asymptotic distributions of the zeros of certain classes of gauss hypergeometric polynomials abstract we study the asymptotic distribution of the zeros of certain classes of gauss hypergeometric polynomials classical analytic methods such as watson s lemma and the method of steepest descents are used to analyze the large n behavior of the zeros of the gauss hypergeometric polynomials 2 f 1 n a kn b z 2 f 1 n kn c kn c z and 2 f 1 n pn z where n is a nonnegative integer the constants k a 0 p 1 and b c r owing to the connection between jacobi polynomials and gauss hypergeometric polynomials we prove a special case of a conjecture made by martinez finkelshtein martinez gonzalez and orive numerical evidence and graphical illustrations of the clustering of zeros on certain curves are generated by mathematica version 4 0

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

analytical approach to network inference investigating degree distribution when the network is reconstructed two types of errors can occur false positive and false negative errors about the presence or absence of links in this paper the influence of these two errors on the vertex degree distribution is analytically analyzed moreover an analytic formula of the density of the biased vertex degree distribution is found in the inverse problem we find a reliable procedure to reconstruct analytically the density of the vertex degree distribution of any network based on the inferred network and estimates for the false positive and false negative errors based on e g simulation studies

real zeros of 2f1 hypergeometric polynomials we use a method based on the division algorithm to determine all the values of the real parameters b and c for which the hypergeometric polynomials 2f 1 n b c z have n real simple zeros furthermore we use the quasi orthogonality of jacobi polynomials to determine the intervals on the real line where the zeros are located

unbiased risk estimates for singular value thresholding and spectral estimators in an increasing number of applications it is of interest to recover an approximately low rank data matrix from noisy observations this paper develops an unbiased risk estimate holding in a gaussian model for any spectral estimator obeying some mild regularity assumptions in particular we give an unbiased risk estimate formula for singular value thresholding svt a popular estimation strategy that applies a soft thresholding rule to the singular values of the noisy observations among other things our formulas offer a principled and automated way of selecting regularization parameters in a variety of problems in particular we demonstrate the utility of the unbiased risk estimation for svt based denoising of real clinical cardiac mri series data we also give new results concerning the differentiability of certain matrix valued functions

the k theory of fields in characteristic p we show that for a field k of characteristic p h i k n is uniquely p divisible for i n we use higher chow groups as our definition of motivic cohomology this implies that the natural map k n m k k n k from milnor k theory to quillen k theory is an isomorphism up to uniquely p divisible groups and that k n m k and k n k are p torsion free as a consequence one can calculate the k theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de rham witt sheaves for example k n x p r 0 for n dimx another consequence is gersten s conjecture with finite coefficients for smooth varieties over discrete valuation rings with residue characteristic p as the last consequence bloch s cycle complexes localized at p satisfy all beilinson lichtenbaum milne axioms for motivic complexes except possibly the vanishing conjecture

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

motivic cohomology over dedekind rings we study properties of bloch s higher chow groups on smooth varieties over dedekind rings we prove the vanishing of open image in new window for i n and the existence of a gersten resolution for open image in new window if the residue characteristic is p we also show that the bloch kato conjecture implies the beilinson lichtenbaum conjecture open image in new window an identification open image in new window for m invertible and a gersten resolution with arbitrary finite coefficients over a complete discrete valuation ring of mixed characteristic 0 p we construct a map from motivic cohomology to syntomic cohomology which is a quasi isomorphism provided the bloch kato conjecture holds

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

bootstrap based regularization for low rank matrix estimation we develop a flexible framework for low rank matrix estimation that allows us to transform noise models into regularization schemes via a simple bootstrap algorithm effectively our procedure seeks an autoencoding basis for the observed matrix that is stable with respect to the specified noise model we call the resulting procedure a stable autoencoder in the simplest case with an isotropic noise model our method is equivalent to a classical singular value shrinkage estimator for non isotropic noise models e g poisson noise the method does not reduce to singular value shrinkage and instead yields new estimators that perform well in experiments moreover by iterating our stable autoencoding scheme we can automatically generate low rank estimates without specifying the target rank as a tuning parameter

separation theorems for the zeros of certain hypergeometric polynomials we study interlacing properties of the zeros of two contiguous 2f1 hypergeometric polynomials we use the connection between hypergeometric 2f1 and jacobi polynomials as well as a monotonicity property of zeros of orthogonal polynomials due to markoff to prove that the zeros of contiguous hypergeometric polynomials separate each other we also discuss interlacing results for the zeros of 2f1 and those of the polynomial obtained by shifting one of the parameters of 2f1 by t where 0 t 1

low complexity algorithms for low rank clutter parameters estimation in radar systems this paper addresses the problem of the clutter subspace projector estimation in the context of a disturbance composed of a low rank heterogeneous compound gaussian clutter and white gaussian noise in such a context adaptive processing based on an estimated orthogonal projector onto the clutter subspace instead of an estimated covariance matrix requires less samples than classical methods the clutter subspace estimate is usually derived from the eigenvalue decomposition of a covariance matrix estimate however it has been previously shown that a direct maximum likelihood estimator of the clutter subspace projector can be obtained for the considered context in this paper we derive two algorithms based on the block majorization minimization framework to reach this estimator these algorithms are shown to be computationally faster than the state of the art with guaranteed convergence finally the performance of the related estimators is illustrated on realistic space time adaptive processing for airborne radar simulations

a novel coupling of noise reduction algorithms for particle flow simulations proper orthogonal decomposition pod and its extension based on time windows have been shown to greatly improve the effectiveness of recovering smooth ensemble solutions from noisy particle data however to successfully de noise any molecular system a large number of measurements still need to be provided in order to achieve a better efficiency in processing time dependent fields we have combined pod with a well established signal processing technique wavelet based thresholding in this novel hybrid procedure the wavelet filtering is applied within the pod domain and referred to as wavinpod the algorithm exhibits promising results when applied to both synthetically generated signals and particle data in this work the simulations compare the performance of our new approach with standard pod or wavelet analysis in extracting smooth profiles from noisy velocity and density fields numerical examples include molecular dynamics and dissipative particle dynamics simulations of unsteady force and shear driven liquid flows as well as phase separation phenomenon simulation results confirm that wavinpod preserves the dimensionality reduction obtained using pod while improving its filtering properties through the sparse representation of data in wavelet basis this paper shows that wavinpod outperforms the other estimators for both synthetically generated signals and particle based measurements achieving a higher signal to noise ratio from a smaller number of samples the new filtering methodology offers significant computational savings particularly for multi scale applications seeking to couple continuum informations with atomistic models it is the first time that a rigorous analysis has compared de noising techniques for particle based fluid simulations

tensor svd statistical and computational limits in this paper we propose a general framework for tensor singular value decomposition tensor singular value decomposition svd which focuses on the methodology and theory for extracting the hidden low rank structure from high dimensional tensor data comprehensive results are developed on both the statistical and computational limits for tensor svd this problem exhibits three different phases according to the signal to noise ratio snr in particular with strong snr we show that the classical higher order orthogonal iteration achieves the minimax optimal rate of convergence in estimation with weak snr the information theoretical lower bound implies that it is impossible to have consistent estimation in general with moderate snr we show that the non convex maximum likelihood estimation provides optimal solution but with np hard computational cost moreover under the hardness hypothesis of hypergraphic planted clique detection there are no polynomial time algorithms performing consistently in general

murthy s conjecture on 0 cycles we show that the levine weibel chow group of 0 cycles mathrm ch d a of a reduced affine algebra a of dimension d ge 2 over an algebraically closed field is torsion free among several applications it implies an affirmative solution to an old conjecture of murthy in classical k theory

polya frequency sequences and real zeros of some f23 polynomials we prove that the zeros of some families of f23 hypergeometric polynomials are all real and negative this result has a connection with the theory of polya frequency sequences and functions as a consequence we establish the asymptotic distribution of these zeros when the degree of the polynomials tends to infinity

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

comparative numerical analysis using reduced order modeling strategies for nonlinear large scale systems we perform a comparative analysis using three reduced order strategies missing point estimation mpe method gappy pod method and discrete empirical interpolation method deim applied to a biological model describing the spatio temporal dynamics of a predator prey community the comparative study is focused on the efficiency of the reduced order approximations and the complexity reduction of the nonlinear terms different variants are discussed related to the projection based model reduction framework combined with selective spatial sampling to efficiently perform the online computations numerical results are presented

torsion and divisibility for reciprocity sheaves and 0 cycles with modulus the notion of modulus is a striking feature of rosenlicht serre s theory of generalized jacobian varieties of curves it was carried over to algebraic cycles on general varieties by bloch esnault park rulling krishna levine recently kerz saito introduced a notion of chow group of 0 cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields we study the non homotopy invariant part of the chow group of 0 cycles with modulus and show their torsion and divisibility properties modulus is being brought to sheaf theory by kahn saito yamazaki in their attempt to construct a generalization of voevodsky suslin friedlander s theory of homotopy invariant presheaves with transfers we prove parallel results about torsion and divisibility properties for them

duality via cycle complexes we show that bloch s complex of relative zero cycles can be used as a dualizing complex over perfect fields and number rings this leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over algebraically closed fields finite fields local fields of mixed characteristic and rings of integers in number rings generalizing results which so far have only been known for smooth schemes or in low dimensions and unifying the p adic and l adic theory as an application we generalize rojtman s theorem to normal projective schemes

joint covariance estimation with mutual linear structure we consider the problem of joint estimation of structured covariance matrices assuming the structure is unknown estimation is achieved using heterogeneous training sets namely given groups of measurements coming from centered populations with different covariances our aim is to determine the mutual structure of these covariance matrices and estimate them supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices we develop a new efficient algorithm discovering the structure and using it to improve the estimation our technique is based on the application of principal component analysis in the matrix space we also derive an upper performance bound of the proposed algorithm in the gaussian scenario and compare it with the cramer rao lower bound numerical simulations are presented to illustrate the performance benefits of the proposed method

large parameter cases of the gauss hypergeometric function we consider the asymptotic behavior of the gauss hypergeometric function when several of the parameters a b c are large we indicate which cases are of interest for orthogonal polynomials jacobi but also meixner krawtchouk etc which results are already available and which cases need more attention we also consider a few examples of 3f2 functions of unit argument to explain which difficulties arise in these cases when standard integrals or differential equations are not available

optimal shrinkage of singular values we consider the recovery of low rank matrices from noisy data by shrinkage of singular values in which a single univariate nonlinearity is applied to each of the empirical singular values we adopt an asymptotic framework in which the matrix size is much larger than the rank of the signal matrix to be recovered and the signal to noise ratio of the low rank piece stays constant for a variety of loss functions including mean square error mse square frobenius norm the nuclear norm loss and the operator norm loss we show that in this framework there is a well defined asymptotic loss that we evaluate precisely in each case in fact each of the loss functions we study admits a unique admissible shrinkage nonlinearity dominating all other nonlinearities we provide a general method for evaluating these optimal nonlinearities and demonstrate our framework by working out simple explicit formulas for the optimal nonlinearities in the frobenius nuclear and operator norm cases for example for a square low rank n by n matrix observed in white noise with level sigma the optimal nonlinearity for mse loss simply shrinks each data singular value y to sqrt y 2 4n sigma 2 or to 0 if y this optimal nonlinearity guarantees an asymptotic mse of 2nr sigma 2 which compares favorably with optimally tuned hard thresholding and optimally tuned soft thresholding providing guarantees of 3nr sigma 2 and 6nr sigma 2 respectively our general method also allows one to evaluate optimal shrinkers numerically to arbitrary precision as an example we compute optimal shrinkers for the schatten p norm loss for any p 0

nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries we demonstrate the synthesis of sparse sampling and dimensionality reduction to characterize and model nonlinear dynamical systems over a range of bifurcation parameters first we construct modal libraries using the classical proper orthogonal decomposition in order to expose the dominant low rank coherent structures here libraries of the nonlinear terms are also constructed in order to take advantage of the discrete empirical interpolation method and projection that allows for the approximation of nonlinear terms from a sparse number of grid points the selected grid points are shown to be effective sensing and measurement locations for characterizing the underlying dynamics stability and bifurcations of nonlinear dynamical systems the use of empirical interpolation points and sparse representation facilitates a family of local reduced order models for each physical regime rather than a higher order global model which has the benefit of physical interpretability of energy transfer between coherent structures the method advocated also allows for orders of magnitude improvement in computational speed and memory requirements to illustrate the method the discrete interpolation points and nonlinear modal libraries are used for sparse representation in order to classify and reconstruct the dynamic bifurcation regimes in the complex ginzburg landau equation it is also shown that point measurements of the nonlinearity are more effective than linear measurements when sensor noise is present

zero cycles and k theory on normal surfaces in this paper we prove a formula conjectured by bloch and srinivas s2 which describes the chow group of zero cycles of a normal quasi projective surface x over a field as an inverse limit of relative chow groups of a desingularisation x relative to multiples of the exceptional divisor we then give several applications of this result a relative version of the famous bloch conjecture on 0 cycles the triviality of the chow group of 0 cycles for any 2 dimensional normal graded q algebra analogue of the bloch beilinson conjecture and the analogue of the roitman theorem for torsion 0 cycles in characteristic p 0 for normal varieties including the case of p torsion

a deterministic theory of low rank matrix completion the problem of completing a large low rank matrix using a subset of revealed entries has received much attention in the last ten years the main result of this paper gives a necessary and sufficient condition stated in the language of graph limit theory for a sequence of matrix completion problems with arbitrary missing patterns to be asymptotically solvable it is then shown that a small modification of the candes recht nuclear norm minimization algorithm provides the required asymptotic solution whenever the sequence of problems is asymptotically solvable the theory is fully deterministic with no assumption of randomness a number of open questions are listed

on the limit of the largest eigenvalue of the large dimensional sample covariance matrix in this paper the authors show that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity the above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products sp matrix exist

the optimal hard threshold for singular values is 4 sqrt 3 we consider recovery of low rank matrices from noisy data by hard thresholding of singular values in which empirical singular values below a threshold are set to 0 we study the asymptotic mean squared error amse in a framework where the matrix size is large compared with the rank of the matrix to be recovered and the signal to noise ratio of the low rank piece stays constant the amse optimal choice of hard threshold in the case of n by n matrix in white noise of level is simply 4 3 n 2 309 n when is known or simply 2 858 y med when is unknown where y med is the median empirical singular value for nonsquare m by n matrices with m n the thresholding coefficients 4 3 and 2 858 are replaced with different provided constants that depend on m n asymptotically this thresholding rule adapts to unknown rank and unknown noise level in an optimal manner it is always better than hard thresholding at any other value and is always better than ideal truncated singular value decomposition tsvd which truncates at the true rank of the low rank matrix we are trying to recover hard thresholding at the recommended value to recover an n by n matrix of rank r guarantees an amse at most 3 nr 2 in comparison the guarantees provided by tsvd optimally tuned singular value soft thresholding and the best guarantee achievable by any shrinkage of the data singular values are 5 nr 2 6 nr 2 and 2 nr 2 respectively the recommended value for hard threshold also offers among hard thresholds the best possible amse guarantees for recovering matrices with bounded nuclear norm empirical evidence suggests that performance improvement over tsvd and other popular shrinkage rules can be substantial for different noise distributions even in relatively small n

deterministic parallel analysis an improved method for selecting factors and principal components factor analysis and principal component analysis are used in many application areas the first step choosing the number of components remains a serious challenge our work proposes improved methods for this important problem one of the most popular state of the art methods is parallel analysis pa which compares the observed factor strengths with simulated strengths under a noise only model the paper proposes improvements to pa we first derandomize it proposing deterministic pa which is faster and more reproducible than pa both pa and deterministic pa are prone to a shadowing phenomenon in which a strong factor makes it difficult to detect smaller but more interesting factors we propose deflation to counter shadowing we also propose to raise the decision threshold to improve estimation accuracy we prove several consistency results for our methods and test them in simulations we also illustrate our methods on data from the human genome diversity project where they significantly improve the accuracy