minimax estimation of linear and quadratic functionals on sparsity classes for the gaussian sequence model we obtain nonasymptotic minimax rates of estimation of the linear quadratic and the ell 2 norm functionals on classes of sparse vectors and construct optimal estimators that attain these rates the main object of interest is the class b 0 s of s sparse vectors theta theta 1 dots theta d for which we also provide completely adaptive estimators independent of s and of the noise variance sigma having logarithmically slower rates than the minimax ones furthermore we obtain the minimax rates on the ell q balls b q r theta in mathbb r d theta q le r where 0 q le2 and theta q sum i 1 d theta i q 1 q this analysis shows that there are in general three zones in the rates of convergence that we call the sparse zone the dense zone and the degenerate zone while a fourth zone appears for estimation of the quadratic functional we show that as opposed to estimation of theta the correct logarithmic terms in the optimal rates for the sparse zone scale as log d s 2 and not as log d s for the class b 0 s the rates of estimation of the linear functional and of the ell 2 norm have a simple elbow at s sqrt d boundary between the sparse and the dense zones and exhibit similar performances whereas the estimation of the quadratic functional q theta reveals more complex effects the minimax risk on b 0 s is infinite and the sparseness assumption needs to be combined with a bound on the ell 2 norm finally we apply our results on estimation of the ell 2 norm to the problem of testing against sparse alternatives in particular we obtain a nonasymptotic analog of the ingster donoho jin theory revealing some effects that were not captured by the previous asymptotic analysis

homogenization of fully nonlinear uniformly elliptic and parabolic partial differential equations in stationary ergodic media we study the homogenization of fully nonlinear uniformly elliptic and parabolic second order partial differential equations in stationary ergodic media c 2004 wiley periodicals inc

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

an upper bound on indices of finite fuzzy relations abstract let r be an arbitrary n n fuzzy matrix which represents a finite fuzzy relation d and k be the period and the index of r respectively in this paper we present some further results about d and k and point out that k n 1 n where n is the least common multiple of 1 2 n

minimax rates in permutation estimation for feature matching the problem of matching two sets of features appears in various tasks of computer vision and can be often formalized as a problem of permutation estimation we address this problem from a statistical point of view and provide a theoretical analysis of the accuracy of several natural estimators to this end the minimax rate of separation is investigated and its expression is obtained as a function of the sample size noise level and dimension of the features we consider the cases of homoscedastic and heteroscedastic noise and establish in each case tight upper bounds on the separation distance of several estimators these upper bounds are shown to be unimprovable both in the homoscedastic and heteroscedastic settings interestingly these bounds demonstrate that a phase transition occurs when the dimension d of the features is of the order of the logarithm of the number of features n for d o log n the rate is dimension free and equals log n 1 2 where is the noise level in contrast when d is larger than c log n for some constant c 0 the minimax rate increases with d and is of the order of d log n 1 4 we also discuss the computational aspects of the estimators and provide empirical evidence of their consistency on synthetic data finally we show that our results extend to more general matching criteria

the periodicity of square fuzzy matrices based on minimal strong components in this paper the notion of minimal strong component of a fuzzy matrix is proposed by using this concept the periodicity of the power sequence of a fuzzy matrix is described on the basis of the relation of periodicity between a fuzzy matrix and its cut matrices further the greatest value max li n li of periodicity of all n n fuzzy matrices is obtained

consistent distribution free k sample and independence tests for univariate random variables a popular approach for testing if two univariate random variables are statistically independent consists of partitioning the sample space into bins and evaluating a test statistic on the binned data the partition size matters and the optimal partition size is data dependent while for detecting simple relationships coarse partitions may be best for detecting complex relationships a great gain in power can be achieved by considering finer partitions we suggest novel consistent distribution free tests that are based on summation or maximization aggregation of scores over all partitions of a fixed size we show that our test statistics based on summation can serve as good estimators of the mutual information moreover we suggest regularized tests that aggregate over all partition sizes and prove those are consistent too we provide polynomial time algorithms which are critical for computing the suggested test statistics efficiently we show that the power of the regularized tests is excellent compared to existing tests and almost as powerful as the tests based on the optimal yet unknown in practice partition size in simulations as well as on a real data example

evolutionary coherence of nonstationary signals coherence is a widely used measure for characterizing linear dependence between a pair of signals for nonstationary signals the autospectrum cross spectrum and coherence between signals may evolve over time a standard approach is to divide the signals into overlapping blocks of fixed width and then smooth over frequency the periodogram matrix at each time block in this paper a consistent estimation procedure is developed using time localized linear filtering the proposed method automatically selects via repeated tests of homogeneity the optimal window width for estimating local coherence it is pointwise adaptive in the sense that the width of the optimal interval is allowed to change across time under the locally stationary process framework we develop a central limit theorem on the fisher z transform of our time localized band coherence we apply our method to a pair of highly dynamic brain waves signals whose coherence is shown to evolve during an epileptic seizure

realigning random states we study how the realignment criterion also called computable cross norm criterion succeeds asymptotically in detecting whether random states are separable or entangled we consider random states on cd cd obtained by partial tracing a haar distributed random pure state on cd cd cs over an ancilla space cs we show that for large d typically the realignment criterion is successful in detecting entanglement if and only if s 83 2d2 in this sense the realignment criterion is asymptotically weaker than the partial transposition criterion

the transitive closure convergence of powers and adjoint of generalized fuzzy matrices generalized fuzzy matrices are considered as matrices over a special type of semiring which is called an incline and their transitive closure powers determinant and adjoint matrices are studied an expression for the transitive closure of a matrix a as a sum of its powers and some su2cient conditions for powers of a matrix to converge are given if the incline is commutative a su2cient condition for nilpotency of a matrix is obtained namely the determinants of the principal submatrices of the matrix are all equal to zero element in addition it is proved that a n 1 is equal to the adjoint matrix of a if the matrix a satis5es a in

indices and periods of incline matrices abstract inclines are the additively idempotent semirings in which products are less than or equal to factors thus inclines generalize boolean algebra fuzzy algebra and distributive lattice and the boolean matrices the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices i e the matrices over inclines this paper studies the power sequence of incline matrices in detail a necessary and sufficient condition for the incline matrix to have index is given and the indices of some incline matrices with indices are estimated it is proved that the period of n n incline matrix with index is a divisor of n and that the set of periods of n n incline matrices with indices is not bounded from above in the sense of a power of n for all n an equivalent condition and some sufficient conditions for the incline matrix to converge in finite steps are established the stability of the orbits of an incline matrix is considered and a theorem in fuzzy sets and systems 81 1996 227 is pointed out being false the results in the present paper include some previous results in the literatures which were obtained for the boolean matrices the fuzzy matrices and the lattice matrices among their special cases

asymptotically optimal discrete time nonlinear filters from stochastically convergent state process approximations we consider the problem of approximating optimal in the mmse sense nonlinear filters in a discrete time setting exploiting properties of stochastically convergent state process approximations more specifically we consider a class of nonlinear partially observable stochastic systems comprised by a possibly nonstationary hidden stochastic process the state observed through another conditionally gaussian stochastic process the observations under general assumptions we show that given an approximating process which for each time step is stochastically convergent to the state process in some appropriate sense an approximate filtering operator can be defined which converges to the true optimal nonlinear filter of the state in a strong and well defined sense i e compactly in time and uniformly in a completely characterized measurable set of probability measure almost unity also providing a purely quantitative justification of egoro s theorem for the problem at hand the results presented in this paper can form a common basis for the analysis and characterization of a number of heuristic approaches for approximating a large class of optimal nonlinear filters such as approximate grid based techniques known to perform well in a variety of applications

homogenization of random semilinear pdes we prove a homogenization result for system of semilinear parabolic pdes of the type where v and a are random ergodic fields we extend to the random case results of buckdahn hu peng 4 for periodic structures the same method involving stability results is applied our main tool is an l p estimate for the gradient of the solution of the auxiliary problems the same type of resultsis obtained for systems of semilinear elliptic pdes

a tight degree 4 sum of squares lower bound for the sherrington kirkpatrick hamiltonian we show that if varvec w is an n times n matrix drawn from the gaussian orthogonal ensemble then with high probability the degree 4 sum of squares relaxation cannot certify an upper bound on the objective n 1 cdot varvec x top varvec w varvec x under the constraints x i 2 1 0 i e varvec x in pm 1 n that is asymptotically smaller than lambda max varvec w approx 2 we also conjecture a proof technique for lower bounds against sum of squares relaxations of any degree held constant as n rightarrow infty by proposing an approximate pseudomoment construction

maximum likelihood estimation for linear gaussian covariance models summary we study parameter estimation in linear gaussian covariance models which are p dimensional gaussian models with linear constraints on the covariance matrix maximum likelihood estimation for this class of models leads to a non convex optimization problem which typically has many local maxima using recent results on the asymptotic distribution of extreme eigenvalues of the wishart distribution we provide sufficient conditions for any hill climbing method to converge to the global maximum although we are primarily interested in the case in which n p the proofs of our results utilize large sample asymptotic theory under the scheme n p 1 remarkably our numerical simulations indicate that our results remain valid for p as small as 2 an important consequence of this analysis is that for sample sizes n 14p maximum likelihood estimation for linear gaussian covariance models behaves as if it were a convex optimization problem

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

sparse phase retrieval via truncated amplitude flow this paper develops a novel algorithm termed sparse truncated amplitude flow sparta to reconstruct a sparse signal from a small number of magnitude only measurements it deals with what is also known as sparse phase retrieval pr which is np hard in general and emerges in many science and engineering applications upon formulating sparse pr as an amplitude based nonconvex optimization task sparta works iteratively in two stages in stage one the support of the underlying sparse signal is recovered using an analytically well justified rule and subsequently a sparse orthogonality promoting initialization is obtained via power iterations restricted on the support and in the second stage the initialization is successively refined by means of hard thresholding based gradient type iterations sparta is a simple yet effective scalable and fast sparse pr solver on the theoretical side for any n dimensional k sparse k ll n signal boldsymbol x with minimum in modulus nonzero entries on the order of 1 sqrt k vert boldsymbol x vert 2 sparta recovers the signal exactly up to a global unimodular constant from about k 2 log n random gaussian measurements with high probability furthermore sparta incurs computational complexity on the order of k 2n log n with total runtime proportional to the time required to read the data which improves upon the state of the art by at least a factor of k finally sparta is robust against additive noise of bounded support extensive numerical tests corroborate markedly improved recovery performance and speedups of sparta relative to existing alternatives

high dimensional analysis of semidefinite relaxations for sparse principal components principal component analysis pca is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix however pca is well known to behave poorly in the large p small n setting in which the problem dimension p is comparable to or larger than the sample size n this paper studies pca in this high dimensional regime but under the additional assumption that the maximal eigenvector is sparse say with at most k nonzero components we consider a spiked covariance model in which a base matrix is perturbed by adding a k sparse maximal eigenvector and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector as follows a a simple diagonal thresholding method which transitions from success to failure as a function of the rescaled sample size dia n p k n k 2 log p k and b a more sophisticated semidefinite programming sdp relaxation which succeeds once the rescaled sample size sdp n p k n klog p k is larger than a critical threshold in addition we prove that no method including the best method which has exponential time complexity can succeed in recovering the support if the order parameter sdp n p k is below a threshold our results thus highlight an interesting trade off between computational and statistical efficiency in high dimensional inference

an invariance principle for non symmetric markov processes and reflecting diffusions in random domains we study an invariance principle for additive functionals of nonsymmetric markov processes with singular mean forward velocities we generalize results of kipnis and varadhan kv and de masi et al de in two directions markov processes are non symmetric and mean forward velocities are distributions we study continuous time markov processes we use our result to homogenize non symmetric reflecting diffusions in random domains

approximation dans les espaces metriques et theorie de l estimation we investigate the relations between the speed of estimation and the metric structure of the parameter space especially in the case when its metric dimension is infinite given some distance d on generally hellinger distance in the case of n i i d variables we consider the minimax risk for n observations r n q mathop inf sup limits t n theta in theta text ie theta d q theta t n t n being any estimate of we shall look for functions r such that for positive constants c 1 q and c 2 q c 1 r q n r n q c2 r q n r n is the speed of estimation and we shall show under fairly general conditions including i i d variables and regular cases of markov chains and stationnary gaussian processes that r n is determined up to multiplicative constants by the metric structure of we shall also give a construction for some sort of universal estimates the risk of which is bounded by c 2 r q n in all cases where the preceding theory applies

estimation of integral functionals of a density let be a smooth function of k 2 variables we shall investigate in this paper the rates of convergence of estimators of t f f f x f x f k x x dx when f belongs to some class of densities of smoothness s we prove that when 2k 1 4 one can define an estimator t n of t f based on n i i d observations of density f on the real line which converges at the semiparametric rate 1 n on the other hand when s 2k 1 4 t f cannot be estimated at a rate faster than n with 4 s k 4s 1 we shall also provide some extensions to the multidimensional case those results extend previous works of levit of bickel and ritov and of donoho and nussbaum on estimation of quadratic functionals

optimal adaptive estimation of a quadratic functional minimax mean squared error estimates of quadratic functionals of smooth functions have been constructed for a variety of smoothness classes in contrast to many nonparametric function estimation problems there are both regular and irregular cases in the regular cases the minimax mean squared error converges at a rate proportional to the inverse of the sample size whereas in the irregular case much slower rates are the rule we investigate the problem of adaptive estimation of a quadratic functional of a smooth function when the degree of smoothness of the underlying function is not known it is shown that estimators cannot achieve the minimax rates of convergence simultaneously over two parameter spaces when at least one of these spaces corresponds to the irregular case a lower bound for the mean squared error is given which shows that any adaptive estimator which is rate optimal for the regular case must lose a logarithmic factor in the irregular case on the other hand we give a rather simple adaptive estimator which is sharp for the regular case and attains this lower bound in the irregular case moreover we explicitly describe a subset of functions where our adaptive estimator loses the logarithmic factor and show that this subset is relatively small

estimating linear dependence between nonstationary time series using the locally stationary wavelet model large volumes of neuroscience data comprise multiple nonstationary electrophysiological or neuroimaging time series recorded from different brain regions accurately estimating the dependence between such neural time series is critical since changes in the dependence structure are presumed to reflect functional interactions between neuronal populations we propose a new dependence measure derived from a bivariate locally stationary wavelet time series model since wavelets are localized in both time and scale this approach leads to a natural local and multi scale estimate of nonstationary dependence our methodology is illustrated by application to a simulated example and to electrophysiological data relating to interactions between the rat hippocampus and prefrontal cortex during working memory and decision making

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

quality gain analysis of the weighted recombination evolution strategy on general convex quadratic functions quality gain is the expected relative improvement of the function value in a single step of a search algorithm quality gain analysis reveals the dependencies of the quality gain on the parameters of a search algorithm based on which one can derive the optimal values for the parameters in this paper we investigate evolution strategies with weighted recombination on general convex quadratic functions we derive a bound for the quality gain and two limit expressions of the quality gain from the limit expressions we derive the optimal recombination weights and the optimal step size and find that the optimal recombination weights are independent of the hessian of the objective function moreover the dependencies of the optimal parameters on the dimension and the population size are revealed differently from previous works where the population size is implicitly assumed to be smaller than the dimension our results cover the population size proportional to or greater than the dimension simulation results show the optimal parameters derived in the limit approximates the optimal values in non asymptotic scenarios

auxiliary sdes for homogenization of quasilinear pdes with periodic coefficients we study the homogenization property of systems of quasi linear pdes of parabolic type with periodic coefficients highly oscillating drift and highly oscillating nonlinear term to this end we propose a probabilistic approach based on the theory of forward backward stochastic differential equations and introduce the new concept of auxiliary sdes

linear matrix period in max plus algebra abstract linear periodicity of matrices in max plus algebra is studied it is proved that the linear period of a matrix can be computed in o n 3 time if the digraph of the matrix is strongly connected the general problem of deciding whether a given matrix is almost linear periodic is shown to be np complete

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

l 1 regularized least squares for support recovery of high dimensional single index models with gaussian designs it is known that for a certain class of single index models sims y f xp 1t 0 e support recovery is impossible when x n 0 ip p and a model complexity adjusted sample size is below a critical threshold recently optimal algorithms based on sliced inverse regression sir were suggested these algorithms work provably under the assumption that the design x comes from an i i d gaussian distribution in the present paper we analyze algorithms based on covariance screening and least squares with l1 penalization i e lasso and demonstrate that they can also enjoy optimal up to a scalar rescaled sample size in terms of support recovery albeit under slightly different assumptions on f and e compared to the sir based algorithms furthermore we show more generally that lasso succeeds in recovering the signed support of 0 if x n 0 and the covariance satisfies the irrepresentable condition our work extends existing results on the support recovery of lasso for the linear model to a more general class of sims

stability of bsdes with random terminal time and homogenization of semilinear elliptic pdes in this paper we extend the probabilistic method for homogenization of semi linear parabolic pdes developed by buckdahn hu and peng to the case of elliptic pdes first we give a stability result for bsdes with random terminal time which are related to elliptic pdes as shown in peng stochastics stochastics rep 37 1991 61 74 in the one dimensional case we also partially relax the monotonicity assumption on the coefficient then we use these stability results for bsdes with random terminal time to study homogenization of systems of semilinear elliptic pdes

automatic statistical analysis of bivariate nonstationary time series we propose a new method for analyzing bivariate nonstationary time series the proposed method is a statistical procedure that automatically segments the time series into approximately stationary blocks and selects the span to be used to obtain the smoothed estimates of the time varying spectra and coherence it is based on the smooth localized complex exponential slex transform which forms a library of orthogonal complex valued transforms that are simultaneously localized in time and frequency we show that the smoothed slex periodograms are consistent estimators report simulation results and apply the method to a two channel electroencephalogram dataset recorded during an epileptic seizure

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

laplace deconvolution on the basis of time domain data and its application to dynamic contrast enhanced imaging in the present paper we consider the problem of laplace deconvolution with noisy discrete non equally spaced observations on a finite time interval which appears in many different contexts we propose a new method for laplace deconvolution which is based on expansions of the convolution kernel the unknown function and the observed signal over laguerre functions basis which acts as a surrogate eigenfunction basis using regression setting the expansion results in a small system of linear equations with the matrix of the system being triangular and toeplitz the number m of the terms in the expansion of the estimator is controlled via complexity penalty the advantage of this methodology is that it leads to very fast computations produces no boundary effects due to extension at zero and cut off at t and provides an estimator with the risk within a logarithmic factor of m of the oracle risk under no assumptions on the model and within a constant factor of the oracle risk under mild assumptions we emphasize that in the present paper we consider the true observational model with possibly non equispaced observations which are available on a finite interval of length t and account for the bias associated with this model the study is motivated by perfusion imaging using a short injection of contrast agent a procedure which is widely applied for medical assessment of blood flows within tissues such as cancerous tumors presence of a tuning parameter a allows to choose the most advantageous time units so that both the kernel and the unknown right hand side of the equation are well represented for the further deconvolution in addition absence of boundary effects allows for an accurate estimator of the tissue blood flow parameter which is of great interest to medical doctors the methodology is illustrated by an extensive simulation study and a real data example which confirms that the proposed technique is fast efficient accurate usable from a practical point of view and competitive

periodicity of powers of fuzzy matrices finite fuzzy relations for an arbitrary n n fuzzy matrix r we define its period of oscillation d and index k we present some properties of d and k and show that d n where n is the least common multiple of 1 2 n then we obtain an algorithm to find d and k the results are useful for examining periodicity of powers of finite fuzzy relations and they also hold for a boolean matrix

shapefit exact location recovery from corrupted pairwise directions let t1 tn r and consider the location recovery problem given a subset of pairwise direction observations ti tj ti tj 2 i j n n where a constant fraction of these observations are arbitrarily corrupted find ti i 1 up to a global translation and scale we propose a novel algorithm for the location recovery problem which consists of a simple convex program over dn real variables we prove that this program recovers a set of n i i d gaussian locations exactly and with high probability if the observations are given by an erdos renyi graph d is large enough and provided that at most a constant fraction of observations involving any particular location are adversarially corrupted

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

tracy widom limit for the largest eigenvalue of a large class of complex sample covariance matrices we consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity more precisely let x be an n x p matrix and let its rows be i i d complex normal vectors with mean 0 and covariance p we show that for a large class of covariance matrices p the largest eigenvalue of x x is asymptotically distributed after recentering and rescaling as the tracy widom distribution that appears in the study of the gaussian unitary ensemble we give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance n and p the main theorem applies to a number of covariance models found in applications for example well behaved toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms under some conditions on the mass of the atoms are among the models the theorem can handle generalizations of the theorem to certain spiked versions of our models and a s results about the largest eigenvalue are given we also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics

near optimal sample complexity bounds for robust learning of gaussian mixtures via compression schemes we introduce a novel technique for distribution learning based on a notion of sample compression any class of distributions that allows such a compression scheme can be learned with few samples m

adaptive tests of linear hypotheses by model selection we propose a new test based on model selection methods for testing that the expectation of a gaussian vector with n independent components belongs to a linear subspace of r n against a nonparametric alternative the testing procedure is available when the variance of the observations is unknown and does not depend on any prior information on the alternative the properties of the test are nonasymptotic and we prove that the test is rate optimal up to a possible log n factor over various classes of alternatives simultaneously we also provide a simulation study in order to evaluate the procedure when the purpose is to test goodness of fit in a regression model 1 introduction we consider the regression model

on the powers of matrices in bottleneck fuzzy algebra abstract powers of square matrices under the operations max and min are studied we show that the powers of a given matrix stabilize if and only if its orbits stabilize for each starting vector and prove a necessary and sufficient condition for this property using the associated graphs of the matrix applications of the obtained results to several special classes of matrices including circulants are given

detecting anomalous activity on networks with the graph fourier scan statistic we consider the problem of deciding based on a single noisy measurement at each vertex of a given graph whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation this problem is relevant to several applications such as surveillance disease outbreak detection biomedical imaging environmental monitoring etc since the activations in these problems often tend to be localized to small groups of vertices in the graphs we model such activity by a class of signals that are elevated over a possibly disconnected cluster with low cut size relative to its size we analyze the corresponding generalized likelihood ratio glr statistics and relate it to the problem of finding a sparsest cut in the graph we develop a convex relaxation of the glr statistic based on spectral graph theory which we call the graph fourier scan statistic gfss in our main theoretical result we show that the performance of the gfss depends explicitly on the spectral properties of the graph to assess the optimality of the gfss we prove an information theoretic lower bound for the detection of anomalous activity on graphs because the gfss requires the specification of a tuning parameter we develop an adaptive version of the gfss using these results we are able to characterize in a very explicit form the performance of the gfss on a few notable graph topologies we demonstrate that the gfss can efficiently detect a simulated arsenic contamination in groundwater

optimal bounds for aggregation of affine estimators we study the problem of aggregation of estimators when the estimators are not independent of the data used for aggregation and no sample splitting is allowed if the estimators are deterministic vectors it is well known that the minimax rate of aggregation is of order log m where m is the number of estimators to aggregate it is proved that for affine estimators the minimax rate of aggregation is unchanged it is possible to handle the linear dependence between the affine estimators and the data used for aggregation at no extra cost the minimax rate is not impacted either by the variance of the affine estimators or any other measure of their statistical complexity the minimax rate is attained with a penalized procedure over the convex hull of the estimators for a penalty that is inspired from the q aggregation procedure the results follow from the interplay between the penalty strong convexity and concentration

convection diffusion equation with space time ergodic random flow we prove the homogenization of convection diffusion in a time dependent ergodic incompressible random flow which has a bounded stream matrix and a constant mean drift we also prove two variational formulas for the effective diffusivity as a consequence we obtain both upper and lower bounds on the effective diffusivity

stochastic homogenization of the keller segel chemotaxis system abstract in this paper we focus on the keller segel chemotaxis system in a random heterogeneous domain we assume that the corresponding diffusion and chemotaxis coefficients are given by stationary ergodic random fields and apply stochastic two scale convergence methods to derive the homogenized macroscopic equations in establishing our results we also derive a priori estimates for the keller segel system that rely only on the boundedness of the coefficients in particular no differentiability assumption on the diffusion and chemotaxis coefficients for the chemotactic species is required finally we prove the convergence of a periodization procedure for approximating the homogenized macroscopic coefficients

optimal estimation of genetic relatedness in high dimensional linear models estimating the genetic relatedness between two traits based on the genome wide association data is an important problem in genetics research in the framework of high dimensional linear models we

powers of matrices over distributive lattices a review in this paper we review the results on powers of matrices over a distributive lattice we stress the significance of the graph theoretical approach and show how several previous results for matrices of special types can be obtained in a unified way

model selection for regression on a fixed design we deal with the problem of estimating some unknown regression function involved in a regression framework with deterministic design points for this end we consider some collection of finite dimensional linear spaces models and the least squares estimator built on a data driven selected model among this collection this data driven choice is performed via the minimization of some penalized model selection criterion that generalizes on mallows c p we provide non asymptotic risk bounds for the so defined estimator from which we deduce adaptivity properties our results hold under mild moment conditions on the errors the statement and the use of a new moment inequality for empirical processes is at the heart of the techniques involved in our approach

on generalized fuzzy matrices with periods in this paper generalized fuzzy matrices are considered as matrices over a special type of semiring which is called path algebra some properties and characterizations for generalized fuzzy matrices with periods are established and an expression for the transitive closure of a generalized fuzzy matrix with a period as a sum of its powers is shown reduction of generalized fuzzy matrices is defined and some properties for reduction of generalized fuzzy matrices with periods are obtained also the structure of all standard eigenvectors for a generalized fuzzy matrix with a period is determined

effect of depth and width on local minima in deep learning in this paper we analyze the effects of depth and width on the quality of local minima without strong overparameterization and simplification assumptions in the literature without any simplification assumption for deep nonlinear neural networks with the squared loss we theoretically show that the quality of local minima tends to improve toward the global minimum value as depth and width increase furthermore with a locally induced structure on deep nonlinear neural networks the values of local minima of neural networks are theoretically proven to be no worse than the globally optimal values of corresponding classical machine learning models we empirically support our theoretical observation with a synthetic data set as well as mnist cifar 10 and svhn data sets when compared to previous studies with strong overparameterization assumptions the results in this letter do not require overparameterization and instead show the gradual effects of overparameterization as consequences of general results

convergence of powers of s transitive fuzzy matrices abstract natural powers via max min multiplication of s transitive fuzzy matrices are considered it is shown that they oscillate with period equal to 2 this is an extension of the convergence property concerning max min transitive fuzzy matrices

a new statistical test based on the wavelet cross spectrum to detect time frequency dependence between non stationary signals application to the analysis of cortico muscular interactions the study of the correlations that may exist between neurophysiological signals is at the heart of modern techniques for data analysis in neuroscience wavelet coherence is a popular method to construct a time frequency map that can be used to analyze the time frequency correlations between two time series coherence is a normalized measure of dependence for which it is possible to construct confidence intervals and that is commonly considered as being more interpretable than the wavelet cross spectrum wcs in this paper we provide empirical and theoretical arguments to show that a significant level of wavelet coherence does not necessarily correspond to a significant level of dependence between random signals especially when the number of trials is small in such cases we demonstrate that the wcs is a much better measure of statistical dependence and a new statistical test to detect significant values of the cross spectrum is proposed this test clearly outperforms the limitations of coherence analysis while still allowing a consistent estimation of the time frequency correlations between two non stationary stochastic processes simulated data are used to investigate the advantages of this new approach over coherence analysis the method is also applied to experimental data sets to analyze the time frequency correlations that may exist between electroencephalogram eeg and surface electromyogram emg

on the powers of matrices over a distributive lattice abstract in this paper the index and the period for a lattice matrix are estimated some necessary and sufficent conditions for the convergence of the powers of a lattice matrix are obtained also some equivalent conditions for a nilpotent lattice matrix are discussed

convergence of powers of a fuzzy transitive matrix convergence of powers of a fuzzy transitive matrix is considered and some conditions for convergence are given under the max min operation on fuzzy matrices a fuzzy transitive matrix is a matrix which represents a fuzzy transitive relation and has many interesting properties the conditions for convergence of fuzzy matrices are examined under a special operation which is essential to reduction of fuzzy matrices or fuzzy systems given results show graph theoretic properties of fuzzy matrices and are useful when we consider convergence of systems represented by fuzzy matrices

low rank positive semidefinite matrix recovery from corrupted rank one measurements we study the problem of estimating a low rank positive semidefinite psd matrix from a set of rank one measurements using sensing vectors composed of i i d standard gaussian entries which are possibly corrupted by arbitrary outliers this problem arises from applications such as phase retrieval covariance sketching quantum space tomography and power spectrum estimation we first propose a convex optimization algorithm that seeks the psd matrix with the minimum ell 1 norm of the observation residual the advantage of our algorithm is that it is free of parameters therefore eliminating the need for tuning parameters and allowing easy implementations we establish that with high probability a low rank psd matrix can be exactly recovered as soon as the number of measurements is large enough even when a fraction of the measurements are corrupted by outliers with arbitrary magnitudes moreover the recovery is also stable against bounded noise with the additional information of an upper bound of the rank of the psd matrix we propose another nonconvex algorithm based on subgradient descent that demonstrates excellent empirical performance in terms of computational efficiency and accuracy

efficient learning and planning with compressed predictive states predictive state representations psrs offer an expressive framework for modelling partially observable systems by compactly representing systems as functions of observable quantities the psr learning approach avoids using local minima prone expectation maximization and instead employs a globally optimal moment based algorithm moreover since psrs do not require a predetermined latent state structure as an input they offer an attractive framework for model based reinforcement learning when agents must plan without a priori access to a system model unfortunately the expressiveness of psrs comes with significant computational cost and this cost is a major factor inhibiting the use of psrs in applications in order to alleviate this shortcoming we introduce the notion of compressed psrs cpsrs the cpsr learning approach combines recent advancements in dimensionality reduction incremental matrix decomposition and compressed sensing we show how this approach provides a principled avenue for learning accurate approximations of psrs drastically reducing the computational costs associated with learning while also providing effective regularization going further we propose a planning framework which exploits these learned models and we show that this approach facilitates model learning and planning in large complex partially observable domains a task that is infeasible without the principled use of compression

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

almost universal codes achieving ergodic mimo capacity within a constant gap this paper addresses the question of achieving capacity with lattice codes in multi antenna block fading channels when the number of fading blocks tends to infinity a design criterion based on the normalized minimum determinant is proposed for division algebra multi block space time codes over fading channels this plays a similar role to the hermite invariant for gaussian channels under maximum likelihood decoding it is shown that this criterion is sufficient to guarantee transmission rates within a constant gap from capacity both for deterministic channels and ergodic fading channels moreover if the number of receive antennas is greater than or equal to the number of transmit antennas the same constant gap is achieved under naive lattice decoding as well in the case of independent identically distributed rayleigh fading the error probability vanishes exponentially fast in contrast to the standard approach in the literature which employs random lattice ensembles the existence results in this paper are derived from the number theory first the gap to capacity is shown to depend on the discriminant of the chosen division algebra then class field theory is applied to build families of algebras with small discriminants the key element in the construction is the choice of a sequence of division algebras whose centers are number fields with small root discriminants

converse bounds on modulation estimation performance for the gaussian multiple access channel this paper focuses on the problem of separately modulating and jointly estimating two independent continuous valued parameters sent over a gaussian multiple access channel mac under the mean square error mse criterion without bandwidth constraints to this end we first improve an existing lower bound on the mse that is obtained using the parameter modulation estimation techniques for the single user additive white gaussian noise awgn channel as for the main contribution of this paper this improved modulation estimation analysis is generalized to the model of the two user gaussian mac we present outer bounds to the achievable region in the plane of the mse s of the two user parameters which provides a trade off between the mse s where we used zero rate lower bounds on the error probability of gaussian channels by shannon and polyanskiy et al numerical results showed that the multi user adaptation of the zero rate lower bound by polyanskiy et al provides a tighter overall lower bound on the mse pairs than the classical shannon bound in addition we introduced upper bounds on the mse exponents namely the exponential decay rates of these mse s in the asymptotic regime of long blocks that could make use of any bound on the error exponent of a single user awgn channel the obtained results are numerically evaluated for three different bounds on the reliability function of the gaussian channel it is shown that the adaptation of the reliability function by ashikhmin et al to the mac provides a significantly tighter characterization than shannon s sphere packing bound and the divergence bound

estimating the intensity of a random measure by histogram type estimators the purpose of this paper is to estimate the intensity of some random measure n on a set mathcal x by a piecewise constant function on a finite partition of mathcal x given a possibly large family mathcal m of candidate partitions we build a piecewise constant estimator histogram on each of them and then use the data to select one estimator in the family choosing the square of a hellinger type distance as our loss function we show that each estimator built on a given partition satisfies an analogue of the classical squared bias plus variance risk bound moreover the selection procedure leads to a final estimator satisfying some oracle type inequality with as usual a possible loss corresponding to the complexity of the family mathcal m when this complexity is not too high the selected estimator has a risk bounded up to a universal constant by the smallest risk bound obtained for the estimators in the family for suitable choices of the family of partitions we deduce uniform risk bounds over various classes of intensities our approach applies to the estimation of the intensity of an inhomogenous poisson process among other counting processes or the estimation of the mean of a random vector with nonnegative components