squared wavefunctions approach to periodic solutions of vector nonlinear schrodinger equation we develop a method of squared wavefunctions for the vector nonlinear schrodinger equation the squared wavefunctions of the octet representation of su 3 group give periodic solutions in terms of weierstrass elliptic functions specific limits of the obtained solution are the plane wave the soliton and cnoidal waves which were previously obtained using the ansatz of stationary motion

covariance matrix estimation from linearly correlated gaussian samples covariance matrix estimation concerns the problem of estimating the covariance matrix from a collection of samples which is of extreme importance in many applications classical results have shown that o n samples are sufficient to accurately estimate the covariance matrix from n dimensional independent gaussian samples however in many practical applications the received signal samples might be correlated which makes the classical analysis inapplicable in this paper we develop a nonasymptotic analysis for the covariance matrix estimation from linearly correlated gaussian samples our theoretical results show that the error bounds are determined by the signal dimension n the sample size m and the shape parameter of the distribution of the correlated sample covariance matrix particularly when the shape parameter is a class of toeplitz matrices which is of great practical interest o n samples are also sufficient to faithfully estimate the covariance matrix from correlated samples simulations are provided to verify the correctness of the theoretical results

on generic chaining and the smallest singular value of random matrices with heavy tails we present a very general chaining method which allows one to control the supremum of the empirical process suph h n 1 i 1nh2 xi eh2 in rather general situations we use this method to establish two main results first a quantitative non asymptotic version of the celebrated bai yin theorem on the singular values of a random matrix with i i d entries that have heavy tails and second a sharp estimate on the quadratic empirical process when h t t t t rn and is an isotropic unconditional log concave measure

a note on the sample complexity of the er spud algorithm by spielman wang and wright for exact recovery of sparsely used dictionaries we consider the problem of recovering an invertible n n matrix a and a sparse n p random matrix x based on the observation of y ax up to a scaling and permutation of columns of a and rows of x using only elementary tools from the theory of empirical processes we show that a version of the er spud algorithm by spielman wang and wright with high probability recovers a and x exactly provided that p cn log n which is optimal up to the constant c

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

new concentration inequalities in product spaces we introduce three new ways to measure the distance from a point to a subset of a product space and we prove corresponding concentration inequalities each of them allows to control the fluctuation of a new class of random variables

quasiperiodic and periodic solutions for vector nonlinear schrodinger equations we consider quasiperiodic and periodic cnoidal wave solutions of a set of n component vector nonlinear schrodinger equations vnlses in a biased photorefractive crystal with a drift mechanism of nonlinear response and kerr type nonlinearity n component nonlinear schrodinger equations can be used to model self trapped mutually incoherent wave packets these equations also model pulse pulse interactions in wavelength division multiplexed channels of optical fiber transmission systems quasiperiodic wave solutions for the vnlses in terms of n dimensional kleinian functions are presented periodic solutions in terms of hermite polynomials and generalized hermite polynomials for n component nonlinear schrodinger equations are found

random tensor theory extending random matrix theory to mixtures of random product states we consider a problem in random matrix theory that is inspired by quantum information theory determining the largest eigenvalue of a sum of p random product states in mathbb c d otimes k where k and p d k are fixed while d when k 1 the marcenko pastur law determines up to small corrections not only the largest eigenvalue 1 sqrt p d k 2 but the smallest eigenvalue min 0 1 sqrt p d k 2 and the spectral density in between we use the method of moments to show that for k 1 the largest eigenvalue is still approximately 1 sqrt p d k 2 and the spectral density approaches that of the marcenko pastur law generalizing the random matrix theory result to the random tensor case our bound on the largest eigenvalue has implications both for sampling from a particular heavy tailed distribution and for a recently proposed quantum data hiding and correlation locking scheme due to leung and winter

symmetry properties and exact patterns in birefringent optical fibers we carry out a group theoretical study of the pair of nonlinear schr o dinger equations describing the propagation of waves in nonlinear birefringent optical fibers we exploit the symmetry algebra associated with these equations to provide examples of specific exact solutions among them we obtain the soliton profile which is related to the coordinate translations and the constant change of phase

the lattice and matroid representations of definable sets in generalized rough sets based on relations abstract the definable set is a core concept in rough set theory it plays an important role in the characterizations of rough sets in this paper we study the lattice and matroid representations of definable sets in generalized rough sets based on relations first we propose the lower definable lattice consisting of all lower definable sets with set inclusion order then we give some conditions under which the lower definable lattice is distributive or geometric or boolean furthermore we discuss the relationship between distributive lattices and the lower definable lattices in generalized approximation spaces based on reflexive and transitive relations on the one hand we show that the lower definable lattice in a generalized approximation space based on reflexive and transitive relation is distributive on the other hand we obtain the result that a distributive lattice can induce a lower definable lattice in a generalized approximation space based on a reflexive and transitive relation finally we investigate the combination of generalized rough sets and matroids in terms of the lower definable lattice we show that if a lower definable lattice is a lattice of closed sets of a matroid then it must be an open closed set lattice of a matroid in addition we prove that some lower definable lattices are not the lattices of closed sets of matroids these results of this paper will benefit to our understanding of the relationship between matroids and generalized rough sets based on relations

covariance estimation for distributions with 2 e moments we study the minimal sample size n n n that suffices to estimate the covariance matrix of an n dimensional distribution by the sample covariance matrix in the operator norm with an arbitrary fixed accuracy we establish the optimal bound n o n for every distribution whose k dimensional marginals have uniformly bounded 2 e moments outside the sphere of radius o k in the specific case of log concave distributions this result provides an alternative approach to the kannan lovasz simonovits problem which was recently solved by adamczak et al j amer math soc 23 2010 535 561 moreover a lower estimate on the covariance matrix holds under a weaker assumption uniformly bounded 2 e moments of one dimensional marginals our argument consists of randomizing the spectral sparsifier a deterministic tool developed recently by batson spielman and srivastava siam j comput 41 2012 1704 1721 the new randomized method allows one to control the spectral edges of the sample covariance matrix via the stieltjes transform evaluated at carefully chosen random points

quantitative estimates of the convergence of the empirical covariance matrix in log concave ensembles let k be an isotropic convex body in rn given e 0 how many independent points xi uniformly distributed on k are neededfor the empirical covariance matrix to approximate the identity up to e with overwhelming probability our paper answers this question from 12 more precisely let x rn be a centered random vector with a log concave distribution and with the identity as covariance matrix an example of such a vector x is a random point in an isotropic convex body we show that for any e 0 there existsc e 0 su ch that if n c e n and xi i n are i i d copies of 1 n x then n i 1 xi xi id e with probability larger than 1 exp c n

the limit of the smallest singular value of random matrices with i i d entries abstract let a i j 1 i j be i i d real valued random variables with zero mean and unit variance and let an integer sequence n m m 1 satisfy m n m z for some z 0 1 for each m n denote by a m the n m m random matrix a i j 1 i n m 1 j m and let s m a m be its smallest singular value we prove that the sequence n m 1 2 s m a m m 1 converges to 1 z almost surely our result does not require boundedness of any moments of a i j s higher than the 2 nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold

lp moments of random vectors via majorizing measures abstract for a random vector x in r n we obtain bounds on the size of a sample for which the empirical pth moments of linear functionals are close to the exact ones uniformly on a convex body k r n we prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis in particular we find a short lewis type decomposition for any finite dimensional subspace of l p we also prove that for an isotropic log concave random vector we only need n p 2 log n sample points so that the empirical pth moments of the linear functionals are almost isometrically the same as the exact ones we obtain a concentration estimate for the empirical moments the main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain gaussian process

concentration around the mean for maxima of empirical processes in this paper we give optimal constants in talagrand s concentration inequalities for maxima of empirical processes associated to independent and eventually nonidentically distributed random variables our approach is based on the entropy method introduced by ledoux

variable metric random pursuit we consider unconstrained randomized optimization of smooth convex objective functions in the gradient free setting we analyze random pursuit rp algorithms with fixed f rp and variable metric v rp the algorithms only use zeroth order information about the objective function and compute an approximate solution by repeated optimization over randomly chosen one dimensional subspaces the distribution of search directions is dictated by the chosen metric variable metric rp uses novel variants of a randomized zeroth order hessian approximation scheme recently introduced by leventhal and lewis optimization 60 3 329 345 2011 doi 10 1080 02331930903100141 we here present 1 a refined analysis of the expected single step progress of rp algorithms and their global convergence on strictly convex functions and 2 novel convergence bounds for v rp on strongly convex functions we also quantify how well the employed metric needs to match the local geometry of the function in order for the rp algorithms to converge with the best possible rate our theoretical results are accompanied by numerical experiments comparing v rp with the derivative free schemes cma es implicit filtering nelder mead newuoa pattern search and nesterov s gradient free algorithms

random vectors in the isotropic position abstract letybe a random vector in r n satisfying e y y id letmbe a natural number and lety1 ymbe independent copies ofy we study the question of approximation of the identity operator by finite sums of the tensorsyi yi we prove that for some absolute constantc e 1 m i 1 m y i y i id c log n m e y log m 1 log m provided that the last expression is smaller than 1 we apply this estimate to improve a result of bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position

tail estimates for norms of sums of log concave random vectors we establish new tail estimates for order statistics and for the euclidean norms of projections of an isotropic log concave random vector more generally we prove tail estimates for the norms of projections of sums of independent log concave random vectors and uniform versions of these in the form of tail estimates for operator norms of matrices and their sub matrices in the setting of a log concave ensemble this is used to study a quantity ak m that controls uniformly the operator norm of the sub matrices with k rows and m columns of a matrix a with independent isotropic log concave random rows we apply our tail estimates of ak m to the study of restricted isometry property that plays a major role in the compressive sensing theory ams classification 46b06 15b52 60e15 60b20

the lower tail of random quadratic forms with applications to ordinary least squares finite sample properties of random covariance type matrices have been the subject of much research in this paper we focus on the lower tail of such a matrix and prove that it is sub gaussian under a simple fourth moment assumption on the one dimensional marginals of the random vectors a similar result holds for more general sums of random positive semidefinite matrices and our relatively simple proof uses a variant of the so called pac bayesian method for bounding empirical processes using this bound we obtain a nearly optimal finite sample result for the ordinary least squares estimator under random design

on weakly bounded empirical processes let f be a class of functions on a probability space and let x 1 x k be independent random variables distributed according to we establish an upper bound that holds with high probability on rm sup f in f i f x i geq t for every t 0 and that depends on a natural geometric parameter associated with f we use this result to analyze the supremum of empirical processes of the form z f left k 1 sum i 1 k f p x i mathbb e f p right for p 1 using the geometry of f we also present some geometric applications of this approach based on properties of the random operator gamma k 1 2 sum i 1 k x i e i where x i i 1 k are sampled according to an isotropic log concave measure on mathbb r n

spectral norm of products of random and deterministic matrices we study the spectral norm of matrices w that can be factored as w ba where a is a random matrix with independent mean zero entries and b is a fixed matrix under the 4 e th moment assumption on the entries of a we show that the spectral norm of such an m n matrix w is bounded by sqrt m sqrt n which is sharp in other words in regard to the spectral norm products of random and deterministic matrices behave similarly to random matrices with independent entries this result along with the previous work of rudelson and the author implies that the smallest singular value of a random m n matrix with i i d mean zero entries and bounded 4 e th moment is bounded below by sqrt m sqrt n 1 with high probability

the smallest singular value of a shifted d regular random square matrix we derive a lower bound on the smallest singular value of a random d regular matrix that is the adjacency matrix of a random d regular directed graph specifically let c 1 d c n log 2 n and let mathcal m n d be the set of all n times n square matrices with 0 1 entries such that each row and each column of every matrix in mathcal m n d has exactly d ones let m be a random matrix uniformly distributed on mathcal m n d then the smallest singular value s n m of m is greater than n 6 with probability at least 1 c 2 log 2 d sqrt d where c c 1 and c 2 are absolute positive constants independent of any other parameters analogous estimates are obtained for matrices of the form m z mathrm id where mathrm id is the identity matrix and z is a fixed complex number