langlands program trace formulas and their geometrization the langlands program relates galois representations and auto morphic representations of reductive algebraic groups the trace formula is a powerful tool in the study of this connection and the langlands functorial ity conjecture after giving an introduction to the langlands program and its geometric version which applies to curves over finite fields and over the complex field i give a survey of my recent joint work with robert langlands and ngo bao chau fln fn on a new approach to proving the functoriality conjecture using the trace formulas and on the geometrization of the trace formulas in particular i discuss the connection of the latter to the categorifi cation of the langlands correspondence

clustering via kernel decomposition spectral clustering methods were proposed recently which rely on the eigenvalue decomposition of an affinity matrix in this letter the affinity matrix is created from the elements of a nonparametric density estimator and then decomposed to obtain posterior probabilities of class membership hyperparameters are selected using standard cross validation methods

on a vanishing conjecture appearing in the geometric langlands correspondence 0 1 this paper should be regarded as a sequel to 7 there it was shown that the geometric langlands conjecture for gln follows from a certain vanishing conjecture the goal of the present paper is to prove this vanishing conjecture let x be a smooth projective curve over a ground field k let e be an m dimensional local system on x and let bunm be the moduli stack of rank m vector bundles on x the geometric langlands conjecture says that to e we can associate a perverse sheaf fe on bunm which is a hecke eigensheaf with respect to e the vanishing conjecture of 7 says that for all integers n m a certain functor ave depending on e and a parameter d z which maps the category d bunn to itself vanishes identically when d is large enough the fact that the vanishing conjecture implies the geometric langlands conjecture may be regarded as a geometric version of the converse theorem moreover as will be explained in the sequel the vanishing of the functor ave is analogous to the condition that the rankin selberg convolution of e viewed as an m dimensional galois representation and an automorphic form on gln with n m is well behaved both the geometric langlands conjecture and the vanishing conjecture can be formulated in any of the sheaf theoretic situations e g q adic sheaves when char k d modules when char k 0 and sheaves with coefficients in a finite field f again when char k when the ground field is the finite field fq and we are working with adic coefficients it was shown in 7 that the vanishing conjecture can be deduced from lafforgue s theorem that establishes the full langlands correspondence for global fields of positive characteristic cf 9

growing scale free networks with tunable clustering we extend the standard scale free network model to include a triad formation step we analyze the geometric properties of networks generated by this algorithm both analytically and by numerical c

orthogonal series density estimation and the kernel eigenvalue problem kernel principal component analysis has been introduced as a method of extracting a set of orthonormal nonlinear features from multivariate data and many impressive applications are being reported within the literature this article presents the view that the eigenvalue decomposition of a kernel matrix can also provide the discrete expansion coefficients required for a nonparametric orthogonal series density estimator in addition to providing novel insights into nonparametric density estimation this article provides an intuitively appealing interpretation for the nonlinear features extracted from data using kernel principal component analysis

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

on the geometric langlands conjecture let x be a smooth projective curve and g be a reductive group over a finite field fq the field kx of rational functions on x is what number theorists call a global field and we can do the theory of automorphic functions over it namely we consider the quotient g kx g a here a is the ring of adeles corresponding to kx and study the space of functions on it as a representation of the adele group g a it was essentially an observation of a weil that the quotient g kx g a is closely related to the set of isomorphism classes of principal g bundles on our curve x however g bundles on x possess a richer structure one can view the above set as a set of fq points of an algebraic variety or rather a stack denoted bung moreover instead of studying the vector space of functions on the discrete set of points of bung we will consider the category of adic sheaves on it basically what people call the geometric langlands program is an attempt to understand a spectral decomposition of the above category under the action of the so called hecke functors the latter will be defined in the talk the answer predicted by the geometric langlands conjecture links this spectral decomposition to the moduli stack of local systems on x with respect to the langlands dual group

optimized kernel entropy components this brief addresses two main issues of the standard kernel entropy component analysis keca algorithm the optimization of the kernel decomposition and the optimization of the gaussian kernel parameter keca roughly reduces to a sorting of the importance of kernel eigenvectors by entropy instead of variance as in the kernel principal components analysis in this brief we propose an extension of the keca method named optimized keca okeca that directly extracts the optimal features retaining most of the data entropy by means of compacting the information in very few features often in just one or two the proposed method produces features which have higher expressive power in particular it is based on the independent component analysis framework and introduces an extra rotation to the eigen decomposition which is optimized via gradient ascent search this maximum entropy preservation suggests that okeca features are more efficient than keca features for density estimation in addition a critical issue in both the methods is the selection of the kernel parameter since it critically affects the resulting performance here we analyze the most common kernel length scale selection criteria the results of both the methods are illustrated in different synthetic and real problems results show that okeca returns projections with more expressive power than keca the most successful rule for estimating the kernel parameter is based on maximum likelihood and okeca is more robust to the selection of the length scale parameter in kernel density estimation

a support vector machine formulation to pca analysis and its kernel version in this paper we present a simple and straightforward primal dual support vector machine formulation to the problem of principal component analysis pca in dual variables by considering a mapping to a high dimensional feature space and application of the kernel trick mercer theorem kernel pca is obtained as introduced by scholkopf et al 2002 while least squares support vector machine classifiers have a natural link with the kernel fisher discriminant analysis minimizing the within class scatter around targets 1 and 1 for pca analysis one can take the interpretation of a one class modeling problem with zero target value around which one maximizes the variance the score variables are interpreted as error variables within the problem formulation in this way primal dual constrained optimization problem interpretations to the linear and kernel pca analysis are obtained in a similar style as for least square support vector machine classifiers

small world graphs by iterated local edge formation we study graphs obtained by successive creation and destruction of edges into small neighborhoods of the vertices starting with a circle graph of large diameter we obtain small world graphs with logarithmic diameter high clustering coefficients and a fat tail distribution for the degree only local edge formation processes are involved and no preferential attachment was used furthermore we found an interesting phase transition with respect to the initial conditions

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

kernel entropy component analysis we introduce kernel entropy component analysis kernel eca as a new method for data transformation and dimensionality reduction kernel eca reveals structure relating to the renyi entropy of the input space data set estimated via a kernel matrix using parzen windowing this is achieved by projections onto a subset of entropy preserving kernel principal component analysis kernel pca axes this subset does not need in general to correspond to the top eigenvalues of the kernel matrix in contrast to the dimensionality reduction using kernel pca we show that kernel eca may produce strikingly different transformed data sets compared to kernel pca with a distinct angle based structure a new spectral clustering algorithm utilizing this structure is developed with positive results furthermore kernel eca is shown to be an useful alternative for pattern denoising

spectral element approximation of fredholm integral eigenvalue problems we consider the numerical approximation of homogeneous fredholm integral equations of second kind with emphasis on computing truncated karhunen loeve expansions we employ the spectral element method with gauss lobatto legendre gll collocation points similar to the piecewise constant finite elements this approach is simple to implement and does not lead to generalized discrete eigenvalue problems numerical experiments confirm the expected convergence rates for some classical kernels and illustrate how this approach can improve the finite element solution of partial differential equations with random input data

growing network with local rules preferential attachment clustering hierarchy and degree correlations the linear preferential attachment hypothesis has been shown to be quite successful in explaining the existence of networks with power law degree distributions it is then quite important to determine if this mechanism is the consequence of a general principle based on local rules in this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules it is also shown that the local models offer an explanation for other properties like the clustering hierarchy and degree correlations recently observed in complex networks these conclusions are based on both analytical and numerical results for different local rules including some models already proposed in the literature

noise level estimation for model selection in kernel pca denoising one of the main challenges in unsupervised learning is to find suitable values for the model parameters in kernel principal component analysis kpca for example these are the number of components the kernel and its parameters this paper presents a model selection criterion based on distance distributions mdds this criterion can be used to find the number of components and the sigma 2 parameter of radial basis function kernels by means of spectral comparison between information and noise the noise content is estimated from the statistical moments of the distribution of distances in the original dataset this allows for a type of randomization of the dataset without actually having to permute the data points or generate artificial datasets after comparing the eigenvalues computed from the estimated noise with the ones from the input dataset information is retained and maximized by a set of model parameters in addition to the model selection criterion this paper proposes a modification to the fixed size method and uses the incomplete cholesky factorization both of which are used to solve kpca in large scale applications these two approaches together with the model selection mdd were tested in toy examples and real life applications and it is shown that they outperform other known algorithms

evolving networks with distance preferences we study evolving networks where new nodes when attached to the network form links with other nodes of preferred distances a particular case is where always the shortest distances are selected make friends with the friends of your present friends we present simulation results for network parameters like the first eigenvalue of the graph laplacian synchronizability clustering coefficients average distances and degree distributions for different distance preferences and compare them with the parameter values for random and scale free networks we find that for the shortest distance rule we obtain a power law degree distribution as in scale free networks while the other parameters are significantly different especially the clustering coefficient

pca based hurst exponent estimator for fbm signals under disturbances in this paper the validity of pca eigenspectrum based hurst exponent estimator proposed in j b gao y cao and j m lee ldquoprincipal component analysis of 1 f alpha noise rdquo phys lett a vol 314 no 5 6 pp 392 400 2003 for single fbm signal is proved moreover how to apply this estimator for fbm signals corrupted with some other signals are discussed theoretical analysis and experiments show that it can also be used for 1 mixed fbm signals with different hurst exponents 2 fbm signals corrupted with additive gaussian white noise when the signal to noise ratio snr is not too small and 3 fbm signals corrupted with additive deterministic sine cosine signals however the estimation accuracy depends on the snr value for the first two situations

recent advances in the langlands program these are the notes for the lecture given by the author at the mathematical current events special session of the ams meeting in baltimore on january 17 2003 topics reviewed include the langlands correspondence for gl n in the function field case and its proof by v drinfeld and l lafforgue the geometric langlands correspondence for gl n and its proof by d gaitsgory k vilonen and the author and the work of a beilinson and v drinfeld on the quantization of the hitchin system and the langlands correspondence for an arbitrary semisimple algebraic group

a multilevel finite element method for fredholm integral eigenvalue problems in this work we proposed a multigrid finite element mfe method for solving the fredholm integral eigenvalue problems the main motivation for such studies is to compute the karhunen loeve expansions of random fields which play an important role in the applications of uncertainty quantification in our mfe framework solving the eigenvalue problem is converted to doing a series of integral iterations and eigenvalue solving in the coarsest mesh then any existing efficient integration scheme can be used for the associated integration process the error estimates are provided and the computational complexity is analyzed it is noticed that the total computational work of our method is comparable with a single integration step in the finest mesh several numerical experiments are presented to validate the efficiency of the proposed numerical method

probability density estimation from optimally condensed data samples the requirement to reduce the computational cost of evaluating a point probability density estimate when employing a parzen window estimator is a well known problem this paper presents the reduced set density estimator that provides a kernel based density estimator which employs a small percentage of the available data sample and is optimal in the l sub 2 sense while only requiring spl oscr n sup 2 optimization routines to estimate the required kernel weighting coefficients the proposed method provides similar levels of performance accuracy and sparseness of representation as support vector machine density estimation which requires spl oscr n sup 3 optimization routines and which has previously been shown to consistently outperform gaussian mixture models it is also demonstrated that the proposed density estimator consistently provides superior density estimates for similar levels of data reduction to that provided by the recently proposed density based multiscale data condensation algorithm and in addition has comparable computational scaling the additional advantage of the proposed method is that no extra free parameters are introduced such as regularization bin width or condensation ratios making this method a very simple and straightforward approach to providing a reduced set density estimator with comparable accuracy to that of the full sample parzen density estimator

the selection of terms in an orthogonal series density estimator abstract we show that kronmal and tarter s well known rule for selecting the terms in an orthogonal series density estimator can lead to poor performance and even inconsistency in certain cases these difficulties arise when the underlying density has a nonmonotone sequence of fourier coefficients as is likely to be the case with sharply peaked or multimodal distributions we suggest a way of overcoming these shortcomings

hall algebras of curves commuting varieties and langlands duality we construct an isomorphism between the universal spherical hall algebra of a smooth projective curve of genus g and a convolution algebra in the equivariant k theory of the genus g commuting varieties c mathfrak gl r big x i y i in mathfrak gl r 2g sum i 1 g x i y i 0 big we can view this isomorphism as a version of the geometric langlands duality in the formal neighborhood of the trivial local system for the group glr we extend this to all reductive groups and we compute the image under our correspondence of the skyscraper sheaf supported on the trivial local system

density dependent quantized least squares support vector machine for large data sets based on the knowledge that input data distribution is important for learning a data density dependent quantization scheme dqs is proposed for sparse input data representation the usefulness of the representation scheme is demonstrated by using it as a data preprocessing unit attached to the well known least squares support vector machine ls svm for application on big data sets essentially the proposed dqs adopts a single shrinkage threshold to obtain a simple quantization scheme which adapts its outputs to input data density with this quantization scheme a large data set is quantized to a small subset where considerable sample size reduction is generally obtained in particular the sample size reduction can save significant computational cost when using the quantized subset for feature approximation via the nystrom method based on the quantized subset the approximated features are incorporated into ls svm to develop a data density dependent quantized ls svm dqls svm where an analytic solution is obtained in the primal solution space the developed dqls svm is evaluated on synthetic and benchmark data with particular emphasis on large data sets extensive experimental results show that the learning machine incorporating dqs attains not only high computational efficiency but also good generalization performance

minimax mutual information approach for independent component analysis minimum output mutual information is regarded as a natural criterion for independent component analysis ica and is used as the performance measure in many ica algorithms two common approaches in information theoretic ica algorithms are minimum mutual information and maximum output entropy approaches in the former approach we substitute some form of probability density function pdf estimate into the mutual information expression and in the latter we incorporate the source pdf assumption in the algorithm through the use of nonlinearities matched to the corresponding cumulative density functions cdf alternative solutions to ica use higher order cumulant based optimization criteria which are related to either one of these approaches through truncated series approximations for densities in this article we propose a new ica algorithm motivated by the maximum entropy principle for estimating signal distributions the optimality criterion is the minimum output mutual information where the estimated pdfs are from the exponential family and are approximate solutions to a constrained entropy maximization problem this approach yields an upper bound for the actual mutual information of the output signals hence the name minimax mutual information ica algorithm in addition we demonstrate that for a specific selection of the constraint functions in the maximum entropy density estimation procedure the algorithm relates strongly to ica methods using higher order cumulants

a physical origin for singular support conditions in geometric langlands theory we explain how the nilpotent singular support condition introduced into the geometric langlands conjecture by arinkin and gaitsgory arises naturally from the point of view of 4 dimensional mathcal n 4 supersymmetric gauge theory we define what it means in topological quantum field theory to restrict a category of boundary conditions to the full subcategory of objects compatible with a fixed choice of vacuum both in functorial field theory and in the language of factorization algebras for b twisted mathcal n 4 gauge theory with gauge group g the moduli space of vacua is equivalent to mathfrak h w and the nilpotent singular support condition arises by restricting to the vacuum 0 in mathfrak h w we then investigate the categories obtained by restricting to points in larger strata and conjecture that these categories are equivalent to the geometric langlands categories with gauge symmetry broken to a levi subgroup and furthermore that by assembling such for the groups mathrm gl n with n ge 1 one finds a hidden factorization algebra structure for the geometric langlands theory

an expectation maximization approach to nonlinear component analysis the proposal of considering nonlinear principal component analysis as a kernel eigenvalue problem has provided an extremely powerful method of extracting nonlinear features for a number of classification and regression applications whereas the utilization of mercer kernels makes the problem of computing principal components in possibly infinite dimensional feature spaces tractable there are still the attendant numerical problems of diagonalizing large matrices in this contribution we propose an expectation maximization approach for performing kernel principal component analysis and show this to be a computationally efficient method especially when the number of data points is large