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

the oberbeck boussinesq problem modified by a thermo absorption term abstract we consider the oberbeck boussinesq problem with an extra coupling establishing a suitable relation between the velocity and the temperature our model involves a system of equations given by the transient navier stokes equations modified by introducing the thermo absorption term the model involves also the transient temperature equation with nonlinear diffusion for the obtained problem we prove the existence of weak solutions for any n 2 and its uniqueness if n 2 then considering a low range of temperature but upper than the phase changing one we study several properties related with vanishing in time of the velocity component of the weak solutions first assuming the buoyancy forces field extinct after a finite time we prove the velocity component will extinct in a later finite time provided the thermo absorption term is sublinear in this case considering a suitable buoyancy forces field which vanishes at some instant of time we prove the velocity component extinct at the same instant we prove also that for non zero buoyancy forces but decaying at a power time rate the velocity component decay at analogous power time rates provided the thermo absorption term is superlinear at last we prove that for a general non zero bounded buoyancy force the velocity component exponentially decay in time whether the thermo absorption term is sub or superlinear

principal component analysis in the local differential privacy model abstract in this paper we study the principal component analysis pca problem under the distributed non interactive local differential privacy model for the low dimensional case i e p n we show the optimal rate of k p n 2 omitting the eigenvalue terms for the private minimax risk of the k dimensional pca using the squared subspace distance as the measurement where n is the sample size and is the privacy parameter for the high dimensional i e p n row sparse case we first give a lower bound of k s log p n 2 on the private minimax risk where s is the underlying sparsity parameter then we provide an efficient algorithm to achieve the upper bound of o s 2 log p n 2 experiments on both synthetic and real world datasets confirm our theoretical guarantees

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

minimax estimation of linear functionals over nonconvex parameter spaces the minimax theory for estimating linear functionals is extended to the case of a finite union of convex parameter spaces upper and lower bounds for the minimax risk can still be described in terms of a modulus of continuity however in contrast to the theory for convex parameter spaces rate optimal procedures are often required to be nonlinear a construction of such nonlinear procedures is given the results developed in this paper have important applications to the theory of adaptation

dirichlet duality and the nonlinear dirichlet problem we study the dirichlet problem for fully nonlinear degenerate elliptic equations of the form f hess u 0 on a smoothly bounded domain n in our approach the equation is replaced by a subset f sym2 n of the symmetric n n matrices with f f 0 we establish the existence and uniqueness of continuous solutions under an explicit geometric f convexity assumption on the boundary we also study the topological structure of f convex domains and prove a theorem of andreotti frankel type two key ingredients in the analysis are the use of subaffine functions and dirichlet duality associated to f is a dirichlet dual set f that gives a dual dirichlet problem this pairing is a true duality in that the dual of f is f and in the analysis the roles of f and f are interchangeable the duality also clarifies many features of the problem including the appropriate conditions on the boundary many interesting examples are covered by these results including all branches of the homogeneous monge ampere equation over and equations appearing naturally in calibrated geometry lagrangian geometry and p convex riemannian geometry and all branches of the special lagrangian potential equation 2008 wiley periodicals inc

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

sharp adaptive estimation of quadratic functionals estimation of a quadratic functional of a function observed in the gaussian white noise model is considered a data dependent method for choosing the amount of smoothing is given the method is based on comparing certain quadratic estimators with each other it is shown that the method is asymptotically sharp or nearly sharp adaptive simultaneously for the regular and irregular region we consider l p bodies and construct bounds for the risk of the estimator which show that for p 4 the estimator is exactly optimal and for example when p 3 100 then the upper bound is at most 1 055 times larger than the lower bound we show the connection of the estimator to the theory of optimal recovery the estimator is a calibration of an estimator which is nearly minimax optimal among quadratic estimators

nonparametric estimation by convex programming the problem we concentrate on is as follows given 1 a convex compact set x in n an affine mapping x a x a parametric family p of probability densities and 2 n i i d observations of the random variable distributed with the density pa x for some unknown x x estimate the value gtx of a given linear form at x for several families p with no additional assumptions on x and a we develop computationally efficient estimation routines which are minimax optimal within an absolute constant factor we then apply these routines to recovering x itself in the euclidean norm

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

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

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

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

a remark on the kolmogorov stein inequality abstract in this paper essentially developing the stein method we prove the kolmogorov stein inequality for any orlicz norm with the same constants

gagliardo nirenberg type inequality for variable exponent lebesgue spaces abstract we prove analogies of the classical gagliardo nirenberg inequalities k f s c f q 1 k m m f p k m when usual l p norms are replaced by variable l p norms

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

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

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

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

sparse pca optimal rates and adaptive estimation principal component analysis pca is one of the most commonly used statistical procedures with a wide range of applications this paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting under mild technical conditions we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate the lower bound is obtained by calculating the local metric entropy and an application of fano s lemma the rate optimal estimator is constructed using aggregation which however might not be computationally feasible we then introduce an adaptive procedure for estimating the principal subspace which is fully data driven and can be computed efficiently it is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces a key idea in our construction is a reduction scheme which reduces the sparse pca problem to a high dimensional multivariate regression problem this method is potentially also useful for other related problems

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

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

tight conditions for consistency of variable selection in the context of high dimensionality we address the issue of variable selection in the regression model with very high ambient dimension textit i e when the number of variables is very large the main focus is on the situation where the number of relevant variables called intrinsic dimension and denoted by d is much smaller than the ambient dimension d without assuming any parametric form of the underlying regression function we get tight conditions making it possible to consistently estimate the set of relevant variables these conditions relate the intrinsic dimension to the ambient dimension and to the sample size the procedure that is provably consistent under these tight conditions is based on comparing quadratic functionals of the empirical fourier coefficients with appropriately chosen threshold values the asymptotic analysis reveals the presence of two quite different re gimes the first regime is when d is fixed in this case the situation in nonparametric regression is the same as in linear regression textit i e consistent variable selection is possible if and only if log d is small compared to the sample size n the picture is different in the second regime d to infty as n to infty where we prove that consistent variable selection in nonparametric set up is possible only if d log log d is small compared to log n we apply these results to derive minimax separation rates for the problem of variable selection

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

interpolation inequalities for derivatives in variable exponent lebesgue sobolev spaces abstract we show the interpolation inequalities for derivatives in variable exponent lebesgue sobolev spaces by applying the boundedness of the hardy littlewood maximal operator on l p as applications we prove a new landau komogorov type inequality for the second order derivative and an embedding theorem and discuss the equivalent norms in the space w 0 1 p w 2 p

structured matrix completion with applications to genomic data integration abstractmatrix completion has attracted significant recent attention in many fields including statistics applied mathematics and electrical engineering current literature on matrix completion focuses primarily on independent sampling models under which the individual observed entries are sampled independently motivated by applications in genomic data integration we propose a new framework of structured matrix completion smc to treat structured missingness by design specifically our proposed method aims at efficient matrix recovery when a subset of the rows and columns of an approximately low rank matrix are observed we provide theoretical justification for the proposed smc method and derive lower bound for the estimation errors which together establish the optimal rate of recovery over certain classes of approximately low rank matrices simulation studies show that the method performs well in finite sample under a variety of configurations the method is applied to integrate several ovarian can

nonquadratic estimators of a quadratic functional estimation of a quadratic functional over parameter spaces that are not quadratically convex is considered it is shown in contrast to the theory for quadratically convex parameter spaces that optimal quadratic rules are often rate suboptimal in such cases minimax rate optimal procedures are constructed based on local thresholding these nonquadratic procedures are sometimes fully efficient even when optimal quadratic rules have slow rates of convergence moreover it is shown that when estimating a quadratic functional nonquadratic procedures may exhibit different elbow phenomena than quadratic procedures

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

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

removable singularities for degenerate elliptic pucci operator on the heisenberg group abstract in this paper we study viscosity solutions to a class of degenerate elliptic pucci operators modelled on the heisenberg group where the second order term is obtained by a composition of degenerate elliptic pucci operator with the degenerate heisenberg hessian matrix we study and answer the question which compact sets have the property that each viscosity subsolution outside this set which is bounded below can be extended to a viscosity subsolution on this set

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

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

random projections for classification a recovery approach random projection has been widely used in data classification it maps high dimensional data into a low dimensional subspace in order to reduce the computational cost in solving the related optimization problem while previous studies are focused on analyzing the classification performance in the low dimensional space in this paper we consider the recovery problem i e how to accurately recover the optimal solution to the original high dimensional optimization problem based on the solution learned after random projection we present a simple algorithm termed dual random projection which uses the dual solution of the low dimensional optimization problem to recover the optimal solution to the original problem our theoretical analysis shows that with a high probability the proposed algorithm is able to accurately recover the optimal solution to the original problem provided that the data matrix is approximately low rank and or optimal solution is approximately sparse we further show that the proposed algorithm can be applied iteratively to reducing the recovery error exponentially

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

on estimation of the lr norm of a regression function be observed with noise in the present paper we study the problem of nonparametric estimation of certain nonsmooth functionals of f specifically l r norms f r of f known from the literature results on functional estimation deal mostly with two extreme cases estimating a smooth differentiable in l 2 functional or estimating a singular functional like the value of f at certain point or the maximum of f in the first case the convergence rate typically is n 1 2 n being the number of observations in the second case the rate of convergence coincides with the one of estimating the function f itself in the corresponding norm we show that the case of estimating f r is in some sense intermediate between the above extremes the optimal rate of convergence is worse than n 1 2 but is better than the rate of convergence of nonparametric estimates of f the results depend on the value of r for r even integer the rate occurs to be n 2 1 1 r where is the degree of smoothness if r is not an even integer then the nonparametric rate n 2 1 can be improved but only by a logarithmic in n factor

goodness of fit testing and quadratic functional estimation from indirect observations we consider the convolution model where i i d random variables x i having unknown density f are observed with additive i i d noise independent of the x s we assume that the density f belongs to either a sobolev class or a class of supersmooth functions the noise distribution is known and its characteristic function decays either polynomially or exponentially asymptotically we consider the problem of goodness of fit testing in the convolution model we prove upper bounds for the risk of a test statistic derived from a kernel estimator of the quadratic functional f f 2 based on indirect observations when the unknown density is smoother enough than the noise density we prove that this estimator is n 1 2 consistent asymptotically normal and efficient for the variance we compute otherwise we give nonparametric upper bounds for the risk of the same estimator we give an approach unifying the proof of nonparametric minimax lower bounds for both problems we establish them for sobolev densities and for supersmooth densities less smooth than exponential noise in the two setups we obtain exact testing constants associated with the asymptotic minimax rates

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

on adaptive estimation of linear functionals adaptive estimation of linear functionals over a collection of parameter spaces is considered a between class modulus of continuity a geometric quantity is shown to be instrumental in characterizing the degree of adaptability over two parameter spaces in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single para meter space a general construction of optimally adaptive estimators based on an ordered modulus of continuity is given the results are complemented by several illustrative examples

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

robust nonparametric nearest neighbor random process clustering we consider the problem of clustering noisy finite length observations of stationary ergodic random processes according to their generative models without prior knowledge of the model statistics and the number of generative models two algorithms both using the l 1 distance between estimated power spectral densities psds as a measure of dissimilarity are analyzed the first one termed nearest neighbor process clustering nnpc relies on partitioning the nearest neighbor graph of the observations via spectral clustering the second algorithm simply referred to as k means consists of a single k means iteration with farthest point initialization and was considered before in the literature albeit with a different dissimilarity measure we prove that both algorithms succeed with high probability in the presence of noise and missing entries and even when the generative process psds overlap significantly all provided that the observation length is sufficiently large our results quantify the tradeoff between the overlap of the generative process psds the observation length the fraction of missing entries and the noise variance finally we provide extensive numerical results for synthetic and real data and find that nnpc outperforms state of the art algorithms in human motion sequence clustering

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

interpolation between holder and lebesgue spaces with applications abstract classical interpolation inequality of the type u x c u y u z 1 is well known in the case when x y z are lebesgue spaces in this paper we show that this result may be extended by replacing norms y or x by suitable holder semi norm we shall even prove sharper version involving weak lorentz norm we apply this result to prove the gagliardo nirenberg inequality for a wider scale of parameters

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

minimax estimation of a functional on a structured high dimensional model we introduce a new method of estimation of parameters in semi parametric and nonparametric models the method is based on estimating equations that are u statistics in the observations the u statistics are based on higher order influence functions that extend ordinary linear influence functions of the parameter of interest and represent higher derivatives of this parameter for parameters for which the representation cannot be perfect the method leads to a bias variance trade off and results in estimators that converge at a slower than n rate in a number of examples the resulting rate can be shown to be optimal we are particularly interested in estimating parameters in models with a nuisance parameter of high dimension or low regularity where the parameter of interest cannot be estimated at n rate but we also consider efficient n estimation using novel nonlinear estimators the general approach is applied in detail to the example of estimating a mean response when the response is not always observed

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

structural inference from reduced forms with many instruments abstract this paper develops exact finite sample and asymptotic distributions for structural equation tests based on partially restricted reduced form estimates particular attention is given to models with large numbers of instruments wherein the use of partially restricted reduced form estimates is shown to be especially advantageous in statistical testing even in cases of uniformly weak instruments comparisons are made with methods based on unrestricted reduced forms and numerical computations showing finite sample performance of the tests are reported some new results are obtained on inequalities between noncentral chi squared distributions with different degrees of freedom that assist in analytic power comparisons

higher criticism for detecting sparse heterogeneous mixtures higher criticism or second level significance testing is a multiplecomparisons concept mentioned in passing by tukey it concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis tukey suggested comparing the fraction of observed significances at a given level to the expected fraction under the joint null in fact he suggested standardizing the difference of the two quantities and forming a z score the resulting z score tests the significance of the body of significance tests we consider a generalization where we maximize this z score over a range of significance levels 0 0 we are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem testing whether n normal means are all zero versus the alternative that a small fraction is nonzero the subtlety of this sparse normal means testing problem can be seen from work of ingster and jin who studied such problems in great detail in their studies they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p values either for the bulk of the tests or for the few most highly significant tests in this range when the amplitude of nonzero means is calibrated with the fraction of nonzero means the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses we show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test

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

sharp adaptive estimation of linear functionals we consider estimation of a linear functional t f where f is an unknown function observed in gaussian white noise we find asymptotically sharp adaptive estimators on various scales of smoothness classes in multidimensional situations the results allow evaluating explicitly the effect of dimension and treating general scales of classes furthermore we establish a connection between sharp adaptation and optimal recovery namely we propose a scheme that reduces the construction of sharp adaptive estimators on a scale of functional classes to a solution of the corresponding optimization problem

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

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

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

the aronsson equation for absolute minimizers of l infty functionals associated with vector fields satisfying hormander s condition given a carnot caratheodory metric space r d x generated by vector fields x i m i 1 satisfying hormander s condition we prove in theorem a that any absolute minimizer u w 1 x to f v q ess sup x f x xv x is a viscosity solution to the aronsson equation x i f x x u x f pi x x u x 0 in under suitable conditions on f in particular any amle is a viscosity solution to the subelliptic oo laplacian equation x u x i ux j ux i x j u 0 in if the carnot caratheodory space is a carnot group g and f is independent of the x variable we establish in theorem c the uniqueness of viscosity solutions to the aronsson equation a xu d 2 u f pi xu f pj xu x i x j u 0 in u on under suitable conditions on f as a consequence the uniqueness of both amle and viscosity solutions to the subelliptic oo laplacian equation is established on any carnot group g

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