on inverse problems with unknown operators consider a problem of recovery of a smooth function signal image f spl isin spl fscr spl isin l sub 2 0 1 sup d passed through an unknown filter and then contaminated by a noise a typical model discussed in the paper is described by a stochastic differential equation dy sub f sup spl epsi t hf t dt spl epsi dw t t spl isin 0 1 sup d spl epsi 0 where h is a linear operator modeling the filter and w is a brownian motion sheet modeling a noise the aim is to recover f with asymptotically as spl epsi spl rarr 0 minimax mean integrated squared error traditionally the problem is studied under the assumption that the operator h is known then the ill posedness of the problem is the main concern in this paper a more complicated and more realistic case is considered where the operator is unknown instead a training set of n pairs e sub l y e sub l sup spl sigma l 1 2 n where e sub l is an orthonormal system in l sub 2 and y e sub l sup spl sigma denote the solutions of stochastic differential equations of the above type with f e sub l and spl epsi spl sigma is available an optimal in a minimax sense over considered operators and signals data driven recovery of the signal is suggested the influence of spl epsi spl sigma and n on the recovery is thoroughly studied in particular we discuss an interesting case of a larger noise during the training and present formulas for threshold levels for n beyond which no improvement in recovery of input signals occurs we also discuss the case where h is an unknown perturbation of a known operator we describe a class of perturbations for which the accuracy of recovery of the signal is asymptotically the same up to a constant as in the case of precisely known operator

searching and pebbling abstract we relate the search number of an undirected graph g with the minimum and maximum of the progressive pebble demands of the directed acyclic graphs obtained by orienting g towards this end we introduce node searching a slight variant of searching in which an edge is cleared by placing searchers on both of its endpoints we also show that the minimum number of searchers necessary to node search a graph equals its vertex separator plus one

spatially inhomogeneous linear inverse problems with possible singularities the objective of the present paper is to introduce the concept of a spatially inhomogeneous linear inverse problem which to the best of the author s knowledge has never been considered previously in statistical framework but is emerging due to a variety of practical applications the special feature of the problem is that the degree of ill posedness depends not only on the scale but also on location in this case the rates of convergence are determined by the interaction of four parameters the smoothness and spatial homogeneity of the unknown function f and degrees of ill posedness and spatial inhomogeneity of operator q an interesting property here is that if operator q is weakly inhomogeneous then the rates of convergence are not influenced by spatial inhomogeneity of operator q and coincide with the rates which are usual for homogeneous linear inverse problems on the other hand if operator q is moderately or strongly inhomogeneous convergence rates are significantly affected by the degree of spatial inhomogeneity estimators obtained in the paper are based either on wavelet vaguelette decomposition if the norms of all vaguelettes are finite or on a hybrid of wavelet vaguelette decomposition and galerkin method if vaguelettes in the neighborhood of the singularity point have infinite norms the hybrid estimator is a combination of a linear part in the vicinity of the singularity point and the nonlinear block thresholding wavelet estimator elsewhere to attain adaptivity an optimal resolution level for the linear singularity affected portion of the estimator is obtained using lepskii 1990 1999 method subsequently this resolution level is used as the lowest resolution level for the nonlinear wavelet estimator we show that convergence rates of the hybrid estimator lie within a logarithmic factor of the optimal minimax convergence rates the theory presented in the paper is supplemented by examples of deconvolution based on irregularly spaced observations or in the presence of a heteroscedastic noise the latter problem is also examined via a limited simulation study which demonstrates advantages of the hybrid estimator when the degree of spatial inhomogeneity is high

a deconvolution approach to estimation of a common shape in a shifted curves model this paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model we show that this problem can be transformed into a linear inverse problem where the density of the random shifts plays the role of a convolution operator an adaptive estimator of the mean pattern based on wavelet thresholding is proposed we study its consistency for the quadratic risk as the number of observed curves tends to infinity and this estimator is shown to achieve a near minimax rate of convergence over a large class of besov balls this rate depends both on the smoothness of the common shape of the curves and on the decay of the fourier coefficients of the density of the random shifts hence this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems which is a new point of view on curve registration and image warping problems we also provide a new method to estimate the unknown random shifts between curves some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature

berry esseen theorems for quadratic forms of gaussian stationary processes letx1 x2 be a gaussian stationary process with spectral densityf and let z j frac 1 sqrt t left sum limits l 1 t j 1 left x l x l j 1 ex l x l j 1 right right j 1 p the present paper gives a berry esseen type theorem for the joint distribution of z1 zp

berry esseen type bounds for signed linear rank statistics with a broad range of scores the berry esseen type bounds of order n 1 2 for the rate of convergence to normality are derived for the signed linear rank statistics under the hypothesis of symmetry the results are obtained with a broad range of regression constants and scores restricted by only mild conditions while almost all previous results are obtained with continuously differentiable score generating functions the proof is very short and elementary based on the conditioning argument

the complexity of searching a graph t parsons originally proposed and studied the following pursuit evasion problem on graphs members of a team of searchers traverse the edges of a graph g in pursuit of a fugitive who moves along the edges of the graph with complete knowledge of the locations of the pursuers what is the smallest number s g of searchers that will suffice for guaranteeing capture of the fugitive it is shown that determining whether s g k for a given integer k is np complete for general graphs but can be solved in linear time for trees we also provide a structural characterization of those graphs g with s g k for k 1 2 3

the edge orientation problem and some of its variants on weighted graphs let g v e be a connected undirected graph with n vertices and m edges where each vertex v is associated with a cost c v and each edge e u v is associated with two weights w u v and w v u the issue of assigning an orientation to each edge so that g becomes a directed graph is resolved in this paper determining a scheme to assign orientations of all edges such that max x vc x x zw x z is minimized is the objective this issue is called the edge orientation problem the eop two variants of the eop the out degree eop and the vertex weighted eop are first proposed and then efficient algorithms for solving them on general graphs are designed ascertaining that the eop is np hard on bipartite graphs and chordal graphs is the second result finally an o nlogn time algorithm for the eop on trees is designed in general the algorithmic results in this paper facilitate the implementation of the weighted fair queuing wfq on real networks the objective of the wfq is to assign an effective weight for each flow to enhance link utilization our findings consequently can be easily extended to other classes of graphs such as cactus graphs block graphs and interval graphs

wavelet deconvolution with noisy eigenvalues over the last decade there has been a lot of interest in wavelet vaguelette methods for the recovery of noisy signals or images in motion blur nonlinear wavelet estimators are known to have good adaptive properties and to outperform linear approximations over a wide range of signals and images see e g the recent waved method of johnstone kerkyacharian picard and raimondo 2004 or the forward method of neelamani choi and baraniuk 2004 and also fan and koo 2002 in the density setting in the deblurring setting wavelet vaguelette methods rely on the complete knowledge of a convolution operator s eigenvalues this is an unlikely situation in practice however a more realistic scenario such as would arise when passing the fourier basis as an input signal through a linear time invariant system is to imagine that one also observes a set of noisy eigenvalues in this paper we define a version of the waved estimator which is near optimal when used with noisy eigenvalues a key feature of our method includes a data driven method for choosing the fine resolution level in waved estimation asymptotic theory is illustrated with a wide range of finite sample examples

estimation of nonparametric conditional moment models with possibly nonsmooth moments this paper studies nonparametric estimation of conditional moment models in which the residual functions could be nonsmooth with respect to the unknown functions of endogenous variables it is a problem of nonparametric nonlinear instrumental variables iv estimation and a difficult nonlinear ill posed inverse problem with an unknown operator we first propose a penalized sieve minimum distance smd estimator of the unknown functions that are identified via the conditional moment models we then establish its consistency and convergence rate in strong metric allowing for possibly non compact function parameter spaces possibly non compact finite or infinite dimensional sieves with flexible lower semicompact or convex penalty or finite dimensional linear sieves without penalty under relatively low level sufficient conditions and for both mildly and severely ill posed problems we show that the convergence rates for the nonlinear ill posed inverse problems coincide with the known minimax optimal rates for the nonparametric mean iv regression we illustrate the theory by two important applications root n asymptotic normality of the plug in penalized smd estimator of a weighted average derivative of a nonparametric nonlinear iv regression and the convergence rate of a nonparametric additive quantile iv regression we also present a simulation study and an empirical estimation of a system of nonparametric quantile iv engel curves

compression techniques for boundary integral equations asymptotically optimal complexity estimates matrix compression techniques in the context of wavelet galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense the fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying galerkin method at a computational expense that is proven to stay proportional to the number of unknowns key issues are the second compression which reduces the near field complexity significantly and an additional a posteriori compression the latter is based on a general result concerning an optimal work balance that applies in particular to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time

the interval merging problem abstract a closed interval is an ordered pair of real numbers x y with x y the interval x y represents the set i r x i y given a set of closed intervals i a 1 b 1 a 2 b 2 a k b k the interval merging problem is to find a minimum cardinality set of intervals m i x 1 y 1 x 2 y 2 x j y j j k such that the real numbers represented by i i 1 k a i b i equal those represented by m i i 1 j x i y i in this paper we show the problem can be solved in o d log d sequential time and in o log d parallel time using o d processors on an erew pram where d is the number of the endpoints of i moreover if the input is given as a set of sorted endpoints then the problem can be solved in o d sequential time and in o log d parallel time using o d log d processors on an erew pram

a decomposition theorem for maximum weight bipartite matchings let g be a bipartite graph with positive integer weights on the edges and without isolated nodes let n n and w be the node count the largest edge weight and the total weight of g let k x y be log x log x2 y we present a new decomposition theorem for maximum weight bipartite matchings and use it to design an o sqrt n w k n w n time algorithm for computing a maximum weight matching of g this algorithm bridges a long standing gap between the best known time complexity of computing a maximum weight matching and that of computing a maximum cardinality matching given g and a maximum weight matching of g we can further compute the weight of a maximum weight matching of g u for all nodes u in o w time

searching for a mobile intruder in a polygonal region the problem of searching for a mobile intruder in a simple polygon by a single mobile searcher is considered this paper investigates the capabilities of searchers having different degrees of visibility by introducing the searcher having k flashlights whose visibility is limited to k rays emanating from his position and the searcher having a point light source who can see in all directions simultaneously this paper presents necessary and sufficient conditions for a polygon to be searchable by various searchers the paper also introduces a class of polygons for which the searcher having two flashlights is as capable as the searcher having a point light source and it gives a simple necessary and sufficient condition for such polygons to be searchable by the searcher having two flashlights the complexity of generating a search schedule under some of these conditions is also discussed many of the results are proved using chord systems that represent the visibility relations among the vertices and edges o

guarding a set of line segments in the plane we consider the following problem given a finite set of straight line segments in the plane find a set of points of minimum size so that every segment contains at least one point in the set this problem can be interpreted as looking for a minimum number of locations of policemen guards cameras or other sensors that can observe a network of streets corridors tunnels tubes etc we show that the problem is strongly np complete even for a set of segments with a cubic graph structure but in p for tree structures

the searchlight scheduling problem the problem of searching for a mobile robber in a simple polygon by a number of searchlights is considered a searchlight is a stationary point which emits a single ray that cannot penetrate the boundary of the polygon the direction of the ray can be changed continuously and a point is detected by a searchlight at a given time if and only if it is on the ray a robber is a point that can move continuously with unbounded speed first it is shown that the problem of obtaining a search schedule for an instance having at least one searchlight on the polygon boundary can be reduced to that for instances having no searchlight on the polygon boundary the reduction is achieved by a recursive search strategy called the one way sweep strategy then various sufficient conditions for the existence of a search schedule are presented by using the concept of a searchlight visibility graph finally a simple necessary and sufficient condition for the existence of a search schedule for instances having exactly two sea

a berry esseen bound for functions of independent random variables the rate of convergence in the central limit theorem for functions of independent random variables is studied in a unifying approach the basic result sharpens and extends a theorem of van zwet applications to u l and r statistics are also given improving or extending the results of helmers and van zwet helmers and huskova does and van es and helmers

an optimal algorithm for solving the searchlight guarding problem on weighted trees abstract in this paper we propose a new graph problem on a connected weighted undirected graph called the searchlight guarding problem consider that there is a fugitive who moves along the edges of a graph at any speed we want to place a set of searchlights at vertices to search the edges of the graph and capture the fugitive suppose that it costs some building cost to place a searchlight at some vertex the searchlight guarding problem is to allocate a set s of searchlights such that the summation of the building costs is minimized if there is more than one set of searchlights with the minimum building cost then find the one with the minimum searching time that is where the time slot needed to capture the fugitive is minimum we first prove that the problem is np hard on weighted bipartite graphs then an o n time optimal algorithm is designed to solve the problem on weighted trees where n is the number of the vertices of the given weighted tree the algorithm is divided into two phases in the first phase we find the set of searchlights with the minimum guarding cost and assign the searching directions of all edges by the dynamic programming strategy in the second phase the searched time slots of each edge are determined both phases take linear time

an optimal algorithm for solving the searchlight guarding problem on weighted interval graphs abstract in this paper a graph problem on connected weighted undirected graphs called the searchlight guarding problem is considered assume that there is a fugitive who moves along the edges of the graph at a random speed the task involves placing a set of searchlights at vertices to search the edges of the graph and to spot the fugitive suppose that placing a searchlight at some vertex incurs some building cost the searchlight guarding problem is to allocate a set s of searchlights at the vertices such that the total cost of the vertices in s is minimized if there is more than one set of searchlights each with a minimum building cost then identify the set with the minimum search time that is where the time slots needed to spot the fugitive is the minimum as is well established the problem is np hard on weighted bipartite graphs but is linear time solvable on weighted trees in this paper the design of a linear time optimal algorithm for the searchlight guarding problem on weighted interval graphs is presented it entails two phases in the first phase a set of searchlights with minimum guarding cost is identified and the search directions of all edges are assigned to achieve this task a new problem called the edge direction assignment problem is first defined and the problem on weighted complete split graphs is solved by the greedy strategy based on this computational result the problem of finding the set of searchlights with minimum guarding cost and assigning the search directions of all edges is solved by the dynamic programming strategy then in the second phase the search time slots of each edge are determined on the basis of the results of the first phase and on some properties of interval graphs

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

data driven density estimation in the presence of additive noise with unknown distribution we study the following model of deconvolution y x varepsilon with i i d observations y 1 dots y n and varepsilon 1 dots varepsilon m the x j 1 leq j leq n are i i d with density f independent of the varepsilon j the aim of the paper is to estimate f without knowing the density f varepsilon of the varepsilon j we first define a projection estimator for which we provide bounds for the pointwise and the integrated l 2 risk we consider ordinary smooth and supersmooth noise varepsilon with regard to ordinary smooth and supersmooth densities f then we present an adaptive estimator of the density of f this estimator is obtained by penalization of the projection contrast which provides model selection lastly we present simulation experiments to illustrate the good performances of our estimator and study from the empirical point of view the importance of theoretical constraints

testing for lack of fit in inverse regression with applications to biophotonic imaging summary we propose two test statistics for use in inverse regression problems y d k c where k is a given linear operator which cannot be continuously inverted thus only noisy indirect observationsy for the function are available both test statistics have a counterpart in classical hypothesis testing where they are called the order selection test and the data driven neyman smooth test we also introduce two model selection criteria which extend the classical akaike information criterion and bayes information criterion to inverse regression problems in a simulation study we show that the inverse order selection and neyman smooth tests outperform their direct counterparts in many cases the theory is motivated by data arising in confocal fluorescence microscopy here images are observed with blurring modelled as convolution and stochastic error at subsequent times the aim is then to reduce the signal to noise ratio by averaging over the distinct images in this context it is relevant to decide whether the images are still equal or have changed by outside influences such as moving of the object table

efficient dual simplex algorithms for the assignment problem efficient algorithms based upon balinski s signature method are described for solving then n assignment problem these algorithms are special variants of the dual simplex method and are shown to have computational bounds of o n3 variants for solving sparse assignment problems withm arcs that require o m space and o mn n2 logn time in the worst case are also presented

combined rank tests for the two sample problem with randomly censored data abstract in the two sample problem under random censorship we consider the uncensored observations and the censored ones as two separate groups for each group a suitable rank statistic is obtained and these two are then combined to a final one this idea which was first employed in akritas 1983 is shown here to produce tests that a closely resemble the ordinary rank tests for the uncensored case and do not require the calculation of the kaplan meier estimator b are comparatively easy to apply and to understand and c allow results on asymptotic normality to follow simply from standard results for the uncensored case it is shown that the loss due to using two separate rankings rather than one complete ranking is asymptotically negligible the optimal score function for each of the two separate rank statistics is seen to depend on the censoring distribution whereas in akritas 1983 no assumption on the form of the censoring distribution was made unrestricted adaptation here we pursue rest

efficient implementation of graph algorithms using contraction the component merging problem is a new graph problem versions of this problem appear as bottlenecks in various graph algorithms a new data structure solves this problem efficiently and two special cases of the problem have even more efficient solutions based on other data structures the performance of the data structures is sped up by introducing a new algorithmic tool called packets the algorithms that use these solutions to the component merging problem also exploit new properties of two existing data structures specifically trees can be used simultaneously as a priority queue and a concatenable queue similarly f heaps support some kinds of split operations with no loss of efficiency an immediate application of the solution to the simplest version of the merging problem is an o t m n algorithm for finding minimum spanning trees in undirected graphs without using f heaps where t m n m log 2 log 2 log d n the graph has n vertices and m edges and d max m n 2 packets also improve the f heap minimum spanning tree algorithm giving the fastest algorithm currently known for this problem the efficient solutions to the merging problem and the new observation about f heaps lead to an o n t m n n log n algorithm for finding a maximum weighted matching in general graphs this settles an open problem posed by tarjan 15 p 123 where the weaker bound of o nm log n 2 m was conjectured

the searchlight problem for road networks we consider the problem of searching for a mobile intruder hiding in a road network given as the union of two or more lines or two or more line segments in the plane some of the intersections of the road network are occupied by stationary guards equipped with a number of searchlights each of which can emit a single ray of light in any direction along the lines or line segments it is on the goal is to detect the intruder that is to illuminate its location guards may alter the direction in which they aim a searchlight but need to switch it off for some finite time interval to effect the change in contrast the intruder may move with arbitrary speed along the network but cannot pass guards and exploit this time interval to recontaminate previously illuminated sections of the network for various classes of road networks characterized by the number n of lines or line segments comprising it and the number g n 1 of possible locations of guards fixed in advance and guaranteed to give complete coverage we present several upper and lower bounds on the worst case number of searchlights each placed at one of the guard positions required to successfully search a given road network in particular we prove the following results 1 min 2 g 1 n 2 searchlights are sometimes necessary and min 7 3 g n 1 are always sufficient for searching a road network given as the union of n lines 2 g log n g searchlights are sometimes necessary and o g 2 log n searchlights are always sufficient for searching a road network given as the union of n line segments and3 at most one searchlight per guard position and hence a total of at most g searchlights is always sufficient for searching a road network given as the union of axis aligned lines or line segments the proofs of the upper bounds induce algorithms for generating a search schedule for detecting the intruder using the claimed number of searchlights the problem of searching for an intruder hiding in a road network is introduced upper and lower bounds on the worst case number of searchlights are presented the proofs of the upper bounds induce algorithms for generating a search schedule

adaptive wavelet galerkin methods for linear inverse problems we introduce and analyze numerical methods for the treatment of inverse problems based on an adaptive wavelet galerkin discretization these methods combine the theoretical advantages of the wavelet vaguelette decomposition wvd in terms of optimally adapting to the unknown smoothness of the solution together with the numerical simplicity of galerkin methods in a first step we simply combine a thresholding algorithm on the data with a galerkin inversion on a fixed linear space in a second step a more elaborate method performs the inversion by an adaptive procedure in which a smaller space adapted to the solution is iteratively constructed this leads to a significant reduction of the computational cost

an annotated bibliography on guaranteed graph searching graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers in this annotated bibliography we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field

adaptive nonparametric instrumental variables estimation empirical choice of the regularization parameter abstract in nonparametric instrumental variables estimation the mapping that identifies the function of interest g is discontinuous and must be regularized to permit consistent estimation the optimal regularization parameter depends on population characteristics that are unknown in applications this paper presents a theoretically justified empirical method for choosing the regularization parameter in series estimation the method adapts to the unknown smoothness of g and other unknown functions the resulting estimator of g converges at least as fast as the optimal rate multiplied by log n 1 2 the asymptotic integrated mean square error aimse of the estimator is within a specified factor of the optimal aimse

incremental assignment problem in this paper we introduce the incremental assignment problem in this problem a new pair of vertices and their incident edges are added to a weighted bipartite graph whose maximum weighted matching is already known and the maximum weighted matching of the extended graph is sought we propose an o v 2 algorithm for the problem

nonlinear estimation for linear inverse problems with error in the operator we study two nonlinear methods for statistical linear inverse problems when the operator is not known the two constructions combine galerkin regularization and wavelet thresholding their performances depend on the underlying structure of the operator quantified by an index of sparsity we prove their rate optimality and adaptivity properties over besov classes

wavelet decomposition approaches to statistical inverse problems summary a wide variety of scientific settings involve indirect noisy measurements where one faces a linear inverse problem in the presence of noise primary interest is in some function f t but data are accessible only about some linear transform corrupted by noise the usual linear methods for such inverse problems do not perform satisfactorily when f t is spatially inhomogeneous one existing nonlinear alternative is the wavelet vaguelette decomposition method based on the expansion of the unknown f t in wavelet series in the vaguelette wavelet decomposition method proposed here the observed data are expanded directly in wavelet series the performances of various methods are compared through exact risk calculations in the context of the estimation of the derivative of a function observed subject to noise a result is proved demonstrating that with a suitable universal threshold somewhat larger than that used for standard denoising problems both the wavelet based approaches have an ideal spatial adaptivity property