this approach produced promising results in 2d object recognition experiments it is invariant to similarity transformations and is relatively insensitive to noise and occlusion potentially it can also be used for 3d object recognition

on the characteristic polynomial of a random unitary matrix we present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial z of a random n n unitary matrix as n first we show that evaluated at a finite set of distinct points is asymptotically a collection of i i d complex normal random variables this leads to a refinement of a recent central limit theorem due to keating and snaith and also explains the covariance structure of the eigenvalue counting function next we obtain a central limit theorem for ln z in a sobolev space of generalised functions on the unit circle in this limiting regime lower order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting gaussian measure large deviations results for ln z a evaluated at a finite set of distinct points can be obtained for for higher order scalings we obtain large deviations results for ln z a evaluated at a single point there is a phase transition at a ln n which only applies to negative deviations of the real part reflecting a switch from global to local conspiracy

bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

on the second mixed moment of the characteristic polynomials of 1d band matrices we consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of 1d gaussian band matrices i e of the hermitian n n matrices h n with independent gaussian entries such that h ij h lk ik jl j ij where j w 2 triangle 1 1 assuming that w 2 n 1 theta 0 theta leq 1 we show that the moment s asymptotic behavior as n to infty in the bulk of the spectrum coincides with that for the gaussian unitary ensemble

universal behavior for averages of characteristic polynomials at the origin of the spectrum it has been shown by strahov and fyodorov that averages of products and ratios of characteristic polynomials corresponding to hermitian matrices of a unitary ensemble involve kernels related to orthogonal polynomials and their cauchy transforms we will show that for the unitary ensemble open image in new window hermitian matrices these kernels have universal behavior at the origin of the spectrum as n in terms of bessel functions our approach is based on the characterization of orthogonal polynomials together with their cauchy transforms via a matrix riemann hilbert problem due to fokas its and kitaev and on an application of the deift zhou steepest descent method for matrix riemann hilbert problems to obtain the asymptotic behavior of the riemann hilbert problem

norm and essential norm of composition followed by differentiation from α bloch spaces to hμ abstract this note calculates the norm of composition followed by differentiation operator from the bloch and the little bloch space to the weighted space h on the unit disk and gives an upper and a lower bound for the essential norm of the operator from the bloch space b 0 to h

averages of characteristic polynomials in random matrix theory we compute averages of products and ratios of characteristic polynomials associated with orthogonal unitary and symplectic ensembles of random matrix theory the pfaffian determinantal formulae for these averages are obtained and the bulk scaling asymptotic limits are found for ensembles with gaussian weights classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation we employ a discrete approximation method the problem is solved for discrete analogues of random matrix ensembles originating from representation theory and then a limit transition is performed exact pfaffian determinantal formulae for the discrete averages are proven using standard tools of linear algebra no application of orthogonal or skew orthogonal polynomials is needed 2005 wiley periodicals inc

iterated differentiation followed by composition from bloch type spaces to weighted bmoa spaces abstract let be a holomorphic self map of the open unit disk d and h d the class of holomorphic functions on d for n n 0 define an operator by c d n f f n f h d we characterize the boundedness and compactness of operator c d n from bloch type spaces to weighted bmoa spaces

an exact formula for general spectral correlation function of random hermitian matrices we have found an exact formula expressing a general correlation function containing both products and ratios of characteristic polynomials of random hermitian matrices the answer is given in the form of a determinant an essential difference from the previously studied correlation functions of products only is the appearance of non polynomial functions along with the orthogonal polynomials these non polynomial functions are the cauchy transforms of the orthogonal polynomials the result is valid for arbitrary ensemble of 2 symmetry class and generalizes recent asymptotic formulae obtained for gaussian unitary ensemble and its chiral counterpart by different methods

eigenvector localization for random band matrices with power law band width it is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors provided the band width w raised to a power remains smaller than the matrix size n for a gaussian band ensemble with matrix elements given by i i d centered gaussians within a band of width w the estimate 8 holds

universality of the second mixed moment of the characteristic polynomials of the 1d band matrices real symmetric case we prove that the asymptotic behavior of the second mixed moment of the characteristic polynomials of the n n 1d gaussian real symmetric band matrices with the width of the band w n1 2 coincides with those for the gaussian orthogonal ensemble here we adapt the approach of shcherbina commun math phys 328 45 82 2014 where the hermitian case was considered

delocalization for a class of random block band matrices we consider n times n hermitian random matrices h consisting of blocks of size m ge n 6 7 the matrix elements are i i d within the blocks close to a gaussian in the four moment matching sense but their distribution varies from block to block to form a block band structure with an essential band width m we show that the entries of the green s function g z h z 1 satisfy the local semicircle law with spectral parameter z e mathbf i eta down to the real axis for any eta gg n 1 using a combination of the supersymmetry method inspired by shcherbina j stat phys 155 3 466 499 2014 and the green s function comparison strategy previous estimates were valid only for eta gg m 1 the new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized

on the second order correlation function of the characteristic polynomial of a hermitian wigner matrix we consider the asymptotics of the second order correlation function of the characteristic polynomial of a random matrix we show that the known result for a random matrix from the gaussian unitary ensemble essentially continues to hold for a general hermitian wigner matrix our proofs rely on an explicit formula for the exponential generating function of the second order correlation function of the characteristic polynomial furthermore we show that the second order correlation function of the characteristic polynomial is closely related to that of the permanental polynomial

robust multidimensional matched subspace classifiers based on weighted least squares we propose and design two classes of robust subspace classifiers for classification of multidimensional signals our classifiers are based on robust m estimators and the least median of squares principle and we show that they may be unified as iterated reweighted oblique subspace classifiers the performance of the proposed classifiers are demonstrated by two examples noncoherent detection of space time frequency shift keying signals and shape classification of partially occluded two dimensional 2 d objects in both cases the proposed robust subspace classifiers outperform the conventional subspace classifiers

on fluctuations of eigenvalues of random band matrices we consider the fluctuations of linear eigenvalue statistics of random band n times n matrices whose entries have the form mathcal m ij b 1 2 u 1 2 i j b tilde w ij with i i d tilde w ij possessing the 4 varepsilon th moment where the function u has a finite support c c so that m has only 2c b 1 nonzero diagonals the parameter b called the bandwidth is assumed to grow with n in a way such that b n rightarrow 0 without any additional assumptions on the growth of b we prove clt for linear eigenvalue statistics for a rather wide class of test functions thus we improve and generalize the results of the previous papers jana et al arxiv 1412 2445 li et al random matrices 2 04 2013 where clt was proven under the assumption n b n 1 2 moreover we develop a method which allows to prove automatically the clt for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory deformed wigner and sample covariance matrices sparse matrices diluted random matrices matrices with heavy tales etc

characteristic polynomials of random matrices number theorists have studied extensively the connections between the distribution of zeros of the riemann function and of some generalizations with the statistics of the eigenvalues of large random matrices it is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices which are the expectation values of the characteristic polynomials of the matrix it turns out that these expectation values are quite interesting for instance the moments of order 2k scale for unitary invariant ensembles as the density of eigenvalues raised to the power k 2 the prefactor turns out to be a universal number i e it is independent of the specific probability distribution an equivalent behaviour and prefactor had been found as a conjecture within number theory the moments of the characteristic determinants of random matrices are computed here as limits at coinciding points of multi point correlators of determinants these correlators are in fact universal in dyson s scaling limit in which the difference between the points goes to zero the size of the matrix goes to infinity and their product remains finite

characteristic polynomials of real symmetric random matrices the correlation functions of the random variables det x in which x is an hermitian n n random matrix are known to exhibit universal local statistics in the large n limit we study here the correlation of those same random variables for real symmetric matrices goe the derivation relies on an exact dual representation of the problem the k point functions are expressed in terms of finite integrals over quaternionic k k matrices however the control of the dyson limit in which the distance of the various parameters s is of the order of the mean spacing requires an integration over the symplectic group it is shown that a generalization of the itzykson zuber method holds for this problem but contrary to the unitary case the semi classical result requires a finite number of corrections to be exact we have also considered the problem of an external matrix source coupled to the random matrix and obtain explicit integral formulae which are useful for the analysis of the large n limit

universal results for correlations of characteristic polynomials riemann hilbert approach we prove that general correlation functions of both ratios and products of characteristic polynomials of hermitian random matrices are governed by integrable kernels of three different types a those constructed from orthogonal polynomials b those constructed from cauchy transforms of the same orthogonal polynomials and finally c those constructed from both orthogonal polynomials and their cauchy transforms these kernels are related with the riemann hilbert problem for orthogonal polynomials for the correlation functions we obtain exact expressions in the form of determinants of these kernels derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via the deift zhou steepest descent stationary phase method for riemann hilbert problems and in particular to find negative moments of characteristic polynomials this reveals the universal parts of the correlation functions and moments of characteristic polynomials for an arbitrary invariant ensemble of 2 symmetry class

on the correlation functions of the characteristic polynomials of the hermitian sample covariance matrices we consider asymptotic behavior of the correlation functions of the characteristic polynomials of the hermitian sample covariance matrices h n n 1 a m n a m n where am n is a m n complex random matrix with independent and identically distributed entries mathfrak r a alpha j and mathfrak i a alpha j we show that for the correlation function of any even order the asymptotic behavior in the bulk and at the edge of the spectrum coincides with those for the gaussian unitary ensemble up to a factor depending only on the fourth moment of the common probability law of entries mathfrak r a alpha j mathfrak i a alpha j i e the higher moments do not contribute to the above limit

partial shape recognition a landmark based approach a method of recognizing partially occluded objects is presented in which each object is represented by a set of landmarks given a scene consisting of partially occluded objects a model object in the scene is hypothesized by matching the landmarks of the model with those in the scene a measure of similarity between two landmarks is needed to perform this matching a local shape measure sphericity is introduced it is shown that any invariant function under a similarity transformation is a function of the sphericity to match landmarks between the model and the scene a table of compatibility is constructed a technique known as hopping dynamic programming is described to guide the landmark matching through the compatibility table the location of the model in the scene is estimated with a least squares fit among the matched landmarks a heuristic measure is then computed to decide if the model is in the scene

the riemannian structure of euclidean shape spaces a novel environment for statistics the riemannian metric structure of the shape space sum k m for k labelled points in mathbb r m was given by kendall for the atypically simple situations in which m 1 or 2 and k geq 2 here we deal with the general case m geq 1 k geq 2 by using the properties of riemannian submersions and warped products as studied by o neill the approach is via the associated size and shape space that is the warped product of the shape space and the half line mathbb r carrying size the warping function being equal to the square of the size when combined with parallel studies by le of the corresponding global geodesic geometry the results obtained here determine the environment in which shape statistical calculations have to be acted out finally three different applications are discussed that illustrate the theory and its use in practice

moments of the characteristic polynomial in the three ensembles of random matrices moments of the characteristic polynomial of a random matrix taken from any of the three ensembles orthogonal unitary or symplectic are given either as a determinant or a pfaffian or as a sum of determinants for gaussian ensembles on comparing two expressions of the same moment one obtains two remarkable identities one between an n n determinant and an m m determinant and another between the pfaffian of a 2n 2n anti symmetric matrix and a sum of m m determinants

products and ratios of characteristic polynomials of random hermitian matrices we present new and streamlined proofs of various formulas for products and ratios of characteristic polynomials of random hermitian matrices that have appeared recently in the literature

recognize objects with three kinds of information in landmarks abstract object recognition is one of the most important tasks in computer vision domain and it has been extensively applied in many areas during the last three decades in this paper a method is proposed to recognize objects in shape spaces in which landmarks have three kinds of information i e landmarks position angle and feature vector by comparing angle and feature vector a landmark in an image can correspond with a few number of landmarks in another image then the pixel s positions of landmarks from two images are used to calculate the procrustean distance in a shape space if the value of the procrustean distance is smaller than a predefined threshold the two sets of landmarks will be considered to match each other applying the proposed method two dimensional objects including occluded objects can be recognized quickly

on the correlation function of the characteristic polynomials of the hermitian wigner ensemble we consider the asymptotic of the correlation functions of the characteristic polynomials of the hermitian wigner matrices h n n 1 2 w n we show that for the correlation function of any even order the asymptotic coincides with this for the gaussian unitary ensemble up to a factor depending only on the fourth moment of the common probability law of entries mathfrak j w jk mathfrak r w jk i e that the higher moments do not contribute to the above limit

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles